Subgroups — #72240545

Order 2048.

At least 6612 conjugacy classes of subgroups; only the 402 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	#72240542	C₂	1	1	yes	maximal, characteristic, complemented
2	#72240541	C₂	1	1	yes	maximal, characteristic, complemented
3	#72240540	C₂	1	1	yes	maximal, characteristic, complemented
4	#72240538	C₂	1	1	yes	maximal, characteristic, complemented
5	#72240530	C₂	1	1	yes	maximal, characteristic, complemented
6	#72240501	C₂	1	1	yes	maximal, characteristic, complemented
7	#72240479	C₂	1	1	yes	maximal, characteristic, complemented
8	#72240457	C₂	1	1	yes	maximal, characteristic, complemented
9	#72238002	C₂	1	1	yes	maximal, characteristic, complemented
10	#72234206	C₂	1	1	yes	maximal, characteristic, complemented
11	#72233775	C₂	1	1	yes	maximal, characteristic
12	#72233637	C₂	1	1	yes	maximal, characteristic, complemented
13	#72233453	C₂	1	1	yes	maximal, characteristic, complemented
14	#72214493	C₂	1	1	yes	maximal, characteristic, complemented
15	#72214449	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586066	C₂²	1	1	yes	characteristic
17	#10574840	C₂²	1	1	yes	characteristic
18	#6628883	C₄	1	2	yes	complemented
19	#6437628	C₂²	1	1	yes	characteristic
20	#6437463	C₂²	1	1	yes	characteristic
21	#6374186	C₂²	1	1	yes	characteristic, complemented
22	#6370693	C₂²	1	1	yes	characteristic
23	#517592	C₂²	1	1	yes	characteristic
24	#493402	C₂²	1	1	yes	characteristic
25	#493396	C₂²	1	1	yes	characteristic
26	#493390	C₂²	1	1	yes	characteristic
27	#493390	C₂²	1	1	yes	characteristic
28	#394086	C₄	1	2	yes	complemented
29	#393986	C₄	1	2	yes	complemented
30	#392729	C₄	1	2	yes	complemented
31	#392132	C₂²	1	1	yes	characteristic
32	#392132	C₂²	1	1	yes	characteristic
33	#392132	C₂²	1	1	yes	characteristic
34	#382550	C₂²	1	1	yes	characteristic, complemented
35	#382440	C₂²	1	1	yes	characteristic, complemented
36	#359562	C₂²	1	1	yes	characteristic
37	#359551	C₂²	1	1	yes	characteristic
38	#355102	C₂²	1	1	yes	characteristic, complemented
39	#187947	C₂²	1	1	yes	characteristic
40	#187947	C₂²	1	1	yes	characteristic
41	#187926	C₂²	1	1	yes	characteristic
42	#187926	C₂²	1	1	yes	characteristic
43	#187926	C₂²	1	1	yes	characteristic, complemented
44	#186840	C₂²	1	1	yes	characteristic, complemented
45	#128042	C₂²	1	1	yes	characteristic, complemented
46	#128042	C₂²	1	1	yes	characteristic, complemented
47	#128034	C₂²	1	1	yes	characteristic, complemented
48	#128034	C₂²	1	1	yes	characteristic, complemented
49	#128018	C₂²	1	1	yes	characteristic, complemented
50	#128016	C₂²	1	1	yes	characteristic, complemented
51	#128012	C₂²	1	1	yes	characteristic, complemented
52	#128008	C₂²	1	1	yes	characteristic, complemented
53	#108243	C₂²	1	1	yes	characteristic, complemented
54	#108225	C₂²	1	1	yes	characteristic, complemented
55	C₂×H(7,𝔽₂)	C₂×C₄	1	1	yes	characteristic, complemented
56	C₂³×H(5,𝔽₂)	C₂³	1	1	yes	characteristic
