Subgroups — C₃×#30440 #10649047

Order 768.

850 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3069	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#3044	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#3020	C₂	1	1	yes	maximal, characteristic
4	C₃×#3011	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2098	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2082	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2080	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2053	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2047	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#2009	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#1912	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#1909	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃×#1905	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃×#1881	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃×#1796	C₂	1	1	yes	maximal, characteristic, complemented
16	#30440	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
18	C₃×#575	C₂²	1	1	yes	characteristic
19	C₃×#575	C₂²	1	1	yes	characteristic
20	C₂×C₃×#137	C₂²	1	1	yes	characteristic
21	C₂×C₃×#137	C₂²	1	1	yes	characteristic
22	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
23	C₃×#568	C₂²	1	1	yes	characteristic
24	C₃×#536	C₂²	1	1	yes	characteristic
25	C₃×#536	C₂²	1	1	yes	characteristic
26	C₃×#519	C₂²	1	1	yes	characteristic
27	C₃×#473	—	2	2	no	—
28	C₃×#471	—	2	2	no	—
29	C₃×#464	—	2	2	no	—
30	C₃×#462	—	2	2	no	—
31	C₃×#456	—	2	2	no	—
32	C₃×#454	—	2	2	no	—
33	C₃×#452	—	2	2	no	—
34	C₃×#451	—	2	2	no	—
35	C₃×#430	C₂²	1	1	yes	characteristic
36	C₃×#424	C₂²	1	1	yes	characteristic
37	C₃×#424	C₂²	1	1	yes	characteristic
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×#419	C₂²	1	1	yes	characteristic
43	C₃×#419	C₂²	1	1	yes	characteristic
44	C₃×#419	C₂²	1	1	yes	characteristic
45	C₃×#419	C₂²	1	1	yes	characteristic
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	C₂×C₃×#101	C₂²	1	1	yes	characteristic
50	C₂×C₃×#101	C₂²	1	1	yes	characteristic
51	C₃×#410	C₂²	1	1	yes	characteristic
52	C₃×#410	C₂²	1	1	yes	characteristic
53	C₃×#408	C₂²	1	1	yes	characteristic
54	C₃×#344	C₂²	1	1	yes	characteristic
55	C₃×#344	C₂²	1	1	yes	characteristic
56	C₃×#341	C₂²	1	1	yes	characteristic
57	C₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
