Subgroups — C₃×#60431 #10679038

Order 768.

2782 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#577	C₂	1	2	yes	maximal, complemented
2	C₃×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#3045	C₂	1	6	yes	maximal, complemented
5	C₃×#3043	C₂	1	6	yes	maximal, complemented
6	C₃×#3038	C₂	1	3	yes	maximal, complemented
7	C₃×#3037	C₂	1	3	yes	maximal, complemented
8	C₃×#3035	C₂	1	3	yes	maximal, complemented
9	C₂×C₃×#460	C₂	1	3	yes	maximal, complemented
10	C₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#2897	C₂	1	2	yes	maximal, complemented
12	#60431	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
14	C₂×C₃×H(5,𝔽₂)	C₂²	1	2	yes	—
15	C₃×#577	—	2	16	no	—
16	C₂×C₃×#137	C₂²	1	6	yes	complemented
17	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
18	C₂×C₃×#135	C₂²	1	6	yes	complemented
19	C₂²×C₃×SD₁₆	C₂²	1	6	yes	complemented
20	C₂×C₃×#131	C₂²	1	2	yes	—
21	C₃×#550	—	2	1	no	—
22	C₃×#550	—	2	1	no	—
23	C₃×D₄²	—	2	3	no	—
24	C₃×D₄²	—	2	3	no	—
25	C₃×#535	C₂²	1	6	yes	—
26	C₃×#532	C₂²	1	3	yes	—
27	C₂×C₃×#127	C₂²	1	3	yes	—
28	C₃×#520	—	2	3	no	—
29	C₃×#520	C₂²	1	1	yes	characteristic
30	C₃×#520	C₂²	1	3	yes	complemented
31	C₃×#519	—	2	1	no	—
32	C₃×#519	C₂²	1	3	yes	complemented
33	C₃×C₄×#41	—	2	3	no	—
34	C₃×C₄×#41	C₂²	1	1	yes	characteristic
35	C₃×C₄×#41	C₂²	1	3	yes	complemented
36	C₂×C₃×C₄×Q₈	—	2	1	no	—
37	C₂×C₃×C₄×Q₈	C₂²	1	3	yes	complemented
38	C₃×#464	C₂²	1	6	yes	complemented
39	C₃×#464	C₂²	1	12	yes	complemented
40	C₃×#463	C₂²	1	6	yes	complemented
41	C₃×#462	C₂²	1	3	yes	complemented
42	C₃×#461	C₂²	1	1	yes	characteristic, complemented
43	C₃×#460	C₂²	1	3	yes	complemented
44	C₃×#460	C₂²	1	6	yes	complemented
45	C₃×#460	C₂²	1	12	yes	complemented
46	C₃×#460	C₂²	1	6	yes	complemented
47	C₃×#459	C₂²	1	3	yes	complemented
48	C₃×#459	C₂²	1	6	yes	complemented
49	C₃×#424	C₂²	1	6	yes	complemented
50	C₃×#423	C₂²	1	3	yes	complemented
51	C₂×C₃×C₄⋊C₈	C₂²	1	3	yes	complemented
52	C₃×#418	C₂²	1	6	yes	complemented
53	C₃×#417	C₂²	1	6	yes	complemented
54	C₃×#416	C₂²	1	6	yes	complemented
55	C₂×C₃×#102	C₂²	1	6	yes	complemented
56	C₂×#577	C₂×C₃	1	2	yes	complemented
57	#3498	C₂×C₃	1	1	yes	characteristic, complemented
58	C₄×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic, complemented
59	#3045	C₂×C₃	1	6	yes	complemented
60	#3043	C₂×C₃	1	6	yes	complemented
61	#3038	C₂×C₃	1	3	yes	complemented
62	#3037	C₂×C₃	1	3	yes	complemented
