Subgroups — C₃×#30442 #10649049

Order 768.

878 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3043	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#3042	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#3022	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#3021	C₂	1	1	yes	maximal, characteristic
5	C₃×#2098	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2080	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2080	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2049	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2048	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#2019	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#1912	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#1911	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃×#1905	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃×#1904	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃×#1794	C₂	1	1	yes	maximal, characteristic, complemented
16	#30442	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₃×#585	C₂²	1	1	yes	characteristic
18	C₂×C₃×#137	C₂²	1	1	yes	characteristic
19	C₂×C₃×#137	C₂²	1	1	yes	characteristic
20	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
21	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
22	C₂²×C₃×SD₁₆	C₂²	1	1	yes	characteristic, complemented
23	C₃×#536	C₂²	1	1	yes	characteristic
24	C₃×#535	C₂²	1	1	yes	characteristic
25	C₃×#519	C₂²	1	1	yes	characteristic, complemented
26	C₃×#518	C₂²	1	1	yes	characteristic, complemented
27	C₃×#464	—	2	2	no	—
28	C₃×#463	—	2	2	no	—
29	C₃×#461	—	2	2	no	—
30	C₃×#460	—	2	2	no	—
31	C₃×#456	—	2	2	no	—
32	C₃×#455	—	2	2	no	—
33	C₃×#454	—	2	2	no	—
34	C₃×#453	—	2	2	no	—
35	C₃×#430	C₂²	1	1	yes	characteristic, complemented
36	C₃×#424	C₂²	1	1	yes	characteristic, complemented
37	C₃×#424	C₂²	1	1	yes	characteristic, complemented
38	C₃×#421	C₂²	1	1	yes	characteristic
39	C₃×#421	C₂²	1	1	yes	characteristic
40	C₃×#421	C₂²	1	1	yes	characteristic
41	C₃×#421	C₂²	1	1	yes	characteristic
42	C₃×#421	C₂²	1	1	yes	characteristic
43	C₃×#421	C₂²	1	1	yes	characteristic
44	C₃×#419	C₂²	1	1	yes	characteristic, complemented
45	C₃×#418	C₂²	1	1	yes	characteristic, complemented
46	C₃×#416	C₂²	1	1	yes	characteristic
47	C₃×#416	C₂²	1	1	yes	characteristic
48	C₃×#413	C₂²	1	1	yes	characteristic
49	C₃×#413	C₂²	1	1	yes	characteristic
50	C₂×C₃×#101	C₂²	1	1	yes	characteristic, complemented
51	C₂×C₃×#100	C₂²	1	1	yes	characteristic, complemented
52	C₃×#410	C₂²	1	1	yes	characteristic
53	C₃×#408	C₂²	1	1	yes	characteristic
54	C₃×#344	C₂²	1	1	yes	characteristic, complemented
55	C₃×#344	C₂²	1	1	yes	characteristic, complemented
56	C₃×#341	C₂²	1	1	yes	characteristic, complemented
57	C₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic, complemented
58	C₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic, complemented
