Subgroups — #72310370

Order 2048.

At least 7137 conjugacy classes of subgroups; only the 486 normal subgroups are shown.

The subgroup lattice has 300 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266863	C₂	1	1	yes	maximal, characteristic, complemented
2	#72260537	C₂	1	1	yes	maximal, characteristic, complemented
3	#72258368	C₂	1	1	yes	maximal, characteristic, complemented
4	#72258334	C₂	1	1	yes	maximal, characteristic, complemented
5	#72258097	C₂	1	1	yes	maximal, characteristic, complemented
6	#72255977	C₂	1	1	yes	maximal, characteristic, complemented
7	#72240419	C₂	1	1	yes	maximal, characteristic, complemented
8	#72238443	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237638	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237638	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237599	C₂	1	1	yes	maximal, characteristic, complemented
12	#72236928	C₂	1	1	yes	maximal, characteristic, complemented
13	#72234693	C₂	1	1	yes	maximal, characteristic
14	#72219160	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#133839	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586404	C₂²	1	1	yes	characteristic
17	#10574873	C₄	1	2	yes	complemented
18	C₂×#60399	C₂²	1	1	yes	characteristic
19	#6451863	C₄	1	2	yes	complemented
20	#6451863	C₄	1	2	yes	complemented
21	C₂×#23895	C₂²	1	1	yes	characteristic
22	C₂×#23895	C₂²	1	1	yes	characteristic
23	#6437621	C₂²	1	1	yes	characteristic
24	#6437621	C₂²	1	1	yes	characteristic
25	#6374251	C₂²	1	1	yes	characteristic, complemented
26	#6374187	C₄	1	2	yes	complemented
27	C₂×#21776	C₂²	1	1	yes	characteristic
28	#552677	C₂²	1	1	yes	characteristic, complemented
29	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
30	C₂×#12707	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, complemented
34	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
35	#493357	C₂²	1	1	yes	characteristic, complemented
36	#493357	C₂²	1	1	yes	characteristic, complemented
37	#382436	C₂²	1	1	yes	characteristic, complemented
38	#382434	C₂²	1	1	yes	characteristic, complemented
39	#187944	C₂²	1	1	yes	characteristic
40	#187944	C₂²	1	1	yes	characteristic, complemented
41	#187944	C₂²	1	1	yes	characteristic, complemented
42	C₂×#8534	C₂²	1	1	yes	characteristic
43	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
44	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
45	#186835	C₂²	1	1	yes	characteristic, complemented
46	C₂×#8453	C₂²	1	1	yes	characteristic, complemented
47	#128033	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
51	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
52	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
53	#128009	C₂²	1	1	yes	characteristic, complemented
54	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
55	C₂×H(7,𝔽₂)	C₂×C₄	1	1	yes	characteristic
56	#62844	C₂×C₄	1	2	yes	—
57	#62844	D₄	1	2	yes	—
58	#62842	C₂×C₄	1	2	yes	—
59	#62842	D₄	1	2	yes	—
60	C₂×#3490	D₄	1	2	yes	—
61	C₂×#3484	D₄	1	2	yes	—
62	C₂×#3484	C₂³	1	1	yes	characteristic
63	C₂×#3484	C₂³	1	1	yes	characteristic
64	C₂×#3484	C₂³	1	1	yes	characteristic
65	C₂×#3484	C₂³	1	1	yes	characteristic
66	C₂²×#534	C₂³	1	1	yes	characteristic
67	C₂×#3026	C₂×C₄	1	2	yes	—
68	C₂×#3026	C₂×C₄	1	2	yes	—
69	C₂×#3026	C₂×C₄	1	2	yes	—
70	C₂×#3026	C₂×C₄	1	2	yes	—
71	C₂×#2883	C₂×C₄	1	1	yes	characteristic
72	C₂×#2883	C₂×C₄	1	1	yes	characteristic
73	C₂×#2883	C₂×C₄	1	1	yes	characteristic