63	#3035	C₂×C₃	1	3	yes	complemented
64	C₂×#460	C₂×C₃	1	3	yes	complemented
65	#2906	C₂×C₃	1	1	yes	characteristic, complemented
66	#2897	C₂×C₃	1	2	yes	complemented
67	C₃×2₋¹⁺⁴	—	2	4	no	—
68	C₃×2₋¹⁺⁴	D₄	1	8	yes	complemented
69	C₃×H(5,𝔽₂)	—	2	8	no	—
70	C₃×H(5,𝔽₂)	—	2	8	no	—
71	C₂×C₃×#41	—	2	2	no	—
72	C₂×C₃×#41	—	2	3	no	—
73	C₂×C₃×#41	C₂³	1	6	yes	—
74	C₂×C₃×#41	C₂³	1	3	yes	—
75	C₂×C₃×#41	C₂³	1	1	yes	characteristic
76	C₂²×C₃×Q₈	—	2	1	no	—
77	C₂²×C₃×Q₈	C₂³	1	3	yes	—
78	C₂²×C₃×D₄	—	2	6	no	—
79	C₂²×C₃×D₄	—	2	3	no	—
80	C₂²×C₃×D₄	—	2	3	no	—
81	C₂²×C₃×D₄	C₂³	1	6	yes	—
82	C₃×#137	—	2	24	no	—
83	C₃×AGL(1,ℤ/8)	—	2	24	no	—
84	C₃×#135	—	2	24	no	—
85	C₂×C₃×Q₁₆	C₂³	1	6	yes	—
86	C₂×C₃×SD₁₆	—	2	24	no	—
87	C₂×C₃×SD₁₆	C₂³	1	6	yes	—
88	C₂×C₃×SD₁₆	C₂³	1	12	yes	—
89	C₂×C₃×SD₁₆	C₂³	1	2	yes	—
90	C₂×C₃×D₈	C₂³	1	6	yes	—
91	C₃×#131	—	2	8	no	—
92	C₂×C₃×M₁₆	C₂³	1	6	yes	—
93	C₂²×C₃×C₈	C₂³	1	6	yes	—
94	C₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
95	C₃×#127	—	2	3	no	—
96	C₃×#127	—	2	3	no	—
97	C₃×#127	C₂³	1	3	yes	—
98	C₃×#127	C₂³	1	6	yes	—
99	C₃×#124	C₂³	1	6	yes	—
100	C₃×#121	—	2	3	no	—
101	C₃×#121	—	2	3	no	—
102	C₃×#121	—	2	6	no	—
103	C₃×#121	—	2	6	no	—
104	C₃×#120	—	2	12	no	—
105	C₃×C₄×Q₈	—	2	1	no	—
106	C₃×C₄×Q₈	—	2	3	no	—
107	C₃×C₄×Q₈	—	2	3	no	—
108	C₃×C₄×Q₈	—	2	3	no	—
109	C₃×C₄×Q₈	—	2	3	no	—
110	C₃×C₄×Q₈	D₄	1	1	yes	characteristic
111	C₃×C₄×Q₈	D₄	1	1	yes	characteristic
112	C₃×C₄×Q₈	C₂³	1	3	yes	—
113	C₃×C₄×Q₈	C₂³	1	6	yes	—
114	C₃×C₄×Q₈	C₂³	1	3	yes	—
115	C₃×C₄×D₄	—	2	3	no	—
116	C₃×C₄×D₄	—	2	3	no	—
117	C₃×C₄×D₄	—	2	3	no	—
118	C₃×C₄×D₄	—	2	6	no	—
119	C₃×C₄×D₄	—	2	3	no	—
120	C₃×C₄×D₄	—	2	3	no	—
121	C₃×C₄×D₄	D₄	1	3	yes	—
122	C₃×C₄×D₄	D₄	1	3	yes	—
123	C₃×C₄×D₄	C₂³	1	1	yes	characteristic
124	C₃×C₄×D₄	C₂³	1	3	yes	—
125	C₃×#117	—	2	3	no	—
126	C₃×#117	—	2	6	no	—
127	C₃×#117	—	2	3	no	—
128	C₃×#117	C₂³	1	3	yes	—
129	C₃×#117	C₂³	1	3	yes	—
130	C₂×C₃×C₄⋊C₄	—	2	3	no	—
131	C₂×C₃×C₄⋊C₄	—	2	3	no	—
132	C₂×C₃×C₄⋊C₄	C₂³	1	3	yes	—
133	C₂×C₃×C₄²	—	2	3	no	—
134	C₂×C₃×C₄²	—	2	3	no	—
135	C₂×C₃×C₄²	C₂³	1	3	yes	—
136	C₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
137	C₃×C₄⋊C₈	C₂³	1	3	yes	—
138	C₃×C₄⋊C₈	C₂³	1	6	yes	—
139	C₃×C₄⋊C₈	C₂³	1	6	yes	—
140	C₃×#103	C₂³	1	2	yes	—
141	C₃×#103	C₂³	1	6	yes	—
142	C₃×#102	C₂³	1	6	yes	—
143	C₃×#102	C₂³	1	12	yes	—
144	C₃×#102	C₂³	1	6	yes	—
145	C₂×2₋¹⁺⁴	C₂²×C₃	1	1	yes	characteristic
146	C₂×H(5,𝔽₂)	C₂²×C₃	1	2	yes	—
147	#577	—	2	16	no	—
