Subgroups — C₂×#72240991 #72343302

Order 6144.

At least 11619 conjugacy classes of subgroups; only the 603 normal subgroups are shown.

The subgroup lattice has 198 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#72236882	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#72236868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#72236905	C₂	1	2	yes	maximal, complemented
4	C₂×#72236878	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#72215718	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#72236864	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#72236846	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11711588	C₂²	1	1	yes	characteristic, complemented
9	C₂×#72236904	C₂	1	2	yes	maximal, complemented
10	C₂×#11711587	C₂²	1	1	yes	characteristic, complemented
11	C₂×#72236850	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11711638	C₂²	1	2	yes	complemented
13	C₂×#21483⋊C₃	C₂²	1	1	yes	characteristic, complemented
14	C₂×#11715736	C₂²	1	1	yes	characteristic, complemented
15	C₂×#32948⋊C₃	C₂²	1	1	yes	characteristic, complemented
16	C₂×#72236763	C₂	1	2	yes	maximal, complemented
17	C₂×#11711643	C₂²	1	2	yes	complemented
18	C₂×#11711642	C₂²	1	2	yes	complemented
19	C₂×#11715709	C₂²	1	2	yes	complemented
20	C₂×#32894⋊C₃	C₂²	1	2	yes	complemented
21	C₂×#11711596	C₂²	1	1	yes	characteristic, complemented
22	C₂×#11715708	C₂²	1	2	yes	complemented
23	C₂×#6573729	D₃	1	1	yes	characteristic, complemented
24	#72240991	C₂	1	16	yes	maximal, direct product factor
25	C₂×#11715687	C₂²	1	1	yes	characteristic, complemented
26	#72236905	C₂²	1	16	yes	complemented
27	#72236904	C₂²	1	16	yes	complemented
28	#72236882	C₂²	1	8	yes	complemented
29	#72236878	C₂²	1	8	yes	complemented
30	#72236868	C₂²	1	8	yes	complemented
31	#72236864	C₂²	1	8	yes	complemented
32	#72236850	C₂²	1	8	yes	complemented
33	#72236846	C₂²	1	8	yes	complemented
34	C₂×#72217027	C₂	1	2	yes	maximal, complemented
35	C₂×#60399⋊C₃	C₂²	1	2	yes	complemented
36	C₂×#11714174	C₂²	1	2	yes	complemented
37	#72236763	C₂²	1	16	yes	complemented
38	C₂×#11714184	C₂²	1	2	yes	complemented
39	C₂×#11714162	C₂²	1	1	yes	characteristic
40	C₂×#11714175	C₂²	1	2	yes	complemented
41	C₂×#11715739	C₂²	1	1	yes	characteristic, complemented
42	C₂×#11710898	C₂²	1	1	yes	characteristic, complemented
43	C₂×#11715681	C₂²	1	1	yes	characteristic, complemented
44	C₂×#11715839	C₂²	1	2	yes	complemented
45	#72217027	C₂²	1	16	yes	complemented
46	#72215718	C₂²	1	8	yes	complemented
47	C₂×#11715828	C₂²	1	2	yes	complemented
