Subgroups — #63849

Order 288.

179 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊Dic₃	C₂	1	2	yes	maximal, complemented
3	#3722	C₂	1	2	yes	maximal
4	Dic₃²	C₂	1	2	yes	maximal, complemented
5	#892	—	3	2	no	maximal
6	C₂²×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
9	C₄×C₃⋊D₃	C₄	1	4	yes	complemented
10	C₂×C₃×Dic₃	C₂²	1	4	yes	—
11	C₂×C₄×D₃	—	3	2	no	—
12	C₂×C₄×D₃	—	6	1	no	—
13	C₃×C₄⋊C₄	—	3	2	no	—
14	(C₂×D₃)⋊C₄	—	3	4	no	—
15	C₄⋊Dic₃	D₃	1	2	yes	complemented
16	C₄×Dic₃	—	3	4	no	—
17	C₂²×C₃²	C₂³	1	1	yes	characteristic
18	C₂×C₃⋊D₃	—	2	2	no	—
19	C₂×C₃⋊D₃	C₂×C₄	1	2	yes	—
20	C₃²×C₄	C₂×C₄	1	2	yes	—
21	C₃⋊Dic₃	C₂×C₄	1	2	yes	—
22	C₃×Dic₃	—	2	4	no	—
23	#117	—	9	1	no	Sylow
24	C₂²×D₃	—	3	2	no	—
25	C₂²×D₃	—	6	1	no	—
26	C₂×C₃×C₄	—	2	1	no	—
27	C₂×C₃×C₄	—	3	4	no	—
28	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
29	C₂×Dic₃	—	3	2	no	—
30	C₂×Dic₃	—	6	1	no	—
31	C₂×Dic₃	C₂×D₃	1	4	yes	—
32	C₄×D₃	—	3	8	no	—
33	C₄×D₃	—	6	4	no	—
34	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₃²	#41	1	2	yes	—
36	C₃⋊D₃	—	2	2	no	—
37	C₂²×C₄	—	9	1	no	—
38	C₄⋊C₄	—	3	2	no	—
39	C₂²⋊C₄	—	9	2	no	—
40	C₄²	—	9	2	no	—
41	C₂²×C₃	—	2	1	no	—
42	C₂²×C₃	C₂²×D₃	1	2	yes	—
43	C₂×D₃	—	3	4	no	—
44	C₂×D₃	—	6	2	no	—
45	C₂×D₃	—	6	4	no	—
46	C₂×D₃	—	6	2	no	—
47	C₂×D₃	—	6	2	no	—
48	C₃×C₄	—	2	2	no	—
49	C₃×C₄	—	6	4	no	—
50	C₃×C₄	C₄×D₃	1	4	yes	—
51	Dic₃	—	2	4	no	—
52	Dic₃	—	3	4	no	—
53	Dic₃	—	6	2	no	—
54	C₃²	#117	1	1	yes	characteristic, complemented, Sylow
55	C₂³	—	9	1	no	—
56	C₂×C₄	—	3	4	no	—
57	C₂×C₄	—	9	1	no	—
58	C₂×C₄	—	9	4	no	—
59	C₂×C₄	D₃²	1	1	yes	characteristic
60	C₂×C₃	—	2	2	no	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	C₂×C₄×D₃	1	2	yes	—
63	C₂×C₃	#237	1	2	yes	—
64	C₂×C₃	#239	1	2	yes	—
65	D₃	—	6	4	no	—
66	D₃	—	6	4	no	—
67	C₂²	—	9	2	no	—
68	C₂²	—	18	2	no	—
69	C₂²	C₂×D₃²	1	1	yes	characteristic
70	C₄	—	6	4	no	—
71	C₄	—	9	2	no	—
72	C₄	#624	1	2	yes	—
73	C₃	—	2	1	no	—
74	C₃	#892	1	2	yes	complemented
75	C₂	—	18	2	no	—
76	C₂	#3796	1	2	yes	—
77	C₂	C₂×#624	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃²
commutator	C₂×C₃²	C₂²×C₄
Fitting	C₂×C₃²×C₄	C₂²
socle, cosocle	C₂²×C₃²	C₂³
prime radical	C₂²×C₃⋊D₃	C₂²
hypercenter	C₂×C₄	D₃²
hypocenter	C₃²	#117
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄⋊Dic₃	C₂	semidirect
4. Dic₃²	C₂	semidirect
9. C₄×C₃⋊D₃	C₄	semidirect
15. C₄⋊Dic₃	D₃	semidirect
54. C₃²	#117	semidirect
74. C₃	#892	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊D₃	2
C₃×C₄⋊Dic₃	2
#3722	2
Dic₃²	2
#892	3