148	C₂×#137	C₂²×C₃	1	6	yes	complemented
149	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	6	yes	complemented
150	C₂×#135	C₂²×C₃	1	6	yes	complemented
151	C₂²×SD₁₆	C₂²×C₃	1	6	yes	complemented
152	C₂×#131	C₂²×C₃	1	2	yes	—
153	#550	—	2	1	no	—
154	#550	—	2	1	no	—
155	D₄²	—	2	3	no	—
156	D₄²	—	2	3	no	—
157	#535	C₂²×C₃	1	6	yes	—
158	#532	C₂²×C₃	1	3	yes	—
159	C₂×#127	C₂²×C₃	1	3	yes	—
160	#520	—	2	3	no	—
161	#520	C₂²×C₃	1	1	yes	characteristic
162	#520	C₂²×C₃	1	3	yes	complemented
163	#519	—	2	1	no	—
164	#519	C₂²×C₃	1	3	yes	complemented
165	C₄×#41	—	2	3	no	—
166	C₄×#41	C₂²×C₃	1	1	yes	characteristic
167	C₄×#41	C₂²×C₃	1	3	yes	complemented
168	C₂×C₄×Q₈	—	2	1	no	—
169	C₂×C₄×Q₈	C₂²×C₃	1	3	yes	complemented
170	#464	C₂²×C₃	1	6	yes	complemented
171	#464	C₂²×C₃	1	12	yes	complemented
172	#463	C₂²×C₃	1	6	yes	complemented
173	#462	C₂²×C₃	1	3	yes	complemented
174	#461	C₂²×C₃	1	1	yes	characteristic, complemented
175	#460	C₂²×C₃	1	3	yes	complemented
176	#460	C₂²×C₃	1	6	yes	complemented
177	#460	C₂²×C₃	1	12	yes	complemented
178	#460	C₂²×C₃	1	6	yes	complemented
179	#459	C₂²×C₃	1	3	yes	complemented
180	#459	C₂²×C₃	1	6	yes	complemented
181	#424	C₂²×C₃	1	6	yes	complemented
182	#423	C₂²×C₃	1	3	yes	complemented
183	C₂×C₄⋊C₈	C₂²×C₃	1	3	yes	complemented
184	#418	C₂²×C₃	1	6	yes	complemented
185	#417	C₂²×C₃	1	6	yes	complemented
186	#416	C₂²×C₃	1	6	yes	complemented
187	C₂×#102	C₂²×C₃	1	6	yes	complemented
188	C₂⁴×C₃	—	2	6	no	—
189	C₃×#41	—	2	24	no	—
190	C₃×#41	—	2	6	no	—
191	C₃×#41	—	2	12	no	—
192	C₃×#41	—	2	12	no	—
193	C₃×#41	—	4	8	no	—
194	C₃×#41	C₂×D₄	1	4	yes	—
195	C₃×#41	C₂×D₄	1	12	yes	complemented
196	C₃×#41	#41	1	4	yes	—
197	C₂×C₃×Q₈	—	2	8	no	—
198	C₂×C₃×Q₈	—	2	3	no	—
199	C₂×C₃×Q₈	—	2	6	no	—
200	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
201	C₂×C₃×Q₈	C₂×D₄	1	12	yes	complemented
202	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
203	C₂×C₃×Q₈	C₂⁴	1	3	yes	—
204	C₂×C₃×D₄	—	2	3	no	—
205	C₂×C₃×D₄	—	2	6	no	—
206	C₂×C₃×D₄	—	2	12	no	—
207	C₂×C₃×D₄	—	2	3	no	—
208	C₂×C₃×D₄	—	2	3	no	—
209	C₂×C₃×D₄	—	2	6	no	—
210	C₂×C₃×D₄	—	2	6	no	—
211	C₂×C₃×D₄	—	2	6	no	—
212	C₂×C₃×D₄	—	2	12	no	—
213	C₂×C₃×D₄	—	4	24	no	—
214	C₂×C₃×D₄	—	8	6	no	—
215	C₂×C₃×D₄	—	8	6	no	—
216	C₂×C₃×D₄	C₂×D₄	1	3	yes	—
217	C₂×C₃×D₄	C₂⁴	1	6	yes	—
218	C₂×C₃×D₄	C₂⁴	1	1	yes	characteristic
219	C₂²×C₃×C₄	—	2	3	no	—
220	C₂²×C₃×C₄	—	2	3	no	—
221	C₂²×C₃×C₄	—	2	3	no	—
222	C₂²×C₃×C₄	—	2	1	no	—
