Subgroups — #72314464

Order 2048.

At least 6585 conjugacy classes of subgroups; only the 486 normal subgroups are shown.

The subgroup lattice has 299 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72314456	C₂	1	1	yes	maximal, characteristic, complemented
2	#72314452	C₂	1	1	yes	maximal, characteristic, complemented
3	#72258097	C₂	1	1	yes	maximal, characteristic, complemented
4	#72258028	C₂	1	1	yes	maximal, characteristic, complemented
5	#72256052	C₂	1	1	yes	maximal, characteristic, complemented
6	#72255830	C₂	1	1	yes	maximal, characteristic, complemented
7	#72238351	C₂	1	1	yes	maximal, characteristic, complemented
8	#72238350	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237608	C₂	1	1	yes	maximal, characteristic, complemented
10	#72234662	C₂	1	1	yes	maximal, characteristic
11	C₂×#139748	C₂	1	1	yes	maximal, characteristic, complemented
12	#72219438	C₂	1	1	yes	maximal, characteristic, complemented
13	#72219340	C₂	1	1	yes	maximal, characteristic, complemented
14	#72219158	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#139750	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586404	C₂²	1	1	yes	characteristic
17	#7630944	C₄	1	2	yes	complemented
18	C₂×#34981	C₂²	1	1	yes	characteristic
19	#6440460	C₄	1	2	yes	complemented
20	#6440460	C₂²	1	1	yes	characteristic
21	#6440460	C₂²	1	1	yes	characteristic
22	C₂×#23900	C₄	1	2	yes	complemented
23	C₂×#23900	C₂²	1	1	yes	characteristic
24	C₂×#23900	C₂²	1	1	yes	characteristic
25	#6374548	C₄	1	2	yes	complemented
26	#6374251	C₂²	1	1	yes	characteristic
27	C₂×#21778	C₂²	1	1	yes	characteristic
28	C₂×#21776	C₂²	1	1	yes	characteristic
29	#493879	C₂²	1	1	yes	characteristic, complemented
30	#493879	C₂²	1	1	yes	characteristic, complemented
31	C₂×#12705	C₂²	1	1	yes	characteristic
32	C₂×#12705	C₂²	1	1	yes	characteristic
33	C₂×#12705	C₂²	1	1	yes	characteristic
34	C₂×#12705	C₂²	1	1	yes	characteristic
35	#493357	C₂²	1	1	yes	characteristic
36	#493357	C₂²	1	1	yes	characteristic
37	#493357	C₂²	1	1	yes	characteristic
38	#493357	C₂²	1	1	yes	characteristic
39	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
40	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
41	#382436	C₂²	1	1	yes	characteristic
42	#382434	C₂²	1	1	yes	characteristic
43	#190147	C₂²	1	1	yes	characteristic, complemented
44	#190116	C₂²	1	1	yes	characteristic, complemented
45	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
46	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
51	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
52	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
53	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
54	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
55	#62844	D₄	1	2	yes	—
56	#62844	D₄	1	2	yes	—
57	#62842	D₄	1	2	yes	—
58	#62842	D₄	1	2	yes	—
59	C₂×#3490	C₂×C₄	1	2	yes	—
60	C₂×#3484	C₂×C₄	1	2	yes	—
61	C₂×#3484	C₂³	1	1	yes	characteristic
62	C₂×#3484	C₂³	1	1	yes	characteristic
63	C₂²×#534	C₂³	1	1	yes	characteristic
64	C₂³×#120	C₂×C₄	1	1	yes	characteristic, complemented
65	C₂³×#120	C₂³	1	1	yes	characteristic
66	C₂⁴×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic, complemented
67	C₂×#2846	C₂×C₄	1	2	yes	—
68	C₂×#2846	C₂×C₄	1	2	yes	complemented
69	#34992	C₂×C₄	1	2	yes	—
70	#34992	C₂×C₄	1	2	yes	complemented
71	#34176	C₂×C₄	1	2	yes	complemented
72	C₂×#2315	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂×#2308	C₂×C₄	1	1	yes	characteristic, complemented
74	C₂×#2308	C₂³	1	1	yes	characteristic
75	#24058	D₄	1	4	yes	—
76	#24058	D₄	1	4	yes	—
77	#23900	D₄	1	2	yes	—
