Subgroups — #72340840

Order 2048.

At least 6884 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

The subgroup lattice has 237 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72340838	C₂	1	1	yes	maximal, characteristic, complemented
2	#72340835	C₂	1	1	yes	maximal, characteristic, complemented
3	#72340833	C₂	1	1	yes	maximal, characteristic, complemented
4	#72286752	C₂	1	1	yes	maximal, characteristic, complemented
5	#72286373	C₂	1	1	yes	maximal, characteristic, complemented
6	#72286264	C₂	1	1	yes	maximal, characteristic, complemented
7	#72284842	C₂	1	1	yes	maximal, characteristic, complemented
8	#72284445	C₂	1	1	yes	maximal, characteristic, complemented
9	#72279638	C₂	1	1	yes	maximal, characteristic, complemented
10	#72279437	C₂	1	1	yes	maximal, characteristic, complemented
11	#72279413	C₂	1	1	yes	maximal, characteristic, complemented
12	#72279318	C₂	1	1	yes	maximal, characteristic, complemented
13	#72251988	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#511819	C₂	1	1	yes	maximal, characteristic, complemented
15	#72215981	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63011	C₂²	1	1	yes	characteristic, complemented
17	#10576177	C₂²	1	1	yes	characteristic, complemented
18	C₂×#33570	C₂²	1	1	yes	characteristic
19	C₂×#33559	C₂²	1	1	yes	characteristic
20	C₂×#33546	C₂²	1	1	yes	characteristic
21	C₂×#33546	C₂²	1	1	yes	characteristic
22	#6443311	C₂²	1	1	yes	characteristic
23	C₂×#23872	C₂²	1	1	yes	characteristic
24	#6437452	C₂²	1	1	yes	characteristic
25	#6351560	C₂²	1	1	yes	characteristic
26	#6348073	C₂²	1	1	yes	characteristic
27	#512601	C₂²	1	1	yes	characteristic, complemented
28	#512594	C₂²	1	1	yes	characteristic, complemented
29	#512578	C₂²	1	1	yes	characteristic, complemented
30	#512571	C₂²	1	1	yes	characteristic, complemented
31	#512564	C₂²	1	1	yes	characteristic, complemented
32	#512564	C₂²	1	1	yes	characteristic, complemented
33	#493630	C₂²	1	1	yes	characteristic, complemented
34	C₂×#12778	C₂²	1	1	yes	characteristic, complemented
35	C₂×#12765	C₂²	1	1	yes	characteristic, complemented
36	#493577	C₂²	1	1	yes	characteristic, complemented
37	#493431	C₂²	1	1	yes	characteristic, complemented
38	C₂×#12718	C₂²	1	1	yes	characteristic, complemented
39	#386050	C₂²	1	1	yes	characteristic, complemented
40	#383066	C₂²	1	1	yes	characteristic, complemented
41	#383057	C₂²	1	1	yes	characteristic, complemented
42	#187899	C₂²	1	1	yes	characteristic, complemented
43	#187081	C₂²	1	1	yes	characteristic, complemented
44	#186576	C₂²	1	1	yes	characteristic, complemented
45	#151864	C₂²	1	1	yes	characteristic, complemented
46	#151854	C₂²	1	1	yes	characteristic, complemented
47	#151816	C₂²	1	1	yes	characteristic, complemented
48	#151806	C₂²	1	1	yes	characteristic, complemented
49	#151786	C₂²	1	1	yes	characteristic, complemented
50	#151782	C₂²	1	1	yes	characteristic, complemented
51	C₂×#3585	D₄	1	1	yes	characteristic, complemented
52	#62852	D₄	1	1	yes	characteristic, complemented
53	C₂×#3498	C₂³	1	1	yes	characteristic
54	C₂×#3549	C₂³	1	1	yes	characteristic
55	C₂×#3481	C₂³	1	1	yes	characteristic
56	#60639	C₂³	1	1	yes	characteristic
57	#60415	D₄	1	1	yes	characteristic, complemented