59	C₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic, complemented
60	#3043	C₂×C₃	1	1	yes	characteristic, complemented
61	#3042	C₂×C₃	1	1	yes	characteristic, complemented
62	#3022	C₂×C₃	1	1	yes	characteristic, complemented
63	#3021	C₂×C₃	1	1	yes	characteristic
64	#2098	C₂×C₃	1	1	yes	characteristic, complemented
65	#2080	C₂×C₃	1	1	yes	characteristic, complemented
66	#2080	C₂×C₃	1	1	yes	characteristic, complemented
67	#2049	C₂×C₃	1	1	yes	characteristic, complemented
68	#2048	C₂×C₃	1	1	yes	characteristic, complemented
69	#2019	C₂×C₃	1	1	yes	characteristic, complemented
70	#1912	C₂×C₃	1	1	yes	characteristic, complemented
71	#1911	C₂×C₃	1	1	yes	characteristic, complemented
72	#1905	C₂×C₃	1	1	yes	characteristic, complemented
73	#1904	C₂×C₃	1	1	yes	characteristic, complemented
74	#1794	C₂×C₃	1	1	yes	characteristic, complemented
75	C₃×2₋¹⁺⁴	—	2	1	no	—
76	C₃×2₋¹⁺⁴	—	2	1	no	—
77	C₃×2₋¹⁺⁴	—	2	1	no	—
78	C₃×H(5,𝔽₂)	—	2	1	no	—
79	C₃×H(5,𝔽₂)	—	2	1	no	—
80	C₃×H(5,𝔽₂)	—	2	1	no	—
81	C₃×H(5,𝔽₂)	—	4	1	no	—
82	C₂×C₃×#41	—	4	1	no	—
83	C₂×C₃×#41	—	4	1	no	—
84	C₂×C₃×#41	—	4	1	no	—
85	C₂×C₃×#41	C₂³	1	1	yes	characteristic
86	C₂×C₃×#41	C₂³	1	1	yes	characteristic
87	C₂×C₃×#41	C₂³	1	1	yes	characteristic
88	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
89	C₂²×C₃×D₄	C₂³	1	1	yes	characteristic
90	C₃×#137	—	2	2	no	—
91	C₃×#137	—	2	2	no	—
92	C₃×#137	—	4	1	no	—
93	C₃×#137	—	4	1	no	—
94	C₃×AGL(1,ℤ/8)	—	2	2	no	—
95	C₃×AGL(1,ℤ/8)	—	2	2	no	—
96	C₃×AGL(1,ℤ/8)	—	4	1	no	—
97	C₃×AGL(1,ℤ/8)	—	4	1	no	—
98	C₂×C₃×Q₁₆	—	2	1	no	—
99	C₂×C₃×Q₁₆	—	2	1	no	—
100	C₂×C₃×SD₁₆	—	2	2	no	—
101	C₂×C₃×SD₁₆	—	2	1	no	—
102	C₂×C₃×SD₁₆	—	2	1	no	—
103	C₂×C₃×SD₁₆	—	2	2	no	—
104	C₂×C₃×SD₁₆	—	2	1	no	—
105	C₂×C₃×SD₁₆	—	2	1	no	—
106	C₂×C₃×SD₁₆	—	4	1	no	—
107	C₂×C₃×D₈	—	2	1	no	—
108	C₂×C₃×D₈	—	2	1	no	—
109	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
110	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
111	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
112	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
113	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
114	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
115	C₃×C₄⋊Q₈	D₄	1	2	yes	complemented
116	C₃×#127	D₄	1	2	yes	—
117	C₃×#124	D₄	1	2	yes	—
118	C₃×#124	D₄	1	2	yes	complemented
119	C₃×#123	—	4	1	no	—
120	C₃×C₂²⋊Q₈	—	4	1	no	—
121	C₃×#121	—	4	1	no	—
122	C₃×#120	—	4	1	no	—
123	C₃×C₄×Q₈	—	2	2	no	—
124	C₃×C₄×Q₈	—	4	1	no	—
125	C₃×C₄×D₄	—	2	2	no	—
126	C₃×C₄×D₄	—	4	1	no	—
127	C₃×#117	—	4	1	no	—
128	C₃×#117	C₂³	1	1	yes	characteristic
129	C₃×#117	C₂³	1	1	yes	characteristic
130	C₃×#117	C₂³	1	1	yes	characteristic
131	C₃×#117	C₂³	1	1	yes	characteristic
132	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
133	C₃×#108	—	4	1	no	—
134	C₃×C₄⋊C₈	—	2	2	no	—
135	C₃×C₄⋊C₈	—	2	2	no	—