78	#23900	D₄	1	2	yes	—
79	#23900	D₄	1	2	yes	—
80	#23900	D₄	1	2	yes	—
81	C₂×#1898	C₂×C₄	1	2	yes	—
82	C₂×#1898	C₂×C₄	1	2	yes	—
83	C₂×#1898	C₂×C₄	1	2	yes	complemented
84	C₂×#1898	C₂×C₄	1	2	yes	complemented
85	C₂×#1783	C₂³	1	1	yes	characteristic
86	C₂×#1783	C₂³	1	1	yes	characteristic
87	C₂×#1783	C₂³	1	1	yes	characteristic
88	C₂×#1783	C₂³	1	1	yes	characteristic
89	C₂×#1781	C₂×C₄	1	2	yes	complemented
90	C₂×#1781	C₂³	1	1	yes	characteristic
91	C₂×#1781	C₂³	1	1	yes	characteristic
92	C₂×#1781	C₂³	1	1	yes	characteristic
93	C₂×#1781	C₂³	1	1	yes	characteristic
94	C₂×#1781	C₂³	1	1	yes	characteristic
95	C₂×#1781	C₂³	1	1	yes	characteristic
96	#12690	D₄	1	4	yes	complemented
97	#12688	D₄	1	2	yes	complemented
98	#12687	D₄	1	2	yes	complemented
99	C₂⁷	C₂²×C₄	1	1	yes	characteristic
100	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	2	yes	—
101	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
102	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
103	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
104	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
105	#3484	C₂×D₄	1	2	yes	—
106	#3484	C₂×D₄	1	2	yes	—
107	#3484	C₂×D₄	1	2	yes	—
108	#3484	C₂×D₄	1	2	yes	—
109	C₂×#543	C₂²⋊C₄	1	2	yes	—
110	C₂×#534	C₂²⋊C₄	1	2	yes	—
111	C₂×#534	C₂²⋊C₄	1	2	yes	—
112	C₂×#534	C₂²⋊C₄	1	2	yes	—
113	C₂×#534	C₂×D₄	1	2	yes	—
114	C₂×#534	C₂×D₄	1	2	yes	—
115	C₂×#534	C₂×D₄	1	2	yes	—
116	C₂×#534	C₂×D₄	1	2	yes	—
117	C₂×#534	C₂×D₄	1	2	yes	—
118	C₂×#534	#41	1	2	yes	—
119	C₂²×#123	C₂×D₄	1	1	yes	characteristic
120	C₂²×#123	#41	1	1	yes	characteristic
121	C₂²×#120	C₂²×C₄	1	2	yes	—
122	C₂²×#120	C₂²×C₄	1	2	yes	—
123	C₂²×#120	C₂×D₄	1	1	yes	characteristic
124	C₂²×#120	C₂×D₄	1	1	yes	characteristic
125	C₂²×#120	C₂×D₄	1	1	yes	characteristic
126	C₂²×#120	C₂×D₄	1	1	yes	characteristic
127	C₂²×#120	C₂×D₄	1	1	yes	characteristic
128	C₂²×#120	#41	1	1	yes	characteristic
129	C₂×#518	C₂²⋊C₄	1	2	yes	—
130	C₂×#518	C₂²⋊C₄	1	2	yes	—
131	C₂×#518	C₂²⋊C₄	1	2	yes	—
132	C₂×#518	C₂×D₄	1	1	yes	characteristic
133	C₂×#518	C₂×D₄	1	1	yes	characteristic
134	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
135	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
136	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
137	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
138	#2846	C₂²⋊C₄	1	4	yes	complemented
139	#2846	C₂²⋊C₄	1	4	yes	complemented
140	#2622	C₂²⋊C₄	1	8	yes	complemented
141	#2308	C₂×D₄	1	2	yes	—
142	#2308	C₂×D₄	1	2	yes	—
143	C₂×#379	C₂²×C₄	1	2	yes	—
144	C₂×#379	C₂²×C₄	1	2	yes	—
145	#1898	C₂²⋊C₄	1	8	yes	complemented
146	#1898	C₂²⋊C₄	1	4	yes	complemented
147	#1898	C₂²⋊C₄	1	4	yes	complemented
148	#1783	C₂×D₄	1	2	yes	—
149	#1783	C₂×D₄	1	2	yes	—
150	#1781	C₂×D₄	1	2	yes	—
151	#1781	C₂×D₄	1	2	yes	—
152	#1781	C₂×D₄	1	2	yes	—
153	#1781	C₂×D₄	1	2	yes	—
154	#1781	C₂×D₄	1	2	yes	—
155	#1781	C₂×D₄	1	2	yes	—
156	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂⁶	C₄×D₄	1	1	yes	characteristic
159	C₂⁶	C₂³×C₄	1	1	yes	characteristic
160	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂×H(5,𝔽₂)	#120	1	2	yes	—
162	C₂×H(5,𝔽₂)	#120	1	2	yes	—
163	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
164	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
165	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
166	C₂³×D₄	C₄×D₄	1	2	yes	—
167	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
168	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
169	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
170	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
171	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
