Subgroups — C₂×#72240979 #72344179

Order 6144.

At least 18949 conjugacy classes of subgroups; only the 691 normal subgroups are shown.

The subgroup lattice has 241 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×#72236838	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#72236796	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#60399⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#72236803	C₂	1	2	yes	maximal, complemented
5	C₂×#11714181	C₂²	1	1	yes	characteristic, complemented
6	C₂×#72236801	C₂	1	2	yes	maximal, complemented
7	C₂×#11711590	C₂²	1	2	yes	complemented
8	C₂×#11714179	C₂²	1	1	yes	characteristic, complemented
9	C₂×#72236903	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#72234214	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#72236905	C₂	1	2	yes	maximal, complemented
12	C₂×#72236914	C₂	1	2	yes	maximal, complemented
13	C₂×#11714177	C₂²	1	1	yes	characteristic, complemented
14	C₂×#11711640	C₂²	1	2	yes	complemented
15	C₂×#11715713	C₂²	1	2	yes	complemented
16	C₂×#11711588	C₂²	1	2	yes	complemented
17	C₂²×#3025⋊C₃	C₂²	1	1	yes	characteristic
18	C₂×#11711638	C₂²	1	2	yes	complemented
19	C₂×#72236759	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂²×#3027⋊C₃	C₂²	1	1	yes	characteristic, complemented
21	C₂×#11711642	C₂²	1	2	yes	complemented
22	C₂×#11715709	C₂²	1	2	yes	complemented
23	C₂×#11711646	C₂²	1	2	yes	complemented
24	C₂×#11715661	C₂²	1	1	yes	characteristic, complemented
25	C₂×#72215726	C₂	1	1	yes	maximal, characteristic, complemented
26	C₂×#11715712	C₂²	1	2	yes	complemented
27	C₂×#11715708	C₂²	1	2	yes	complemented
28	C₂×#11715667	C₂²	1	1	yes	characteristic, complemented
29	C₂×#11715838	C₂²	1	1	yes	characteristic, complemented
30	C₂×#11715827	C₂²	1	1	yes	characteristic, complemented
31	C₂²×#87547	C₂²	1	1	yes	characteristic, complemented
32	#72240979	C₂	1	16	yes	maximal, direct product factor
33	#72236914	C₂²	1	16	yes	complemented
34	#72236905	C₂²	1	16	yes	complemented
35	#72236903	C₂²	1	8	yes	complemented
36	#72236838	C₂²	1	8	yes	complemented
37	#72236803	C₂²	1	16	yes	complemented
38	#72236801	C₂²	1	16	yes	complemented
39	#72236796	C₂²	1	8	yes	complemented
40	C₂×#60399⋊C₃	C₂²	1	2	yes	—