58	C₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
59	C₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
60	#3069	C₂×C₃	1	1	yes	characteristic, complemented
61	#3044	C₂×C₃	1	1	yes	characteristic, complemented
62	#3020	C₂×C₃	1	1	yes	characteristic
63	#3011	C₂×C₃	1	1	yes	characteristic, complemented
64	#2098	C₂×C₃	1	1	yes	characteristic, complemented
65	#2082	C₂×C₃	1	1	yes	characteristic, complemented
66	#2080	C₂×C₃	1	1	yes	characteristic, complemented
67	#2053	C₂×C₃	1	1	yes	characteristic, complemented
68	#2047	C₂×C₃	1	1	yes	characteristic, complemented
69	#2009	C₂×C₃	1	1	yes	characteristic, complemented
70	#1912	C₂×C₃	1	1	yes	characteristic, complemented
71	#1909	C₂×C₃	1	1	yes	characteristic, complemented
72	#1905	C₂×C₃	1	1	yes	characteristic, complemented
73	#1881	C₂×C₃	1	1	yes	characteristic, complemented
74	#1796	C₂×C₃	1	1	yes	characteristic, complemented
75	C₃×2₋¹⁺⁴	—	2	2	no	—
76	C₃×2₋¹⁺⁴	—	4	1	no	—
77	C₃×2₋¹⁺⁴	—	4	1	no	—
78	C₃×2₋¹⁺⁴	—	4	1	no	—
79	C₂×C₃×#41	—	2	1	no	—
80	C₂×C₃×#41	—	2	1	no	—
81	C₂×C₃×#41	—	4	1	no	—
82	C₂×C₃×#41	C₂³	1	1	yes	characteristic
83	C₂×C₃×#41	C₂³	1	1	yes	characteristic
84	C₂×C₃×#41	C₂³	1	1	yes	characteristic
85	C₂²×C₃×Q₈	—	2	1	no	—
86	C₂²×C₃×Q₈	—	2	1	no	—
87	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
88	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
89	C₃×#137	—	2	2	no	—
90	C₃×#137	—	2	1	no	—
91	C₃×#137	—	2	1	no	—
92	C₃×#137	—	2	1	no	—
93	C₃×#137	—	2	1	no	—
94	C₃×#137	—	4	1	no	—
95	C₃×#137	—	4	1	no	—
96	C₃×#137	—	4	1	no	—
97	C₃×AGL(1,ℤ/8)	—	2	1	no	—
98	C₃×AGL(1,ℤ/8)	—	2	1	no	—
99	C₃×AGL(1,ℤ/8)	—	2	1	no	—
100	C₃×AGL(1,ℤ/8)	—	2	1	no	—
101	C₃×#135	—	4	1	no	—
102	C₃×#135	—	4	1	no	—
103	C₂×C₃×Q₁₆	—	2	2	no	—
104	C₂×C₃×Q₁₆	—	2	2	no	—
105	C₂×C₃×Q₁₆	—	2	1	no	—
106	C₂×C₃×Q₁₆	—	4	1	no	—
107	C₂×C₃×Q₁₆	D₄	1	2	yes	—
108	C₂×C₃×SD₁₆	—	2	1	no	—
109	C₂×C₃×SD₁₆	D₄	1	2	yes	—
110	C₃×#131	—	2	2	no	—
111	C₃×#131	—	4	1	no	—
112	C₂×C₃×M₁₆	—	4	1	no	—
113	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
114	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
115	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
116	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
117	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
118	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
119	C₃×C₄⋊Q₈	—	2	1	no	—
120	C₃×C₄⋊Q₈	D₄	1	2	yes	—
121	C₃×#124	—	2	1	no	—