48	#11715839	C₂³	1	8	yes	complemented
49	C₂×#87521	C₂³	1	2	yes	complemented
50	C₂×#87519	C₂³	1	1	yes	characteristic
51	#11715828	C₂³	1	8	yes	complemented
52	C₂×#87522	C₂³	1	2	yes	complemented
53	C₂×#87520	C₂³	1	1	yes	characteristic
54	#11715739	C₂³	1	4	yes	complemented
55	#11715736	C₂³	1	4	yes	complemented
56	#11715709	C₂³	1	8	yes	complemented
57	#11715708	C₂³	1	8	yes	complemented
58	#11715687	C₂³	1	4	yes	complemented
59	#11715681	C₂³	1	4	yes	complemented
60	C₂×#85567	C₂³	1	1	yes	characteristic, complemented
61	C₂×#85566	C₂³	1	1	yes	characteristic, complemented
62	#11714184	C₂³	1	8	yes	complemented
63	#11714175	C₂³	1	8	yes	complemented
64	#11714174	C₂³	1	8	yes	complemented
65	#11714162	D₄	1	4	yes	complemented
66	#11714162	C₂³	1	4	yes	—
67	C₂×#73031	D₄	1	1	yes	characteristic, complemented
68	C₂×#73025	D₄	1	1	yes	characteristic, complemented
69	C₂×#73030	D₄	1	1	yes	characteristic, complemented
70	C₂×#73026	D₄	1	1	yes	characteristic, complemented
71	#11711643	C₂³	1	8	yes	complemented
72	#11711642	C₂³	1	8	yes	complemented
73	#11711638	C₂³	1	8	yes	complemented
74	#11711596	C₂³	1	4	yes	complemented
75	#11711588	C₂³	1	4	yes	complemented
76	#11711587	C₂³	1	4	yes	complemented
77	C₂×#87547	C₂³	1	2	yes	complemented
78	#11710898	C₂³	1	4	yes	complemented
79	C₂×#85666	C₂³	1	1	yes	characteristic, complemented
80	#60399⋊C₃	C₂³	1	8	yes	complemented
81	C₂×#3027⋊C₃	C₂³	1	2	yes	complemented
82	C₂×#3025⋊C₃	C₂³	1	1	yes	characteristic
83	#32948⋊C₃	C₂³	1	4	yes	complemented
84	#32894⋊C₃	C₂³	1	8	yes	complemented
85	#21483⋊C₃	C₂³	1	4	yes	complemented
86	C₂×#1791⋊C₃	C₂³	1	1	yes	characteristic, complemented
87	C₂×#63091	C₂×D₃	1	1	yes	characteristic, complemented
88	C₂×#60399	C₂×D₃	1	2	yes	complemented
89	C₂×#32948	C₂×D₃	1	1	yes	characteristic, complemented
90	C₂×#32894	C₂×D₃	1	2	yes	complemented
91	#6573729	C₂×D₃	1	8	yes	complemented
92	C₂×#21483	C₂×D₃	1	1	yes	characteristic, complemented
93	#87547	C₂⁴	1	4	yes	complemented
94	C₂×#5950	C₂⁴	1	1	yes	characteristic
95	#87522	C₂⁴	1	4	yes	complemented
96	#87521	C₂⁴	1	4	yes	complemented
97	#87520	C₂×D₄	1	2	yes	—
98	#87520	C₂⁴	1	2	yes	—
99	#87519	C₂×D₄	1	2	yes	—
100	#87519	C₂⁴	1	2	yes	—
101	C₂×#5935	C₂×D₄	1	1	yes	characteristic
102	C₂×#5934	C₂×D₄	1	1	yes	characteristic
103	C₂×#5933	C₂×D₄	1	1	yes	characteristic
104	C₂×#5932	C₂×D₄	1	1	yes	characteristic
105	#85666	C₂⁴	1	2	yes	complemented
106	#85567	C₂⁴	1	2	yes	complemented
107	#85566	C₂⁴	1	2	yes	complemented
108	#3027⋊C₃	C₂⁴	1	4	yes	complemented
109	#3025⋊C₃	C₂×D₄	1	2	yes	—
110	#3025⋊C₃	C₂⁴	1	2	yes	—