58	C₂×#3027	D₄	1	1	yes	characteristic, complemented
59	C₂×#3021	D₄	1	1	yes	characteristic, complemented
60	C₂×#3018	C₂³	1	1	yes	characteristic
61	C₂×#3018	C₂³	1	1	yes	characteristic
62	#35274	D₄	1	1	yes	characteristic, complemented
63	#33570	D₄	1	4	yes	complemented
64	C₂×#2199	D₄	1	4	yes	complemented
65	#33546	D₄	1	4	yes	complemented
66	C₂×#2198	D₄	1	2	yes	complemented
67	C₂×#2196	D₄	1	2	yes	complemented
68	C₂×#2132	C₂³	1	1	yes	characteristic
69	C₂×#2126	C₂³	1	1	yes	characteristic
70	C₂×#2126	C₂³	1	1	yes	characteristic
71	C₂×#2126	C₂³	1	1	yes	characteristic
72	C₂×#1965	D₄	1	1	yes	characteristic, complemented
73	#21209	C₂³	1	1	yes	characteristic
74	C₂×#1914	D₄	1	1	yes	characteristic, complemented
75	#20956	D₄	1	1	yes	characteristic, complemented
76	#20952	C₂³	1	1	yes	characteristic
77	C₂×#1884	C₂³	1	1	yes	characteristic
78	C₂×#1795	D₄	1	1	yes	characteristic, complemented
79	#20592	D₄	1	1	yes	characteristic, complemented
80	C₄×#421	D₄	1	1	yes	characteristic, complemented
81	#19074	D₄	1	2	yes	complemented
82	#19073	D₄	1	2	yes	complemented
83	#18528	D₄	1	2	yes	complemented
84	#18527	D₄	1	2	yes	complemented
85	#17738	C₂³	1	1	yes	characteristic
86	#17663	C₂³	1	1	yes	characteristic
87	C₂×#585	C₂×D₄	1	1	yes	characteristic
88	C₂×#585	C₂×D₄	1	1	yes	characteristic
89	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
90	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
91	C₂×#568	C₂×D₄	1	1	yes	characteristic
92	#3549	C₂×D₄	1	2	yes	—
93	#3549	C₂×D₄	1	2	yes	—
94	C₂×H(3,𝔽₄)	C₂×D₄	1	2	yes	—
95	#3481	C₂×D₄	1	2	yes	—
96	#3481	C₂×D₄	1	2	yes	—
97	C₂×#559	C₂×D₄	1	1	yes	characteristic
98	C₂×D₄²	C₂×D₄	1	1	yes	characteristic
99	C₂×#535	C₂×D₄	1	1	yes	characteristic
100	C₂×#535	C₂×D₄	1	1	yes	characteristic
101	C₂×#532	C₂×D₄	1	1	yes	characteristic
102	C₂×#532	C₂⁴	1	1	yes	characteristic
103	C₂²×#127	C₂×D₄	1	1	yes	characteristic
104	#3428	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×#41	C₂×D₄	1	1	yes	characteristic
106	#3152	C₂×D₄	1	1	yes	characteristic
107	#3152	C₂×D₄	1	1	yes	characteristic
108	#3151	C₂×D₄	1	1	yes	characteristic
109	#3151	C₂×D₄	1	1	yes	characteristic
110	#3018	C₂×D₄	1	2	yes	—
111	#3018	C₂×D₄	1	2	yes	—
112	C₂×#455	C₂×D₄	1	2	yes	—
113	C₂×#453	C₂×D₄	1	2	yes	—
114	C₂×#423	C₂×D₄	1	1	yes	characteristic
115	C₂×#413	C₂×D₄	1	1	yes	characteristic
116	C₂×#413	C₂×D₄	1	1	yes	characteristic
117	C₂×#413	C₂×D₄	1	1	yes	characteristic
118	C₂×#410	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×M₁₆	C₂×D₄	1	1	yes	characteristic
120	C₄×#117	C₂×D₄	1	1	yes	characteristic
121	#2132	C₂×D₄	1	2	yes	—
122	#2126	C₂×D₄	1	2	yes	—
123	C₂×#354	C₂×D₄	1	2	yes	—
124	C₂×#353	C₂×D₄	1	2	yes	—
125	#1560	C₂×D₄	1	2	yes	—
126	#1525	C₂×D₄	1	2	yes	—
127	#585	#120	1	2	yes	complemented
128	C₂×H(5,𝔽₂)	#120	1	2	yes	—
129	C₂²×#41	#120	1	1	yes	characteristic
130	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
131	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
132	C₂³×D₄	#120	1	1	yes	characteristic
133	C₂³×D₄	#120	1	1	yes	characteristic
134	H(3,𝔽₄)	#120	1	4	yes	—
135	#559	#120	1	2	yes	—
136	D₄²	#120	1	2	yes	—
137	#532	C₂²×D₄	1	1	yes	characteristic
138	#532	C₂²×D₄	1	1	yes	characteristic