122	C₃×#124	D₄	1	2	yes	—
123	C₃×#123	—	4	1	no	—
124	C₃×#123	—	4	1	no	—
125	C₃×C₂²⋊Q₈	—	2	2	no	—
126	C₃×C₂²⋊Q₈	—	4	1	no	—
127	C₃×C₄×Q₈	—	2	2	no	—
128	C₃×C₄×Q₈	—	4	1	no	—
129	C₃×#117	—	4	1	no	—
130	C₃×#117	C₂³	1	1	yes	characteristic
131	C₃×#117	C₂³	1	1	yes	characteristic
132	C₃×#117	C₂³	1	1	yes	characteristic
133	C₃×#117	C₂³	1	1	yes	characteristic
134	C₃×#108	—	2	2	no	—
135	C₃×C₄⋊C₈	—	2	2	no	—
136	C₃×C₄⋊C₈	—	4	1	no	—
137	C₃×#104	—	2	2	no	—
138	C₃×#104	—	2	2	no	—
139	C₃×#104	—	4	1	no	—
140	C₃×#104	—	4	1	no	—
141	C₃×#103	—	2	1	no	—
142	C₃×#103	—	2	1	no	—
143	C₃×#103	—	2	2	no	—
144	C₃×#103	—	2	2	no	—
145	C₃×#103	—	4	1	no	—
146	C₃×#103	—	4	1	no	—
147	C₃×#102	—	2	1	no	—
148	C₃×#102	—	2	1	no	—
149	C₃×#101	—	2	2	no	—
150	C₃×#101	—	2	2	no	—
151	C₃×#101	—	2	1	no	—
152	C₃×#100	—	2	1	no	—
153	C₃×#99	—	4	1	no	—
154	C₃×#99	—	4	1	no	—
155	C₃×C₂²⋊C₈	—	2	2	no	—
156	C₂×2₋¹⁺⁴	C₂²×C₃	1	1	yes	characteristic
157	#575	C₂²×C₃	1	1	yes	characteristic
158	#575	C₂²×C₃	1	1	yes	characteristic
159	C₂×#137	C₂²×C₃	1	1	yes	characteristic
160	C₂×#137	C₂²×C₃	1	1	yes	characteristic
161	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
162	#568	C₂²×C₃	1	1	yes	characteristic
163	#536	C₂²×C₃	1	1	yes	characteristic
164	#536	C₂²×C₃	1	1	yes	characteristic
165	#519	C₂²×C₃	1	1	yes	characteristic
166	#473	—	2	2	no	—
167	#471	—	2	2	no	—
168	#464	—	2	2	no	—
169	#462	—	2	2	no	—
170	#456	—	2	2	no	—
171	#454	—	2	2	no	—
172	#452	—	2	2	no	—
173	#451	—	2	2	no	—
174	#430	C₂²×C₃	1	1	yes	characteristic
175	#424	C₂²×C₃	1	1	yes	characteristic
176	#424	C₂²×C₃	1	1	yes	characteristic
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	#419	C₂²×C₃	1	1	yes	characteristic
182	#419	C₂²×C₃	1	1	yes	characteristic
183	#419	C₂²×C₃	1	1	yes	characteristic
184	#419	C₂²×C₃	1	1	yes	characteristic
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	C₂×#101	C₂²×C₃	1	1	yes	characteristic
189	C₂×#101	C₂²×C₃	1	1	yes	characteristic
190	#410	C₂²×C₃	1	1	yes	characteristic
191	#410	C₂²×C₃	1	1	yes	characteristic
192	#408	C₂²×C₃	1	1	yes	characteristic
193	#344	C₂²×C₃	1	1	yes	characteristic
194	#344	C₂²×C₃	1	1	yes	characteristic
195	#341	C₂²×C₃	1	1	yes	characteristic
196	H(3,ℤ/4)	C₂²×C₃	1	1	yes	characteristic
197	H(3,ℤ/4)	C₂²×C₃	1	1	yes	characteristic
198	H(3,ℤ/4)	C₂²×C₃	1	1	yes	characteristic
199	C₃×#41	—	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	—	4	4	no	—