223	C₂²×C₃×C₄	—	2	6	no	—
224	C₂²×C₃×C₄	—	2	3	no	—
225	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
226	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
227	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
228	C₃×Q₁₆	—	2	12	no	—
229	C₃×SD₁₆	—	2	12	no	—
230	C₃×SD₁₆	—	2	24	no	—
231	C₃×SD₁₆	—	2	4	no	—
232	C₃×D₈	—	2	12	no	—
233	C₃×M₁₆	—	2	12	no	—
234	C₂×C₃×C₈	—	2	12	no	—
235	C₂×C₃×C₈	C₂⁴	1	2	yes	—
236	C₂×C₃×C₈	C₂⁴	1	6	yes	—
237	C₃×C₄⋊C₄	—	2	6	no	—
238	C₃×C₄⋊C₄	—	2	3	no	—
239	C₃×C₄⋊C₄	—	2	3	no	—
240	C₃×C₄⋊C₄	—	2	3	no	—
241	C₃×C₄⋊C₄	—	2	1	no	—
242	C₃×C₄⋊C₄	—	2	1	no	—
243	C₃×C₄⋊C₄	C₂×D₄	1	3	yes	—
244	C₃×C₄⋊C₄	C₂×D₄	1	3	yes	—
245	C₃×C₄⋊C₄	C₂⁴	1	3	yes	—
246	C₃×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
247	C₃×C₂²⋊C₄	—	2	3	no	—
248	C₃×C₂²⋊C₄	—	2	1	no	—
249	C₃×C₂²⋊C₄	—	2	3	no	—
250	C₃×C₂²⋊C₄	—	2	6	no	—
251	C₃×C₂²⋊C₄	—	2	6	no	—
252	C₃×C₂²⋊C₄	C₂×D₄	1	6	yes	—
253	C₃×C₄²	—	2	3	no	—
254	C₃×C₄²	—	2	3	no	—
255	C₃×C₄²	—	2	6	no	—
256	C₃×C₄²	—	2	3	no	—
257	C₃×C₄²	C₂×D₄	1	3	yes	—
258	C₃×C₄²	C₂×D₄	1	3	yes	—
259	C₃×C₄²	C₂⁴	1	1	yes	characteristic
260	C₃×C₄²	C₂⁴	1	3	yes	—
261	2₋¹⁺⁴	—	2	4	no	—
262	2₋¹⁺⁴	C₃×D₄	1	8	yes	complemented
263	H(5,𝔽₂)	—	2	8	no	—
264	H(5,𝔽₂)	—	2	8	no	—
265	C₂×#41	—	2	2	no	—
266	C₂×#41	—	2	3	no	—
267	C₂×#41	C₂³×C₃	1	6	yes	—
268	C₂×#41	C₂³×C₃	1	3	yes	—
269	C₂×#41	C₂³×C₃	1	1	yes	characteristic
270	C₂²×Q₈	—	2	1	no	—
271	C₂²×Q₈	C₂³×C₃	1	3	yes	—
272	C₂²×D₄	—	2	6	no	—
273	C₂²×D₄	—	2	3	no	—
274	C₂²×D₄	—	2	3	no	—
275	C₂²×D₄	C₂³×C₃	1	6	yes	—
276	#137	—	2	24	no	—
277	AGL(1,ℤ/8)	—	2	24	no	—
278	#135	—	2	24	no	—
279	C₂×Q₁₆	C₂³×C₃	1	6	yes	—
280	C₂×SD₁₆	—	2	24	no	—
281	C₂×SD₁₆	C₂³×C₃	1	6	yes	—
282	C₂×SD₁₆	C₂³×C₃	1	12	yes	—
283	C₂×SD₁₆	C₂³×C₃	1	2	yes	—
284	C₂×D₈	C₂³×C₃	1	6	yes	—
285	#131	—	2	8	no	—
286	C₂×M₁₆	C₂³×C₃	1	6	yes	—
287	C₂²×C₈	C₂³×C₃	1	6	yes	—
288	C₄⋊Q₈	C₂³×C₃	1	1	yes	characteristic
289	#127	—	2	3	no	—
290	#127	—	2	3	no	—
291	#127	C₂³×C₃	1	3	yes	—
292	#127	C₂³×C₃	1	6	yes	—
293	#124	C₂³×C₃	1	6	yes	—
294	#121	—	2	3	no	—
295	#121	—	2	3	no	—
296	#121	—	2	6	no	—
297	#121	—	2	6	no	—
298	#120	—	2	12	no	—
299	C₄×Q₈	—	2	1	no	—
300	C₄×Q₈	—	2	3	no	—
301	C₄×Q₈	—	2	3	no	—
302	C₄×Q₈	—	2	3	no	—
303	C₄×Q₈	—	2	3	no	—
304	C₄×Q₈	C₃×D₄	1	1	yes	characteristic
305	C₄×Q₈	C₃×D₄	1	1	yes	characteristic
306	C₄×Q₈	C₂³×C₃	1	3	yes	—
307	C₄×Q₈	C₂³×C₃	1	6	yes	—
308	C₄×Q₈	C₂³×C₃	1	3	yes	—
309	C₄×D₄	—	2	3	no	—
310	C₄×D₄	—	2	3	no	—
311	C₄×D₄	—	2	3	no	—
