Subgroups — #64139

Order 288.

282 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#638	C₂	1	2	yes	maximal, complemented
3	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	#3752	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
6	#930	—	3	4	no	maximal
7	C₂³×C₃²	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊D₃	C₂²	1	2	yes	—
9	C₂×C₃²×C₄	—	2	2	no	—
10	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
12	#638	—	2	4	no	—
13	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
14	C₄⋊(C₃⋊D₃)	—	2	2	no	—
15	C₂²×C₃×C₄	D₃	1	4	yes	complemented
16	C₂×#67	—	3	8	no	—
17	C₂×D₁₂	—	3	4	no	—
18	(C₂×D₃)⋊C₄	—	3	8	no	—
19	C₄⋊Dic₃	—	3	4	no	—
20	C₂²×C₃²	—	2	1	no	—
21	C₂²×C₃²	—	2	1	no	—
22	C₂²×C₃²	D₄	1	2	yes	complemented
23	C₂²×C₃²	C₂³	1	1	yes	characteristic
24	C₂×C₃⋊D₃	—	2	2	no	—
25	C₂×C₃⋊D₃	—	2	2	no	—
26	C₂×C₃⋊D₃	—	2	2	no	—
27	C₃²×C₄	—	2	1	no	—
28	C₃²×C₄	D₄	1	2	yes	complemented
29	C₃⋊Dic₃	—	2	2	no	—
30	#121	—	9	1	no	Sylow
31	C₂³×C₃	C₂×D₃	1	4	yes	—
32	C₂²×D₃	—	3	8	no	—
33	C₂×C₃×C₄	—	2	8	no	—
34	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
35	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
36	#67	—	6	16	no	—
37	C₂×Dic₃	—	3	8	no	—
38	D₁₂	—	6	8	no	—
39	C₂×C₃²	—	2	2	no	—
40	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
41	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
42	C₂×C₃²	#41	1	1	yes	characteristic
43	C₃⋊D₃	—	4	2	no	—
44	C₂×D₄	—	9	2	no	—
45	C₂×D₄	—	9	1	no	—
46	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic, complemented
47	C₄⋊C₄	—	9	1	no	—
48	C₂²⋊C₄	—	9	2	no	—
49	C₂²×C₃	—	2	4	no	—
50	C₂²×C₃	—	2	4	no	—
51	C₂²×C₃	D₁₂	1	8	yes	complemented
52	C₂²×C₃	C₂²×D₃	1	4	yes	—
53	C₂×D₃	—	6	8	no	—
54	C₂×D₃	—	6	8	no	—
55	C₂×D₃	—	6	8	no	—
56	C₃×C₄	—	2	4	no	—
57	C₃×C₄	#67	1	8	yes	complemented
58	Dic₃	—	6	8	no	—
59	C₃²	#121	1	1	yes	characteristic, complemented, Sylow
60	C₂³	—	9	2	no	—
61	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
62	D₄	—	18	4	no	—
63	D₄	—	18	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	9	2	no	—
66	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
67	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
68	C₂×C₃	—	2	8	no	—
69	C₂×C₃	C₂×D₁₂	1	4	yes	—
70	C₂×C₃	#235	1	4	yes	—
71	C₂×C₃	C₂×#67	1	4	yes	—
72	D₃	—	12	8	no	—
73	C₂²	—	2	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	18	2	no	—
76	C₂²	—	18	2	no	—
77	C₂²	—	18	2	no	—
78	C₂²	C₄⋊(C₃⋊D₃)	1	2	yes	complemented
79	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	18	2	no	—
82	C₄	#638	1	2	yes	complemented
83	C₃	#930	1	4	yes	complemented
84	C₂	—	2	2	no	—
85	C₂	—	36	2	no	—
86	C₂	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
87	C₂	#3828	1	1	yes	characteristic
88	C₂	C₂×#638	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₃
commutator, socle, cosocle	C₂²×C₃²	C₂³
Fitting	C₂²×C₃²×C₄	C₂
hypercenter	C₂²×C₄	C₃⋊D₃
hypocenter	C₃²	#121
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₄	C₂	semidirect
2. C₂×#638	C₂	semidirect
3. C₂×C₄⋊(C₃⋊D₃)	C₂	semidirect
4. #3752	C₂	semidirect
5. C₄⋊(C₃⋊Dic₃)	C₂	semidirect
15. C₂²×C₃×C₄	D₃	semidirect
22. C₂²×C₃²	D₄	semidirect
28. C₃²×C₄	D₄	semidirect
46. C₂²×C₄	C₃⋊D₃	semidirect
51. C₂²×C₃	D₁₂	semidirect
57. C₃×C₄	#67	semidirect
59. C₃²	#121	semidirect
78. C₂²	C₄⋊(C₃⋊D₃)	semidirect
82. C₄	#638	semidirect
83. C₃	#930	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄	2
C₂×#638	2
C₂×C₄⋊(C₃⋊D₃)	2
#3752	2
C₄⋊(C₃⋊Dic₃)	2
#930	3