41	C₂×#60399⋊C₃	C₂²	1	2	yes	complemented
42	C₂×#60399⋊C₃	C₂²	1	8	yes	complemented
43	#72236759	C₂²	1	8	yes	complemented
44	C₂×#11714175	C₂²	1	1	yes	characteristic, complemented
45	C₂²×#60399	D₃	1	1	yes	characteristic, complemented
46	#72234214	C₂²	1	8	yes	complemented
47	#72215726	C₂²	1	8	yes	complemented
48	#11715838	C₂³	1	4	yes	complemented
49	C₂×#87517	C₂³	1	1	yes	characteristic, complemented
50	C₂×#87521	C₂³	1	1	yes	characteristic, complemented
51	#11715827	C₂³	1	4	yes	complemented
52	C₂×#87518	C₂³	1	1	yes	characteristic, complemented
53	C₂×#87522	C₂³	1	1	yes	characteristic, complemented
54	#11715713	C₂³	1	8	yes	complemented
55	#11715712	C₂³	1	8	yes	complemented
56	#11715709	C₂³	1	8	yes	complemented
57	#11715708	C₂³	1	8	yes	complemented
58	#11715667	C₂³	1	4	yes	complemented
59	C₂×#85567	C₂³	1	2	yes	complemented
60	#11715661	C₂³	1	4	yes	complemented
61	C₂×#85566	C₂³	1	2	yes	complemented
62	#11714181	C₂³	1	4	yes	complemented
63	#11714179	C₂³	1	4	yes	complemented
64	#11714177	C₂³	1	4	yes	complemented
65	#11714175	C₂³	1	4	yes	complemented
66	#11711646	C₂³	1	8	yes	complemented
67	#11711642	C₂³	1	8	yes	complemented
68	#11711640	C₂³	1	8	yes	complemented
69	#11711638	C₂³	1	8	yes	complemented
70	#11711590	C₂³	1	8	yes	complemented
71	#11711588	C₂³	1	8	yes	complemented
72	C₂×#87547	C₂³	1	2	yes	—
73	C₂×#87547	C₂³	1	4	yes	complemented
74	C₂²×#5950	C₂³	1	1	yes	characteristic
75	#60399⋊C₃	C₂³	1	8	yes	—
76	#60399⋊C₃	C₂³	1	8	yes	complemented
77	C₂×#3027⋊C₃	C₂³	1	2	yes	—
78	C₂×#3027⋊C₃	C₂³	1	4	yes	complemented
79	C₂×#3025⋊C₃	D₄	1	1	yes	characteristic, complemented
80	C₂×#3025⋊C₃	D₄	1	1	yes	characteristic, complemented
81	C₂×#3025⋊C₃	D₄	1	4	yes	complemented
82	C₂×#3025⋊C₃	C₂³	1	1	yes	characteristic
83	C₂×#3025⋊C₃	C₂³	1	1	yes	characteristic
84	C₂×#3025⋊C₃	C₂³	1	4	yes	—
85	C₂²×#458⋊C₃	D₄	1	1	yes	characteristic, complemented
86	C₂²×B(4,𝔽₂)⋊C₃	D₄	1	1	yes	characteristic, complemented
87	C₂²×H(7,𝔽₂)	C₂×D₃	1	1	yes	characteristic, complemented
88	C₂²×#3027	C₂×D₃	1	1	yes	characteristic, complemented
89	C₂²×#3025	C₂×D₃	1	1	yes	characteristic
90	C₂×#60399	C₂×D₃	1	2	yes	—
91	C₂×#60399	C₂×D₃	1	2	yes	complemented
92	C₂×#60399	C₂×D₃	1	8	yes	complemented
93	#87547	C₂⁴	1	4	yes	—
94	C₂×#5950	C₂×D₄	1	1	yes	characteristic