57	C₂×#3585	D₄	1	1	yes	characteristic
58	C₂×#3585	D₄	1	1	yes	characteristic
59	#62852	C₂×C₄	1	1	yes	characteristic, complemented
60	#62852	C₂×C₄	1	1	yes	characteristic, complemented
61	#62811	C₂³	1	1	yes	characteristic
62	#62811	C₂³	1	1	yes	characteristic
63	#62811	C₂³	1	1	yes	characteristic
64	#62652	C₂×C₄	1	1	yes	characteristic, complemented
65	#60398	C₂×C₄	1	2	yes	complemented
66	#60387	C₂×C₄	1	2	yes	complemented
67	#60367	C₂×C₄	1	2	yes	—
68	#60367	C₂×C₄	1	2	yes	—
69	#60359	D₄	1	2	yes	—
70	C₂×#3003	D₄	1	2	yes	—
71	#60087	D₄	1	1	yes	characteristic
72	#60087	D₄	1	1	yes	characteristic
73	#60064	C₂³	1	1	yes	characteristic
74	#34171	D₄	1	2	yes	complemented
75	#21014	C₂×C₄	1	2	yes	complemented
76	#21012	C₂×C₄	1	2	yes	complemented
77	C₂×#1795	C₂×C₄	1	2	yes	complemented
78	C₂×#1795	C₂×C₄	1	2	yes	complemented
79	C₂×#1795	D₄	1	2	yes	complemented
80	C₂×#1795	D₄	1	2	yes	complemented
81	#20620	C₂×C₄	1	2	yes	complemented
82	#20612	C₂×C₄	1	2	yes	complemented
83	#20591	C₂³	1	1	yes	characteristic
84	#20591	C₂³	1	1	yes	characteristic
85	#20591	C₂³	1	1	yes	characteristic
86	#20584	D₄	1	2	yes	complemented
87	#20584	C₂³	1	1	yes	characteristic
88	#20584	C₂³	1	1	yes	characteristic
89	#20584	C₂³	1	1	yes	characteristic
90	#20584	C₂³	1	1	yes	characteristic
91	#20584	C₂³	1	1	yes	characteristic
92	#14337	C₂³	1	1	yes	characteristic
93	#13851	C₂³	1	1	yes	characteristic
94	#10756	C₂×C₄	1	2	yes	complemented
95	#10743	C₂×C₄	1	2	yes	complemented
96	#8534	D₄	1	2	yes	—
97	#8534	D₄	1	2	yes	—
98	#8534	D₄	1	4	yes	complemented
99	C₂×#585	C₂²×C₄	1	1	yes	characteristic
100	C₂×#585	C₂²×C₄	1	1	yes	characteristic
101	C₂×#585	C₂×D₄	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	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
106	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
107	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
108	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
109	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
110	C₂³×#41	C₂×D₄	1	1	yes	characteristic
111	C₂³×#41	C₂×D₄	1	1	yes	characteristic
112	C₂³×#41	#41	1	1	yes	characteristic
113	C₂³×#41	C₂⁴	1	1	yes	characteristic
114	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
115	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
116	C₂⁴×D₄	#41	1	1	yes	characteristic
117	C₂×#575	C₂²⋊C₄	1	2	yes	—
118	C₂×#575	C₂²⋊C₄	1	2	yes	—
119	C₂²×AGL(1,ℤ/8)	C₂²⋊C₄	1	2	yes	—
120	C₂²×AGL(1,ℤ/8)	C₂²⋊C₄	1	2	yes	—
121	C₂×#568	C₂×D₄	1	1	yes	characteristic
122	C₂×#568	C₂×D₄	1	1	yes	characteristic
123	#3485	#41	1	2	yes	—
124	#3485	#41	1	2	yes	—
125	#3484	C₂×D₄	1	2	yes	—
126	#3484	C₂×D₄	1	2	yes	—
127	#3449	C₂²×C₄	1	2	yes	—
128	C₂×#525	C₂×D₄	1	2	yes	—
129	#3428	C₂²×C₄	1	1	yes	characteristic
130	#3428	C₂²×C₄	1	1	yes	characteristic
131	#3428	C₂×D₄	1	1	yes	characteristic
132	#2896	C₂²⋊C₄	1	2	yes	—
133	#2896	C₂²⋊C₄	1	2	yes	—
134	C₂×#418	C₂²⋊C₄	1	2	yes	—
