Subgroups — #64098

Order 288.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
3	#3750	C₂	1	1	yes	maximal, characteristic
4	#3750	C₂	1	1	yes	maximal, characteristic
5	#3750	C₂	1	1	yes	maximal, characteristic
6	#3750	C₂	1	1	yes	maximal, characteristic
7	C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
8	#889	—	3	4	no	maximal
9	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊C₄	D₃	1	4	yes	—
17	C₄⋊Dic₃	—	3	4	no	—
18	Dic₃⋊C₄	—	3	4	no	—
19	Dic₃⋊C₄	—	3	4	no	—
20	Dic₃⋊C₄	—	3	4	no	—
21	Dic₃⋊C₄	—	3	4	no	—
22	C₄×Dic₃	—	3	4	no	—
23	C₂²×C₃²	C₂³	1	1	yes	characteristic
24	C₃²×C₄	—	2	1	no	—
25	C₃²×C₄	—	2	1	no	—
26	C₃²×C₄	—	2	1	no	—
27	C₃⋊Dic₃	—	2	1	no	—
28	C₃⋊Dic₃	—	2	1	no	—
29	C₃⋊Dic₃	—	2	1	no	—
30	C₃⋊Dic₃	Q₈	1	2	yes	—
31	#125	—	9	1	no	Sylow
32	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
33	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
34	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
35	C₂×Dic₃	—	3	4	no	—
36	C₂×Dic₃	—	3	4	no	—
37	C₂×Dic₃	—	3	4	no	—
38	C₂×Dic₃	—	3	4	no	—
39	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₃²	#41	1	1	yes	characteristic
41	C₂×C₃²	#41	1	1	yes	characteristic
42	C₄⋊C₄	—	9	1	no	—
43	C₄⋊C₄	—	9	1	no	—
44	C₄⋊C₄	—	9	1	no	—
45	C₄⋊C₄	—	9	1	no	—
46	C₄⋊C₄	—	9	1	no	—
47	C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
48	C₄²	—	9	1	no	—
49	C₂²×C₃	C₂²×D₃	1	4	yes	—
50	C₃×C₄	—	2	4	no	—
51	C₃×C₄	—	2	4	no	—
52	C₃×C₄	—	2	4	no	—
53	Dic₃	—	3	8	no	—
54	Dic₃	—	6	4	no	—
55	Dic₃	—	6	4	no	—
56	Dic₃	—	6	4	no	—
57	C₃²	#125	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄	—	9	1	no	—
59	C₂×C₄	—	9	1	no	—
60	C₂×C₄	—	9	1	no	—
61	C₂×C₄	—	9	1	no	—
62	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
63	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
64	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
65	C₂×C₃	#235	1	4	yes	—
66	C₂×C₃	#237	1	4	yes	—
67	C₂×C₃	D₃×Q₈	1	4	yes	—
68	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	—	9	2	no	—
73	C₄	—	18	1	no	—
74	C₄	—	18	1	no	—
75	C₄	—	18	1	no	—
76	C₃	#889	1	4	yes	complemented
77	C₂	#3828	1	1	yes	characteristic
78	C₂	#3830	1	1	yes	characteristic
79	C₂	Q₈×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₃
commutator, socle, prime radical, cosocle	C₂²×C₃²	C₂³
Fitting	C₃²×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	C₃⋊D₃
hypocenter	C₃²	#125
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
57. C₃²	#125	semidirect
76. C₃	#889	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄⋊C₄	2
C₄⋊(C₃⋊Dic₃)	2
#3750	2
C₄×C₃⋊Dic₃	2
#889	3