95	C₂×#5950	C₂×D₄	1	2	yes	—
96	C₂×#5950	C₂⁴	1	1	yes	characteristic
97	C₂×#5950	C₂⁴	1	2	yes	—
98	C₂²×#997	C₂×D₄	1	1	yes	characteristic
99	#87522	C₂⁴	1	2	yes	complemented
100	#87521	C₂⁴	1	2	yes	complemented
101	#87518	C₂⁴	1	2	yes	complemented
102	#87517	C₂⁴	1	2	yes	complemented
103	#85567	C₂⁴	1	4	yes	complemented
104	#85566	C₂⁴	1	4	yes	complemented
105	#3027⋊C₃	C₂⁴	1	4	yes	—
106	#3025⋊C₃	C₂×D₄	1	2	yes	—
107	#3025⋊C₃	C₂×D₄	1	2	yes	—
108	#3025⋊C₃	C₂×D₄	1	4	yes	complemented
109	#3025⋊C₃	C₂×D₄	1	4	yes	complemented
110	#3025⋊C₃	C₂⁴	1	2	yes	—
111	#3025⋊C₃	C₂⁴	1	2	yes	—
112	C₂×#458⋊C₃	C₂×D₄	1	1	yes	characteristic
113	C₂×#458⋊C₃	C₂×D₄	1	1	yes	characteristic
114	C₂×#458⋊C₃	C₂×D₄	1	4	yes	complemented
115	C₂×B(4,𝔽₂)⋊C₃	C₂×D₄	1	1	yes	characteristic
116	C₂×B(4,𝔽₂)⋊C₃	C₂×D₄	1	1	yes	characteristic
117	C₂×B(4,𝔽₂)⋊C₃	C₂×D₄	1	4	yes	complemented
118	C₂×H(7,𝔽₂)	C₂²×D₃	1	2	yes	—
119	C₂×H(7,𝔽₂)	C₂²×D₃	1	4	yes	complemented
120	C₂²×#585	C₂²×D₃	1	1	yes	characteristic
121	C₂⁵×D₄	S₄	1	1	yes	characteristic, complemented
122	#60399	C₂²×D₃	1	8	yes	—
123	#60399	C₂²×D₃	1	8	yes	complemented
124	C₂×#3027	C₂²×D₃	1	2	yes	—
125	C₂×#3027	C₂²×D₃	1	4	yes	complemented
126	C₂×#3025	#67	1	1	yes	characteristic
127	C₂×#3025	#67	1	1	yes	characteristic
128	C₂×#3025	#67	1	4	yes	—
129	C₂×#3025	C₂²×D₃	1	1	yes	characteristic
130	C₂×#3025	C₂²×D₃	1	1	yes	characteristic
131	C₂×#3025	C₂²×D₃	1	4	yes	—
132	C₂²×#458	#67	1	1	yes	characteristic
133	C₂²×B(4,𝔽₂)	#67	1	1	yes	characteristic
134	#5950	C₂²×D₄	1	1	yes	characteristic
135	#5950	C₂²×D₄	1	2	yes	—
136	#5950	C₂⁵	1	1	yes	characteristic
137	C₂×#997	C₂²×D₄	1	1	yes	characteristic
138	C₂×#997	C₂²×D₄	1	2	yes	—
139	#458⋊C₃	C₂²×D₄	1	2	yes	—
140	#458⋊C₃	C₂²×D₄	1	2	yes	—
141	B(4,𝔽₂)⋊C₃	C₂²×D₄	1	2	yes	—
142	B(4,𝔽₂)⋊C₃	C₂²×D₄	1	2	yes	—
143	H(7,𝔽₂)	C₂³×D₃	1	4	yes	—
144	C₂×#585	C₂×#67	1	1	yes	characteristic
145	C₂×#585	C₂×#67	1	2	yes	—
146	C₂×#585	C₂³×D₃	1	1	yes	characteristic
147	C₂×#585	C₂³×D₃	1	2	yes	—
148	C₂²×H(5,𝔽₂)	C₂×#67	1	1	yes	characteristic
149	C₂⁴×D₄	C₂×S₄	1	1	yes	characteristic
150	C₂⁴×D₄	C₂×S₄	1	1	yes	characteristic
151	C₂⁴×D₄	C₂×S₄	1	2	yes	—
152	C₂⁴×D₄	C₂×S₄	1	2	yes	—
153	C₂⁴×D₄	C₂×S₄	1	8	yes	complemented
154	C₂⁵×C₄	C₂×S₄	1	1	yes	characteristic
