Subgroups — #72339877

Order 2048.

At least 6564 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

The subgroup lattice has 194 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72339876	C₂	1	1	yes	maximal, characteristic, complemented
2	#72339875	C₂	1	1	yes	maximal, characteristic, complemented
3	#72339874	C₂	1	1	yes	maximal, characteristic, complemented
4	#72339873	C₂	1	1	yes	maximal, characteristic, complemented
5	#72339872	C₂	1	1	yes	maximal, characteristic, complemented
6	#72339871	C₂	1	1	yes	maximal, characteristic, complemented
7	#72339870	C₂	1	1	yes	maximal, characteristic, complemented
8	#72339869	C₂	1	1	yes	maximal, characteristic, complemented
9	#72339836	C₂	1	1	yes	maximal, characteristic, complemented
10	#72339825	C₂	1	1	yes	maximal, characteristic, complemented
11	#72306894	C₂	1	1	yes	maximal, characteristic, complemented
12	#72306589	C₂	1	1	yes	maximal, characteristic, complemented
13	#72283261	C₂	1	1	yes	maximal, characteristic, complemented
14	#72231260	C₂	1	1	yes	maximal, characteristic, complemented
15	#72231250	C₂	1	1	yes	maximal, characteristic, complemented
16	#7487876	C₂²	1	1	yes	characteristic, complemented
17	#6368904	C₂²	1	1	yes	characteristic
18	#6365636	C₂²	1	1	yes	characteristic
19	#6363896	C₂²	1	1	yes	characteristic
20	#6363073	C₂²	1	1	yes	characteristic
21	#6358059	C₂²	1	1	yes	characteristic
22	#6350173	C₂²	1	1	yes	characteristic
23	#6350079	C₂²	1	1	yes	characteristic
24	#512903	C₂²	1	1	yes	characteristic, complemented
25	#512866	C₂²	1	1	yes	characteristic, complemented
26	#512864	C₂²	1	1	yes	characteristic
27	#512863	C₂²	1	1	yes	characteristic, complemented
28	#489351	C₂²	1	1	yes	characteristic, complemented
29	#489349	C₂²	1	1	yes	characteristic, complemented
30	#153549	C₂²	1	1	yes	characteristic, complemented
31	#153547	C₂²	1	1	yes	characteristic, complemented
32	#153539	C₂²	1	1	yes	characteristic, complemented
33	#153538	C₂²	1	1	yes	characteristic, complemented
34	#153469	C₂²	1	1	yes	characteristic, complemented
35	#153469	C₂²	1	1	yes	characteristic, complemented
36	#153468	C₂²	1	1	yes	characteristic, complemented
37	#153465	C₂²	1	1	yes	characteristic, complemented
38	#153462	C₂²	1	1	yes	characteristic, complemented
39	#153460	C₂²	1	1	yes	characteristic
40	#153460	C₂²	1	1	yes	characteristic
41	#153459	C₂²	1	1	yes	characteristic, complemented
42	#152619	C₂²	1	1	yes	characteristic, complemented
43	#152616	C₂²	1	1	yes	characteristic, complemented
44	#152604	C₂²	1	1	yes	characteristic
45	#152603	C₂²	1	1	yes	characteristic, complemented
46	#152123	C₂²	1	1	yes	characteristic, complemented
47	#152108	C₂²	1	1	yes	characteristic, complemented
48	#139431	C₂²	1	1	yes	characteristic, complemented
49	#139425	C₂²	1	1	yes	characteristic, complemented
50	#133268	C₂²	1	1	yes	characteristic, complemented
51	#63030	D₄	1	1	yes	characteristic, complemented
52	#63011	D₄	1	1	yes	characteristic, complemented
53	#61110	C₂³	1	1	yes	characteristic
54	#61015	D₄	1	1	yes	characteristic, complemented
55	#60817	C₂³	1	1	yes	characteristic
56	#23361	C₂³	1	1	yes	characteristic
57	#20319	D₄	1	1	yes	characteristic, complemented
58	#20294	D₄	1	1	yes	characteristic, complemented
59	#20294	D₄	1	1	yes	characteristic, complemented
60	#20294	D₄	1	1	yes	characteristic, complemented
61	#20157	D₄	1	1	yes	characteristic, complemented
62	#20156	D₄	1	1	yes	characteristic, complemented
63	#20152	D₄	1	1	yes	characteristic, complemented
64	#20138	D₄	1	1	yes	characteristic, complemented
65	#20138	D₄	1	1	yes	characteristic, complemented
66	#20131	D₄	1	1	yes	characteristic, complemented
67	#20067	D₄	1	1	yes	characteristic, complemented
