Subgroups — #72342852

Order 2048.

At least 6174 conjugacy classes of subgroups; only the 374 normal subgroups are shown.

The subgroup lattice has 250 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72342836	C₂	1	1	yes	maximal, characteristic, complemented
2	#72342832	C₂	1	1	yes	maximal, characteristic, complemented
3	#72342824	C₂	1	1	yes	maximal, characteristic, complemented
4	#72342814	C₂	1	1	yes	maximal, characteristic, complemented
5	#72342797	C₂	1	1	yes	maximal, characteristic, complemented
6	#72342786	C₂	1	1	yes	maximal, characteristic, complemented
7	#72342770	C₂	1	1	yes	maximal, characteristic, complemented
8	#72317728	C₂	1	1	yes	maximal, characteristic, complemented
9	#72317394	C₂	1	1	yes	maximal, characteristic, complemented
10	#72317374	C₂	1	1	yes	maximal, characteristic, complemented
11	#72317357	C₂	1	1	yes	maximal, characteristic, complemented
12	#72317353	C₂	1	1	yes	maximal, characteristic, complemented
13	#72240519	C₂	1	1	yes	maximal, characteristic, complemented
14	#72233407	C₂	1	1	yes	maximal, characteristic, complemented
15	#72218460	C₂	1	1	yes	maximal, characteristic
16	#10586169	C₂²	1	1	yes	characteristic
17	#10574299	C₂²	1	1	yes	characteristic, complemented
18	#6437628	C₂²	1	1	yes	characteristic
19	#6437616	C₂²	1	1	yes	characteristic
20	#6437463	C₂²	1	1	yes	characteristic
21	#6374194	C₂²	1	1	yes	characteristic
22	#6371562	C₂²	1	1	yes	characteristic
23	#511791	C₄	1	4	yes	—
24	#511790	C₄	1	4	yes	—
25	#493402	C₂²	1	1	yes	characteristic, complemented
26	#493396	C₂²	1	1	yes	characteristic
27	#493390	C₂²	1	1	yes	characteristic
28	#493352	C₂²	1	1	yes	characteristic
29	#373690	C₂²	1	1	yes	characteristic, complemented
30	#359556	C₂²	1	1	yes	characteristic
31	#359556	C₂²	1	1	yes	characteristic
32	#359551	C₂²	1	1	yes	characteristic
33	#359551	C₂²	1	1	yes	characteristic
34	#359551	C₂²	1	1	yes	characteristic, complemented
35	#359507	C₂²	1	1	yes	characteristic
36	#359507	C₂²	1	1	yes	characteristic
37	#359426	C₂²	1	1	yes	characteristic
38	#355102	C₂²	1	1	yes	characteristic, complemented
39	#188071	C₂²	1	1	yes	characteristic
40	#188004	C₂²	1	1	yes	characteristic
41	#188004	C₂²	1	1	yes	characteristic
42	#188004	C₂²	1	1	yes	characteristic, complemented
43	#146483	C₂²	1	1	yes	characteristic, complemented
44	#146478	C₂²	1	1	yes	characteristic, complemented
45	#146463	C₂²	1	1	yes	characteristic, complemented
46	#128040	C₂²	1	1	yes	characteristic, complemented
47	#128040	C₂²	1	1	yes	characteristic, complemented
48	#128036	C₂²	1	1	yes	characteristic, complemented
49	#128026	C₂²	1	1	yes	characteristic, complemented
50	#108236	C₂²	1	1	yes	characteristic, complemented
51	#107152	C₂²	1	1	yes	characteristic, complemented
52	#106146	C₂²	1	1	yes	characteristic, complemented
53	#63076	D₄	1	1	yes	characteristic, complemented
54	#62820	C₂×C₄	1	2	yes	—
55	#62817	C₂×C₄	1	2	yes	—