136	C₃×#104	—	2	2	no	—
137	C₃×#104	—	2	2	no	—
138	C₃×#104	—	2	2	no	—
139	C₃×#104	—	2	2	no	—
140	C₃×#104	—	4	1	no	—
141	C₃×#104	—	4	1	no	—
142	C₃×#103	—	2	1	no	—
143	C₃×#103	—	2	1	no	—
144	C₃×#103	—	2	2	no	—
145	C₃×#102	—	2	1	no	—
146	C₃×#102	—	2	1	no	—
147	C₃×#102	—	2	2	no	—
148	C₃×#101	—	2	1	no	—
149	C₃×#101	—	2	1	no	—
150	C₃×#101	—	4	1	no	—
151	C₃×#100	—	2	1	no	—
152	C₃×#100	—	2	1	no	—
153	C₃×#100	—	4	1	no	—
154	C₃×#99	—	4	1	no	—
155	C₃×C₂²⋊C₈	—	4	1	no	—
156	#585	C₂²×C₃	1	1	yes	characteristic
157	C₂×#137	C₂²×C₃	1	1	yes	characteristic
158	C₂×#137	C₂²×C₃	1	1	yes	characteristic
159	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	1	yes	characteristic
160	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	1	yes	characteristic
161	C₂²×SD₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
162	#536	C₂²×C₃	1	1	yes	characteristic
163	#535	C₂²×C₃	1	1	yes	characteristic
164	#519	C₂²×C₃	1	1	yes	characteristic, complemented
165	#518	C₂²×C₃	1	1	yes	characteristic, complemented
166	#464	—	2	2	no	—
167	#463	—	2	2	no	—
168	#461	—	2	2	no	—
169	#460	—	2	2	no	—
170	#456	—	2	2	no	—
171	#455	—	2	2	no	—
172	#454	—	2	2	no	—
173	#453	—	2	2	no	—
174	#430	C₂²×C₃	1	1	yes	characteristic, complemented
175	#424	C₂²×C₃	1	1	yes	characteristic, complemented
176	#424	C₂²×C₃	1	1	yes	characteristic, complemented
177	#421	C₂²×C₃	1	1	yes	characteristic
178	#421	C₂²×C₃	1	1	yes	characteristic
179	#421	C₂²×C₃	1	1	yes	characteristic
180	#421	C₂²×C₃	1	1	yes	characteristic
181	#421	C₂²×C₃	1	1	yes	characteristic
182	#421	C₂²×C₃	1	1	yes	characteristic
183	#419	C₂²×C₃	1	1	yes	characteristic, complemented
184	#418	C₂²×C₃	1	1	yes	characteristic, complemented
185	#416	C₂²×C₃	1	1	yes	characteristic
186	#416	C₂²×C₃	1	1	yes	characteristic
187	#413	C₂²×C₃	1	1	yes	characteristic
188	#413	C₂²×C₃	1	1	yes	characteristic
189	C₂×#101	C₂²×C₃	1	1	yes	characteristic, complemented
190	C₂×#100	C₂²×C₃	1	1	yes	characteristic, complemented
191	#410	C₂²×C₃	1	1	yes	characteristic
192	#408	C₂²×C₃	1	1	yes	characteristic
193	#344	C₂²×C₃	1	1	yes	characteristic, complemented
194	#344	C₂²×C₃	1	1	yes	characteristic, complemented
195	#341	C₂²×C₃	1	1	yes	characteristic, complemented
196	H(3,ℤ/4)	C₂²×C₃	1	1	yes	characteristic, complemented
197	H(3,ℤ/4)	C₂²×C₃	1	1	yes	characteristic, complemented
198	H(3,ℤ/4)	C₂²×C₃	1	1	yes	characteristic, complemented
199	C₂⁴×C₃	—	2	1	no	—
200	C₃×#41	—	2	1	no	—
201	C₃×#41	—	2	1	no	—
202	C₃×#41	—	2	1	no	—
203	C₃×#41	—	2	1	no	—
204	C₃×#41	—	2	1	no	—
205	C₃×#41	—	2	1	no	—
206	C₃×#41	—	2	1	no	—
207	C₃×#41	—	2	1	no	—
208	C₃×#41	—	2	1	no	—
209	C₃×#41	—	2	1	no	—
210	C₃×#41	—	2	1	no	—
211	C₃×#41	—	2	1	no	—
212	C₃×#41	—	4	2	no	—
213	C₃×#41	—	4	2	no	—
214	C₃×#41	—	4	2	no	—
215	C₃×#41	—	4	1	no	—
216	C₃×#41	—	4	1	no	—
217	C₃×#41	—	4	1	no	—
218	C₃×#41	—	4	1	no	—
219	C₃×#41	—	4	1	no	—
