Subgroups — #72306628

Order 2048.

At least 6596 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	#72306626	C₂	1	1	yes	maximal, characteristic, complemented
2	#72306624	C₂	1	1	yes	maximal, characteristic, complemented
3	#72306623	C₂	1	1	yes	maximal, characteristic, complemented
4	#72306621	C₂	1	1	yes	maximal, characteristic, complemented
5	#72306619	C₂	1	1	yes	maximal, characteristic, complemented
6	#72306530	C₂	1	1	yes	maximal, characteristic, complemented
7	#72286331	C₂	1	1	yes	maximal, characteristic, complemented
8	#72240967	C₂	1	1	yes	maximal, characteristic, complemented
9	#72240966	C₂	1	1	yes	maximal, characteristic, complemented
10	#72231263	C₂	1	1	yes	maximal, characteristic, complemented
11	#72231262	C₂	1	1	yes	maximal, characteristic, complemented
12	#72231261	C₂	1	1	yes	maximal, characteristic, complemented
13	#72231251	C₂	1	1	yes	maximal, characteristic, complemented
14	#72231245	C₂	1	1	yes	maximal, characteristic, complemented
15	#72231241	C₂	1	1	yes	maximal, characteristic, complemented
16	#7487867	C₂²	1	1	yes	characteristic, complemented
17	#6363008	C₂²	1	1	yes	characteristic
18	#6362982	C₂²	1	1	yes	characteristic
19	#6362966	C₂²	1	1	yes	characteristic
20	#6362952	C₂²	1	1	yes	characteristic
21	#6357797	C₂²	1	1	yes	characteristic
22	#6349118	C₂²	1	1	yes	characteristic
23	#6349101	C₂²	1	1	yes	characteristic
24	#512900	C₂²	1	1	yes	characteristic, complemented
25	#512864	C₂²	1	1	yes	characteristic, complemented
26	#512862	C₂²	1	1	yes	characteristic, complemented
27	#512862	C₂²	1	1	yes	characteristic, complemented
28	#489349	C₂²	1	1	yes	characteristic, complemented
29	#489347	C₂²	1	1	yes	characteristic
30	#153547	C₂²	1	1	yes	characteristic, complemented
31	#153545	C₂²	1	1	yes	characteristic
32	#153538	C₂²	1	1	yes	characteristic, complemented
33	#153537	C₂²	1	1	yes	characteristic
34	#153472	C₂²	1	1	yes	characteristic, complemented
35	#153472	C₂²	1	1	yes	characteristic, complemented
36	#153467	C₂²	1	1	yes	characteristic, complemented
37	#153467	C₂²	1	1	yes	characteristic, complemented
38	#153461	C₂²	1	1	yes	characteristic, complemented
39	#153461	C₂²	1	1	yes	characteristic, complemented
40	#153460	C₂²	1	1	yes	characteristic, complemented
41	#153460	C₂²	1	1	yes	characteristic, complemented
42	#152606	C₂²	1	1	yes	characteristic, complemented
43	#152604	C₂²	1	1	yes	characteristic, complemented
44	#152602	C₂²	1	1	yes	characteristic, complemented
45	#152602	C₂²	1	1	yes	characteristic, complemented
46	#152119	C₂²	1	1	yes	characteristic, complemented
47	#152104	C₂²	1	1	yes	characteristic, complemented
48	#139427	C₂²	1	1	yes	characteristic
49	#139425	C₂²	1	1	yes	characteristic, complemented
50	#133262	C₂²	1	1	yes	characteristic, complemented
51	#62762	D₄	1	1	yes	characteristic, complemented
52	C₂×#3549	D₄	1	1	yes	characteristic, complemented
53	#61110	C₂³	1	1	yes	characteristic
54	#60972	D₄	1	1	yes	characteristic, complemented
55	#60801	C₂³	1	1	yes	characteristic
56	#23361	C₂³	1	1	yes	characteristic
57	#20314	D₄	1	1	yes	characteristic, complemented
58	#20310	D₄	1	1	yes	characteristic, complemented
59	#20302	D₄	1	1	yes	characteristic, complemented
60	#20289	D₄	1	1	yes	characteristic, complemented
61	#20152	D₄	1	1	yes	characteristic, complemented
62	#20131	D₄	1	1	yes	characteristic, complemented
63	#20084	D₄	1	1	yes	characteristic, complemented
64	#20067	D₄	1	1	yes	characteristic, complemented
65	#20053	D₄	1	1	yes	characteristic, complemented
66	#20040	D₄	1	1	yes	characteristic, complemented
67	#20023	D₄	1	1	yes	characteristic, complemented
68	#20023	D₄	1	1	yes	characteristic, complemented
69	#19996	D₄	1	1	yes	characteristic, complemented
70	#19987	D₄	1	1	yes	characteristic, complemented
71	SD₁₆²	D₄	1	1	yes	characteristic, complemented
72	D₈²	D₄	1	1	yes	characteristic, complemented
73	#19754	D₄	1	1	yes	characteristic, complemented
74	#19751	D₄	1	1	yes	characteristic, complemented
75	#19749	C₂³	1	1	yes	characteristic
76	#19745	C₂³	1	1	yes	characteristic
77	#18549	D₄	1	1	yes	characteristic, complemented
78	#18548	C₂³	1	1	yes	characteristic
79	#18529	D₄	1	1	yes	characteristic, complemented
80	#18529	D₄	1	1	yes	characteristic, complemented
81	#18528	D₄	1	1	yes	characteristic, complemented
82	#18527	C₂³	1	1	yes	characteristic
83	#18524	C₂³	1	1	yes	characteristic