74	C₂×#2883	C₂×C₄	1	2	yes	—
75	#23903	D₄	1	4	yes	—
76	#23903	D₄	1	4	yes	—
77	#23901	D₄	1	2	yes	—
78	#23901	D₄	1	2	yes	—
79	#23900	D₄	1	2	yes	—
80	#23900	D₄	1	2	yes	—
81	C₂×#1913	C₂×C₄	1	2	yes	—
82	C₂×#1913	C₂×C₄	1	2	yes	—
83	#20960	C₂×C₄	1	2	yes	—
84	#20960	C₂×C₄	1	2	yes	—
85	C₂×#1828	C₂×C₄	1	2	yes	—
86	C₂×#1783	C₂³	1	1	yes	characteristic
87	C₂×#1783	C₂³	1	1	yes	characteristic
88	C₂×#1781	C₂³	1	1	yes	characteristic
89	C₂×#1781	C₂³	1	1	yes	characteristic
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	#14335	C₂³	1	1	yes	characteristic
95	C₂×#1438	C₂³	1	1	yes	characteristic
96	#8534	D₄	1	2	yes	complemented
97	#8534	D₄	1	2	yes	complemented
98	#8534	D₄	1	2	yes	complemented
99	#8534	D₄	1	2	yes	complemented
100	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	2	yes	—
101	C₂²×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
102	C₂²×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
103	C₂²×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
104	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
105	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
106	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
107	#3484	C₂×D₄	1	2	yes	—
108	#3484	C₂×D₄	1	2	yes	—
109	#3484	C₂×D₄	1	2	yes	—
110	#3484	C₂×D₄	1	2	yes	—
111	#3484	C₂×D₄	1	2	yes	—
112	#3484	C₂×D₄	1	2	yes	—
113	#3484	#41	1	2	yes	—
114	#3484	#41	1	2	yes	—
115	C₂×#543	C₂²⋊C₄	1	2	yes	—
116	C₂×#534	C₂²⋊C₄	1	2	yes	—
117	C₂×#534	C₂²⋊C₄	1	2	yes	—
118	C₂×#534	C₂²⋊C₄	1	2	yes	—
119	C₂×#534	C₂²×C₄	1	2	yes	—
120	C₂×#534	C₂²×C₄	1	2	yes	—
121	C₂×#534	C₂×D₄	1	2	yes	—
122	C₂×#534	C₂×D₄	1	2	yes	—
123	C₂×#534	C₂×D₄	1	2	yes	—
124	C₂×#534	C₂×D₄	1	2	yes	—
125	C₂²×#123	C₂×D₄	1	1	yes	characteristic
126	C₂²×#123	C₂×D₄	1	1	yes	characteristic
127	C₂²×#120	C₂×D₄	1	2	yes	—
128	C₂²×#120	C₂×D₄	1	2	yes	—
129	C₂²×#120	C₂×D₄	1	1	yes	characteristic
130	C₂²×#120	C₂×D₄	1	1	yes	characteristic
131	C₂×#518	C₂²⋊C₄	1	2	yes	—
132	C₂×#518	C₂²⋊C₄	1	2	yes	—
133	C₂×#518	C₂²⋊C₄	1	2	yes	—
134	C₂×#518	C₂²×C₄	1	1	yes	characteristic
135	C₂×#518	C₂²×C₄	1	1	yes	characteristic
136	C₂×#518	C₂²×C₄	1	1	yes	characteristic
137	C₂×#518	C₂×D₄	1	1	yes	characteristic
138	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
139	C₂×#458	C₂²⋊C₄	1	4	yes	complemented
140	C₂×#458	C₂²⋊C₄	1	4	yes	complemented
141	C₂×B(4,𝔽₂)	C₂²⋊C₄	1	4	yes	complemented
142	C₂×B(4,𝔽₂)	C₂²⋊C₄	1	4	yes	complemented
143	C₂²×#99	C₂²×C₄	1	2	yes	—
144	C₂²×#99	C₂²×C₄	1	2	yes	—
145	#1890	C₂²⋊C₄	1	4	yes	complemented
146	#1890	C₂²⋊C₄	1	4	yes	complemented
147	#1889	C₂²⋊C₄	1	4	yes	complemented
148	#1889	C₂²⋊C₄	1	4	yes	complemented
149	#1781	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	complemented
154	#1781	C₂×D₄	1	2	yes	complemented
155	#1438	C₂×D₄	1	1	yes	characteristic, complemented
156	#1438	C₂×D₄	1	1	yes	characteristic, complemented
157	#1438	C₂×D₄	1	1	yes	characteristic, complemented
158	#1438	C₂×D₄	1	1	yes	characteristic, complemented
159	C₂⁶	#120	1	1	yes	characteristic
160	C₂⁶	C₂²×D₄	1	1	yes	characteristic
161	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
164	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
165	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
166	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
167	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
168	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂³×D₄	#120	1	2	yes	—