56	#62811	C₂³	1	1	yes	characteristic
57	#62811	C₂³	1	1	yes	characteristic
58	#62811	C₂³	1	1	yes	characteristic
59	C₂²×#525	C₂³	1	1	yes	characteristic
60	#60398	D₄	1	2	yes	—
61	#60076	D₄	1	1	yes	characteristic, complemented
62	#60064	C₂³	1	1	yes	characteristic
63	#60064	C₂³	1	1	yes	characteristic
64	C₂²×#408	C₂³	1	1	yes	characteristic
65	#23901	C₂×C₄	1	2	yes	—
66	#23900	C₂×C₄	1	2	yes	—
67	#23897	C₂×C₄	1	4	yes	—
68	#23896	D₄	1	4	yes	—
69	#23895	C₂×C₄	1	4	yes	—
70	#21014	D₄	1	2	yes	—
71	#20989	D₄	1	2	yes	complemented
72	#20895	D₄	1	2	yes	complemented
73	#20612	D₄	1	2	yes	complemented
74	#20591	C₂³	1	1	yes	characteristic
75	#20591	C₂³	1	1	yes	characteristic
76	#20591	C₂³	1	1	yes	characteristic
77	#20591	C₂³	1	1	yes	characteristic
78	#20591	C₂³	1	1	yes	characteristic
79	#20584	C₂³	1	1	yes	characteristic
80	#20584	C₂³	1	1	yes	characteristic
81	#20579	D₄	1	2	yes	complemented
82	C₂×#1780	C₂³	1	1	yes	characteristic
83	#20360	D₄	1	1	yes	characteristic, complemented
84	#20360	D₄	1	1	yes	characteristic, complemented
85	#12707	C₂×C₄	1	4	yes	—
86	#12705	C₂×C₄	1	4	yes	—
87	#12688	C₂×C₄	1	2	yes	—
88	#12687	C₂×C₄	1	2	yes	—
89	#10756	D₄	1	2	yes	complemented
90	#10738	D₄	1	2	yes	complemented
91	C₂×#585	C₂×D₄	1	1	yes	characteristic
92	C₂×#585	C₂×D₄	1	1	yes	characteristic
93	C₂×#585	C₂×D₄	1	1	yes	characteristic
94	C₂³×#41	C₂⁴	1	1	yes	characteristic
95	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×#568	C₂×D₄	1	1	yes	characteristic
97	C₂×#568	C₂×D₄	1	1	yes	characteristic
98	C₂×#568	C₂×D₄	1	1	yes	characteristic
99	C₂×#568	C₂×D₄	1	1	yes	characteristic
100	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
101	C₂⁴×C₈	C₂×D₄	1	1	yes	characteristic
102	#3485	C₂×D₄	1	2	yes	—
103	#3485	C₂×D₄	1	2	yes	—
104	#3485	#41	1	2	yes	—
105	#3485	#41	1	2	yes	—
106	#3484	C₂²×C₄	1	2	yes	—
107	#3484	C₂²×C₄	1	2	yes	—
108	#3484	C₂²×C₄	1	2	yes	—
109	#3484	#41	1	2	yes	—
110	#3449	C₂×D₄	1	2	yes	—
111	C₂×#525	C₂×D₄	1	2	yes	—
112	C₂²×#123	C₂²×C₄	1	1	yes	characteristic
113	C₂²×#123	#41	1	1	yes	characteristic
114	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
115	C₂²×#120	#41	1	1	yes	characteristic
116	#3428	C₂×D₄	1	1	yes	characteristic
117	#3428	C₂×D₄	1	1	yes	characteristic
118	#3428	C₂×D₄	1	1	yes	characteristic
119	C₂²×#117	C₂×D₄	1	1	yes	characteristic
120	#3026	C₂²⋊C₄	1	8	yes	—
121	C₂×#413	C₂×D₄	1	2	yes	—
122	C₂×#410	C₂×D₄	1	2	yes	—
123	#2879	C₂×D₄	1	2	yes	—
124	C₂×#407	C₂×D₄	1	2	yes	—
125	C₂×#407	C₂×D₄	1	2	yes	—
126	C₂×#407	C₂×D₄	1	2	yes	—
127	#1913	C₂²⋊C₄	1	8	yes	—
128	#1783	C₂²×C₄	1	2	yes	—
129	#1782	C₂×D₄	1	2	yes	—
130	#1781	C₂²×C₄	1	2	yes	—
131	#1781	C₂²×C₄	1	2	yes	—
132	C₂⁶	C₄×D₄	1	1	yes	characteristic
133	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
135	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
136	C₂×H(5,𝔽₂)	#121	1	2	yes	—
137	C₂²×#41	#120	1	2	yes	—
138	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