172	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
173	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
174	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
175	C₂³×D₄	C₂×#41	1	1	yes	characteristic
176	C₂³×D₄	C₂×#41	1	1	yes	characteristic
177	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
178	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
179	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
180	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
181	#534	#120	1	4	yes	—
182	#534	#120	1	4	yes	—
183	C₂×#123	#99	1	4	yes	complemented
184	C₂×#123	#120	1	2	yes	—
185	C₂×#123	#121	1	2	yes	—
186	C₂×#123	#121	1	2	yes	—
187	C₂×#123	#123	1	2	yes	—
188	C₂×#120	#99	1	4	yes	complemented
189	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
190	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
191	C₂×#120	#120	1	2	yes	—
192	C₂×#120	#121	1	2	yes	—
193	C₂×#120	#121	1	2	yes	—
194	C₂×#120	#123	1	2	yes	—
195	#518	#120	1	2	yes	—
196	#518	#120	1	2	yes	—
197	C₂×#117	#99	1	4	yes	complemented
198	C₂²×C₂²⋊C₄	#99	1	4	yes	complemented
199	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
200	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
201	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
202	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
203	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
204	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×C₂²⋊C₄	C₄×D₄	1	2	yes	—
207	C₂²×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
208	C₂²×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
209	#379	C₂×C₂²⋊C₄	1	4	yes	—
210	#379	C₂×C₂²⋊C₄	1	4	yes	—
211	C₂⁵	#386	1	2	yes	—
212	C₂⁵	C₂×#99	1	2	yes	—
213	C₂⁵	C₂×#99	1	2	yes	—
214	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
215	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
216	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
217	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂⁵	#518	1	1	yes	characteristic
220	C₂⁵	#518	1	1	yes	characteristic
221	C₂⁵	#534	1	1	yes	characteristic
222	C₂⁵	#534	1	1	yes	characteristic
223	C₂⁵	D₄²	1	1	yes	characteristic
224	C₂⁵	D₄²	1	1	yes	characteristic
225	C₂⁵	#546	1	1	yes	characteristic
226	C₂⁵	#546	1	1	yes	characteristic
227	C₂²×D₄	#379	1	4	yes	—
228	C₂²×D₄	C₂×#99	1	4	yes	—
229	C₂²×D₄	C₂×#120	1	1	yes	characteristic
230	C₂²×D₄	C₂×#120	1	1	yes	characteristic
231	C₂²×D₄	C₂×#120	1	1	yes	characteristic
232	C₂²×D₄	C₂×#120	1	1	yes	characteristic
233	C₂²×D₄	C₂×#120	1	1	yes	characteristic
234	C₂²×D₄	C₂×#121	1	1	yes	characteristic
235	C₂²×D₄	C₂×#121	1	1	yes	characteristic
236	C₂²×D₄	C₂×#123	1	1	yes	characteristic
237	C₂²×D₄	D₄²	1	1	yes	characteristic
238	C₂²×D₄	D₄²	1	1	yes	characteristic
239	C₂²×D₄	D₄²	1	1	yes	characteristic
240	C₂²×D₄	D₄²	1	1	yes	characteristic
241	C₂²×D₄	#546	1	1	yes	characteristic
242	C₂²×D₄	#546	1	1	yes	characteristic
243	C₂²×D₄	#546	1	1	yes	characteristic
244	C₂²×D₄	#546	1	1	yes	characteristic
245	C₂³×C₄	#386	1	2	yes	—
246	C₂³×C₄	C₂×#99	1	2	yes	—
247	C₂³×C₄	C₂×#99	1	2	yes	—
248	C₂³×C₄	#534	1	1	yes	characteristic
249	C₂³×C₄	#534	1	1	yes	characteristic
250	C₂³×C₄	D₄²	1	1	yes	characteristic
251	C₂³×C₄	D₄²	1	1	yes	characteristic
252	C₂³×C₄	#546	1	1	yes	characteristic
253	C₂³×C₄	#546	1	1	yes	characteristic
254	C₂×C₂²⋊C₄	#379	1	4	yes	—
255	C₂×C₂²⋊C₄	#386	1	2	yes	—
256	C₂×C₂²⋊C₄	#386	1	2	yes	—
257	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
258	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
259	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