139	#532	C₂²×D₄	1	1	yes	characteristic
140	#532	C₂²×D₄	1	1	yes	characteristic
141	#532	H(5,𝔽₂)	1	1	yes	characteristic
142	#532	H(5,𝔽₂)	1	1	yes	characteristic
143	#532	H(5,𝔽₂)	1	1	yes	characteristic
144	#532	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄⋊Q₈	#120	1	1	yes	characteristic, complemented
146	C₂×C₄⋊Q₈	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×#127	#120	1	1	yes	characteristic, complemented
148	C₂×#127	C₂²×D₄	1	1	yes	characteristic
149	C₂×#124	C₂²×D₄	1	1	yes	characteristic
150	C₂×#124	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×#117	#120	1	1	yes	characteristic
152	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
153	#413	#120	1	2	yes	complemented
154	C₂×#100	#120	1	2	yes	—
155	C₄×M₁₆	#120	1	1	yes	characteristic, complemented
156	C₄×M₁₆	#120	1	1	yes	characteristic, complemented
157	C₂⁵	B(4,𝔽₂)	1	2	yes	—
158	C₂×#41	C₂×#120	1	1	yes	characteristic
159	C₂×#41	#534	1	1	yes	characteristic
160	C₂×#41	D₄²	1	1	yes	characteristic
161	C₂×#41	D₄²	1	1	yes	characteristic
162	C₂×#41	D₄²	1	1	yes	characteristic
163	C₂×#41	D₄²	1	1	yes	characteristic
164	C₂×#41	D₄²	1	1	yes	characteristic
165	C₂×#41	D₄²	1	1	yes	characteristic
166	C₂²×Q₈	B(4,𝔽₂)	1	1	yes	characteristic
167	C₂²×Q₈	#534	1	1	yes	characteristic
168	C₂²×Q₈	D₄²	1	1	yes	characteristic
169	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
170	C₂²×D₄	B(4,𝔽₂)	1	1	yes	characteristic
171	C₂²×D₄	C₂×#120	1	1	yes	characteristic
172	C₂²×D₄	D₄²	1	1	yes	characteristic
173	C₂³×C₄	C₂×#120	1	1	yes	characteristic
174	C₄⋊Q₈	#534	1	1	yes	characteristic
175	C₄⋊Q₈	#534	1	1	yes	characteristic
176	C₄⋊Q₈	D₄²	1	1	yes	characteristic
177	C₄⋊Q₈	H(3,𝔽₄)	1	1	yes	characteristic
178	#127	C₂×#120	1	1	yes	characteristic
179	#127	#534	1	1	yes	characteristic
180	#127	D₄²	1	1	yes	characteristic
181	#127	D₄²	1	1	yes	characteristic
182	#124	C₂×#120	1	1	yes	characteristic
183	#124	#534	1	1	yes	characteristic
184	#124	#534	1	1	yes	characteristic
185	#124	#534	1	1	yes	characteristic
186	#124	D₄²	1	1	yes	characteristic
187	#124	D₄²	1	1	yes	characteristic
188	#124	D₄²	1	1	yes	characteristic
189	#124	H(3,𝔽₄)	1	1	yes	characteristic
190	#117	B(4,𝔽₂)	1	2	yes	—
191	C₂×C₄²	C₂×#120	1	1	yes	characteristic
192	C₂×C₄²	#534	1	1	yes	characteristic
193	C₂×C₄²	D₄²	1	1	yes	characteristic
194	C₂×C₄²	D₄²	1	1	yes	characteristic
195	C₂×C₄²	D₄²	1	1	yes	characteristic
196	C₂×C₄²	D₄²	1	1	yes	characteristic
197	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
198	C₂×Q₈	#2403	1	1	yes	characteristic
199	C₂×Q₈	#2679	1	1	yes	characteristic
200	C₂×Q₈	#2679	1	1	yes	characteristic
201	C₂×Q₈	#2679	1	1	yes	characteristic
202	C₂×D₄	#2196	1	2	yes	—
203	C₂×D₄	#2403	1	1	yes	characteristic
204	C₂×D₄	#2403	1	1	yes	characteristic
205	C₂×D₄	#2403	1	1	yes	characteristic
206	C₂×D₄	#2679	1	1	yes	characteristic
207	C₂²×C₄	#2403	1	1	yes	characteristic
208	C₂²×C₄	#2403	1	1	yes	characteristic
209	C₂²×C₄	#2403	1	1	yes	characteristic
210	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
211	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
212	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
213	C₄²	#2403	1	1	yes	characteristic
214	C₄²	#2403	1	1	yes	characteristic