139	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
140	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
141	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
142	C₂²×#41	C₂×#41	1	1	yes	characteristic
143	C₂²×#41	C₂×#41	1	1	yes	characteristic
144	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂³×Q₈	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
148	C₂³×D₄	C₂×#41	1	1	yes	characteristic
149	C₂³×D₄	C₂×#41	1	1	yes	characteristic
150	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
151	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
153	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂²×M₁₆	#120	1	2	yes	—
155	C₂²×M₁₆	#120	1	2	yes	—
156	#534	C₂×C₂²⋊C₄	1	4	yes	—
157	#534	#123	1	4	yes	—
158	C₂×#123	#121	1	4	yes	—
159	C₂×#120	C₄×D₄	1	4	yes	—
160	#518	C₂×C₂²⋊C₄	1	2	yes	—
161	#518	C₄×D₄	1	2	yes	—
162	#518	C₄×D₄	1	2	yes	—
163	#518	#121	1	2	yes	—
164	C₂×#117	#120	1	2	yes	—
165	C₂²×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
166	C₂²×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
167	C₂×#99	C₂×C₂²⋊C₄	1	4	yes	—
168	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂⁵	#551	1	1	yes	characteristic
171	C₂²×Q₈	B(4,𝔽₂)	1	2	yes	—
172	C₂²×Q₈	#535	1	1	yes	characteristic
173	C₂²×Q₈	#538	1	1	yes	characteristic
174	C₂²×Q₈	D₄²	1	1	yes	characteristic
175	C₂²×Q₈	#546	1	1	yes	characteristic
176	C₂²×D₄	#386	1	4	yes	—
177	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
178	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
181	C₂²×D₄	#518	1	1	yes	characteristic
182	C₂²×D₄	C₂×#121	1	1	yes	characteristic
183	C₂²×D₄	C₂×#123	1	1	yes	characteristic
184	C₂²×D₄	#525	1	1	yes	characteristic
185	C₂²×D₄	#525	1	1	yes	characteristic
186	C₂²×D₄	#546	1	1	yes	characteristic
187	C₂²×D₄	#546	1	1	yes	characteristic
188	C₂²×D₄	#546	1	1	yes	characteristic
189	C₂²×D₄	#551	1	1	yes	characteristic
190	C₂³×C₄	B(4,𝔽₂)	1	4	yes	—
191	C₂³×C₄	#518	1	1	yes	characteristic
192	C₂³×C₄	C₂×#120	1	1	yes	characteristic
193	C₂³×C₄	C₂×#120	1	1	yes	characteristic
194	C₂³×C₄	C₂×#121	1	1	yes	characteristic
195	C₂³×C₄	#525	1	1	yes	characteristic
196	C₂³×C₄	#534	1	1	yes	characteristic
197	C₂³×C₄	#538	1	1	yes	characteristic
198	C₂³×C₄	D₄²	1	1	yes	characteristic
199	C₂³×C₄	D₄²	1	1	yes	characteristic
200	C₂³×C₄	D₄²	1	1	yes	characteristic
201	C₂³×C₄	D₄²	1	1	yes	characteristic
202	C₂³×C₄	D₄²	1	1	yes	characteristic
203	C₂³×C₄	#546	1	1	yes	characteristic
204	C₂³×C₄	#546	1	1	yes	characteristic
205	C₂³×C₄	#546	1	1	yes	characteristic
206	C₂³×C₄	#546	1	1	yes	characteristic
207	C₂²×C₈	B(4,𝔽₂)	1	4	yes	—
208	C₂×C₂²⋊C₄	#386	1	4	yes	—
209	C₂⁴	#1889	1	4	yes	—
210	C₂⁴	C₂×#386	1	1	yes	characteristic
211	C₂⁴	C₄×#120	1	1	yes	characteristic
212	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
213	C₂⁴	#2358	1	1	yes	characteristic
214	C₂⁴	#2358	1	1	yes	characteristic
215	C₂⁴	#2418	1	1	yes	characteristic