155	#3027	C₂³×D₃	1	4	yes	—
156	#3025	C₂×#67	1	4	yes	—
157	#3025	C₂×#67	1	4	yes	—
158	#3025	C₂×#67	1	2	yes	—
159	#3025	C₂×#67	1	2	yes	—
160	#3025	C₂³×D₃	1	2	yes	—
161	#3025	C₂³×D₃	1	2	yes	—
162	C₂×#458	C₂×#67	1	1	yes	characteristic
163	C₂×#458	C₂×#67	1	1	yes	characteristic
164	C₂×#458	C₂×#67	1	4	yes	—
165	C₂×B(4,𝔽₂)	C₂×#67	1	1	yes	characteristic
166	C₂×B(4,𝔽₂)	C₂×#67	1	1	yes	characteristic
167	C₂×B(4,𝔽₂)	C₂×#67	1	4	yes	—
168	#997	C₂³×D₄	1	1	yes	characteristic
169	C₂⁶	GL(2,ℤ/4)	1	1	yes	characteristic
170	#585	C₂²×#67	1	2	yes	—
171	#585	C₂²×#67	1	1	yes	characteristic
172	#585	C₂⁴×D₃	1	1	yes	characteristic
173	C₂×H(5,𝔽₂)	C₂²×#67	1	1	yes	characteristic
174	C₂×H(5,𝔽₂)	C₂²×#67	1	2	yes	—
175	C₂³×D₄	C₂²×S₄	1	2	yes	—
176	C₂³×D₄	C₂²×S₄	1	4	yes	—
177	C₂³×D₄	C₂²×S₄	1	2	yes	—
178	C₂³×D₄	C₂²×S₄	1	4	yes	—
179	C₂³×D₄	C₂²×S₄	1	8	yes	—
180	C₂³×D₄	C₂²×S₄	1	8	yes	—
181	C₂⁴×C₄	GL(2,ℤ/4)	1	1	yes	characteristic
182	C₂⁴×C₄	GL(2,ℤ/4)	1	1	yes	characteristic
183	C₂⁴×C₄	GL(2,ℤ/4)	1	1	yes	characteristic
184	C₂⁴×C₄	GL(2,ℤ/4)	1	4	yes	—
185	C₂⁴×C₄	C₂²×S₄	1	1	yes	characteristic
186	C₂⁴×C₄	C₂²×S₄	1	1	yes	characteristic
187	C₂⁴×C₄	C₂²×S₄	1	1	yes	characteristic
188	C₂⁴×C₄	C₂²×S₄	1	4	yes	—
189	#458	C₂²×#67	1	2	yes	—
190	#458	C₂²×#67	1	2	yes	—
191	B(4,𝔽₂)	C₂²×#67	1	2	yes	—
192	B(4,𝔽₂)	C₂²×#67	1	2	yes	—
193	C₂⁵	C₂×GL(2,ℤ/4)	1	1	yes	characteristic
194	C₂⁵	C₂×GL(2,ℤ/4)	1	1	yes	characteristic
195	C₂⁵	C₂×GL(2,ℤ/4)	1	1	yes	characteristic
196	C₂⁵	C₂×GL(2,ℤ/4)	1	4	yes	—
197	H(5,𝔽₂)	C₂³×#67	1	1	yes	characteristic
198	C₂²×D₄	#1020⋊C₂	1	1	yes	characteristic
199	C₂²×D₄	C₂³×S₄	1	4	yes	—
200	C₂²×D₄	C₂³×S₄	1	4	yes	—
201	C₂³×C₄	C₂×GL(2,ℤ/4)	1	1	yes	characteristic
202	C₂³×C₄	C₂×GL(2,ℤ/4)	1	2	yes	—
203	C₂³×C₄	C₂×GL(2,ℤ/4)	1	1	yes	characteristic
204	C₂³×C₄	C₂×GL(2,ℤ/4)	1	4	yes	—
205	C₂³×C₄	C₂×GL(2,ℤ/4)	1	1	yes	characteristic
206	C₂³×C₄	C₂×GL(2,ℤ/4)	1	2	yes	—
207	C₂³×C₄	C₂×GL(2,ℤ/4)	1	4	yes	—
208	C₂³×C₄	C₂×GL(2,ℤ/4)	1	4	yes	—
209	C₂³×C₄	C₂×GL(2,ℤ/4)	1	2	yes	—
210	C₂³×C₄	C₂³×S₄	1	1	yes	characteristic
211	C₂³×C₄	C₂³×S₄	1	2	yes	—
212	C₂³×C₄	C₂³×S₄	1	2	yes	—
213	C₂³×C₄	C₂³×S₄	1	2	yes	—
214	C₂⁴	C₂²×GL(2,ℤ/4)	1	1	yes	characteristic
215	C₂⁴	C₂²×GL(2,ℤ/4)	1	2	yes	—
216	C₂⁴	C₂²×GL(2,ℤ/4)	1	2	yes	—