220	C₃×#41	—	4	1	no	—
221	C₃×#41	—	4	2	no	—
222	C₂×C₃×Q₈	—	2	1	no	—
223	C₂×C₃×Q₈	—	2	1	no	—
224	C₂×C₃×Q₈	—	2	1	no	—
225	C₂×C₃×Q₈	—	2	1	no	—
226	C₂×C₃×Q₈	—	2	1	no	—
227	C₂×C₃×Q₈	—	4	1	no	—
228	C₂×C₃×Q₈	—	4	1	no	—
229	C₂×C₃×Q₈	—	4	1	no	—
230	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
231	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
232	C₂×C₃×Q₈	C₂×D₄	1	2	yes	—
233	C₂×C₃×Q₈	#41	1	1	yes	characteristic
234	C₂×C₃×D₄	—	2	1	no	—
235	C₂×C₃×D₄	—	2	1	no	—
236	C₂×C₃×D₄	—	2	1	no	—
237	C₂×C₃×D₄	—	2	1	no	—
238	C₂×C₃×D₄	—	2	1	no	—
239	C₂×C₃×D₄	—	2	1	no	—
240	C₂×C₃×D₄	—	2	1	no	—
241	C₂×C₃×D₄	—	2	1	no	—
242	C₂×C₃×D₄	—	4	2	no	—
243	C₂×C₃×D₄	—	4	1	no	—
244	C₂×C₃×D₄	—	4	1	no	—
245	C₂×C₃×D₄	—	4	1	no	—
246	C₂×C₃×D₄	—	4	2	no	—
247	C₂×C₃×D₄	—	4	2	no	—
248	C₂×C₃×D₄	—	4	2	no	—
249	C₂×C₃×D₄	C₂×D₄	1	1	yes	characteristic
250	C₂×C₃×D₄	C₂×D₄	1	1	yes	characteristic
251	C₂×C₃×D₄	C₂×D₄	1	2	yes	—
252	C₂×C₃×D₄	#41	1	1	yes	characteristic
253	C₂²×C₃×C₄	—	2	1	no	—
254	C₂²×C₃×C₄	—	2	1	no	—
255	C₂²×C₃×C₄	—	2	1	no	—
256	C₂²×C₃×C₄	—	2	1	no	—
257	C₂²×C₃×C₄	—	4	2	no	—
258	C₂²×C₃×C₄	—	4	2	no	—
259	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
260	C₃×Q₁₆	—	4	2	no	—
261	C₃×Q₁₆	—	4	2	no	—
262	C₃×SD₁₆	—	4	4	no	—
263	C₃×SD₁₆	—	4	2	no	—
264	C₃×SD₁₆	—	4	2	no	—
265	C₃×SD₁₆	—	4	2	no	—
266	C₃×SD₁₆	—	4	2	no	—
267	C₃×D₈	—	4	2	no	—
268	C₃×D₈	—	4	2	no	—
269	C₃×M₁₆	—	2	1	no	—
270	C₃×M₁₆	—	2	1	no	—
271	C₃×M₁₆	—	2	1	no	—
272	C₃×M₁₆	—	2	1	no	—
273	C₃×M₁₆	—	2	1	no	—
274	C₃×M₁₆	—	2	1	no	—
275	C₃×M₁₆	—	2	1	no	—
276	C₃×M₁₆	—	2	1	no	—
277	C₃×M₁₆	—	2	1	no	—
278	C₃×M₁₆	—	2	1	no	—
279	C₂×C₃×C₈	—	2	1	no	—
280	C₂×C₃×C₈	—	2	1	no	—
281	C₂×C₃×C₈	—	2	1	no	—
282	C₂×C₃×C₈	—	2	1	no	—
283	C₂×C₃×C₈	—	2	1	no	—
284	C₂×C₃×C₈	—	2	1	no	—
285	C₂×C₃×C₈	—	2	1	no	—
286	C₂×C₃×C₈	—	2	1	no	—
287	C₃×C₄⋊C₄	—	2	1	no	—
288	C₃×C₄⋊C₄	—	2	1	no	—
289	C₃×C₄⋊C₄	—	2	1	no	—
290	C₃×C₄⋊C₄	—	2	1	no	—
291	C₃×C₄⋊C₄	—	4	2	no	—
292	C₃×C₄⋊C₄	—	4	2	no	—
293	C₃×C₂²⋊C₄	—	2	1	no	—
294	C₃×C₂²⋊C₄	—	2	1	no	—
295	C₃×C₂²⋊C₄	—	2	1	no	—
296	C₃×C₂²⋊C₄	—	2	1	no	—
297	C₃×C₂²⋊C₄	—	4	2	no	—
298	C₃×C₂²⋊C₄	—	4	2	no	—
299	C₃×C₂²⋊C₄	—	4	2	no	—
300	C₃×C₄²	—	2	1	no	—
301	C₃×C₄²	—	2	1	no	—
302	C₃×C₄²	—	4	2	no	—
303	C₃×C₄²	C₂×D₄	1	2	yes	—
304	C₃×C₄²	C₂×D₄	1	2	yes	—
305	2₋¹⁺⁴	—	2	1	no	—
306	2₋¹⁺⁴	—	2	1	no	—
307	2₋¹⁺⁴	—	2	1	no	—
308	H(5,𝔽₂)	—	2	1	no	—
309	H(5,𝔽₂)	—	2	1	no	—
310	H(5,𝔽₂)	—	2	1	no	—
311	H(5,𝔽₂)	—	4	1	no	—
312	C₂×#41	—	4	1	no	—
313	C₂×#41	—	4	1	no	—
314	C₂×#41	—	4	1	no	—
315	C₂×#41	C₂³×C₃	1	1	yes	characteristic
316	C₂×#41	C₂³×C₃	1	1	yes	characteristic
317	C₂×#41	C₂³×C₃	1	1	yes	characteristic
318	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
319	C₂²×D₄	C₂³×C₃	1	1	yes	characteristic
320	#137	—	2	2	no	—
321	#137	—	2	2	no	—
322	#137	—	4	1	no	—
323	#137	—	4	1	no	—
324	AGL(1,ℤ/8)	—	2	2	no	—
325	AGL(1,ℤ/8)	—	2	2	no	—
326	AGL(1,ℤ/8)	—	4	1	no	—
