Subgroups — #63579

Order 288.

179 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₃	C₂	1	2	yes	maximal
3	C₃⋊#95	—	3	2	no	maximal
4	C₂³×C₃²	C₂²	1	1	yes	characteristic
5	C₂×C₃⋊Dic₃	—	2	2	no	—
6	C₂×C₃⋊Dic₃	C₄	1	2	yes	complemented
7	C₂×C₃×Dic₃	—	2	2	no	—
8	C₂×C₃×Dic₃	—	2	2	no	—
9	C₂×C₃×Dic₃	C₄	1	4	yes	complemented
10	C₂²×C₃×C₄	—	3	2	no	—
11	C₂²×Dic₃	—	3	2	no	—
12	C₂²×Dic₃	—	6	1	no	—
13	C₂²×Dic₃	D₃	1	2	yes	—
14	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₃²	C₂×C₄	1	2	yes	—
16	C₂²×C₃²	D₄	1	2	yes	—
17	C₂²×C₃²	D₄	1	1	yes	characteristic
18	C₂²×C₃²	Q₈	1	1	yes	characteristic
19	C₃⋊Dic₃	—	2	2	no	—
20	C₃×Dic₃	—	2	4	no	—
21	#95	—	9	1	no	Sylow
22	C₂³×C₃	—	2	1	no	—
23	C₂³×C₃	C₂×D₃	1	2	yes	—
24	C₂×C₃×C₄	—	3	4	no	—
25	C₂×C₃×C₄	—	6	2	no	—
26	C₂×C₃×C₄	—	6	2	no	—
27	C₂×Dic₃	—	2	2	no	—
28	C₂×Dic₃	—	2	2	no	—
29	C₂×Dic₃	—	3	4	no	—
30	C₂×Dic₃	—	6	2	no	—
31	C₂×Dic₃	—	6	4	no	—
32	C₂×Dic₃	—	6	2	no	—
33	C₂×Dic₃	—	6	2	no	—
34	C₂×Dic₃	Dic₃	1	4	yes	complemented
35	C₂×C₃²	C₄²	1	1	yes	characteristic
36	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₃²	C₂²⋊C₄	1	2	yes	—
38	C₂×C₃²	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₃²	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄	—	3	2	no	—
41	C₂²×C₄	—	9	1	no	—
42	C₂²×C₃	—	2	2	no	—
43	C₂²×C₃	—	2	1	no	—
44	C₂²×C₃	—	2	1	no	—
45	C₂²×C₃	—	2	2	no	—
46	C₂²×C₃	—	2	1	no	—
47	C₂²×C₃	Dic₆	1	2	yes	—
48	C₂²×C₃	C₄×D₃	1	2	yes	—
49	C₂²×C₃	C₄×D₃	1	2	yes	—
50	C₂²×C₃	D₁₂	1	2	yes	—
51	C₂²×C₃	C₂×Dic₃	1	2	yes	—
52	C₂²×C₃	#67	1	2	yes	—
53	C₂²×C₃	#67	1	2	yes	—
54	C₃×C₄	—	6	4	no	—
55	Dic₃	—	2	4	no	—
56	Dic₃	—	6	4	no	—
57	Dic₃	—	6	4	no	—
58	C₃²	#95	1	1	yes	characteristic, complemented, Sylow
59	C₂³	D₃²	1	1	yes	characteristic
60	C₂×C₄	—	3	4	no	—
61	C₂×C₄	—	6	2	no	—
62	C₂×C₄	—	6	2	no	—
63	C₂×C₄	—	9	2	no	—
64	C₂×C₄	—	18	2	no	—
65	C₂×C₃	—	2	1	no	—
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	—	2	1	no	—
68	C₂×C₃	—	2	2	no	—
69	C₂×C₃	—	2	1	no	—
70	C₂×C₃	C₄×Dic₃	1	2	yes	—
71	C₂×C₃	Dic₃⋊C₄	1	2	yes	—
72	C₂×C₃	Dic₃⋊C₄	1	2	yes	—
73	C₂×C₃	C₄⋊Dic₃	1	2	yes	—
74	C₂×C₃	(C₂×D₃)⋊C₄	1	2	yes	—
75	C₂×C₃	(C₂×D₃)⋊C₄	1	2	yes	—
76	C₂×C₃	#217	1	2	yes	—
77	C₂²	D₃×Dic₃	1	2	yes	—
78	C₂²	#624	1	1	yes	characteristic
79	C₂²	#625	1	1	yes	characteristic
80	C₂²	#626	1	2	yes	—
81	C₂²	#627	1	1	yes	characteristic
82	C₄	—	6	4	no	—
83	C₄	—	18	2	no	—
84	C₃	—	2	1	no	—
85	C₃	C₃⋊#95	1	2	yes	complemented
86	C₂	Dic₃²	1	1	yes	characteristic
87	C₂	(C₂×D₃)⋊Dic₃	1	2	yes	—
88	C₂	#3722	1	1	yes	characteristic
89	C₂	Dic₃⋊Dic₃	1	2	yes	—
90	C₂	#3724	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂³	D₃²
commutator	C₂×C₃²	C₄²
Fitting, socle, prime radical, cosocle	C₂³×C₃²	C₂²
hypocenter	C₃²	#95
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
6. C₂×C₃⋊Dic₃	C₄	semidirect
9. C₂×C₃×Dic₃	C₄	semidirect
34. C₂×Dic₃	Dic₃	semidirect
58. C₃²	#95	semidirect
85. C₃	C₃⋊#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊Dic₃	2
C₂²×C₃×Dic₃	2
C₃⋊#95	3