212	C₃×#41	—	4	2	no	—
213	C₃×#41	—	4	2	no	—
214	C₃×#41	—	4	2	no	—
215	C₃×#41	—	4	2	no	—
216	C₃×#41	—	4	4	no	—
217	C₂×C₃×Q₈	—	2	1	no	—
218	C₂×C₃×Q₈	—	2	1	no	—
219	C₂×C₃×Q₈	—	2	1	no	—
220	C₂×C₃×Q₈	—	2	1	no	—
221	C₂×C₃×Q₈	—	2	1	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	2	no	—
229	C₂×C₃×Q₈	—	4	2	no	—
230	C₂×C₃×Q₈	—	4	2	no	—
231	C₂×C₃×Q₈	—	4	2	no	—
232	C₂×C₃×Q₈	C₂×D₄	1	2	yes	—
233	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
234	C₂×C₃×Q₈	C₂×D₄	1	2	yes	—
235	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
236	C₂×C₃×Q₈	#41	1	1	yes	characteristic
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	—	4	1	no	—
241	C₂×C₃×D₄	C₂×D₄	1	1	yes	characteristic
242	C₂×C₃×D₄	C₂×D₄	1	1	yes	characteristic
243	C₂×C₃×D₄	#41	1	1	yes	characteristic
244	C₂²×C₃×C₄	—	2	1	no	—
245	C₂²×C₃×C₄	—	2	1	no	—
246	C₂²×C₃×C₄	—	2	1	no	—
247	C₂²×C₃×C₄	—	2	1	no	—
248	C₂²×C₃×C₄	—	2	1	no	—
249	C₂²×C₃×C₄	—	4	1	no	—
250	C₂²×C₃×C₄	—	4	1	no	—
251	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
252	C₃×Q₁₆	—	4	2	no	—
253	C₃×Q₁₆	—	4	4	no	—
254	C₃×Q₁₆	—	4	2	no	—
255	C₃×Q₁₆	—	4	1	no	—
256	C₃×Q₁₆	—	4	1	no	—
257	C₃×SD₁₆	—	4	2	no	—
258	C₃×SD₁₆	—	4	2	no	—
259	C₃×SD₁₆	—	4	1	no	—
260	C₃×SD₁₆	—	4	1	no	—
261	C₃×SD₁₆	—	4	1	no	—
262	C₃×SD₁₆	—	4	1	no	—
263	C₃×D₈	—	4	1	no	—
264	C₃×D₈	—	4	1	no	—
265	C₃×M₁₆	—	2	1	no	—
266	C₃×M₁₆	—	2	1	no	—
267	C₃×M₁₆	—	2	1	no	—
268	C₃×M₁₆	—	2	1	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	—	4	2	no	—
276	C₂×C₃×C₈	—	2	1	no	—
277	C₂×C₃×C₈	—	2	1	no	—
278	C₂×C₃×C₈	—	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	—	4	2	no	—
284	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
285	C₃×C₄⋊C₄	—	2	1	no	—
286	C₃×C₄⋊C₄	—	2	1	no	—
287	C₃×C₄⋊C₄	—	2	1	no	—
288	C₃×C₄⋊C₄	—	2	1	no	—
289	C₃×C₄⋊C₄	—	4	2	no	—
290	C₃×C₄⋊C₄	—	4	2	no	—
291	C₃×C₄⋊C₄	—	4	2	no	—
292	C₃×C₂²⋊C₄	—	2	1	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	—	4	2	no	—
297	C₃×C₂²⋊C₄	—	4	2	no	—
298	C₃×C₄²	—	2	1	no	—
299	C₃×C₄²	—	2	1	no	—
300	C₃×C₄²	—	2	1	no	—
301	C₃×C₄²	—	4	2	no	—
302	C₃×C₄²	C₂×D₄	1	2	yes	—
303	2₋¹⁺⁴	—	2	2	no	—
304	2₋¹⁺⁴	—	4	1	no	—
305	2₋¹⁺⁴	—	4	1	no	—
306	2₋¹⁺⁴	—	4	1	no	—
307	C₂×#41	—	2	1	no	—
308	C₂×#41	—	2	1	no	—
309	C₂×#41	—	4	1	no	—
310	C₂×#41	C₂³×C₃	1	1	yes	characteristic
311	C₂×#41	C₂³×C₃	1	1	yes	characteristic
312	C₂×#41	C₂³×C₃	1	1	yes	characteristic
313	C₂²×Q₈	—	2	1	no	—