312	C₄×D₄	—	2	6	no	—
313	C₄×D₄	—	2	3	no	—
314	C₄×D₄	—	2	3	no	—
315	C₄×D₄	C₃×D₄	1	3	yes	—
316	C₄×D₄	C₃×D₄	1	3	yes	—
317	C₄×D₄	C₂³×C₃	1	1	yes	characteristic
318	C₄×D₄	C₂³×C₃	1	3	yes	—
319	#117	—	2	3	no	—
320	#117	—	2	6	no	—
321	#117	—	2	3	no	—
322	#117	C₂³×C₃	1	3	yes	—
323	#117	C₂³×C₃	1	3	yes	—
324	C₂×C₄⋊C₄	—	2	3	no	—
325	C₂×C₄⋊C₄	—	2	3	no	—
326	C₂×C₄⋊C₄	C₂³×C₃	1	3	yes	—
327	C₂×C₄²	—	2	3	no	—
328	C₂×C₄²	—	2	3	no	—
329	C₂×C₄²	C₂³×C₃	1	3	yes	—
330	C₄⋊C₈	C₂³×C₃	1	1	yes	characteristic
331	C₄⋊C₈	C₂³×C₃	1	3	yes	—
332	C₄⋊C₈	C₂³×C₃	1	6	yes	—
333	C₄⋊C₈	C₂³×C₃	1	6	yes	—
334	#103	C₂³×C₃	1	2	yes	—
335	#103	C₂³×C₃	1	6	yes	—
336	#102	C₂³×C₃	1	6	yes	—
337	#102	C₂³×C₃	1	12	yes	—
338	#102	C₂³×C₃	1	6	yes	—
339	C₂³×C₃	—	2	1	no	—
340	C₂³×C₃	—	2	6	no	—
341	C₂³×C₃	—	4	12	no	—
342	C₂³×C₃	—	8	6	no	—
343	C₂³×C₃	—	8	6	no	—
344	C₂³×C₃	C₂²×D₄	1	3	yes	—
345	C₃×Q₈	—	2	12	no	—
346	C₃×Q₈	—	2	3	no	—
347	C₃×Q₈	—	2	2	no	—
348	C₃×Q₈	—	2	2	no	—
349	C₃×Q₈	#121	1	4	yes	—
350	C₃×Q₈	C₂²×D₄	1	6	yes	—
351	C₃×D₄	—	2	12	no	—
352	C₃×D₄	—	2	6	no	—
353	C₃×D₄	—	2	1	no	—
354	C₃×D₄	—	2	6	no	—
355	C₃×D₄	—	4	6	no	—
356	C₃×D₄	—	4	12	no	—
357	C₃×D₄	—	4	12	no	—
358	C₃×D₄	—	4	6	no	—
359	C₃×D₄	—	8	6	no	—
360	C₃×D₄	—	8	6	no	—
361	C₃×D₄	#121	1	12	yes	—
362	C₃×D₄	C₂²×D₄	1	2	yes	—
363	C₂×C₃×C₄	—	2	12	no	—
364	C₂×C₃×C₄	—	2	3	no	—
365	C₂×C₃×C₄	—	2	12	no	—
366	C₂×C₃×C₄	—	2	6	no	—
367	C₂×C₃×C₄	—	2	1	no	—
368	C₂×C₃×C₄	—	2	3	no	—
369	C₂×C₃×C₄	—	2	2	no	—
370	C₂×C₃×C₄	—	4	3	no	—
371	C₂×C₃×C₄	—	4	12	no	—
372	C₂×C₃×C₄	—	4	2	no	—
373	C₂×C₃×C₄	—	8	3	no	—
374	C₂×C₃×C₄	—	8	3	no	—
375	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
376	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
377	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
378	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
379	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
380	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
381	C₂×C₃×C₄	C₂×#41	1	6	yes	—
382	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
383	C₃×C₈	—	2	2	no	—
384	C₃×C₈	—	2	6	no	—
385	C₂⁴	—	2	6	no	—
386	#41	—	2	24	no	—
387	#41	—	2	6	no	—
388	#41	—	2	12	no	—
389	#41	—	2	12	no	—
390	#41	—	4	8	no	—
391	#41	C₂×C₃×D₄	1	4	yes	—
392	#41	C₂×C₃×D₄	1	12	yes	complemented
393	#41	C₃×#41	1	4	yes	—
394	C₂×Q₈	—	2	8	no	—
395	C₂×Q₈	—	2	3	no	—
396	C₂×Q₈	—	2	6	no	—
397	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
398	C₂×Q₈	C₂×C₃×D₄	1	12	yes	complemented