111	C₂×#458⋊C₃	C₂×D₄	1	1	yes	characteristic
112	C₂×B(4,𝔽₂)⋊C₃	C₂×D₄	1	1	yes	characteristic
113	#73031	C₂×D₄	1	4	yes	complemented
114	#73030	C₂×D₄	1	4	yes	complemented
115	#73026	C₂×D₄	1	4	yes	complemented
116	#73025	C₂×D₄	1	4	yes	complemented
117	#1791⋊C₃	C₂⁴	1	2	yes	complemented
118	C₂×H(7,𝔽₂)	C₂²×D₃	1	2	yes	complemented
119	#63091	C₂²×D₃	1	4	yes	complemented
120	C₂⁴×D₈	S₄	1	1	yes	characteristic, complemented
121	C₂×#3573	C₂²×D₃	1	1	yes	characteristic, complemented
122	#60399	C₂²×D₃	1	8	yes	complemented
123	C₂×#3027	C₂²×D₃	1	2	yes	complemented
124	C₂×#3025	C₂²×D₃	1	1	yes	characteristic
125	#32948	C₂²×D₃	1	4	yes	complemented
126	#32894	C₂²×D₃	1	8	yes	complemented
127	#21483	C₂²×D₃	1	4	yes	complemented
128	C₂×#1791	C₂²×D₃	1	1	yes	characteristic, complemented
129	#5950	C₂²×D₄	1	1	yes	characteristic
130	#5950	C₂⁵	1	1	yes	characteristic
131	C₂×#997	C₂²×D₄	1	1	yes	characteristic
132	#5935	C₂²×D₄	1	2	yes	—
133	#5934	C₂²×D₄	1	2	yes	—
134	#5933	C₂²×D₄	1	2	yes	—
135	#5932	C₂²×D₄	1	2	yes	—
136	#458⋊C₃	C₂²×D₄	1	2	yes	—
137	B(4,𝔽₂)⋊C₃	C₂²×D₄	1	2	yes	—
138	H(7,𝔽₂)	C₂³×D₃	1	4	yes	complemented
139	C₂×#585	C₂³×D₃	1	1	yes	characteristic
140	C₂⁴×D₄	C₂×S₄	1	2	yes	complemented
141	C₂³×D₈	C₂×S₄	1	1	yes	characteristic
142	C₂³×D₈	C₂×S₄	1	2	yes	complemented
143	C₂³×D₈	C₂×S₄	1	1	yes	characteristic, complemented
144	C₂³×D₈	C₂×S₄	1	8	yes	complemented
145	#3573	C₂³×D₃	1	2	yes	complemented
146	C₂⁴×C₈	C₂×S₄	1	1	yes	characteristic, complemented
147	#3027	C₂³×D₃	1	4	yes	complemented
148	#3025	D₃×D₄	1	2	yes	—
149	#3025	C₂³×D₃	1	2	yes	—
150	C₂×#458	D₃×D₄	1	1	yes	characteristic
151	C₂×B(4,𝔽₂)	D₃×D₄	1	1	yes	characteristic
152	#1791	C₂³×D₃	1	2	yes	complemented
153	#997	C₂³×D₄	1	1	yes	characteristic
154	#585	C₂×D₃×D₄	1	1	yes	characteristic
155	#585	C₂⁴×D₃	1	1	yes	characteristic
156	C₂×H(5,𝔽₂)	C₂×D₃×D₄	1	1	yes	characteristic
157	C₂³×D₄	C₂²×S₄	1	2	yes	—
158	C₂³×D₄	C₂²×S₄	1	2	yes	—
159	C₂³×D₄	C₂²×S₄	1	8	yes	complemented
160	C₂⁴×C₄	C₂²×S₄	1	1	yes	characteristic
161	C₂²×D₈	C₂²×S₄	1	4	yes	—
162	C₂²×D₈	C₂²×S₄	1	4	yes	complemented
163	C₂²×D₈	C₂²×S₄	1	8	yes	complemented
164	C₂³×C₈	C₂²×S₄	1	1	yes	characteristic
165	C₂³×C₈	C₂²×S₄	1	1	yes	characteristic
166	C₂³×C₈	C₂²×S₄	1	4	yes	complemented
167	#458	C₂×D₃×D₄	1	2	yes	—
168	B(4,𝔽₂)	C₂×D₃×D₄	1	2	yes	—
169	C₂⁵	D₄×S₄	1	1	yes	characteristic
170	H(5,𝔽₂)	C₂²×D₃×D₄	1	1	yes	characteristic
171	C₂²×D₄	C₂³×S₄	1	4	yes	—
172	C₂²×D₄	C₂³×S₄	1	4	yes	—
173	C₂³×C₄	D₄×S₄	1	1	yes	characteristic
174	C₂³×C₄	D₄×S₄	1	2	yes	—
175	C₂³×C₄	C₂³×S₄	1	1	yes	characteristic
176	C₂³×C₄	C₂³×S₄	1	2	yes	—