327	AGL(1,ℤ/8)	—	4	1	no	—
328	C₂×Q₁₆	—	2	1	no	—
329	C₂×Q₁₆	—	2	1	no	—
330	C₂×SD₁₆	—	2	2	no	—
331	C₂×SD₁₆	—	2	1	no	—
332	C₂×SD₁₆	—	2	1	no	—
333	C₂×SD₁₆	—	2	2	no	—
334	C₂×SD₁₆	—	2	1	no	—
335	C₂×SD₁₆	—	2	1	no	—
336	C₂×SD₁₆	—	4	1	no	—
337	C₂×D₈	—	2	1	no	—
338	C₂×D₈	—	2	1	no	—
339	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
340	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
341	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
342	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
343	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
345	C₄⋊Q₈	C₃×D₄	1	2	yes	complemented
346	#127	C₃×D₄	1	2	yes	—
347	#124	C₃×D₄	1	2	yes	—
348	#124	C₃×D₄	1	2	yes	complemented
349	#123	—	4	1	no	—
350	C₂²⋊Q₈	—	4	1	no	—
351	#121	—	4	1	no	—
352	#120	—	4	1	no	—
353	C₄×Q₈	—	2	2	no	—
354	C₄×Q₈	—	4	1	no	—
355	C₄×D₄	—	2	2	no	—
356	C₄×D₄	—	4	1	no	—
357	#117	—	4	1	no	—
358	#117	C₂³×C₃	1	1	yes	characteristic
359	#117	C₂³×C₃	1	1	yes	characteristic
360	#117	C₂³×C₃	1	1	yes	characteristic
361	#117	C₂³×C₃	1	1	yes	characteristic
362	C₂×C₂²⋊C₄	—	4	1	no	—
363	#108	—	4	1	no	—
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	2	2	no	—
366	#104	—	2	2	no	—
367	#104	—	2	2	no	—
368	#104	—	2	2	no	—
369	#104	—	2	2	no	—
370	#104	—	4	1	no	—
371	#104	—	4	1	no	—
372	#103	—	2	1	no	—
373	#103	—	2	1	no	—
374	#103	—	2	2	no	—
375	#102	—	2	1	no	—
376	#102	—	2	1	no	—
377	#102	—	2	2	no	—
378	#101	—	2	1	no	—
379	#101	—	2	1	no	—
380	#101	—	4	1	no	—
381	#100	—	2	1	no	—
382	#100	—	2	1	no	—
383	#100	—	4	1	no	—
384	#99	—	4	1	no	—
385	C₂²⋊C₈	—	4	1	no	—
386	C₂³×C₃	—	2	1	no	—
387	C₂³×C₃	—	2	2	no	—
388	C₂³×C₃	—	2	1	no	—
389	C₂³×C₃	—	2	1	no	—
390	C₂³×C₃	—	4	2	no	—
391	C₂³×C₃	—	4	1	no	—
392	C₂³×C₃	—	4	1	no	—
393	C₂³×C₃	—	16	1	no	—
394	C₂³×C₃	H(5,𝔽₂)	1	1	yes	characteristic
395	C₃×Q₈	—	2	2	no	—
396	C₃×Q₈	—	2	1	no	—
397	C₃×Q₈	—	2	1	no	—
398	C₃×Q₈	—	2	1	no	—
399	C₃×Q₈	—	2	2	no	—
400	C₃×Q₈	—	2	1	no	—
401	C₃×Q₈	—	2	1	no	—
402	C₃×Q₈	—	2	1	no	—
403	C₃×Q₈	—	4	2	no	—
404	C₃×Q₈	—	4	1	no	—
405	C₃×Q₈	—	4	1	no	—
406	C₃×D₄	—	2	1	no	—
407	C₃×D₄	—	2	2	no	—
408	C₃×D₄	—	2	1	no	—
409	C₃×D₄	—	2	1	no	—
410	C₃×D₄	—	2	2	no	—
411	C₃×D₄	—	2	1	no	—
412	C₃×D₄	—	2	1	no	—
413	C₃×D₄	—	2	1	no	—
414	C₃×D₄	—	4	2	no	—
415	C₃×D₄	—	4	2	no	—
416	C₃×D₄	—	4	2	no	—
417	C₃×D₄	—	4	1	no	—
418	C₃×D₄	—	4	1	no	—
419	C₃×D₄	—	4	1	no	—
420	C₃×D₄	—	4	1	no	—
421	C₃×D₄	—	4	1	no	—
422	C₃×D₄	—	4	1	no	—
423	C₃×D₄	—	4	1	no	—
424	C₃×D₄	—	4	1	no	—
425	C₃×D₄	—	16	2	no	—
426	C₂×C₃×C₄	—	2	2	no	—
427	C₂×C₃×C₄	—	2	2	no	—
428	C₂×C₃×C₄	—	2	1	no	—
429	C₂×C₃×C₄	—	2	2	no	—
430	C₂×C₃×C₄	—	2	1	no	—
431	C₂×C₃×C₄	—	2	1	no	—
432	C₂×C₃×C₄	—	4	2	no	—
433	C₂×C₃×C₄	—	4	2	no	—
434	C₂×C₃×C₄	—	4	2	no	—
435	C₂×C₃×C₄	—	4	1	no	—
436	C₂×C₃×C₄	—	4	1	no	—
437	C₂×C₃×C₄	—	4	1	no	—
438	C₂×C₃×C₄	—	4	1	no	—
439	C₂×C₃×C₄	—	4	1	no	—
440	C₂×C₃×C₄	—	4	1	no	—
