Subgroups — C₂×#89080 #11724681

Order 864.

2296 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#89080	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂	1	3	yes	maximal, complemented
5	C₂×#3832	—	3	13	no	maximal
6	C₂²×C₃⋊(C₃⋊D₃)	C₂²	1	3	yes	—
7	C₃³×Q₈	C₂²	1	4	yes	complemented
8	C₂×C₃³×C₄	C₂²	1	3	yes	—
9	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
10	C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	12	yes	complemented
11	C₄×C₃⋊(C₃⋊D₃)	C₂²	1	12	yes	complemented
12	C₂×C₃²×Q₈	D₃	1	13	yes	complemented
13	#3832	—	3	104	no	—
14	C₂×C₄⋊(C₃⋊D₃)	—	3	39	no	—
15	C₂×C₄×C₃⋊D₃	—	3	39	no	—
16	C₂²×C₃³	C₂³	1	1	yes	characteristic
17	C₂×C₃⋊(C₃⋊D₃)	—	2	6	no	—
18	C₂×C₃⋊(C₃⋊D₃)	C₂³	1	6	yes	—
19	C₃³×C₄	C₂³	1	6	yes	—
20	C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
21	C₂×#239	—	9	13	no	—
22	C₂²×C₃⋊D₃	—	3	39	no	—
23	C₃²×Q₈	C₂×D₃	1	52	yes	complemented
24	C₂×C₃²×C₄	C₂×D₃	1	39	yes	—
25	C₂×C₃⋊Dic₃	—	3	13	no	—
26	C₄⋊(C₃⋊D₃)	—	3	156	no	—
27	C₄×C₃⋊D₃	—	3	156	no	—
28	C₂×C₃³	#41	1	2	yes	complemented
29	C₂×C₃³	C₂⁴	1	1	yes	characteristic
30	C₃⋊(C₃⋊D₃)	—	2	6	no	—
31	C₂×C₃×Q₈	C₃⋊D₃	1	13	yes	complemented
32	#239	—	9	104	no	—
33	C₂×D₁₂	—	9	39	no	—
34	C₂×C₄×D₃	—	9	39	no	—
35	C₂²×C₃²	C₂²×D₃	1	13	yes	—
36	C₂×C₃⋊D₃	—	3	78	no	—
37	C₂×C₃⋊D₃	—	6	78	no	—
38	C₃²×C₄	C₂²×D₃	1	78	yes	—
39	C₃⋊Dic₃	—	3	26	no	—
40	C₂×#41	—	27	1	no	Sylow
41	C₃³	C₂×#41	1	1	yes	characteristic, complemented, Sylow
42	C₂²×D₃	—	9	39	no	—
43	C₃×Q₈	C₂×C₃⋊D₃	1	52	yes	complemented
44	C₂×C₃×C₄	C₂×C₃⋊D₃	1	39	yes	—
45	C₂×Dic₃	—	9	13	no	—
46	D₁₂	—	9	156	no	—
47	C₄×D₃	—	9	156	no	—
48	C₂×C₃²	#239	1	26	yes	complemented
49	C₂×C₃²	C₂³×D₃	1	13	yes	—
50	C₃⋊D₃	—	6	78	no	—
51	#41	—	27	8	no	—
52	C₂×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
53	C₂×D₄	—	27	3	no	—
54	C₂²×C₄	—	27	3	no	—
55	C₂²×C₃	C₂²×C₃⋊D₃	1	13	yes	—
56	C₂×D₃	—	9	78	no	—
57	C₂×D₃	—	18	78	no	—
58	C₃×C₄	C₂²×C₃⋊D₃	1	78	yes	—
59	Dic₃	—	9	26	no	—
60	C₃²	C₂×#239	1	13	yes	complemented
61	C₂³	—	27	3	no	—
62	Q₈	C₂×C₃⋊(C₃⋊D₃)	1	4	yes	complemented
63	D₄	—	27	12	no	—
64	C₂×C₄	—	27	1	no	—
65	C₂×C₄	—	27	12	no	—
66	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	3	yes	—
67	C₂×C₃	#3832	1	26	yes	complemented
68	C₂×C₃	C₂³×C₃⋊D₃	1	13	yes	—
69	D₃	—	18	78	no	—
70	C₂²	—	27	6	no	—
71	C₂²	—	54	6	no	—
72	C₂²	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
73	C₄	—	27	2	no	—
74	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	6	yes	—
75	C₃	C₂×#3832	1	13	yes	complemented
76	C₂	—	54	6	no	—
77	C₂	#89080	1	2	yes	direct product factor
78	C₂	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃³	C₂⁴
Frattini	C₂	C₂³×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×Q₈	C₂
socle	C₂²×C₃³	C₂³
hypercenter	C₂×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×#41
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×Q₈	C₂	semidirect
2. #89080	C₂	direct (N × H)
3. C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
4. C₂×C₄×C₃⋊(C₃⋊D₃)	C₂	semidirect
7. C₃³×Q₈	C₂²	semidirect
10. C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	semidirect
11. C₄×C₃⋊(C₃⋊D₃)	C₂²	semidirect
12. C₂×C₃²×Q₈	D₃	semidirect
23. C₃²×Q₈	C₂×D₃	semidirect
28. C₂×C₃³	#41	semidirect
31. C₂×C₃×Q₈	C₃⋊D₃	semidirect
41. C₃³	C₂×#41	semidirect
43. C₃×Q₈	C₂×C₃⋊D₃	semidirect
48. C₂×C₃²	#239	semidirect
52. C₂×Q₈	C₃⋊(C₃⋊D₃)	semidirect
60. C₃²	C₂×#239	semidirect
62. Q₈	C₂×C₃⋊(C₃⋊D₃)	semidirect
67. C₂×C₃	#3832	semidirect
75. C₃	C₂×#3832	semidirect
77. C₂	#89080	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×C₃³×Q₈	2
#89080	2
C₂×C₄⋊(C₃⋊(C₃⋊D₃))	2
C₂×C₄×C₃⋊(C₃⋊D₃)	2
C₂×#3832	3

Subgroups — C2×#89080 #11724681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#89080 #11724681