68	#20067	D₄	1	1	yes	characteristic, complemented
69	#20021	D₄	1	1	yes	characteristic, complemented
70	#19996	D₄	1	1	yes	characteristic, complemented
71	#19987	D₄	1	1	yes	characteristic, complemented
72	D₈²	D₄	1	1	yes	characteristic, complemented
73	#19749	C₂³	1	1	yes	characteristic
74	#19747	D₄	1	1	yes	characteristic, complemented
75	#19727	C₂³	1	1	yes	characteristic
76	#19717	D₄	1	1	yes	characteristic, complemented
77	#18714	C₂³	1	1	yes	characteristic
78	#18712	C₂³	1	1	yes	characteristic
79	#18691	D₄	1	1	yes	characteristic, complemented
80	#18682	D₄	1	1	yes	characteristic, complemented
81	#18527	C₂³	1	1	yes	characteristic
82	#18526	C₂³	1	1	yes	characteristic
83	#18485	D₄	1	1	yes	characteristic, complemented
84	C₂×#1620	D₄	1	1	yes	characteristic, complemented
85	#17782	D₄	1	1	yes	characteristic, complemented
86	#17771	D₄	1	1	yes	characteristic, complemented
87	#17342	D₄	1	1	yes	characteristic, complemented
88	#13591	C₂³	1	1	yes	characteristic
89	#13585	C₂³	1	1	yes	characteristic
90	#13584	C₂³	1	1	yes	characteristic
91	#13584	C₂³	1	1	yes	characteristic
92	#11992	C₂³	1	1	yes	characteristic
93	#11992	C₂³	1	1	yes	characteristic
94	#3549	C₂×D₄	1	1	yes	characteristic
95	#3543	C₂×D₄	1	1	yes	characteristic
96	#3508	C₂×D₄	1	1	yes	characteristic
97	#3498	C₂×D₄	1	1	yes	characteristic
98	#3481	C₂×D₄	1	1	yes	characteristic
99	C₂×#559	C₂×D₄	1	1	yes	characteristic
100	#3185	C₂×D₄	1	1	yes	characteristic
101	#3184	C₂×D₄	1	1	yes	characteristic
102	#3151	C₂⁴	1	1	yes	characteristic
103	C₂×#493	C₂×D₄	1	1	yes	characteristic
104	#3138	C₂×D₄	1	1	yes	characteristic
105	#3131	C₂×D₄	1	1	yes	characteristic
106	#2922	C₂×D₄	1	1	yes	characteristic
107	#1722	C₂×D₄	1	1	yes	characteristic
108	#1721	C₂×D₄	1	1	yes	characteristic
109	#1715	C₂×D₄	1	1	yes	characteristic
110	#1715	C₂×D₄	1	1	yes	characteristic
111	#1712	C₂×D₄	1	1	yes	characteristic
112	#1709	C₂×D₄	1	1	yes	characteristic
113	#1708	C₂×D₄	1	1	yes	characteristic
114	#1708	C₂×D₄	1	1	yes	characteristic
115	#1630	C₂×D₄	1	1	yes	characteristic
116	#1630	C₂×D₄	1	1	yes	characteristic
117	#1630	C₂×D₄	1	1	yes	characteristic
118	#1629	C₂×D₄	1	1	yes	characteristic
119	#1624	C₂×D₄	1	1	yes	characteristic
120	#1624	C₂×D₄	1	1	yes	characteristic
121	#1624	C₂×D₄	1	1	yes	characteristic
122	#1624	C₂×D₄	1	1	yes	characteristic
123	#1620	C₂×D₄	1	1	yes	characteristic
124	#1620	C₂×D₄	1	1	yes	characteristic
125	#1620	C₂×D₄	1	1	yes	characteristic
126	#1620	C₂×D₄	1	1	yes	characteristic
127	#1620	C₂×D₄	1	1	yes	characteristic
128	#1619	C₂×D₄	1	1	yes	characteristic
129	#1619	C₂×D₄	1	1	yes	characteristic
130	#1619	C₂×D₄	1	1	yes	characteristic
131	#1560	C₂×D₄	1	1	yes	characteristic
132	#1555	C₂×D₄	1	1	yes	characteristic
133	#1543	C₂×D₄	1	1	yes	characteristic
134	C₂×#331	C₂×D₄	1	1	yes	characteristic
135	#1455	C₂×D₄	1	1	yes	characteristic
136	#1453	C₂×D₄	1	1	yes	characteristic
137	#559	#120	1	1	yes	characteristic
138	#559	#120	1	1	yes	characteristic
139	#559	#120	1	1	yes	characteristic
140	#559	#120	1	1	yes	characteristic
141	#555	#120	1	1	yes	characteristic
142	#555	#120	1	1	yes	characteristic
143	D₄²	#120	1	1	yes	characteristic
144	D₄²	#120	1	1	yes	characteristic
145	#533	#120	1	1	yes	characteristic
146	#532	C₂²×D₄	1	1	yes	characteristic
147	C₂×#127	#120	1	1	yes	characteristic
148	C₂×#127	C₂²×D₄	1	1	yes	characteristic
149	C₄×#41	#120	1	1	yes	characteristic
150	#498	#120	1	1	yes	characteristic
151	#498	#120	1	1	yes	characteristic
152	#494	#120	1	1	yes	characteristic
153	#493	#120	1	1	yes	characteristic
154	#493	C₂²×D₄	1	1	yes	characteristic
155	#493	C₂²×D₄	1	1	yes	characteristic
156	#492	C₂²×D₄	1	1	yes	characteristic
157	#492	C₂²×D₄	1	1	yes	characteristic