170	C₂³×D₄	C₂³×C₄	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	C₂²×D₄	1	1	yes	characteristic
176	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
177	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
178	C₂³×D₄	C₂×#41	1	1	yes	characteristic
179	C₂³×D₄	C₂×#41	1	1	yes	characteristic
180	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
182	C₂⁴×C₄	#120	1	1	yes	characteristic
183	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
184	#534	#121	1	4	yes	—
185	#534	#121	1	4	yes	—
186	C₂×#123	#99	1	4	yes	complemented
187	C₂×#123	C₂×C₂²⋊C₄	1	2	yes	—
188	C₂×#123	C₂×C₂²⋊C₄	1	2	yes	—
189	C₂×#123	#120	1	2	yes	—
190	C₂×#123	#120	1	2	yes	—
191	C₂×#120	#99	1	4	yes	complemented
192	C₂×#120	C₂×C₂²⋊C₄	1	2	yes	—
193	C₂×#120	C₂×C₂²⋊C₄	1	2	yes	—
194	C₂×#120	#120	1	2	yes	—
195	C₂×#120	#120	1	2	yes	—
196	C₂×#120	#120	1	4	yes	—
197	C₂×#120	#123	1	4	yes	—
198	#518	C₂×C₂²⋊C₄	1	2	yes	—
199	#518	C₂×C₂²⋊C₄	1	2	yes	—
200	#518	C₄×D₄	1	2	yes	—
201	#518	C₄×D₄	1	2	yes	—
202	C₂×#117	#99	1	4	yes	complemented
203	C₂²×C₂²⋊C₄	#99	1	4	yes	complemented
204	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
205	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
206	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
207	C₂²×C₂²⋊C₄	#120	1	2	yes	—
208	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
209	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
210	C₂×#99	C₂×C₂²⋊C₄	1	4	yes	—
211	C₂×#99	C₂×C₂²⋊C₄	1	4	yes	—
212	C₂⁵	#379	1	2	yes	—
213	C₂⁵	C₂×#99	1	2	yes	—
214	C₂⁵	C₂×#99	1	2	yes	—
215	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
216	C₂⁵	#518	1	1	yes	characteristic
217	C₂⁵	C₂×#120	1	1	yes	characteristic
218	C₂⁵	C₂×#120	1	1	yes	characteristic
219	C₂⁵	C₂×#120	1	1	yes	characteristic
220	C₂⁵	C₂×#123	1	1	yes	characteristic
221	C₂⁵	#534	1	1	yes	characteristic
222	C₂⁵	#534	1	1	yes	characteristic
223	C₂⁵	#534	1	1	yes	characteristic
224	C₂⁵	D₄²	1	1	yes	characteristic
225	C₂⁵	D₄²	1	1	yes	characteristic
226	C₂⁵	#546	1	1	yes	characteristic
227	C₂⁵	#546	1	1	yes	characteristic
228	C₂²×D₄	#386	1	4	yes	—
229	C₂²×D₄	C₂×#99	1	4	yes	—
230	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
231	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
232	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
233	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
234	C₂²×D₄	C₂×#120	1	1	yes	characteristic
235	C₂²×D₄	C₂×#120	1	1	yes	characteristic
236	C₂²×D₄	C₂×#121	1	1	yes	characteristic
237	C₂²×D₄	C₂×#121	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	D₄²	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	C₂²×D₄	#546	1	1	yes	characteristic
246	C₂³×C₄	#379	1	2	yes	—
247	C₂³×C₄	C₂×#99	1	2	yes	—
248	C₂³×C₄	C₂×#99	1	2	yes	—
249	C₂³×C₄	#518	1	1	yes	characteristic
250	C₂³×C₄	#534	1	1	yes	characteristic
251	C₂³×C₄	D₄²	1	1	yes	characteristic
252	C₂³×C₄	D₄²	1	1	yes	characteristic
253	C₂³×C₄	#546	1	1	yes	characteristic
254	C₂³×C₄	#546	1	1	yes	characteristic
255	C₂×C₂²⋊C₄	#379	1	2	yes	—
256	C₂×C₂²⋊C₄	#379	1	2	yes	—
257	C₂×C₂²⋊C₄	#386	1	4	yes	—
258	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
259	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
260	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
261	C₂⁴	#1898	1	4	yes	—
262	C₂⁴	C₂×#379	1	1	yes	characteristic
263	C₂⁴	C₂×#386	1	1	yes	characteristic
264	C₂⁴	#2308	1	1	yes	characteristic
265	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
