Subgroups — #11723977

Order 864.

1064 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#6308	C₂	1	1	yes	maximal, characteristic, complemented
3	#88864	C₂	1	4	yes	maximal, complemented
4	C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
5	#64145	—	3	13	no	maximal
6	C₂³×C₃³	C₂²	1	2	yes	—
7	C₃³×D₄	—	2	2	no	—
8	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	#6308	—	2	2	no	—
12	C₂×C₃²×D₄	D₃	1	13	yes	—
13	C₂×#634	—	3	13	no	—
14	#3757	—	3	52	no	—
15	C₄×C₃⋊Dic₃	—	3	13	no	—
16	C₂²×C₃³	—	2	2	no	—
17	C₂²×C₃³	—	2	4	no	—
18	C₂²×C₃³	C₂³	1	1	yes	characteristic
19	C₃³×C₄	D₄	1	2	yes	—
20	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
21	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
22	#936	—	9	13	no	—
23	C₂³×C₃²	C₂×D₃	1	26	yes	—
24	C₃²×D₄	—	2	26	no	—
25	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
26	C₂×C₃⋊Dic₃	—	3	26	no	—
27	C₂×C₃⋊Dic₃	—	3	26	no	—
28	#634	—	6	26	no	—
29	C₂×C₃³	—	4	2	no	—
30	C₂×C₃³	C₂×D₄	1	1	yes	characteristic
31	C₂×C₃³	#41	1	2	yes	—
32	C₂×C₃×D₄	C₃⋊D₃	1	13	yes	—
33	C₂×Dic₆	—	9	13	no	—
34	#217	—	9	52	no	—
35	C₄×Dic₃	—	9	13	no	—
36	C₂²×C₃²	—	2	52	no	—
37	C₂²×C₃²	—	2	26	no	—
38	C₂²×C₃²	C₂²×D₃	1	13	yes	—
39	C₃²×C₄	#67	1	26	yes	—
40	C₃⋊Dic₃	—	6	26	no	—
41	C₃⋊Dic₃	—	6	26	no	—
42	#124	—	27	1	no	Sylow
43	C₃³	#124	1	1	yes	characteristic, complemented, Sylow
44	C₂³×C₃	C₂×C₃⋊D₃	1	26	yes	—
45	C₃×D₄	—	2	26	no	—
46	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
47	C₂×Dic₃	—	9	26	no	—
48	C₂×Dic₃	—	9	26	no	—
49	Dic₆	—	18	26	no	—
50	C₂×C₃²	—	4	26	no	—
51	C₂×C₃²	#237	1	26	yes	—
52	C₂×C₃²	C₂×#67	1	13	yes	—
53	C₂×Q₈	—	27	1	no	—
54	C₂×D₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
55	C₂²⋊C₄	—	27	4	no	—
56	C₄²	—	27	1	no	—
57	C₂²×C₃	—	2	52	no	—
58	C₂²×C₃	—	2	26	no	—
59	C₂²×C₃	C₂²×C₃⋊D₃	1	13	yes	—
60	C₃×C₄	#638	1	26	yes	—
61	Dic₃	—	18	26	no	—
62	Dic₃	—	18	26	no	—
63	C₃²	#936	1	13	yes	complemented
64	C₂³	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
65	Q₈	—	54	2	no	—
66	D₄	—	2	2	no	—
67	C₂×C₄	—	27	2	no	—
68	C₂×C₄	—	27	2	no	—
69	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
70	C₂×C₃	—	4	26	no	—
71	C₂×C₃	#3830	1	26	yes	—
72	C₂×C₃	C₂×#638	1	13	yes	—
73	C₂²	—	2	4	no	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
76	C₄	—	54	2	no	—
77	C₄	—	54	2	no	—
78	C₄	#6312	1	2	yes	—
79	C₃	#64145	1	13	yes	complemented
80	C₂	—	4	2	no	—
81	C₂	#89078	1	2	yes	—
82	C₂	C₂×#6312	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂³
Fitting, prime radical	C₂×C₃³×D₄	C₂
hypercenter	C₂×D₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#124
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#6308	C₂	semidirect
3. #88864	C₂	semidirect
4. C₄×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
43. C₃³	#124	semidirect
63. C₃²	#936	semidirect
79. C₃	#64145	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×D₄	2
C₂×#6308	2
#88864	2
C₄×C₃⋊(C₃⋊Dic₃)	2
#64145	3