441	C₂×C₃×C₄	—	4	1	no	—
442	C₂×C₃×C₄	—	4	1	no	—
443	C₂×C₃×C₄	—	4	1	no	—
444	C₂×C₃×C₄	—	4	1	no	—
445	C₂×C₃×C₄	—	16	2	no	—
446	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
447	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
448	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
449	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
450	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
451	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
452	C₃×C₈	—	4	1	no	—
453	C₃×C₈	—	4	1	no	—
454	C₃×C₈	—	4	1	no	—
455	C₃×C₈	—	4	1	no	—
456	C₃×C₈	—	4	1	no	—
457	C₃×C₈	—	4	1	no	—
458	C₂⁴	—	2	1	no	—
459	#41	—	2	1	no	—
460	#41	—	2	1	no	—
461	#41	—	2	1	no	—
462	#41	—	2	1	no	—
463	#41	—	2	1	no	—
464	#41	—	2	1	no	—
465	#41	—	2	1	no	—
466	#41	—	2	1	no	—
467	#41	—	2	1	no	—
468	#41	—	2	1	no	—
469	#41	—	2	1	no	—
470	#41	—	2	1	no	—
471	#41	—	4	2	no	—
472	#41	—	4	2	no	—
473	#41	—	4	2	no	—
474	#41	—	4	1	no	—
475	#41	—	4	1	no	—
476	#41	—	4	1	no	—
477	#41	—	4	1	no	—
478	#41	—	4	1	no	—
479	#41	—	4	1	no	—
480	#41	—	4	2	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	4	1	no	—
487	C₂×Q₈	—	4	1	no	—
488	C₂×Q₈	—	4	1	no	—
489	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
490	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
491	C₂×Q₈	C₂×C₃×D₄	1	2	yes	—
492	C₂×Q₈	C₃×#41	1	1	yes	characteristic
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	1	no	—
496	C₂×D₄	—	2	1	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	4	2	no	—
502	C₂×D₄	—	4	1	no	—
503	C₂×D₄	—	4	1	no	—
504	C₂×D₄	—	4	1	no	—
505	C₂×D₄	—	4	2	no	—
506	C₂×D₄	—	4	2	no	—
507	C₂×D₄	—	4	2	no	—
508	C₂×D₄	C₂×C₃×D₄	1	1	yes	characteristic
509	C₂×D₄	C₂×C₃×D₄	1	1	yes	characteristic
510	C₂×D₄	C₂×C₃×D₄	1	2	yes	—
511	C₂×D₄	C₃×#41	1	1	yes	characteristic
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	—	4	2	no	—
517	C₂²×C₄	—	4	2	no	—
518	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
519	Q₁₆	—	4	2	no	—
520	Q₁₆	—	4	2	no	—
521	SD₁₆	—	4	4	no	—
522	SD₁₆	—	4	2	no	—
523	SD₁₆	—	4	2	no	—
524	SD₁₆	—	4	2	no	—
525	SD₁₆	—	4	2	no	—
526	D₈	—	4	2	no	—
527	D₈	—	4	2	no	—
528	M₁₆	—	2	1	no	—
529	M₁₆	—	2	1	no	—
530	M₁₆	—	2	1	no	—
531	M₁₆	—	2	1	no	—
532	M₁₆	—	2	1	no	—
533	M₁₆	—	2	1	no	—
534	M₁₆	—	2	1	no	—
535	M₁₆	—	2	1	no	—
536	M₁₆	—	2	1	no	—
537	M₁₆	—	2	1	no	—
538	C₂×C₈	—	2	1	no	—
539	C₂×C₈	—	2	1	no	—
540	C₂×C₈	—	2	1	no	—
541	C₂×C₈	—	2	1	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	1	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	1	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	4	2	no	—
551	C₄⋊C₄	—	4	2	no	—
552	C₂²⋊C₄	—	2	1	no	—
553	C₂²⋊C₄	—	2	1	no	—
554	C₂²⋊C₄	—	2	1	no	—
555	C₂²⋊C₄	—	2	1	no	—
556	C₂²⋊C₄	—	4	2	no	—
557	C₂²⋊C₄	—	4	2	no	—
558	C₂²⋊C₄	—	4	2	no	—
559	C₄²	—	2	1	no	—
560	C₄²	—	2	1	no	—
561	C₄²	—	4	2	no	—
562	C₄²	C₂×C₃×D₄	1	2	yes	—
563	C₄²	C₂×C₃×D₄	1	2	yes	—