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
316	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
317	#137	—	2	2	no	—
318	#137	—	2	1	no	—
319	#137	—	2	1	no	—
320	#137	—	2	1	no	—
321	#137	—	2	1	no	—
322	#137	—	4	1	no	—
323	#137	—	4	1	no	—
324	#137	—	4	1	no	—
325	AGL(1,ℤ/8)	—	2	1	no	—
326	AGL(1,ℤ/8)	—	2	1	no	—
327	AGL(1,ℤ/8)	—	2	1	no	—
328	AGL(1,ℤ/8)	—	2	1	no	—
329	#135	—	4	1	no	—
330	#135	—	4	1	no	—
331	C₂×Q₁₆	—	2	2	no	—
332	C₂×Q₁₆	—	2	2	no	—
333	C₂×Q₁₆	—	2	1	no	—
334	C₂×Q₁₆	—	4	1	no	—
335	C₂×Q₁₆	C₃×D₄	1	2	yes	—
336	C₂×SD₁₆	—	2	1	no	—
337	C₂×SD₁₆	C₃×D₄	1	2	yes	—
338	#131	—	2	2	no	—
339	#131	—	4	1	no	—
340	C₂×M₁₆	—	4	1	no	—
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	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
345	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
346	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
347	C₄⋊Q₈	—	2	1	no	—
348	C₄⋊Q₈	C₃×D₄	1	2	yes	—
349	#124	—	2	1	no	—
350	#124	C₃×D₄	1	2	yes	—
351	#123	—	4	1	no	—
352	#123	—	4	1	no	—
353	C₂²⋊Q₈	—	2	2	no	—
354	C₂²⋊Q₈	—	4	1	no	—
355	C₄×Q₈	—	2	2	no	—
356	C₄×Q₈	—	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	#108	—	2	2	no	—
363	C₄⋊C₈	—	2	2	no	—
364	C₄⋊C₈	—	4	1	no	—
365	#104	—	2	2	no	—
366	#104	—	2	2	no	—
367	#104	—	4	1	no	—
368	#104	—	4	1	no	—
369	#103	—	2	1	no	—
370	#103	—	2	1	no	—
371	#103	—	2	2	no	—
372	#103	—	2	2	no	—
373	#103	—	4	1	no	—
374	#103	—	4	1	no	—
375	#102	—	2	1	no	—
376	#102	—	2	1	no	—
377	#101	—	2	2	no	—
378	#101	—	2	2	no	—
379	#101	—	2	1	no	—
380	#100	—	2	1	no	—
381	#99	—	4	1	no	—
382	#99	—	4	1	no	—
383	C₂²⋊C₈	—	2	2	no	—
384	C₂³×C₃	—	2	1	no	—
385	C₂³×C₃	—	2	1	no	—
386	C₂³×C₃	—	2	1	no	—
387	C₂³×C₃	H(5,𝔽₂)	1	1	yes	characteristic
388	C₃×Q₈	—	2	2	no	—
389	C₃×Q₈	—	2	1	no	—
390	C₃×Q₈	—	2	1	no	—
391	C₃×Q₈	—	2	2	no	—
392	C₃×Q₈	—	2	1	no	—
393	C₃×Q₈	—	2	1	no	—
394	C₃×Q₈	—	2	2	no	—
395	C₃×Q₈	—	2	1	no	—
396	C₃×Q₈	—	2	1	no	—
397	C₃×Q₈	—	2	2	no	—
398	C₃×Q₈	—	4	4	no	—
399	C₃×Q₈	—	4	1	no	—
400	C₃×Q₈	—	4	1	no	—
401	C₃×Q₈	—	4	1	no	—
402	C₃×Q₈	—	4	1	no	—
403	C₃×D₄	—	2	1	no	—
404	C₃×D₄	—	2	1	no	—
405	C₃×D₄	—	2	1	no	—
406	C₃×D₄	—	2	1	no	—
407	C₃×D₄	—	2	1	no	—
408	C₃×D₄	—	2	1	no	—
409	C₃×D₄	—	4	4	no	—
410	C₃×D₄	—	4	1	no	—
411	C₃×D₄	—	4	1	no	—
412	C₃×D₄	—	4	1	no	—
413	C₃×D₄	—	4	1	no	—
414	C₃×D₄	—	4	1	no	—
415	C₃×D₄	—	4	1	no	—
416	C₂×C₃×C₄	—	2	2	no	—
417	C₂×C₃×C₄	—	2	2	no	—
418	C₂×C₃×C₄	—	2	1	no	—