84	#18524	C₂³	1	1	yes	characteristic
85	#17754	D₄	1	1	yes	characteristic, complemented
86	#17752	D₄	1	1	yes	characteristic, complemented
87	M₁₆²	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	#3483	C₂×D₄	1	1	yes	characteristic
95	#3481	C₂×D₄	1	1	yes	characteristic
96	C₂×D₄²	C₂×D₄	1	1	yes	characteristic
97	C₂×D₄²	C₂×D₄	1	1	yes	characteristic
98	#3444	C₂×D₄	1	1	yes	characteristic
99	C₂×#532	C₂×D₄	1	1	yes	characteristic
100	#3185	C₂×D₄	1	1	yes	characteristic
101	#3184	C₂×D₄	1	1	yes	characteristic
102	#3152	C₂×D₄	1	1	yes	characteristic
103	#3151	C₂⁴	1	1	yes	characteristic
104	#3139	C₂×D₄	1	1	yes	characteristic
105	#3138	C₂×D₄	1	1	yes	characteristic
106	#2920	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	#1705	C₂×D₄	1	1	yes	characteristic
115	#1629	C₂×D₄	1	1	yes	characteristic
116	#1629	C₂×D₄	1	1	yes	characteristic
117	#1629	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	#1623	C₂×D₄	1	1	yes	characteristic
122	#1623	C₂×D₄	1	1	yes	characteristic
123	#1620	C₂×D₄	1	1	yes	characteristic
124	#1620	C₂×D₄	1	1	yes	characteristic
125	#1619	C₂×D₄	1	1	yes	characteristic
126	#1619	C₂×D₄	1	1	yes	characteristic
127	#1619	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	#1560	C₂×D₄	1	1	yes	characteristic
133	#1555	C₂×D₄	1	1	yes	characteristic
134	#1555	C₂×D₄	1	1	yes	characteristic
135	#1455	C₂×D₄	1	1	yes	characteristic
136	#1455	C₂×D₄	1	1	yes	characteristic
137	#559	#120	1	1	yes	characteristic
138	Q₈²	#120	1	1	yes	characteristic
139	#555	#120	1	1	yes	characteristic
140	D₄²	#120	1	1	yes	characteristic
141	D₄²	#120	1	1	yes	characteristic
142	D₄²	#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×C₄⋊Q₈	#120	1	1	yes	characteristic
148	C₂×#127	C₂²×D₄	1	1	yes	characteristic
149	C₂²×C₄²	#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	#120	1	1	yes	characteristic
159	C₄×M₁₆	C₂²×D₄	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	C₂³×C₄	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	C₂×C₄²	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	C₂⁴	#2196	1	1	yes	characteristic
182	C₂²×C₄	#2196	1	1	yes	characteristic
183	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
184	C₄²	#2196	1	1	yes	characteristic
185	C₄²	#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 ◁ C₂²×C₄² ◁ #60801 ◁ 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. #72306626	C₂	semidirect
2. #72306624	C₂	semidirect
3. #72306623	C₂	semidirect
4. #72306621	C₂	semidirect
5. #72306619	C₂	semidirect
6. #72306530	C₂	semidirect
7. #72286331	C₂	semidirect
8. #72240967	C₂	semidirect
9. #72240966	C₂	semidirect
10. #72231263	C₂	semidirect
11. #72231262	C₂	semidirect
12. #72231261	C₂	semidirect
13. #72231251	C₂	semidirect
14. #72231245	C₂	semidirect
15. #72231241	C₂	semidirect
16. #7487867	C₂²	semidirect
24. #512900	C₂²	semidirect
25. #512864	C₂²	semidirect
26. #512862	C₂²	semidirect
27. #512862	C₂²	semidirect
28. #489349	C₂²	semidirect
30. #153547	C₂²	semidirect
32. #153538	C₂²	semidirect
34. #153472	C₂²	semidirect
35. #153472	C₂²	semidirect
36. #153467	C₂²	semidirect
37. #153467	C₂²	semidirect
38. #153461	C₂²	semidirect
39. #153461	C₂²	semidirect
40. #153460	C₂²	semidirect
41. #153460	C₂²	semidirect
42. #152606	C₂²	semidirect
43. #152604	C₂²	semidirect
44. #152602	C₂²	semidirect
45. #152602	C₂²	semidirect
46. #152119	C₂²	semidirect
47. #152104	C₂²	semidirect
49. #139425	C₂²	semidirect
50. #133262	C₂²	semidirect
51. #62762	D₄	semidirect
52. C₂×#3549	D₄	semidirect
54. #60972	D₄	semidirect
57. #20314	D₄	semidirect
58. #20310	D₄	semidirect
59. #20302	D₄	semidirect
60. #20289	D₄	semidirect
61. #20152	D₄	semidirect
62. #20131	D₄	semidirect
63. #20084	D₄	semidirect
64. #20067	D₄	semidirect
65. #20053	D₄	semidirect
66. #20040	D₄	semidirect
67. #20023	D₄	semidirect
68. #20023	D₄	semidirect
69. #19996	D₄	semidirect
70. #19987	D₄	semidirect
71. SD₁₆²	D₄	semidirect
72. D₈²	D₄	semidirect
73. #19754	D₄	semidirect
74. #19751	D₄	semidirect
77. #18549	D₄	semidirect
79. #18529	D₄	semidirect
80. #18529	D₄	semidirect
81. #18528	D₄	semidirect
85. #17754	D₄	semidirect
86. #17752	D₄	semidirect
87. M₁₆²	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72306626	2
#72306624	2
#72306623	2
#72306621	2
#72306619	2
#72306530	2
#72286331	2
#72240967	2
#72240966	2
#72231263	2
#72231262	2
#72231261	2
#72231251	2
#72231245	2
#72231241	2