564	C₂²×C₃	—	4	1	no	—
565	C₂²×C₃	—	4	1	no	—
566	C₂²×C₃	—	4	1	no	—
567	C₂²×C₃	—	4	1	no	—
568	C₂²×C₃	—	4	2	no	—
569	C₂²×C₃	—	8	1	no	—
570	C₂²×C₃	—	8	2	no	—
571	C₂²×C₃	—	8	1	no	—
572	C₂²×C₃	—	8	1	no	—
573	C₂²×C₃	—	8	2	no	—
574	C₂²×C₃	—	16	1	no	—
575	C₂²×C₃	—	16	2	no	—
576	C₂²×C₃	#534	1	1	yes	characteristic
577	C₂²×C₃	#534	1	1	yes	characteristic
578	C₂²×C₃	#538	1	1	yes	characteristic
579	C₃×C₄	—	4	2	no	—
580	C₃×C₄	—	4	2	no	—
581	C₃×C₄	—	4	1	no	—
582	C₃×C₄	—	4	1	no	—
583	C₃×C₄	—	4	1	no	—
584	C₃×C₄	—	4	2	no	—
585	C₃×C₄	—	8	1	no	—
586	C₃×C₄	—	8	1	no	—
587	C₃×C₄	D₄²	1	1	yes	characteristic
588	C₃×C₄	#546	1	1	yes	characteristic
589	C₃×C₄	#546	1	1	yes	characteristic
590	C₃×C₄	#560	1	1	yes	characteristic
591	C₂³	—	2	1	no	—
592	C₂³	—	2	2	no	—
593	C₂³	—	2	1	no	—
594	C₂³	—	2	1	no	—
595	C₂³	—	4	2	no	—
596	C₂³	—	4	1	no	—
597	C₂³	—	4	1	no	—
598	C₂³	—	16	1	no	—
599	C₂³	C₃×H(5,𝔽₂)	1	1	yes	characteristic
600	Q₈	—	2	2	no	—
601	Q₈	—	2	1	no	—
602	Q₈	—	2	1	no	—
603	Q₈	—	2	1	no	—
604	Q₈	—	2	2	no	—
605	Q₈	—	2	1	no	—
606	Q₈	—	2	1	no	—
607	Q₈	—	2	1	no	—
608	Q₈	—	4	2	no	—
609	Q₈	—	4	1	no	—
610	Q₈	—	4	1	no	—
611	D₄	—	2	1	no	—
612	D₄	—	2	2	no	—
613	D₄	—	2	1	no	—
614	D₄	—	2	1	no	—
615	D₄	—	2	2	no	—
616	D₄	—	2	1	no	—
617	D₄	—	2	1	no	—
618	D₄	—	2	1	no	—
619	D₄	—	4	2	no	—
620	D₄	—	4	2	no	—
621	D₄	—	4	2	no	—
622	D₄	—	4	1	no	—
623	D₄	—	4	1	no	—
624	D₄	—	4	1	no	—
625	D₄	—	4	1	no	—
626	D₄	—	4	1	no	—
627	D₄	—	4	1	no	—
628	D₄	—	4	1	no	—
629	D₄	—	4	1	no	—
630	D₄	—	16	2	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	1	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	1	no	—
636	C₂×C₄	—	2	1	no	—
637	C₂×C₄	—	4	2	no	—
638	C₂×C₄	—	4	2	no	—
639	C₂×C₄	—	4	2	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	4	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	16	2	no	—
651	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
652	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
653	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
654	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
655	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
656	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
657	C₈	—	4	1	no	—
658	C₈	—	4	1	no	—
659	C₈	—	4	1	no	—
660	C₈	—	4	1	no	—
661	C₈	—	4	1	no	—
662	C₈	—	4	1	no	—
663	C₂×C₃	—	2	1	no	—
664	C₂×C₃	—	2	1	no	—
665	C₂×C₃	—	2	1	no	—
666	C₂×C₃	—	8	2	no	—
667	C₂×C₃	—	8	1	no	—
668	C₂×C₃	—	8	1	no	—
669	C₂×C₃	—	8	1	no	—
670	C₂×C₃	#2668	1	1	yes	characteristic
671	C₂²	—	4	1	no	—
672	C₂²	—	4	1	no	—
673	C₂²	—	4	1	no	—
674	C₂²	—	4	1	no	—
675	C₂²	—	4	2	no	—
676	C₂²	—	8	1	no	—
677	C₂²	—	8	2	no	—
678	C₂²	—	8	1	no	—
679	C₂²	—	8	1	no	—
680	C₂²	—	8	2	no	—
681	C₂²	—	16	1	no	—
682	C₂²	—	16	2	no	—
683	C₂²	C₃×#534	1	1	yes	characteristic
684	C₂²	C₃×#534	1	1	yes	characteristic
685	C₂²	C₃×#538	1	1	yes	characteristic
686	C₄	—	4	2	no	—
687	C₄	—	4	2	no	—