158	C₄×M₁₆	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₈	#120	1	1	yes	characteristic
160	#333	#120	1	1	yes	characteristic
161	#333	#120	1	1	yes	characteristic
162	#331	#120	1	1	yes	characteristic
163	#331	#120	1	1	yes	characteristic
164	#331	#120	1	1	yes	characteristic
165	#331	#120	1	1	yes	characteristic
166	#331	#120	1	1	yes	characteristic
167	#331	#120	1	1	yes	characteristic
168	#322	#120	1	1	yes	characteristic
169	#322	#120	1	1	yes	characteristic
170	C₈²	#120	1	1	yes	characteristic
171	C₈²	#120	1	1	yes	characteristic
172	C₂×#41	B(4,𝔽₂)	1	1	yes	characteristic
173	C₄⋊Q₈	C₂×#120	1	1	yes	characteristic
174	#127	C₂×#120	1	1	yes	characteristic
175	#127	C₂×#120	1	1	yes	characteristic
176	#127	C₂×#120	1	1	yes	characteristic
177	#117	B(4,𝔽₂)	1	1	yes	characteristic
178	C₂×C₄²	C₂×#120	1	1	yes	characteristic
179	C₄×C₈	C₂×#120	1	1	yes	characteristic
180	C₄×C₈	C₂×#120	1	1	yes	characteristic
181	C₂×Q₈	#2196	1	1	yes	characteristic
182	C₂×D₄	#2196	1	1	yes	characteristic
183	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
184	C₂²⋊C₄	#2196	1	1	yes	characteristic
185	C₂²⋊C₄	#2196	1	1	yes	characteristic
186	C₄²	#2846	1	1	yes	characteristic
187	C₄²	#3026	1	1	yes	characteristic
188	C₂³	C₂×#2196	1	1	yes	characteristic
189	C₂×C₄	#32898	1	1	yes	characteristic
190	C₂×C₄	C₂×#2196	1	1	yes	characteristic
191	C₂²	#511771	1	1	yes	characteristic
192	C₂	#72213671	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72213671
commutator, Frattini, cosocle	#3151	C₂⁴
LCS	G ▷ #3151 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#41 ◁ #60817 ◁ G
LECS	G ▷ #3151 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72339876	C₂	semidirect
2. #72339875	C₂	semidirect
3. #72339874	C₂	semidirect
4. #72339873	C₂	semidirect
5. #72339872	C₂	semidirect
6. #72339871	C₂	semidirect
7. #72339870	C₂	semidirect
8. #72339869	C₂	semidirect
9. #72339836	C₂	semidirect
10. #72339825	C₂	semidirect
11. #72306894	C₂	semidirect
12. #72306589	C₂	semidirect
13. #72283261	C₂	semidirect
14. #72231260	C₂	semidirect
15. #72231250	C₂	semidirect
16. #7487876	C₂²	semidirect
24. #512903	C₂²	semidirect
25. #512866	C₂²	semidirect
27. #512863	C₂²	semidirect
28. #489351	C₂²	semidirect
29. #489349	C₂²	semidirect
30. #153549	C₂²	semidirect
31. #153547	C₂²	semidirect
32. #153539	C₂²	semidirect
33. #153538	C₂²	semidirect
34. #153469	C₂²	semidirect
35. #153469	C₂²	semidirect
36. #153468	C₂²	semidirect
37. #153465	C₂²	semidirect
38. #153462	C₂²	semidirect
41. #153459	C₂²	semidirect
42. #152619	C₂²	semidirect
43. #152616	C₂²	semidirect
45. #152603	C₂²	semidirect
46. #152123	C₂²	semidirect
47. #152108	C₂²	semidirect
48. #139431	C₂²	semidirect
49. #139425	C₂²	semidirect
50. #133268	C₂²	semidirect
51. #63030	D₄	semidirect
52. #63011	D₄	semidirect
54. #61015	D₄	semidirect
57. #20319	D₄	semidirect
58. #20294	D₄	semidirect
59. #20294	D₄	semidirect
60. #20294	D₄	semidirect
61. #20157	D₄	semidirect
62. #20156	D₄	semidirect
63. #20152	D₄	semidirect
64. #20138	D₄	semidirect
65. #20138	D₄	semidirect
66. #20131	D₄	semidirect
67. #20067	D₄	semidirect
68. #20067	D₄	semidirect
69. #20021	D₄	semidirect
70. #19996	D₄	semidirect
71. #19987	D₄	semidirect
72. D₈²	D₄	semidirect
74. #19747	D₄	semidirect
76. #19717	D₄	semidirect
79. #18691	D₄	semidirect
80. #18682	D₄	semidirect
83. #18485	D₄	semidirect
84. C₂×#1620	D₄	semidirect
85. #17782	D₄	semidirect
86. #17771	D₄	semidirect
87. #17342	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72339876	2
#72339875	2
#72339874	2
#72339873	2
#72339872	2
#72339871	2
#72339870	2
#72339869	2
#72339836	2
#72339825	2
#72306894	2
#72306589	2
#72283261	2
#72231260	2
#72231250	2