217	C₂⁴	C₂²×GL(2,ℤ/4)	1	2	yes	—
218	C₂×D₄	C₂×#1020⋊C₂	1	2	yes	—
219	C₂×D₄	C₂×#1020⋊C₂	1	4	yes	—
220	C₂²×C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
221	C₂²×C₄	C₂²×GL(2,ℤ/4)	1	2	yes	—
222	C₂²×C₄	C₂²×GL(2,ℤ/4)	1	1	yes	characteristic
223	C₂²×C₄	C₂²×GL(2,ℤ/4)	1	2	yes	—
224	C₂²×C₄	C₂²×GL(2,ℤ/4)	1	2	yes	—
225	C₂²×C₄	C₂⁴×S₄	1	1	yes	characteristic
226	C₂³	#11713201	1	1	yes	characteristic
227	C₂³	C₂³×GL(2,ℤ/4)	1	1	yes	characteristic
228	D₄	C₂²×#1020⋊C₂	1	4	yes	—
229	C₂×C₄	#11713201	1	2	yes	—
230	C₂×C₄	#11713201	1	1	yes	characteristic
231	C₂×C₄	C₂²×#1020⋊C₂	1	2	yes	—
232	C₂×C₄	C₂²×#1020⋊C₂	1	1	yes	characteristic
233	C₂²	C₂×#11713201	1	2	yes	—
234	C₂²	C₂×#11713201	1	1	yes	characteristic
235	C₄	C₂³×#1020⋊C₂	1	1	yes	characteristic
236	C₄	C₂×#11713201	1	2	yes	—
237	C₄	C₂×#11713201	1	1	yes	characteristic
238	C₂	#72240979	1	2	yes	direct product factor
239	C₂	C₂²×#11713201	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#11713201
commutator, cosocle	#5950	C₂⁵
Frattini	C₂²×C₄	C₂⁴×S₄
Fitting	C₂²×#60399	D₃
hypercenter	C₂²×D₄	#1020⋊C₂
hypocenter	#997	C₂³×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#72236838	C₂	semidirect
2. C₂×#72236796	C₂	semidirect
3. C₂²×#60399⋊C₃	C₂	semidirect
4. C₂×#72236803	C₂	semidirect
5. C₂×#11714181	C₂²	semidirect
6. C₂×#72236801	C₂	semidirect
7. C₂×#11711590	C₂²	semidirect
8. C₂×#11714179	C₂²	semidirect
9. C₂×#72236903	C₂	semidirect
10. C₂×#72234214	C₂	semidirect
11. C₂×#72236905	C₂	semidirect
12. C₂×#72236914	C₂	semidirect
13. C₂×#11714177	C₂²	semidirect
14. C₂×#11711640	C₂²	semidirect
15. C₂×#11715713	C₂²	semidirect
16. C₂×#11711588	C₂²	semidirect
18. C₂×#11711638	C₂²	semidirect
19. C₂×#72236759	C₂	semidirect
20. C₂²×#3027⋊C₃	C₂²	semidirect
21. C₂×#11711642	C₂²	semidirect
22. C₂×#11715709	C₂²	semidirect
23. C₂×#11711646	C₂²	semidirect
24. C₂×#11715661	C₂²	semidirect
25. C₂×#72215726	C₂	semidirect
26. C₂×#11715712	C₂²	semidirect
27. C₂×#11715708	C₂²	semidirect
28. C₂×#11715667	C₂²	semidirect
29. C₂×#11715838	C₂²	semidirect
30. C₂×#11715827	C₂²	semidirect
31. C₂²×#87547	C₂²	semidirect
32. #72240979	C₂	direct (N × H)
33. #72236914	C₂²	semidirect
34. #72236905	C₂²	semidirect
35. #72236903	C₂²	semidirect
36. #72236838	C₂²	semidirect
