Subgroups — #88994

Order 432.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊#157	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#157	—	3	1	no	maximal
4	C₂²×C₃³	C₄	1	1	yes	characteristic, complemented
5	C₂×C₃×C₃⋊D₃	—	2	2	no	—
6	C₂×C₃×C₃⋊D₃	C₄	1	1	yes	characteristic, complemented
7	C₂×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
8	C₃⋊#157	—	2	2	no	—
9	C₂²×C₃⋊D₃	D₃	1	1	yes	characteristic
10	C₂²×C₃×D₃	—	6	2	no	—
11	C₂×#157	—	3	2	no	—
12	C₂×C₃³	—	2	1	no	—
13	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
14	C₃×C₃⋊D₃	—	2	1	no	—
15	C₃×C₃⋊D₃	D₄	1	2	yes	—
16	#217	—	9	1	no	maximal
17	C₂²×C₃²	—	2	2	no	—
18	C₂²×C₃²	—	4	2	no	—
19	C₂²×C₃²	Dic₃	1	1	yes	characteristic, complemented
20	C₂×C₃⋊D₃	—	2	2	no	—
21	C₂×C₃⋊D₃	Dic₃	1	1	yes	characteristic, complemented
22	C₂×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
23	C₂×C₃×D₃	—	6	2	no	—
24	C₂×C₃×D₃	—	6	2	no	—
25	C₂×C₃×D₃	—	6	8	no	—
26	#157	—	6	2	no	—
27	C₃³	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
28	C₂³×C₃	—	9	1	no	—
29	C₂²×D₃	—	6	2	no	—
30	C₂×Dic₃	—	9	2	no	—
31	C₂×C₃²	—	2	1	no	—
32	C₂×C₃²	—	2	4	no	—
33	C₂×C₃²	—	2	2	no	—
34	C₂×C₃²	—	4	4	no	—
35	C₂×C₃²	—	4	2	no	—
36	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
37	C₃⋊D₃	—	2	1	no	—
38	C₃⋊D₃	#67	1	2	yes	—
39	C₃×D₃	—	6	4	no	—
40	C₃×D₃	—	6	4	no	—
41	C₂²⋊C₄	—	27	1	no	Sylow
42	C₂²×C₃	—	2	2	no	—
43	C₂²×C₃	—	4	2	no	—
44	C₂²×C₃	—	9	1	no	—
45	C₂²×C₃	—	9	1	no	—
46	C₂²×C₃	—	18	2	no	—
47	C₂²×C₃	#157	1	1	yes	characteristic, complemented
48	C₂×D₃	—	6	2	no	—
49	C₂×D₃	—	6	2	no	—
50	C₂×D₃	—	6	8	no	—
51	Dic₃	—	18	2	no	—
52	C₃²	—	2	2	no	—
53	C₃²	—	4	2	no	—
54	C₃²	#217	1	1	yes	characteristic, complemented
55	C₂³	—	9	1	no	—
56	C₂×C₄	—	27	2	no	—
57	C₂×C₃	—	2	4	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	2	2	no	—
60	C₂×C₃	—	4	4	no	—
61	C₂×C₃	—	4	2	no	—
62	C₂×C₃	—	9	2	no	—
63	C₂×C₃	—	18	1	no	—
64	C₂×C₃	C₂×#157	1	1	yes	characteristic
65	D₃	—	6	4	no	—
66	D₃	—	6	4	no	—
67	C₂²	—	9	1	no	—
68	C₂²	—	9	1	no	—
69	C₂²	—	18	2	no	—
70	C₂²	C₃⋊#157	1	1	yes	characteristic, complemented
71	C₄	—	54	2	no	—
72	C₃	—	2	2	no	—
73	C₃	—	4	2	no	—
74	C₃	C₂²⋊#157	1	1	yes	characteristic, complemented
75	C₂	—	2	1	no	—
76	C₂	—	9	2	no	—
77	C₂	—	18	1	no	—
78	C₂	C₂×C₃⋊#157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊#157
commutator, socle	C₂×C₃³	C₂×C₄
Fitting	C₂²×C₃³	C₄
prime radical	C₂²×C₃×C₃⋊D₃	C₂
cosocle	C₂×C₃×C₃⋊D₃	C₂²
hypercenter	C₂²	C₃⋊#157
hypocenter	C₃³	C₂²⋊C₄
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊#157	C₂	semidirect
4. C₂²×C₃³	C₄	semidirect
6. C₂×C₃×C₃⋊D₃	C₄	semidirect
19. C₂²×C₃²	Dic₃	semidirect
21. C₂×C₃⋊D₃	Dic₃	semidirect
27. C₃³	C₂²⋊C₄	semidirect
47. C₂²×C₃	#157	semidirect
54. C₃²	#217	semidirect
70. C₂²	C₃⋊#157	semidirect
74. C₃	C₂²⋊#157	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₃⋊D₃	2
C₂×C₃⋊#157	2
C₂²⋊#157	3
#217	9