260	C₂⁴	#1781	1	4	yes	—
261	C₂⁴	C₂×#379	1	1	yes	characteristic
262	C₂⁴	C₂×#386	1	1	yes	characteristic
263	C₂⁴	#2308	1	1	yes	characteristic
264	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
265	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
266	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
267	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
268	C₂⁴	#2403	1	1	yes	characteristic
269	C₂⁴	#2405	1	1	yes	characteristic
270	C₂⁴	#2410	1	1	yes	characteristic
271	C₂⁴	#2410	1	1	yes	characteristic
272	C₂⁴	C₂²×#99	1	2	yes	—
273	C₂⁴	C₂²×#99	1	2	yes	—
274	C₂²×C₄	#1781	1	4	yes	—
275	C₂²×C₄	#2403	1	1	yes	characteristic
276	C₂²×C₄	#2405	1	1	yes	characteristic
277	C₂²×C₄	#2410	1	1	yes	characteristic
278	C₂²×C₄	#2410	1	1	yes	characteristic
279	C₂²⋊C₄	#1898	1	4	yes	—
280	C₂²⋊C₄	#1898	1	4	yes	—
281	C₂³	#13867	1	1	yes	characteristic
282	C₂³	C₂×#1781	1	2	yes	—
283	C₂³	C₂×#1898	1	2	yes	—
284	C₂³	#21746	1	1	yes	characteristic
285	C₂³	#21746	1	1	yes	characteristic
286	C₂³	#21776	1	1	yes	characteristic
287	C₂³	#21776	1	1	yes	characteristic
288	C₂³	#21777	1	1	yes	characteristic
289	C₂³	#21777	1	1	yes	characteristic
290	C₂²	#213832	1	1	yes	characteristic
291	C₂²	#213832	1	1	yes	characteristic
292	C₂²	#213852	1	1	yes	characteristic
293	C₂²	#213852	1	1	yes	characteristic
294	C₂²	#359428	1	1	yes	characteristic
295	C₂²	#359591	1	1	yes	characteristic
296	C₂²	#359592	1	1	yes	characteristic
297	C₂	#72241165	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241165
commutator	C₂⁶	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72234662	C₂
LCS	G ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72314456	C₂	semidirect
2. #72314452	C₂	semidirect
3. #72258097	C₂	semidirect
4. #72258028	C₂	semidirect
5. #72256052	C₂	semidirect
6. #72255830	C₂	semidirect
7. #72238351	C₂	semidirect
8. #72238350	C₂	semidirect
9. #72237608	C₂	semidirect
11. C₂×#139748	C₂	semidirect
12. #72219438	C₂	semidirect
13. #72219340	C₂	semidirect
14. #72219158	C₂	semidirect
15. C₂×#139750	C₂	semidirect
17. #7630944	C₄	semidirect
19. #6440460	C₄	semidirect
22. C₂×#23900	C₄	semidirect
25. #6374548	C₄	semidirect
29. #493879	C₂²	semidirect
30. #493879	C₂²	semidirect
39. C₂×#12687	C₂²	semidirect
40. C₂×#12687	C₂²	semidirect
43. #190147	C₂²	semidirect
44. #190116	C₂²	semidirect
45. C₂×#8547	C₂²	semidirect
46. C₂×#8541	C₂²	semidirect
47. C₂×#7316	C₂²	semidirect
48. C₂×#7316	C₂²	semidirect
49. C₂×#7316	C₂²	semidirect
50. C₂×#7316	C₂²	semidirect
51. C₂×#7311	C₂²	semidirect
52. C₂×#7311	C₂²	semidirect
53. C₂×#7310	C₂²	semidirect
54. C₂×#7310	C₂²	semidirect
64. C₂³×#120	C₂×C₄	semidirect
66. C₂⁴×C₂²⋊C₄	C₂×C₄	semidirect
68. C₂×#2846	C₂×C₄	semidirect
70. #34992	C₂×C₄	semidirect
71. #34176	C₂×C₄	semidirect
72. C₂×#2315	C₂×C₄	semidirect
73. C₂×#2308	C₂×C₄	semidirect
83. C₂×#1898	C₂×C₄	semidirect
84. C₂×#1898	C₂×C₄	semidirect
89. C₂×#1781	C₂×C₄	semidirect
96. #12690	D₄	semidirect
97. #12688	D₄	semidirect
98. #12687	D₄	semidirect
138. #2846	C₂²⋊C₄	semidirect
139. #2846	C₂²⋊C₄	semidirect
140. #2622	C₂²⋊C₄	semidirect
145. #1898	C₂²⋊C₄	semidirect
146. #1898	C₂²⋊C₄	semidirect
147. #1898	C₂²⋊C₄	semidirect
183. C₂×#123	#99	semidirect
188. C₂×#120	#99	semidirect
197. C₂×#117	#99	semidirect
198. C₂²×C₂²⋊C₄	#99	semidirect

Maximal subgroups

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

subgroup	index
#72314456	2
#72314452	2
#72258097	2
#72258028	2
#72256052	2
#72255830	2
#72238351	2
#72238350	2
#72237608	2
#72234662	2
C₂×#139748	2
#72219438	2
#72219340	2
#72219158	2
C₂×#139750	2