216	C₂⁴	#2534	1	1	yes	characteristic
217	C₂²×C₄	#1889	1	4	yes	—
218	C₂²×C₄	#2321	1	1	yes	characteristic
219	C₂²×C₄	D₄×C₂²⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	#2403	1	1	yes	characteristic
221	C₂²×C₄	#2404	1	1	yes	characteristic
222	C₂²×C₄	#2408	1	1	yes	characteristic
223	C₂²×C₄	#2418	1	1	yes	characteristic
224	C₂²×C₄	#2424	1	1	yes	characteristic
225	C₂²×C₄	#2424	1	1	yes	characteristic
226	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
227	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
228	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
229	C₂²×C₄	#3025	1	2	yes	—
230	C₂³	#13967	1	1	yes	characteristic
231	C₂³	C₂×#1889	1	2	yes	—
232	C₂³	C₄×B(4,𝔽₂)	1	2	yes	—
233	C₂×C₄	#23895	1	1	yes	characteristic
234	C₂×C₄	#23896	1	1	yes	characteristic
235	C₂×C₄	#23900	1	1	yes	characteristic
236	C₂×C₄	#23900	1	1	yes	characteristic
237	C₂×C₄	#23900	1	1	yes	characteristic
238	C₂×C₄	#23902	1	1	yes	characteristic
239	C₂×C₄	#23984	1	2	yes	—
240	C₂×C₄	#24260	1	2	yes	—
241	C₂²	#364410	1	1	yes	characteristic
242	C₂²	#364500	1	1	yes	characteristic
243	C₂²	#364513	1	1	yes	characteristic
244	C₄	#245225	1	1	yes	characteristic
245	C₄	#245227	1	1	yes	characteristic
246	C₄	#245229	1	1	yes	characteristic
247	C₄	#245231	1	1	yes	characteristic
248	C₂	#72237497	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#245225
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂³×#41	C₂⁴
socle	C₂	#72237497
prime radical	#72218460	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ 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. #72342836	C₂	semidirect
2. #72342832	C₂	semidirect
3. #72342824	C₂	semidirect
4. #72342814	C₂	semidirect
5. #72342797	C₂	semidirect
6. #72342786	C₂	semidirect
7. #72342770	C₂	semidirect
8. #72317728	C₂	semidirect
9. #72317394	C₂	semidirect
10. #72317374	C₂	semidirect
11. #72317357	C₂	semidirect
12. #72317353	C₂	semidirect
13. #72240519	C₂	semidirect
14. #72233407	C₂	semidirect
17. #10574299	C₂²	semidirect
25. #493402	C₂²	semidirect
29. #373690	C₂²	semidirect
34. #359551	C₂²	semidirect
38. #355102	C₂²	semidirect
42. #188004	C₂²	semidirect
43. #146483	C₂²	semidirect
44. #146478	C₂²	semidirect
45. #146463	C₂²	semidirect
46. #128040	C₂²	semidirect
47. #128040	C₂²	semidirect
48. #128036	C₂²	semidirect
49. #128026	C₂²	semidirect
50. #108236	C₂²	semidirect
51. #107152	C₂²	semidirect
52. #106146	C₂²	semidirect
53. #63076	D₄	semidirect
61. #60076	D₄	semidirect
71. #20989	D₄	semidirect
72. #20895	D₄	semidirect
73. #20612	D₄	semidirect
81. #20579	D₄	semidirect
83. #20360	D₄	semidirect
84. #20360	D₄	semidirect
89. #10756	D₄	semidirect
90. #10738	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72342836	2
#72342832	2
#72342824	2
#72342814	2
#72342797	2
#72342786	2
#72342770	2
#72317728	2
#72317394	2
#72317374	2
#72317357	2
#72317353	2
#72240519	2
#72233407	2
#72218460	2