Subgroups — C₂×#88863 #11723984

Order 864.

840 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
2	C₂×#6308	C₂	1	1	yes	maximal, characteristic
3	#88863	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
5	C₂×#3756	—	3	13	no	maximal
6	C₃³×Q₈	—	2	1	no	—
7	C₃³×Q₈	C₂²	1	2	yes	—
8	C₂×C₃³×C₄	—	2	1	no	—
9	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
11	#6308	—	2	1	no	—
12	#6308	C₂²	1	2	yes	—
13	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
14	C₂×C₃²×Q₈	D₃	1	13	yes	—
15	C₂×#634	—	3	13	no	—
16	#3756	—	3	52	no	—
17	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
18	C₂²×C₃³	D₄	1	1	yes	characteristic
19	C₃³×C₄	—	2	2	no	—
20	C₃³×C₄	D₄	1	1	yes	characteristic
21	C₃³×C₄	C₂³	1	1	yes	characteristic
22	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
23	C₂×#216	—	9	13	no	—
24	C₃²×Q₈	—	2	13	no	—
25	C₃²×Q₈	C₂×D₃	1	26	yes	—
26	C₂×C₃²×C₄	—	2	13	no	—
27	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
28	C₂×C₃⋊Dic₃	—	6	13	no	—
29	#634	—	3	26	no	—
30	#634	—	6	13	no	—
31	C₃⋊(C₃⋊C₈)	—	3	26	no	—
32	C₂×C₃³	Q₁₆	1	2	yes	complemented
33	C₂×C₃³	C₂×D₄	1	1	yes	characteristic
34	C₂×C₃×Q₈	C₃⋊D₃	1	13	yes	—
35	C₂×Dic₆	—	9	13	no	—
36	#216	—	9	52	no	—
37	C₂×C₃⋊C₈	—	9	13	no	—
38	C₂²×C₃²	#67	1	13	yes	—
39	C₃²×C₄	—	2	26	no	—
40	C₃²×C₄	#67	1	13	yes	—
41	C₃²×C₄	C₂²×D₃	1	13	yes	—
42	C₃⋊Dic₃	—	6	26	no	—
43	C₂×Q₁₆	—	27	1	no	Sylow
44	C₃³	C₂×Q₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₃×Q₈	—	2	13	no	—
46	C₃×Q₈	C₂×C₃⋊D₃	1	26	yes	—
47	C₂×C₃×C₄	—	2	13	no	—
48	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
49	C₂×Dic₃	—	18	13	no	—
50	Dic₆	—	9	26	no	—
51	Dic₆	—	18	13	no	—
52	C₃⋊C₈	—	9	26	no	—
53	C₂×C₃²	#216	1	26	yes	complemented
54	C₂×C₃²	C₂×#67	1	13	yes	—
55	C₂×Q₈	—	27	1	no	—
56	C₂×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
57	Q₁₆	—	27	4	no	—
58	C₂×C₈	—	27	1	no	—
59	C₂²×C₃	#638	1	13	yes	—
60	C₃×C₄	—	2	26	no	—
61	C₃×C₄	#638	1	13	yes	—
62	C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
63	Dic₃	—	18	26	no	—
64	C₃²	C₂×#216	1	13	yes	complemented
65	Q₈	—	2	1	no	—
66	Q₈	—	27	2	no	—
67	Q₈	—	54	1	no	—
68	Q₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	54	1	no	—
71	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
72	C₈	—	27	2	no	—
73	C₂×C₃	#3756	1	26	yes	complemented
74	C₂×C₃	C₂×#638	1	13	yes	—
75	C₂²	#6312	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	54	2	no	—
78	C₄	#6312	1	1	yes	characteristic
79	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
80	C₃	C₂×#3756	1	13	yes	complemented
81	C₂	#88863	1	2	yes	direct product factor
82	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, prime radical	C₂²×C₃³	D₄
hypercenter	C₂×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×Q₁₆
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
3. #88863	C₂	direct (N × H)
32. C₂×C₃³	Q₁₆	semidirect
44. C₃³	C₂×Q₁₆	semidirect
53. C₂×C₃²	#216	semidirect
64. C₃²	C₂×#216	semidirect
73. C₂×C₃	#3756	semidirect
80. C₃	C₂×#3756	semidirect
81. C₂	#88863	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×#6308	2
#88863	2
C₂×C₃⋊(C₃⋊(C₃⋊C₈))	2
C₂×#3756	3

Subgroups — C2×#88863 #11723984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#88863 #11723984