37. #72236803	C₂²	semidirect
38. #72236801	C₂²	semidirect
39. #72236796	C₂²	semidirect
41. C₂×#60399⋊C₃	C₂²	semidirect
42. C₂×#60399⋊C₃	C₂²	semidirect
43. #72236759	C₂²	semidirect
44. C₂×#11714175	C₂²	semidirect
45. C₂²×#60399	D₃	semidirect
46. #72234214	C₂²	semidirect
47. #72215726	C₂²	semidirect
48. #11715838	C₂³	semidirect
49. C₂×#87517	C₂³	semidirect
50. C₂×#87521	C₂³	semidirect
51. #11715827	C₂³	semidirect
52. C₂×#87518	C₂³	semidirect
53. C₂×#87522	C₂³	semidirect
54. #11715713	C₂³	semidirect
55. #11715712	C₂³	semidirect
56. #11715709	C₂³	semidirect
57. #11715708	C₂³	semidirect
58. #11715667	C₂³	semidirect
59. C₂×#85567	C₂³	semidirect
60. #11715661	C₂³	semidirect
61. C₂×#85566	C₂³	semidirect
62. #11714181	C₂³	semidirect
63. #11714179	C₂³	semidirect
64. #11714177	C₂³	semidirect
65. #11714175	C₂³	semidirect
66. #11711646	C₂³	semidirect
67. #11711642	C₂³	semidirect
68. #11711640	C₂³	semidirect
69. #11711638	C₂³	semidirect
70. #11711590	C₂³	semidirect
71. #11711588	C₂³	semidirect
73. C₂×#87547	C₂³	semidirect
76. #60399⋊C₃	C₂³	semidirect
78. C₂×#3027⋊C₃	C₂³	semidirect
79. C₂×#3025⋊C₃	D₄	semidirect
80. C₂×#3025⋊C₃	D₄	semidirect
81. C₂×#3025⋊C₃	D₄	semidirect
85. C₂²×#458⋊C₃	D₄	semidirect
86. C₂²×B(4,𝔽₂)⋊C₃	D₄	semidirect
87. C₂²×H(7,𝔽₂)	C₂×D₃	semidirect
88. C₂²×#3027	C₂×D₃	semidirect
91. C₂×#60399	C₂×D₃	semidirect
92. C₂×#60399	C₂×D₃	semidirect
99. #87522	C₂⁴	semidirect
100. #87521	C₂⁴	semidirect
101. #87518	C₂⁴	semidirect
102. #87517	C₂⁴	semidirect
103. #85567	C₂⁴	semidirect
104. #85566	C₂⁴	semidirect
108. #3025⋊C₃	C₂×D₄	semidirect
109. #3025⋊C₃	C₂×D₄	semidirect
114. C₂×#458⋊C₃	C₂×D₄	semidirect
117. C₂×B(4,𝔽₂)⋊C₃	C₂×D₄	semidirect
119. C₂×H(7,𝔽₂)	C₂²×D₃	semidirect
121. C₂⁵×D₄	S₄	semidirect
123. #60399	C₂²×D₃	semidirect
125. C₂×#3027	C₂²×D₃	semidirect
153. C₂⁴×D₄	C₂×S₄	semidirect
238. C₂	#72240979	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×#72236838	2
C₂×#72236796	2
C₂²×#60399⋊C₃	2
C₂×#72236803	2
C₂×#72236801	2
C₂×#72236903	2
C₂×#72234214	2
C₂×#72236905	2
C₂×#72236914	2
C₂×#72236759	2
C₂×#72215726	2
C₂×#72231746	3
#72240979	2
C₂²×#85489	4

Subgroups — C2×#72240979 #72344179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72240979 #72344179