135	C₂×#418	C₂²⋊C₄	1	2	yes	—
136	C₂×#413	C₂²×C₄	1	2	yes	—
137	C₂×#413	C₂×D₄	1	2	yes	—
138	C₂×#410	C₂²×C₄	1	2	yes	—
139	C₂×#410	C₂×D₄	1	2	yes	—
140	#2879	C₂²×C₄	1	2	yes	—
141	C₂×#407	C₂×D₄	1	2	yes	—
142	#2280	C₂×D₄	1	2	yes	—
143	#2280	C₂×D₄	1	2	yes	—
144	#1781	C₂×D₄	1	2	yes	—
145	#1781	C₂×D₄	1	2	yes	—
146	#1438	C₂×D₄	1	2	yes	—
147	#1438	C₂×D₄	1	1	yes	characteristic
148	#1438	C₂×D₄	1	1	yes	characteristic
149	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
150	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
151	C₂²×#41	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×#41	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×#41	C₂×C₂²⋊C₄	1	2	yes	—
154	C₂²×#41	C₄×D₄	1	1	yes	characteristic
155	C₂²×#41	C₄×D₄	1	1	yes	characteristic
156	C₂²×#41	C₄×D₄	1	1	yes	characteristic
157	C₂²×#41	C₄×D₄	1	1	yes	characteristic
158	C₂²×#41	#120	1	1	yes	characteristic
159	C₂²×#41	#120	1	1	yes	characteristic
160	C₂²×#41	#121	1	1	yes	characteristic
161	C₂²×#41	#121	1	1	yes	characteristic
162	C₂²×#41	#123	1	1	yes	characteristic
163	C₂²×#41	#123	1	1	yes	characteristic
164	C₂²×#41	C₂³×C₄	1	1	yes	characteristic
165	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
166	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
167	C₂²×#41	C₂×#41	1	1	yes	characteristic
168	C₂²×#41	C₂×#41	1	1	yes	characteristic
169	C₂²×#41	C₂×#41	1	1	yes	characteristic
170	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂³×Q₈	#121	1	1	yes	characteristic
173	C₂³×Q₈	H(5,𝔽₂)	1	1	yes	characteristic
174	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
176	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
177	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
178	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
179	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
180	C₂³×D₄	#120	1	1	yes	characteristic
181	C₂³×D₄	#120	1	1	yes	characteristic
182	C₂³×D₄	#121	1	1	yes	characteristic
183	C₂³×D₄	#121	1	1	yes	characteristic
184	C₂³×D₄	#121	1	1	yes	characteristic
185	C₂³×D₄	#121	1	1	yes	characteristic
186	C₂³×D₄	#121	1	1	yes	characteristic
187	C₂³×D₄	#123	1	1	yes	characteristic
188	C₂³×D₄	#123	1	1	yes	characteristic
189	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
190	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
191	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
192	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
193	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
194	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
195	C₂²×SD₁₆	#99	1	4	yes	complemented
196	C₂²×D₈	#99	1	4	yes	complemented
197	C₂²×M₁₆	C₂×C₂²⋊C₄	1	2	yes	—
198	C₂²×M₁₆	C₂×C₂²⋊C₄	1	2	yes	—
199	#518	C₄×D₄	1	2	yes	—
200	#518	C₄×D₄	1	2	yes	—
201	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
202	#379	#120	1	4	yes	—
203	C₂⁵	C₂×#120	1	1	yes	characteristic
204	C₂⁵	D₄²	1	1	yes	characteristic
