Subgroups — C₂×#88862 #11723982

Order 864.

952 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	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic
3	#88862	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#3755	—	3	13	no	maximal
6	C₂²×C₃⋊(C₃⋊D₃)	—	2	1	no	—
7	C₃³×Q₈	—	2	1	no	—
8	C₃³×Q₈	C₂²	1	2	yes	complemented
9	C₂×C₃³×C₄	—	2	1	no	—
10	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	1	no	—
12	C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	2	yes	—
13	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	complemented
14	C₂×C₃²×Q₈	D₃	1	13	yes	complemented
15	C₂×C₄⋊(C₃⋊D₃)	—	3	13	no	—
16	#3755	—	3	52	no	—
17	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
18	C₂²×C₃³	D₄	1	1	yes	characteristic
19	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
20	C₂×C₃⋊(C₃⋊D₃)	—	4	2	no	—
21	C₃³×C₄	—	2	2	no	—
22	C₃³×C₄	D₄	1	1	yes	characteristic
23	C₃³×C₄	C₂³	1	1	yes	characteristic
24	C₂×#215	—	9	13	no	—
25	C₂²×C₃⋊D₃	—	6	13	no	—
26	C₃²×Q₈	—	2	13	no	—
27	C₃²×Q₈	C₂×D₃	1	26	yes	complemented
28	C₂×C₃²×C₄	—	2	13	no	—
29	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
30	C₄⋊(C₃⋊D₃)	—	3	26	no	—
31	C₄⋊(C₃⋊D₃)	—	6	13	no	—
32	C₃⋊(C₃⋊C₈)	—	3	26	no	—
33	C₂×C₃³	SD₁₆	1	2	yes	complemented
34	C₂×C₃³	C₂×D₄	1	1	yes	characteristic
35	C₃⋊(C₃⋊D₃)	—	4	2	no	—
36	C₂×C₃×Q₈	C₃⋊D₃	1	13	yes	complemented
37	C₂×D₁₂	—	9	13	no	—
38	#215	—	9	52	no	—
39	C₂×C₃⋊C₈	—	9	13	no	—
40	C₂²×C₃²	#67	1	13	yes	—
41	C₂×C₃⋊D₃	—	6	26	no	—
42	C₂×C₃⋊D₃	—	12	26	no	—
43	C₃²×C₄	—	2	26	no	—
44	C₃²×C₄	#67	1	13	yes	—
45	C₃²×C₄	C₂²×D₃	1	13	yes	—
46	C₂×SD₁₆	—	27	1	no	Sylow
47	C₃³	C₂×SD₁₆	1	1	yes	characteristic, complemented, Sylow
48	C₂²×D₃	—	18	13	no	—
49	C₃×Q₈	—	2	13	no	—
50	C₃×Q₈	C₂×C₃⋊D₃	1	26	yes	complemented
51	C₂×C₃×C₄	—	2	13	no	—
52	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
53	D₁₂	—	9	26	no	—
54	D₁₂	—	18	13	no	—
55	C₃⋊C₈	—	9	26	no	—
56	C₂×C₃²	#215	1	26	yes	complemented
57	C₂×C₃²	C₂×#67	1	13	yes	—
58	C₃⋊D₃	—	12	26	no	—
59	C₂×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
60	C₂×D₄	—	27	1	no	—
61	SD₁₆	—	27	4	no	—
62	C₂×C₈	—	27	1	no	—
63	C₂²×C₃	#638	1	13	yes	—
64	C₂×D₃	—	18	26	no	—
65	C₂×D₃	—	36	26	no	—
66	C₃×C₄	—	2	26	no	—
67	C₃×C₄	#638	1	13	yes	—
68	C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
69	C₃²	C₂×#215	1	13	yes	complemented
70	C₂³	—	54	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
73	D₄	—	27	2	no	—
74	D₄	—	54	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
77	C₈	—	27	2	no	—
78	C₂×C₃	#3755	1	26	yes	complemented
79	C₂×C₃	C₂×#638	1	13	yes	—
80	D₃	—	36	26	no	—
81	C₂²	—	54	2	no	—
82	C₂²	—	108	2	no	—
83	C₂²	#6312	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	#6312	1	1	yes	characteristic
86	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
87	C₃	C₂×#3755	1	13	yes	complemented
88	C₂	—	108	2	no	—
89	C₂	#88862	1	2	yes	direct product factor
90	C₂	C₂×#6312	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#6312
commutator, cosocle	C₃³×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×Q₈	C₂
socle	C₂²×C₃³	D₄
prime radical	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂
hypercenter	C₂×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×SD₁₆
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×Q₈	C₂	semidirect
3. #88862	C₂	direct (N × H)
4. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
8. C₃³×Q₈	C₂²	semidirect
13. C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	semidirect
14. C₂×C₃²×Q₈	D₃	semidirect
27. C₃²×Q₈	C₂×D₃	semidirect
33. C₂×C₃³	SD₁₆	semidirect
36. C₂×C₃×Q₈	C₃⋊D₃	semidirect
47. C₃³	C₂×SD₁₆	semidirect
50. C₃×Q₈	C₂×C₃⋊D₃	semidirect
56. C₂×C₃²	#215	semidirect
59. C₂×Q₈	C₃⋊(C₃⋊D₃)	semidirect
69. C₃²	C₂×#215	semidirect
72. Q₈	C₂×C₃⋊(C₃⋊D₃)	semidirect
78. C₂×C₃	#3755	semidirect
87. C₃	C₂×#3755	semidirect
89. C₂	#88862	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
C₂×C₄⋊(C₃⋊(C₃⋊D₃))	2
#88862	2
C₂×C₃⋊(C₃⋊(C₃⋊C₈))	2
C₂×#3755	3

Subgroups — C2×#88862 #11723982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#88862 #11723982