Subgroups — #63838

Order 288.

155 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#627	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃×Dic₃⋊C₄	C₂	1	2	yes	maximal
4	#3724	C₂	1	1	yes	maximal, characteristic
5	Dic₃²	C₂	1	2	yes	maximal
6	#887	—	3	2	no	maximal
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
10	C₂×C₃×Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×Dic₃	C₂²	1	2	yes	—
12	#627	C₄	1	4	yes	complemented
13	C₂×Dic₆	—	3	2	no	—
14	C₃×C₄⋊C₄	—	3	2	no	—
15	Dic₃⋊C₄	—	3	2	no	—
16	Dic₃⋊C₄	D₃	1	2	yes	—
17	C₄×Dic₃	—	3	2	no	—
18	C₄×Dic₃	—	3	2	no	—
19	C₄×Dic₃	—	3	2	no	—
20	C₄×Dic₃	—	6	1	no	—
21	C₂²×C₃²	C₂³	1	1	yes	characteristic
22	C₃²×C₄	—	2	1	no	—
23	C₃⋊Dic₃	C₂×C₄	1	2	yes	—
24	C₃⋊Dic₃	Q₈	1	2	yes	complemented
25	C₃×Dic₃	—	2	2	no	—
26	C₃×Dic₃	C₂×C₄	1	4	yes	—
27	C₄×Q₈	—	9	1	no	Sylow
28	C₂×C₃×C₄	—	2	1	no	—
29	C₂×C₃×C₄	—	3	2	no	—
30	C₂×C₃×C₄	—	3	2	no	—
31	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
32	C₂×Dic₃	—	3	2	no	—
33	C₂×Dic₃	—	3	2	no	—
34	C₂×Dic₃	—	6	1	no	—
35	C₂×Dic₃	—	6	1	no	—
36	C₂×Dic₃	C₂×D₃	1	2	yes	—
37	C₂×Dic₃	C₂×D₃	1	2	yes	—
38	Dic₆	—	3	8	no	—
39	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₃²	#41	1	1	yes	characteristic
42	C₂×Q₈	—	9	1	no	—
43	C₄⋊C₄	—	3	2	no	—
44	C₄⋊C₄	—	9	1	no	—
45	C₄²	—	9	2	no	—
46	C₄²	—	9	1	no	—
47	C₂²×C₃	—	2	1	no	—
48	C₂²×C₃	C₂²×D₃	1	2	yes	—
49	C₃×C₄	—	2	2	no	—
50	C₃×C₄	—	2	2	no	—
51	C₃×C₄	—	3	4	no	—
52	C₃×C₄	—	6	2	no	—
53	Dic₃	—	2	2	no	—
54	Dic₃	—	3	4	no	—
55	Dic₃	—	3	4	no	—
56	Dic₃	—	6	2	no	—
57	Dic₃	—	6	2	no	—
58	Dic₃	C₄×D₃	1	4	yes	—
59	C₃²	C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
60	Q₈	—	9	4	no	—
61	C₂×C₄	—	3	2	no	—
62	C₂×C₄	—	3	2	no	—
63	C₂×C₄	—	9	1	no	—
64	C₂×C₄	—	9	1	no	—
65	C₂×C₄	D₃²	1	1	yes	characteristic
66	C₂×C₃	—	2	1	no	—
67	C₂×C₃	—	2	1	no	—
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	C₂×C₄×D₃	1	2	yes	—
70	C₂×C₃	#237	1	2	yes	—
71	C₂×C₃	D₃×Q₈	1	2	yes	—
72	C₂²	C₂×D₃²	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	3	4	no	—
75	C₄	—	6	2	no	—
76	C₄	—	9	2	no	—
77	C₄	—	9	2	no	—
78	C₃	—	2	1	no	—
79	C₃	#887	1	2	yes	complemented
80	C₂	#3797	1	1	yes	characteristic
81	C₂	C₄×D₃²	1	1	yes	characteristic
82	C₂	#3805	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃²
commutator	C₂×C₃²	C₂²×C₄
Fitting	C₂×C₃²×C₄	C₂²
socle, prime radical, cosocle	C₂²×C₃²	C₂³
hypercenter	C₂×C₄	D₃²
hypocenter	C₃²	C₄×Q₈
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
12. #627	C₄	semidirect
24. C₃⋊Dic₃	Q₈	semidirect
59. C₃²	C₄×Q₈	semidirect
79. C₃	#887	semidirect

Maximal subgroups

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

subgroup	index
C₂×#627	2
C₄×C₃⋊Dic₃	2
C₃×Dic₃⋊C₄	2
#3724	2
Dic₃²	2
#887	3