205	C₂⁵	#546	1	1	yes	characteristic
206	C₂²×Q₈	#386	1	1	yes	characteristic
207	C₂²×Q₈	#386	1	1	yes	characteristic
208	C₂²×Q₈	C₂×#99	1	2	yes	complemented
209	C₂²×Q₈	#518	1	1	yes	characteristic
210	C₂²×Q₈	#538	1	1	yes	characteristic
211	C₂²×Q₈	#546	1	1	yes	characteristic
212	C₂²×Q₈	#546	1	1	yes	characteristic
213	C₂²×D₄	#386	1	1	yes	characteristic
214	C₂²×D₄	#386	1	1	yes	characteristic
215	C₂²×D₄	#386	1	1	yes	characteristic
216	C₂²×D₄	#386	1	1	yes	characteristic
217	C₂²×D₄	#386	1	1	yes	characteristic
218	C₂²×D₄	#386	1	1	yes	characteristic
219	C₂²×D₄	C₂×#99	1	4	yes	—
220	C₂²×D₄	C₂×#99	1	2	yes	complemented
221	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
222	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
223	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
224	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂²×D₄	#518	1	1	yes	characteristic
226	C₂²×D₄	#525	1	1	yes	characteristic
227	C₂²×D₄	#525	1	1	yes	characteristic
228	C₂²×D₄	#538	1	1	yes	characteristic
229	C₂²×D₄	D₄²	1	1	yes	characteristic
230	C₂²×D₄	D₄²	1	1	yes	characteristic
231	C₂²×D₄	D₄²	1	1	yes	characteristic
232	C₂²×D₄	D₄²	1	1	yes	characteristic
233	C₂²×D₄	#546	1	1	yes	characteristic
234	C₂²×D₄	#546	1	1	yes	characteristic
235	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
236	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
237	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
238	C₂³×C₄	C₂×#120	1	1	yes	characteristic
239	C₂³×C₄	C₂×#121	1	1	yes	characteristic
240	C₂³×C₄	C₂×#121	1	1	yes	characteristic
241	C₂³×C₄	C₂×#123	1	1	yes	characteristic
242	C₂³×C₄	#525	1	1	yes	characteristic
243	C₂³×C₄	#534	1	1	yes	characteristic
244	C₂³×C₄	#535	1	1	yes	characteristic
245	C₂³×C₄	D₄²	1	1	yes	characteristic
246	C₂³×C₄	#546	1	1	yes	characteristic
247	C₂³×C₄	#546	1	1	yes	characteristic
248	C₂³×C₄	#546	1	1	yes	characteristic
249	C₂³×C₄	#551	1	1	yes	characteristic
250	C₂³×C₄	#551	1	1	yes	characteristic
251	C₂²×C₈	C₂×#99	1	4	yes	complemented
252	C₂⁴	C₄×#120	1	1	yes	characteristic
253	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
254	C₂⁴	#2358	1	1	yes	characteristic
255	C₂⁴	#2403	1	1	yes	characteristic
256	C₂⁴	#2408	1	1	yes	characteristic
257	C₂⁴	#2418	1	1	yes	characteristic
258	C₂⁴	#2424	1	1	yes	characteristic
259	C₂×Q₈	#1913	1	2	yes	—
260	C₂×D₄	#1889	1	2	yes	—
261	C₂×D₄	#1889	1	2	yes	—
262	C₂×D₄	#1889	1	2	yes	—
263	C₂×D₄	#1889	1	2	yes	—
264	C₂×D₄	#1898	1	2	yes	—
265	C₂×D₄	#1898	1	2	yes	—
266	C₂×D₄	#1913	1	2	yes	—
267	C₂²×C₄	C₂×#386	1	1	yes	characteristic
268	C₂²×C₄	#2321	1	1	yes	characteristic
269	C₂²×C₄	D₄×C₂²⋊C₄	1	1	yes	characteristic
270	C₂²×C₄	#2358	1	1	yes	characteristic
271	C₂²×C₄	#2404	1	1	yes	characteristic
272	C₂²×C₄	#2418	1	1	yes	characteristic
273	C₂²×C₄	#2424	1	1	yes	characteristic
274	C₂²×C₄	#2534	1	1	yes	characteristic
275	C₂²×C₄	C₂²×#99	1	2	yes	—
276	C₂²×C₄	#2883	1	2	yes	—
277	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