177	C₂×D₈	#1020⋊C₂	1	1	yes	characteristic
178	C₂²×C₈	C₂³×S₄	1	2	yes	—
179	C₂²×C₈	C₂³×S₄	1	2	yes	—
180	C₂⁴	C₂×D₄×S₄	1	1	yes	characteristic
181	C₂⁴	C₂×D₄×S₄	1	2	yes	—
182	C₂×D₄	C₂×#1020⋊C₂	1	2	yes	—
183	C₂²×C₄	C₂×D₄×S₄	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×S₄	1	2	yes	—
185	C₂²×C₄	C₂⁴×S₄	1	1	yes	characteristic
186	D₈	C₂×#1020⋊C₂	1	4	yes	—
187	C₂×C₈	C₂×#1020⋊C₂	1	1	yes	characteristic
188	C₂³	C₂²×D₄×S₄	1	1	yes	characteristic
189	D₄	C₂²×#1020⋊C₂	1	4	yes	—
190	C₂×C₄	C₂²×#1020⋊C₂	1	1	yes	characteristic
191	C₈	C₂²×#1020⋊C₂	1	2	yes	—
192	C₂²	D₄×#1020⋊C₂	1	1	yes	characteristic
193	C₄	D₄×#1020⋊C₂	1	1	yes	characteristic
194	C₄	C₂³×#1020⋊C₂	1	1	yes	characteristic
195	C₂	#72240991	1	2	yes	direct product factor
196	C₂	C₂×D₄×#1020⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄×#1020⋊C₂
commutator, cosocle	#5950	C₂⁵
Frattini	C₂²×C₄	C₂⁴×S₄
Fitting	C₂×#6573729	D₃
hypercenter	C₂×D₈	#1020⋊C₂
hypocenter	#997	C₂³×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#72236882	C₂	semidirect
2. C₂×#72236868	C₂	semidirect
3. C₂×#72236905	C₂	semidirect
4. C₂×#72236878	C₂	semidirect
5. C₂×#72215718	C₂	semidirect
6. C₂×#72236864	C₂	semidirect
7. C₂×#72236846	C₂	semidirect
8. C₂×#11711588	C₂²	semidirect
9. C₂×#72236904	C₂	semidirect
10. C₂×#11711587	C₂²	semidirect
11. C₂×#72236850	C₂	semidirect
12. C₂×#11711638	C₂²	semidirect
13. C₂×#21483⋊C₃	C₂²	semidirect
14. C₂×#11715736	C₂²	semidirect
15. C₂×#32948⋊C₃	C₂²	semidirect
16. C₂×#72236763	C₂	semidirect
17. C₂×#11711643	C₂²	semidirect
18. C₂×#11711642	C₂²	semidirect
19. C₂×#11715709	C₂²	semidirect
20. C₂×#32894⋊C₃	C₂²	semidirect
21. C₂×#11711596	C₂²	semidirect
22. C₂×#11715708	C₂²	semidirect
23. C₂×#6573729	D₃	semidirect
24. #72240991	C₂	direct (N × H)
25. C₂×#11715687	C₂²	semidirect
26. #72236905	C₂²	semidirect
27. #72236904	C₂²	semidirect
28. #72236882	C₂²	semidirect
29. #72236878	C₂²	semidirect
30. #72236868	C₂²	semidirect
31. #72236864	C₂²	semidirect
32. #72236850	C₂²	semidirect
33. #72236846	C₂²	semidirect
34. C₂×#72217027	C₂	semidirect
35. C₂×#60399⋊C₃	C₂²	semidirect
36. C₂×#11714174	C₂²	semidirect
37. #72236763	C₂²	semidirect
38. C₂×#11714184	C₂²	semidirect
40. C₂×#11714175	C₂²	semidirect
41. C₂×#11715739	C₂²	semidirect
42. C₂×#11710898	C₂²	semidirect
43. C₂×#11715681	C₂²	semidirect
44. C₂×#11715839	C₂²	semidirect
45. #72217027	C₂²	semidirect
46. #72215718	C₂²	semidirect
47. C₂×#11715828	C₂²	semidirect
48. #11715839	C₂³	semidirect
49. C₂×#87521	C₂³	semidirect
51. #11715828	C₂³	semidirect