266	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
267	C₂⁴	#2403	1	1	yes	characteristic
268	C₂⁴	#2403	1	1	yes	characteristic
269	C₂⁴	#2405	1	1	yes	characteristic
270	C₂⁴	#2405	1	1	yes	characteristic
271	C₂⁴	#2410	1	1	yes	characteristic
272	C₂⁴	#2410	1	1	yes	characteristic
273	C₂⁴	C₂²×#99	1	2	yes	—
274	C₂⁴	C₂²×#99	1	2	yes	—
275	C₂²×C₄	#1898	1	4	yes	—
276	C₂²×C₄	D₄×C₂²⋊C₄	1	1	yes	characteristic
277	C₂²×C₄	D₄×C₂²⋊C₄	1	1	yes	characteristic
278	C₂²×C₄	#2410	1	1	yes	characteristic
279	C₂²×C₄	#2410	1	1	yes	characteristic
280	C₂²⋊C₄	#1781	1	4	yes	—
281	C₂²⋊C₄	#1781	1	4	yes	—
282	C₂³	#13867	1	1	yes	characteristic
283	C₂³	C₂×#1781	1	2	yes	—
284	C₂³	C₂×#1898	1	2	yes	—
285	C₂³	#21746	1	1	yes	characteristic
286	C₂³	#21746	1	1	yes	characteristic
287	C₂³	#21776	1	1	yes	characteristic
288	C₂³	#21776	1	1	yes	characteristic
289	C₂³	#21776	1	1	yes	characteristic
290	C₂³	#21776	1	1	yes	characteristic
291	C₂²	#213829	1	1	yes	characteristic
292	C₂²	#213829	1	1	yes	characteristic
293	C₂²	#213854	1	1	yes	characteristic
294	C₂²	#213854	1	1	yes	characteristic
295	C₂²	#359428	1	1	yes	characteristic
296	C₂²	#359591	1	1	yes	characteristic
297	C₂²	#359591	1	1	yes	characteristic
298	C₂	#72241167	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241167
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72234693	C₂
LCS	G ▷ C₂³×D₄ ▷ 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. #72266863	C₂	semidirect
2. #72260537	C₂	semidirect
3. #72258368	C₂	semidirect
4. #72258334	C₂	semidirect
5. #72258097	C₂	semidirect
6. #72255977	C₂	semidirect
7. #72240419	C₂	semidirect
8. #72238443	C₂	semidirect
9. #72237638	C₂	semidirect
10. #72237638	C₂	semidirect
11. #72237599	C₂	semidirect
12. #72236928	C₂	semidirect
14. #72219160	C₂	semidirect
15. C₂×#133839	C₂	semidirect
17. #10574873	C₄	semidirect
19. #6451863	C₄	semidirect
20. #6451863	C₄	semidirect
25. #6374251	C₂²	semidirect
26. #6374187	C₄	semidirect
28. #552677	C₂²	semidirect
29. C₂×#12707	C₂²	semidirect
30. C₂×#12707	C₂²	semidirect
33. C₂×#12705	C₂²	semidirect
34. C₂×#12705	C₂²	semidirect
35. #493357	C₂²	semidirect
36. #493357	C₂²	semidirect
37. #382436	C₂²	semidirect
38. #382434	C₂²	semidirect
40. #187944	C₂²	semidirect
41. #187944	C₂²	semidirect
43. C₂×#8534	C₂²	semidirect
44. C₂×#8534	C₂²	semidirect
45. #186835	C₂²	semidirect
46. C₂×#8453	C₂²	semidirect
47. #128033	C₂²	semidirect
48. C₂×#7304	C₂²	semidirect
49. C₂×#7314	C₂²	semidirect
50. C₂×#7314	C₂²	semidirect
51. C₂×#7310	C₂²	semidirect
52. C₂×#7310	C₂²	semidirect
53. #128009	C₂²	semidirect
54. C₂×#7307	C₂²	semidirect
96. #8534	D₄	semidirect
97. #8534	D₄	semidirect
98. #8534	D₄	semidirect
99. #8534	D₄	semidirect
139. C₂×#458	C₂²⋊C₄	semidirect
140. C₂×#458	C₂²⋊C₄	semidirect
141. C₂×B(4,𝔽₂)	C₂²⋊C₄	semidirect
142. C₂×B(4,𝔽₂)	C₂²⋊C₄	semidirect
145. #1890	C₂²⋊C₄	semidirect
146. #1890	C₂²⋊C₄	semidirect
147. #1889	C₂²⋊C₄	semidirect
148. #1889	C₂²⋊C₄	semidirect
153. #1781	C₂×D₄	semidirect
154. #1781	C₂×D₄	semidirect
155. #1438	C₂×D₄	semidirect
156. #1438	C₂×D₄	semidirect
157. #1438	C₂×D₄	semidirect
158. #1438	C₂×D₄	semidirect
186. C₂×#123	#99	semidirect
191. C₂×#120	#99	semidirect
202. C₂×#117	#99	semidirect
203. 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
#72266863	2
#72260537	2
#72258368	2
#72258334	2
#72258097	2
#72255977	2
#72240419	2
#72238443	2
#72237638	2
#72237599	2
#72236928	2
#72234693	2
#72219160	2
C₂×#133839	2