278	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
279	C₂³	#13967	1	1	yes	characteristic
280	C₂³	D₄×#99	1	2	yes	—
281	C₂³	#23900	1	2	yes	—
282	D₄	#9495	1	4	yes	—
283	C₂×C₄	C₂×#1889	1	1	yes	characteristic
284	C₂×C₄	C₂×#1889	1	1	yes	characteristic
285	C₂×C₄	C₂×#1898	1	1	yes	characteristic
286	C₂×C₄	#20984	1	1	yes	characteristic
287	C₂×C₄	C₂×#1913	1	1	yes	characteristic
288	C₂×C₄	C₂×#1913	1	1	yes	characteristic
289	C₂×C₄	D₄×#99	1	2	yes	—
290	C₂×C₄	#23984	1	2	yes	—
291	C₂²	#364410	1	1	yes	characteristic
292	C₂²	#364440	1	1	yes	characteristic
293	C₂²	#364471	1	1	yes	characteristic
294	C₄	C₂×#9495	1	1	yes	characteristic
295	C₄	#201664	1	1	yes	characteristic
296	C₄	#201678	1	1	yes	characteristic
297	C₄	#201678	1	1	yes	characteristic
298	C₂	#72240426	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72240426
commutator	C₂²×#41	C₂³×C₄
Frattini, cosocle	C₂³×#41	C₂⁴
prime radical	#72233775	C₂
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72240542	C₂	semidirect
2. #72240541	C₂	semidirect
3. #72240540	C₂	semidirect
4. #72240538	C₂	semidirect
5. #72240530	C₂	semidirect
6. #72240501	C₂	semidirect
7. #72240479	C₂	semidirect
8. #72240457	C₂	semidirect
9. #72238002	C₂	semidirect
10. #72234206	C₂	semidirect
12. #72233637	C₂	semidirect
13. #72233453	C₂	semidirect
14. #72214493	C₂	semidirect
15. #72214449	C₂	semidirect
18. #6628883	C₄	semidirect
21. #6374186	C₂²	semidirect
28. #394086	C₄	semidirect
29. #393986	C₄	semidirect
30. #392729	C₄	semidirect
34. #382550	C₂²	semidirect
35. #382440	C₂²	semidirect
38. #355102	C₂²	semidirect
43. #187926	C₂²	semidirect
44. #186840	C₂²	semidirect
45. #128042	C₂²	semidirect
46. #128042	C₂²	semidirect
47. #128034	C₂²	semidirect
48. #128034	C₂²	semidirect
49. #128018	C₂²	semidirect
50. #128016	C₂²	semidirect
51. #128012	C₂²	semidirect
52. #128008	C₂²	semidirect
53. #108243	C₂²	semidirect
54. #108225	C₂²	semidirect
55. C₂×H(7,𝔽₂)	C₂×C₄	semidirect
59. #62852	C₂×C₄	semidirect
60. #62852	C₂×C₄	semidirect
64. #62652	C₂×C₄	semidirect
65. #60398	C₂×C₄	semidirect
66. #60387	C₂×C₄	semidirect
74. #34171	D₄	semidirect
75. #21014	C₂×C₄	semidirect
76. #21012	C₂×C₄	semidirect
77. C₂×#1795	C₂×C₄	semidirect
78. C₂×#1795	C₂×C₄	semidirect
79. C₂×#1795	D₄	semidirect
80. C₂×#1795	D₄	semidirect
81. #20620	C₂×C₄	semidirect
82. #20612	C₂×C₄	semidirect
86. #20584	D₄	semidirect
94. #10756	C₂×C₄	semidirect
95. #10743	C₂×C₄	semidirect
98. #8534	D₄	semidirect
195. C₂²×SD₁₆	#99	semidirect
196. C₂²×D₈	#99	semidirect
208. C₂²×Q₈	C₂×#99	semidirect
220. C₂²×D₄	C₂×#99	semidirect
251. C₂²×C₈	C₂×#99	semidirect

Maximal subgroups

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

subgroup	index
#72240542	2
#72240541	2
#72240540	2
#72240538	2
#72240530	2
#72240501	2
#72240479	2
#72240457	2
#72238002	2
#72234206	2
#72233775	2
#72233637	2
#72233453	2
#72214493	2
#72214449	2