399	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
400	C₂×Q₈	C₂⁴×C₃	1	3	yes	—
401	C₂×D₄	—	2	3	no	—
402	C₂×D₄	—	2	6	no	—
403	C₂×D₄	—	2	12	no	—
404	C₂×D₄	—	2	3	no	—
405	C₂×D₄	—	2	3	no	—
406	C₂×D₄	—	2	6	no	—
407	C₂×D₄	—	2	6	no	—
408	C₂×D₄	—	2	6	no	—
409	C₂×D₄	—	2	12	no	—
410	C₂×D₄	—	4	24	no	—
411	C₂×D₄	—	8	6	no	—
412	C₂×D₄	—	8	6	no	—
413	C₂×D₄	C₂×C₃×D₄	1	3	yes	—
414	C₂×D₄	C₂⁴×C₃	1	6	yes	—
415	C₂×D₄	C₂⁴×C₃	1	1	yes	characteristic
416	C₂²×C₄	—	2	3	no	—
417	C₂²×C₄	—	2	3	no	—
418	C₂²×C₄	—	2	3	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	6	no	—
421	C₂²×C₄	—	2	3	no	—
422	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
423	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
424	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
425	Q₁₆	—	2	12	no	—
426	SD₁₆	—	2	12	no	—
427	SD₁₆	—	2	24	no	—
428	SD₁₆	—	2	4	no	—
429	D₈	—	2	12	no	—
430	M₁₆	—	2	12	no	—
431	C₂×C₈	—	2	12	no	—
432	C₂×C₈	C₂⁴×C₃	1	2	yes	—
433	C₂×C₈	C₂⁴×C₃	1	6	yes	—
434	C₄⋊C₄	—	2	6	no	—
435	C₄⋊C₄	—	2	3	no	—
436	C₄⋊C₄	—	2	3	no	—
437	C₄⋊C₄	—	2	3	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	C₂×C₃×D₄	1	3	yes	—
441	C₄⋊C₄	C₂×C₃×D₄	1	3	yes	—
442	C₄⋊C₄	C₂⁴×C₃	1	3	yes	—
443	C₄⋊C₄	C₂⁴×C₃	1	1	yes	characteristic
444	C₂²⋊C₄	—	2	3	no	—
445	C₂²⋊C₄	—	2	1	no	—
446	C₂²⋊C₄	—	2	3	no	—
447	C₂²⋊C₄	—	2	6	no	—
448	C₂²⋊C₄	—	2	6	no	—
449	C₂²⋊C₄	C₂×C₃×D₄	1	6	yes	—
450	C₄²	—	2	3	no	—
451	C₄²	—	2	3	no	—
452	C₄²	—	2	6	no	—
453	C₄²	—	2	3	no	—
454	C₄²	C₂×C₃×D₄	1	3	yes	—
455	C₄²	C₂×C₃×D₄	1	3	yes	—
456	C₄²	C₂⁴×C₃	1	1	yes	characteristic
457	C₄²	C₂⁴×C₃	1	3	yes	—
458	C₂²×C₃	—	2	3	no	—
459	C₂²×C₃	—	2	3	no	—
460	C₂²×C₃	—	2	2	no	—
461	C₂²×C₃	—	4	1	no	—
462	C₂²×C₃	—	4	1	no	—
463	C₂²×C₃	—	4	6	no	—
464	C₂²×C₃	—	4	6	no	—
465	C₂²×C₃	—	4	6	no	—
466	C₂²×C₃	—	16	12	no	—
467	C₂²×C₃	C₂×#121	1	6	yes	—
468	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
469	C₃×C₄	—	2	1	no	—
470	C₃×C₄	—	2	3	no	—
471	C₃×C₄	—	2	6	no	—
472	C₃×C₄	—	2	3	no	—
473	C₃×C₄	—	4	3	no	—
474	C₃×C₄	—	4	2	no	—
475	C₃×C₄	—	4	1	no	—
476	C₃×C₄	C₂×#121	1	6	yes	—
477	C₃×C₄	#577	1	2	yes	complemented
478	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
479	C₃×C₄	C₂²×#41	1	1	yes	characteristic
480	C₂³	—	2	1	no	—
481	C₂³	—	2	6	no	—
482	C₂³	—	4	12	no	—
483	C₂³	—	8	6	no	—
484	C₂³	—	8	6	no	—
485	C₂³	C₂²×C₃×D₄	1	3	yes	—
486	Q₈	—	2	3	no	—
487	Q₈	—	2	12	no	—
488	Q₈	—	2	2	no	—
489	Q₈	—	2	2	no	—
490	Q₈	C₃×#121	1	4	yes	—