688	C₄	—	4	1	no	—
689	C₄	—	4	1	no	—
690	C₄	—	4	1	no	—
691	C₄	—	4	2	no	—
692	C₄	—	8	1	no	—
693	C₄	—	8	1	no	—
694	C₄	C₃×D₄²	1	1	yes	characteristic
695	C₄	C₃×#546	1	1	yes	characteristic
696	C₄	C₃×#546	1	1	yes	characteristic
697	C₄	C₃×#560	1	1	yes	characteristic
698	C₃	#30442	1	1	yes	characteristic, direct product factor, Sylow
699	C₂	—	2	1	no	—
700	C₂	—	2	1	no	—
701	C₂	—	2	1	no	—
702	C₂	—	8	2	no	—
703	C₂	—	8	1	no	—
704	C₂	—	8	1	no	—
705	C₂	—	8	1	no	—
706	C₂	C₃×#2668	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#2668
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₃×#3021	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×#3043	C₂	semidirect
2. C₃×#3042	C₂	semidirect
3. C₃×#3022	C₂	semidirect
5. C₃×#2098	C₂	semidirect
6. C₃×#2080	C₂	semidirect
7. C₃×#2080	C₂	semidirect
8. C₃×#2049	C₂	semidirect
9. C₃×#2048	C₂	semidirect
10. C₃×#2019	C₂	semidirect
11. C₃×#1912	C₂	semidirect
12. C₃×#1911	C₂	semidirect
13. C₃×#1905	C₂	semidirect
14. C₃×#1904	C₂	semidirect
15. C₃×#1794	C₂	semidirect
16. #30442	C₃	direct (N × H)
22. C₂²×C₃×SD₁₆	C₂²	semidirect
25. C₃×#519	C₂²	semidirect
26. C₃×#518	C₂²	semidirect
35. C₃×#430	C₂²	semidirect
36. C₃×#424	C₂²	semidirect
37. C₃×#424	C₂²	semidirect
44. C₃×#419	C₂²	semidirect
45. C₃×#418	C₂²	semidirect
50. C₂×C₃×#101	C₂²	semidirect
51. C₂×C₃×#100	C₂²	semidirect
54. C₃×#344	C₂²	semidirect
55. C₃×#344	C₂²	semidirect
56. C₃×#341	C₂²	semidirect
57. C₃×H(3,ℤ/4)	C₂²	semidirect
58. C₃×H(3,ℤ/4)	C₂²	semidirect
59. C₃×H(3,ℤ/4)	C₂²	semidirect
60. #3043	C₂×C₃	semidirect
61. #3042	C₂×C₃	semidirect
62. #3022	C₂×C₃	semidirect
64. #2098	C₂×C₃	semidirect
65. #2080	C₂×C₃	semidirect
66. #2080	C₂×C₃	semidirect
67. #2049	C₂×C₃	semidirect
68. #2048	C₂×C₃	semidirect
69. #2019	C₂×C₃	semidirect
70. #1912	C₂×C₃	semidirect
71. #1911	C₂×C₃	semidirect
72. #1905	C₂×C₃	semidirect
73. #1904	C₂×C₃	semidirect
74. #1794	C₂×C₃	semidirect
115. C₃×C₄⋊Q₈	D₄	semidirect
118. C₃×#124	D₄	semidirect
161. C₂²×SD₁₆	C₂²×C₃	semidirect
164. #519	C₂²×C₃	semidirect
165. #518	C₂²×C₃	semidirect
174. #430	C₂²×C₃	semidirect
175. #424	C₂²×C₃	semidirect
176. #424	C₂²×C₃	semidirect
183. #419	C₂²×C₃	semidirect
184. #418	C₂²×C₃	semidirect
189. C₂×#101	C₂²×C₃	semidirect
190. C₂×#100	C₂²×C₃	semidirect
193. #344	C₂²×C₃	semidirect
194. #344	C₂²×C₃	semidirect
195. #341	C₂²×C₃	semidirect
196. H(3,ℤ/4)	C₂²×C₃	semidirect
197. H(3,ℤ/4)	C₂²×C₃	semidirect
198. H(3,ℤ/4)	C₂²×C₃	semidirect
345. C₄⋊Q₈	C₃×D₄	semidirect
348. #124	C₃×D₄	semidirect
698. C₃	#30442	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#3043	2
C₃×#3042	2
C₃×#3022	2
C₃×#3021	2
C₃×#2098	2
C₃×#2080	2
C₃×#2049	2
C₃×#2048	2
C₃×#2019	2
C₃×#1912	2
C₃×#1911	2
C₃×#1905	2
C₃×#1904	2
C₃×#1794	2
#30442	3

Subgroups — C3×#30442 #10649049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#30442 #10649049