419	C₂×C₃×C₄	—	2	2	no	—
420	C₂×C₃×C₄	—	2	1	no	—
421	C₂×C₃×C₄	—	2	1	no	—
422	C₂×C₃×C₄	—	4	4	no	—
423	C₂×C₃×C₄	—	4	4	no	—
424	C₂×C₃×C₄	—	4	1	no	—
425	C₂×C₃×C₄	—	4	1	no	—
426	C₂×C₃×C₄	—	4	1	no	—
427	C₂×C₃×C₄	—	4	1	no	—
428	C₂×C₃×C₄	—	4	1	no	—
429	C₂×C₃×C₄	—	4	1	no	—
430	C₂×C₃×C₄	—	4	1	no	—
431	C₂×C₃×C₄	—	4	1	no	—
432	C₂×C₃×C₄	—	4	1	no	—
433	C₂×C₃×C₄	—	4	1	no	—
434	C₂×C₃×C₄	—	4	1	no	—
435	C₂×C₃×C₄	—	4	1	no	—
436	C₂×C₃×C₄	—	4	1	no	—
437	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
438	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
439	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
440	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
441	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
442	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
443	C₃×C₈	—	2	2	no	—
444	C₃×C₈	—	4	1	no	—
445	C₃×C₈	—	4	1	no	—
446	C₃×C₈	—	4	1	no	—
447	C₃×C₈	—	4	1	no	—
448	C₃×C₈	—	4	1	no	—
449	#41	—	2	1	no	—
450	#41	—	2	1	no	—
451	#41	—	2	1	no	—
452	#41	—	2	1	no	—
453	#41	—	2	1	no	—
454	#41	—	2	1	no	—
455	#41	—	2	1	no	—
456	#41	—	2	1	no	—
457	#41	—	2	1	no	—
458	#41	—	2	1	no	—
459	#41	—	2	1	no	—
460	#41	—	2	1	no	—
461	#41	—	4	4	no	—
462	#41	—	4	2	no	—
463	#41	—	4	2	no	—
464	#41	—	4	2	no	—
465	#41	—	4	2	no	—
466	#41	—	4	4	no	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	—	2	1	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	4	1	no	—
478	C₂×Q₈	—	4	2	no	—
479	C₂×Q₈	—	4	2	no	—
480	C₂×Q₈	—	4	2	no	—
481	C₂×Q₈	—	4	2	no	—
482	C₂×Q₈	C₂×C₃×D₄	1	2	yes	—
483	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
484	C₂×Q₈	C₂×C₃×D₄	1	2	yes	—
485	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
486	C₂×Q₈	C₃×#41	1	1	yes	characteristic
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	2	1	no	—
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	4	1	no	—
491	C₂×D₄	C₂×C₃×D₄	1	1	yes	characteristic
492	C₂×D₄	C₂×C₃×D₄	1	1	yes	characteristic
493	C₂×D₄	C₃×#41	1	1	yes	characteristic
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	1	no	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	4	1	no	—
500	C₂²×C₄	—	4	1	no	—
501	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
502	Q₁₆	—	4	2	no	—
503	Q₁₆	—	4	4	no	—
504	Q₁₆	—	4	2	no	—
505	Q₁₆	—	4	1	no	—
506	Q₁₆	—	4	1	no	—
507	SD₁₆	—	4	2	no	—
508	SD₁₆	—	4	2	no	—
509	SD₁₆	—	4	1	no	—
510	SD₁₆	—	4	1	no	—
511	SD₁₆	—	4	1	no	—