491	Q₈	C₂²×C₃×D₄	1	6	yes	—
492	D₄	—	2	12	no	—
493	D₄	—	2	6	no	—
494	D₄	—	2	1	no	—
495	D₄	—	2	6	no	—
496	D₄	—	4	6	no	—
497	D₄	—	4	12	no	—
498	D₄	—	4	6	no	—
499	D₄	—	4	12	no	—
500	D₄	—	8	6	no	—
501	D₄	—	8	6	no	—
502	D₄	C₃×#121	1	12	yes	—
503	D₄	C₂²×C₃×D₄	1	2	yes	—
504	C₂×C₄	—	2	3	no	—
505	C₂×C₄	—	2	12	no	—
506	C₂×C₄	—	2	6	no	—
507	C₂×C₄	—	2	12	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	3	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	4	3	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	12	no	—
514	C₂×C₄	—	8	3	no	—
515	C₂×C₄	—	8	3	no	—
516	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
517	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
518	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
519	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
520	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
521	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
522	C₂×C₄	C₂×C₃×#41	1	6	yes	—
523	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
524	C₈	—	2	2	no	—
525	C₈	—	2	6	no	—
526	C₂×C₃	—	2	6	no	—
527	C₂×C₃	—	4	2	no	—
528	C₂×C₃	—	8	6	no	—
529	C₂×C₃	C₂²×#121	1	1	yes	characteristic
530	C₂×C₃	C₂×#577	1	1	yes	characteristic
531	C₂×C₃	#3585	1	1	yes	characteristic
532	C₂²	—	2	3	no	—
533	C₂²	—	2	3	no	—
534	C₂²	—	2	2	no	—
535	C₂²	—	4	1	no	—
536	C₂²	—	4	1	no	—
537	C₂²	—	4	6	no	—
538	C₂²	—	4	6	no	—
539	C₂²	—	4	6	no	—
540	C₂²	—	16	12	no	—
541	C₂²	C₂×C₃×#121	1	6	yes	—
542	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
543	C₄	—	2	6	no	—
544	C₄	—	2	3	no	—
545	C₄	—	2	1	no	—
546	C₄	—	2	3	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	3	no	—
550	C₄	C₂×C₃×#121	1	6	yes	—
551	C₄	C₃×#577	1	2	yes	complemented
552	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
553	C₄	C₂²×C₃×#41	1	1	yes	characteristic
554	C₃	#60431	1	1	yes	characteristic, direct product factor, Sylow
555	C₂	—	2	6	no	—
556	C₂	—	4	2	no	—
557	C₂	—	8	6	no	—
558	C₂	C₂²×C₃×#121	1	1	yes	characteristic
559	C₂	C₂×C₃×#577	1	1	yes	characteristic
560	C₂	C₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#577	C₂	semidirect
2. C₃×#3498	C₂	semidirect
3. C₃×C₄×2₋¹⁺⁴	C₂	semidirect
4. C₃×#3045	C₂	semidirect
5. C₃×#3043	C₂	semidirect
6. C₃×#3038	C₂	semidirect
7. C₃×#3037	C₂	semidirect
8. C₃×#3035	C₂	semidirect
9. C₂×C₃×#460	C₂	semidirect
10. C₃×#2906	C₂	semidirect
11. C₃×#2897	C₂	semidirect
12. #60431	C₃	direct (N × H)