215	C₄²	#2679	1	1	yes	characteristic
216	C₄²	#2679	1	1	yes	characteristic
217	C₂³	#23900	1	1	yes	characteristic
218	C₂³	#23900	1	1	yes	characteristic
219	C₂³	#23900	1	1	yes	characteristic
220	C₂×C₄	#15947	1	1	yes	characteristic
221	C₂×C₄	#23900	1	1	yes	characteristic
222	C₂×C₄	#23900	1	1	yes	characteristic
223	C₂×C₄	#23900	1	1	yes	characteristic
224	C₂×C₄	C₂×#2196	1	1	yes	characteristic
225	C₂×C₄	#33551	1	1	yes	characteristic
226	C₂×C₄	#33551	1	1	yes	characteristic
227	C₂×C₄	#33557	1	1	yes	characteristic
228	C₂²	#449475	1	1	yes	characteristic
229	C₂²	#499787	1	1	yes	characteristic
230	C₂²	#499804	1	1	yes	characteristic
231	C₄	#499787	1	1	yes	characteristic
232	C₄	#499804	1	1	yes	characteristic
233	C₄	#499832	1	1	yes	characteristic
234	C₄	#499843	1	1	yes	characteristic
235	C₂	#72318263	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72318263
commutator, Frattini, cosocle	C₂×#532	C₂⁴
LCS	G ▷ C₂×#532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#3498 ◁ G
LECS	G ▷ C₂×#532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72340838	C₂	semidirect
2. #72340835	C₂	semidirect
3. #72340833	C₂	semidirect
4. #72286752	C₂	semidirect
5. #72286373	C₂	semidirect
6. #72286264	C₂	semidirect
7. #72284842	C₂	semidirect
8. #72284445	C₂	semidirect
9. #72279638	C₂	semidirect
10. #72279437	C₂	semidirect
11. #72279413	C₂	semidirect
12. #72279318	C₂	semidirect
13. #72251988	C₂	semidirect
14. C₂×#511819	C₂	semidirect
15. #72215981	C₂	semidirect
16. C₂×#63011	C₂²	semidirect
17. #10576177	C₂²	semidirect
27. #512601	C₂²	semidirect
28. #512594	C₂²	semidirect
29. #512578	C₂²	semidirect
30. #512571	C₂²	semidirect
31. #512564	C₂²	semidirect
32. #512564	C₂²	semidirect
33. #493630	C₂²	semidirect
34. C₂×#12778	C₂²	semidirect
35. C₂×#12765	C₂²	semidirect
36. #493577	C₂²	semidirect
37. #493431	C₂²	semidirect
38. C₂×#12718	C₂²	semidirect
39. #386050	C₂²	semidirect
40. #383066	C₂²	semidirect
41. #383057	C₂²	semidirect
42. #187899	C₂²	semidirect
43. #187081	C₂²	semidirect
44. #186576	C₂²	semidirect
45. #151864	C₂²	semidirect
46. #151854	C₂²	semidirect
47. #151816	C₂²	semidirect
48. #151806	C₂²	semidirect
49. #151786	C₂²	semidirect
50. #151782	C₂²	semidirect
51. C₂×#3585	D₄	semidirect
52. #62852	D₄	semidirect
57. #60415	D₄	semidirect
58. C₂×#3027	D₄	semidirect
59. C₂×#3021	D₄	semidirect
62. #35274	D₄	semidirect
63. #33570	D₄	semidirect
64. C₂×#2199	D₄	semidirect
65. #33546	D₄	semidirect
66. C₂×#2198	D₄	semidirect
67. C₂×#2196	D₄	semidirect
72. C₂×#1965	D₄	semidirect
74. C₂×#1914	D₄	semidirect
75. #20956	D₄	semidirect
78. C₂×#1795	D₄	semidirect
79. #20592	D₄	semidirect
80. C₄×#421	D₄	semidirect
81. #19074	D₄	semidirect
82. #19073	D₄	semidirect
83. #18528	D₄	semidirect
84. #18527	D₄	semidirect
127. #585	#120	semidirect
145. C₂×C₄⋊Q₈	#120	semidirect
147. C₂×#127	#120	semidirect
153. #413	#120	semidirect
155. C₄×M₁₆	#120	semidirect
156. C₄×M₁₆	#120	semidirect

Maximal subgroups

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

subgroup	index
#72340838	2
#72340835	2
#72340833	2
#72286752	2
#72286373	2
#72286264	2
#72284842	2
#72284445	2
#72279638	2
#72279437	2
#72279413	2
#72279318	2
#72251988	2
C₂×#511819	2
#72215981	2