Subgroups — #64107

Order 288.

180 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, complemented
2	#3752	C₂	1	1	yes	maximal, characteristic
3	#3752	C₂	1	1	yes	maximal, characteristic
4	#3752	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
6	#3750	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#898	—	3	4	no	maximal
9	C₂²×C₃⋊D₃	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×C₃²×C₄	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	complemented
17	(C₂×D₃)⋊C₄	—	3	4	no	—
18	(C₂×D₃)⋊C₄	—	3	4	no	—
19	(C₂×D₃)⋊C₄	—	3	4	no	—
20	C₄⋊Dic₃	—	3	4	no	—
21	Dic₃⋊C₄	—	3	4	no	—
22	C₄×Dic₃	—	3	4	no	—
23	C₂²×C₃²	C₂³	1	1	yes	characteristic
24	C₂×C₃⋊D₃	—	2	1	no	—
25	C₂×C₃⋊D₃	—	2	1	no	—
26	C₂×C₃⋊D₃	—	2	1	no	—
27	C₃²×C₄	—	2	1	no	—
28	C₃²×C₄	—	2	1	no	—
29	C₃²×C₄	—	2	1	no	—
30	C₃⋊Dic₃	—	2	1	no	—
31	C₃⋊Dic₃	—	2	1	no	—
32	C₃⋊Dic₃	—	2	1	no	—
33	#126	—	9	1	no	Sylow
34	C₂²×D₃	—	3	4	no	—
35	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
36	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
37	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
38	C₂×Dic₃	—	3	4	no	—
39	C₂×Dic₃	—	3	4	no	—
40	C₂×Dic₃	—	3	4	no	—
41	C₂×C₃²	#41	1	1	yes	characteristic
42	C₂×C₃²	#41	1	1	yes	characteristic
43	C₂×C₃²	#41	1	1	yes	characteristic
44	C₃⋊D₃	—	4	1	no	—
45	C₄⋊C₄	—	9	1	no	—
46	C₄⋊C₄	—	9	1	no	—
47	C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	—	9	1	no	—
49	C₂²⋊C₄	—	9	1	no	—
50	C₂²⋊C₄	—	9	1	no	—
51	C₄²	—	9	1	no	—
52	C₂²×C₃	C₂²×D₃	1	4	yes	—
53	C₂×D₃	—	6	4	no	—
54	C₂×D₃	—	6	4	no	—
55	C₂×D₃	—	6	4	no	—
56	C₃×C₄	—	2	4	no	—
57	C₃×C₄	—	2	4	no	—
58	C₃×C₄	—	2	4	no	—
59	Dic₃	—	6	4	no	—
60	Dic₃	—	6	4	no	—
61	Dic₃	—	6	4	no	—
62	C₃²	#126	1	1	yes	characteristic, complemented, Sylow
63	C₂³	—	9	1	no	—
64	C₂×C₄	—	9	1	no	—
65	C₂×C₄	—	9	1	no	—
66	C₂×C₄	—	9	1	no	—
67	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
68	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
69	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
70	C₂×C₃	#235	1	4	yes	—
71	C₂×C₃	#237	1	4	yes	—
72	C₂×C₃	#239	1	4	yes	—
73	D₃	—	12	4	no	—
74	C₂²	—	18	1	no	—
75	C₂²	—	18	1	no	—
76	C₂²	—	18	1	no	—
77	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	2	1	no	—
81	C₄	—	18	1	no	—
82	C₄	—	18	1	no	—
83	C₄	—	18	1	no	—
84	C₃	#898	1	4	yes	complemented
85	C₂	—	36	1	no	—
86	C₂	#3828	1	1	yes	characteristic
87	C₂	#3830	1	1	yes	characteristic
88	C₂	#3832	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₃
commutator, socle, cosocle	C₂²×C₃²	C₂³
Fitting	C₃²×C₄⋊C₄	C₂
prime radical	C₂²×C₃⋊D₃	C₂²
hypercenter	C₄⋊C₄	C₃⋊D₃
hypocenter	C₃²	#126
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
1. C₃²×C₄⋊C₄	C₂	semidirect
5. C₄⋊(C₃⋊Dic₃)	C₂	semidirect
6. #3750	C₂	semidirect
7. C₄×C₃⋊Dic₃	C₂	semidirect
16. C₃×C₄⋊C₄	D₃	semidirect
47. C₄⋊C₄	C₃⋊D₃	semidirect
62. C₃²	#126	semidirect
84. C₃	#898	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
#3752	2
C₄⋊(C₃⋊Dic₃)	2
#3750	2
C₄×C₃⋊Dic₃	2
#898	3