512	SD₁₆	—	4	1	no	—
513	D₈	—	4	1	no	—
514	D₈	—	4	1	no	—
515	M₁₆	—	2	1	no	—
516	M₁₆	—	2	1	no	—
517	M₁₆	—	2	1	no	—
518	M₁₆	—	2	1	no	—
519	M₁₆	—	2	1	no	—
520	M₁₆	—	2	1	no	—
521	M₁₆	—	2	1	no	—
522	M₁₆	—	2	1	no	—
523	M₁₆	—	2	1	no	—
524	M₁₆	—	2	1	no	—
525	M₁₆	—	4	2	no	—
526	C₂×C₈	—	2	1	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	1	no	—
532	C₂×C₈	—	2	1	no	—
533	C₂×C₈	—	4	2	no	—
534	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	2	1	no	—
538	C₄⋊C₄	—	2	1	no	—
539	C₄⋊C₄	—	4	2	no	—
540	C₄⋊C₄	—	4	2	no	—
541	C₄⋊C₄	—	4	2	no	—
542	C₂²⋊C₄	—	2	1	no	—
543	C₂²⋊C₄	—	2	1	no	—
544	C₂²⋊C₄	—	2	1	no	—
545	C₂²⋊C₄	—	2	1	no	—
546	C₂²⋊C₄	—	4	2	no	—
547	C₂²⋊C₄	—	4	2	no	—
548	C₄²	—	2	1	no	—
549	C₄²	—	2	1	no	—
550	C₄²	—	2	1	no	—
551	C₄²	—	4	2	no	—
552	C₄²	C₂×C₃×D₄	1	2	yes	—
553	C₂²×C₃	—	4	1	no	—
554	C₂²×C₃	—	4	2	no	—
555	C₂²×C₃	—	4	1	no	—
556	C₂²×C₃	—	4	1	no	—
557	C₂²×C₃	—	4	1	no	—
558	C₂²×C₃	—	8	1	no	—
559	C₂²×C₃	—	8	1	no	—
560	C₂²×C₃	#534	1	1	yes	characteristic
561	C₂²×C₃	#546	1	1	yes	characteristic
562	C₂²×C₃	#560	1	1	yes	characteristic
563	C₃×C₄	—	4	2	no	—
564	C₃×C₄	—	4	2	no	—
565	C₃×C₄	—	4	1	no	—
566	C₃×C₄	—	4	1	no	—
567	C₃×C₄	—	4	1	no	—
568	C₃×C₄	—	4	2	no	—
569	C₃×C₄	—	8	1	no	—
570	C₃×C₄	—	8	1	no	—
571	C₃×C₄	—	8	1	no	—
572	C₃×C₄	#534	1	1	yes	characteristic
573	C₃×C₄	#538	1	1	yes	characteristic
574	C₃×C₄	D₄²	1	1	yes	characteristic
575	C₃×C₄	#546	1	1	yes	characteristic
576	C₂³	—	2	1	no	—
577	C₂³	—	2	1	no	—
578	C₂³	—	2	1	no	—
579	C₂³	C₃×H(5,𝔽₂)	1	1	yes	characteristic
580	Q₈	—	2	2	no	—
581	Q₈	—	2	1	no	—
582	Q₈	—	2	1	no	—
583	Q₈	—	2	2	no	—
584	Q₈	—	2	1	no	—
585	Q₈	—	2	1	no	—
586	Q₈	—	2	2	no	—
587	Q₈	—	2	1	no	—
588	Q₈	—	2	1	no	—
589	Q₈	—	2	2	no	—
590	Q₈	—	4	4	no	—
591	Q₈	—	4	1	no	—
592	Q₈	—	4	1	no	—
593	Q₈	—	4	1	no	—
594	Q₈	—	4	1	no	—
595	D₄	—	2	1	no	—
596	D₄	—	2	1	no	—
597	D₄	—	2	1	no	—
598	D₄	—	2	1	no	—
599	D₄	—	2	1	no	—
600	D₄	—	2	1	no	—
601	D₄	—	4	4	no	—
602	D₄	—	4	1	no	—
603	D₄	—	4	1	no	—
604	D₄	—	4	1	no	—
605	D₄	—	4	1	no	—
606	D₄	—	4	1	no	—
607	D₄	—	4	1	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	4	4	no	—
615	C₂×C₄	—	4	4	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
630	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
631	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
632	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
633	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