52. C₂×#87522	C₂³	semidirect
54. #11715739	C₂³	semidirect
55. #11715736	C₂³	semidirect
56. #11715709	C₂³	semidirect
57. #11715708	C₂³	semidirect
58. #11715687	C₂³	semidirect
59. #11715681	C₂³	semidirect
60. C₂×#85567	C₂³	semidirect
61. C₂×#85566	C₂³	semidirect
62. #11714184	C₂³	semidirect
63. #11714175	C₂³	semidirect
64. #11714174	C₂³	semidirect
65. #11714162	D₄	semidirect
67. C₂×#73031	D₄	semidirect
68. C₂×#73025	D₄	semidirect
69. C₂×#73030	D₄	semidirect
70. C₂×#73026	D₄	semidirect
71. #11711643	C₂³	semidirect
72. #11711642	C₂³	semidirect
73. #11711638	C₂³	semidirect
74. #11711596	C₂³	semidirect
75. #11711588	C₂³	semidirect
76. #11711587	C₂³	semidirect
77. C₂×#87547	C₂³	semidirect
78. #11710898	C₂³	semidirect
79. C₂×#85666	C₂³	semidirect
80. #60399⋊C₃	C₂³	semidirect
81. C₂×#3027⋊C₃	C₂³	semidirect
83. #32948⋊C₃	C₂³	semidirect
84. #32894⋊C₃	C₂³	semidirect
85. #21483⋊C₃	C₂³	semidirect
86. C₂×#1791⋊C₃	C₂³	semidirect
87. C₂×#63091	C₂×D₃	semidirect
88. C₂×#60399	C₂×D₃	semidirect
89. C₂×#32948	C₂×D₃	semidirect
90. C₂×#32894	C₂×D₃	semidirect
91. #6573729	C₂×D₃	semidirect
92. C₂×#21483	C₂×D₃	semidirect
93. #87547	C₂⁴	semidirect
95. #87522	C₂⁴	semidirect
96. #87521	C₂⁴	semidirect
105. #85666	C₂⁴	semidirect
106. #85567	C₂⁴	semidirect
107. #85566	C₂⁴	semidirect
108. #3027⋊C₃	C₂⁴	semidirect
113. #73031	C₂×D₄	semidirect
114. #73030	C₂×D₄	semidirect
115. #73026	C₂×D₄	semidirect
116. #73025	C₂×D₄	semidirect
117. #1791⋊C₃	C₂⁴	semidirect
118. C₂×H(7,𝔽₂)	C₂²×D₃	semidirect
119. #63091	C₂²×D₃	semidirect
120. C₂⁴×D₈	S₄	semidirect
121. C₂×#3573	C₂²×D₃	semidirect
122. #60399	C₂²×D₃	semidirect
123. C₂×#3027	C₂²×D₃	semidirect
125. #32948	C₂²×D₃	semidirect
126. #32894	C₂²×D₃	semidirect
127. #21483	C₂²×D₃	semidirect
128. C₂×#1791	C₂²×D₃	semidirect
138. H(7,𝔽₂)	C₂³×D₃	semidirect
140. C₂⁴×D₄	C₂×S₄	semidirect
142. C₂³×D₈	C₂×S₄	semidirect
143. C₂³×D₈	C₂×S₄	semidirect
144. C₂³×D₈	C₂×S₄	semidirect
145. #3573	C₂³×D₃	semidirect
146. C₂⁴×C₈	C₂×S₄	semidirect
147. #3027	C₂³×D₃	semidirect
152. #1791	C₂³×D₃	semidirect
159. C₂³×D₄	C₂²×S₄	semidirect
162. C₂²×D₈	C₂²×S₄	semidirect
163. C₂²×D₈	C₂²×S₄	semidirect
166. C₂³×C₈	C₂²×S₄	semidirect
195. C₂	#72240991	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#72236882	2
C₂×#72236868	2
C₂²×D₈×S₄	4
C₂×#72236905	2
C₂×#72236878	2
C₂×#72215718	2
C₂×#72236864	2
C₂×#72236846	2
C₂×#72236904	2
C₂×#72236850	2
C₂×#72236763	2
C₂×#72237526	3
#72240991	2
C₂×#72217027	2

Subgroups — C2×#72240991 #72343302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72240991 #72343302