16. C₂×C₃×#137	C₂²	semidirect
17. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
18. C₂×C₃×#135	C₂²	semidirect
19. C₂²×C₃×SD₁₆	C₂²	semidirect
30. C₃×#520	C₂²	semidirect
32. C₃×#519	C₂²	semidirect
35. C₃×C₄×#41	C₂²	semidirect
37. C₂×C₃×C₄×Q₈	C₂²	semidirect
38. C₃×#464	C₂²	semidirect
39. C₃×#464	C₂²	semidirect
40. C₃×#463	C₂²	semidirect
41. C₃×#462	C₂²	semidirect
42. C₃×#461	C₂²	semidirect
43. C₃×#460	C₂²	semidirect
44. C₃×#460	C₂²	semidirect
45. C₃×#460	C₂²	semidirect
46. C₃×#460	C₂²	semidirect
47. C₃×#459	C₂²	semidirect
48. C₃×#459	C₂²	semidirect
49. C₃×#424	C₂²	semidirect
50. C₃×#423	C₂²	semidirect
51. C₂×C₃×C₄⋊C₈	C₂²	semidirect
52. C₃×#418	C₂²	semidirect
53. C₃×#417	C₂²	semidirect
54. C₃×#416	C₂²	semidirect
55. C₂×C₃×#102	C₂²	semidirect
56. C₂×#577	C₂×C₃	semidirect
57. #3498	C₂×C₃	semidirect
58. C₄×2₋¹⁺⁴	C₂×C₃	semidirect
59. #3045	C₂×C₃	semidirect
60. #3043	C₂×C₃	semidirect
61. #3038	C₂×C₃	semidirect
62. #3037	C₂×C₃	semidirect
63. #3035	C₂×C₃	semidirect
64. C₂×#460	C₂×C₃	semidirect
65. #2906	C₂×C₃	semidirect
66. #2897	C₂×C₃	semidirect
68. C₃×2₋¹⁺⁴	D₄	semidirect
148. C₂×#137	C₂²×C₃	semidirect
149. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
150. C₂×#135	C₂²×C₃	semidirect
151. C₂²×SD₁₆	C₂²×C₃	semidirect
162. #520	C₂²×C₃	semidirect
164. #519	C₂²×C₃	semidirect
167. C₄×#41	C₂²×C₃	semidirect
169. C₂×C₄×Q₈	C₂²×C₃	semidirect
170. #464	C₂²×C₃	semidirect
171. #464	C₂²×C₃	semidirect
172. #463	C₂²×C₃	semidirect
173. #462	C₂²×C₃	semidirect
174. #461	C₂²×C₃	semidirect
175. #460	C₂²×C₃	semidirect
176. #460	C₂²×C₃	semidirect
177. #460	C₂²×C₃	semidirect
178. #460	C₂²×C₃	semidirect
179. #459	C₂²×C₃	semidirect
180. #459	C₂²×C₃	semidirect
181. #424	C₂²×C₃	semidirect
182. #423	C₂²×C₃	semidirect
183. C₂×C₄⋊C₈	C₂²×C₃	semidirect
184. #418	C₂²×C₃	semidirect
185. #417	C₂²×C₃	semidirect
186. #416	C₂²×C₃	semidirect
187. C₂×#102	C₂²×C₃	semidirect
195. C₃×#41	C₂×D₄	semidirect
201. C₂×C₃×Q₈	C₂×D₄	semidirect
262. 2₋¹⁺⁴	C₃×D₄	semidirect
392. #41	C₂×C₃×D₄	semidirect
398. C₂×Q₈	C₂×C₃×D₄	semidirect
477. C₃×C₄	#577	semidirect
551. C₄	C₃×#577	semidirect
554. C₃	#60431	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#577	2
C₃×#3498	2
C₃×C₄×2₋¹⁺⁴	2
C₃×#3045	2
C₃×#3043	2
C₃×#3038	2
C₃×#3037	2
C₃×#3035	2
C₂×C₃×#460	2
C₃×#2906	2
C₃×#2897	2
#60431	3

Subgroups — C3×#60431 #10679038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#60431 #10679038