634	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
635	C₈	—	2	2	no	—
636	C₈	—	4	1	no	—
637	C₈	—	4	1	no	—
638	C₈	—	4	1	no	—
639	C₈	—	4	1	no	—
640	C₈	—	4	1	no	—
641	C₂×C₃	—	2	1	no	—
642	C₂×C₃	—	2	1	no	—
643	C₂×C₃	—	2	1	no	—
644	C₂×C₃	—	8	1	no	—
645	C₂×C₃	—	8	1	no	—
646	C₂×C₃	—	8	1	no	—
647	C₂×C₃	#2668	1	1	yes	characteristic
648	C₂²	—	4	1	no	—
649	C₂²	—	4	2	no	—
650	C₂²	—	4	1	no	—
651	C₂²	—	4	1	no	—
652	C₂²	—	4	1	no	—
653	C₂²	—	8	1	no	—
654	C₂²	—	8	1	no	—
655	C₂²	C₃×#534	1	1	yes	characteristic
656	C₂²	C₃×#546	1	1	yes	characteristic
657	C₂²	C₃×#560	1	1	yes	characteristic
658	C₄	—	4	2	no	—
659	C₄	—	4	2	no	—
660	C₄	—	4	1	no	—
661	C₄	—	4	1	no	—
662	C₄	—	4	1	no	—
663	C₄	—	4	2	no	—
664	C₄	—	8	1	no	—
665	C₄	—	8	1	no	—
666	C₄	—	8	1	no	—
667	C₄	C₃×#534	1	1	yes	characteristic
668	C₄	C₃×#538	1	1	yes	characteristic
669	C₄	C₃×D₄²	1	1	yes	characteristic
670	C₄	C₃×#546	1	1	yes	characteristic
671	C₃	#30440	1	1	yes	characteristic, direct product factor, Sylow
672	C₂	—	2	1	no	—
673	C₂	—	2	1	no	—
674	C₂	—	2	1	no	—
675	C₂	—	8	1	no	—
676	C₂	—	8	1	no	—
677	C₂	—	8	1	no	—
678	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×#3020	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×#3069	C₂	semidirect
2. C₃×#3044	C₂	semidirect
4. C₃×#3011	C₂	semidirect
5. C₃×#2098	C₂	semidirect
6. C₃×#2082	C₂	semidirect
7. C₃×#2080	C₂	semidirect
8. C₃×#2053	C₂	semidirect
9. C₃×#2047	C₂	semidirect
10. C₃×#2009	C₂	semidirect
11. C₃×#1912	C₂	semidirect
12. C₃×#1909	C₂	semidirect
13. C₃×#1905	C₂	semidirect
14. C₃×#1881	C₂	semidirect
15. C₃×#1796	C₂	semidirect
16. #30440	C₃	direct (N × H)
60. #3069	C₂×C₃	semidirect
61. #3044	C₂×C₃	semidirect
63. #3011	C₂×C₃	semidirect
64. #2098	C₂×C₃	semidirect
65. #2082	C₂×C₃	semidirect
66. #2080	C₂×C₃	semidirect
67. #2053	C₂×C₃	semidirect
68. #2047	C₂×C₃	semidirect
69. #2009	C₂×C₃	semidirect
70. #1912	C₂×C₃	semidirect
71. #1909	C₂×C₃	semidirect
72. #1905	C₂×C₃	semidirect
73. #1881	C₂×C₃	semidirect
74. #1796	C₂×C₃	semidirect
671. C₃	#30440	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×#3069	2
C₃×#3044	2
C₃×#3020	2
C₃×#3011	2
C₃×#2098	2
C₃×#2082	2
C₃×#2080	2
C₃×#2053	2
C₃×#2047	2
C₃×#2009	2
C₃×#1912	2
C₃×#1909	2
C₃×#1905	2
C₃×#1881	2
C₃×#1796	2
#30440	3

Subgroups — C3×#30440 #10649047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#30440 #10649047