Subgroups — #63562

Order 288.

111 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#832	—	3	1	no	maximal
5	#809	—	3	1	no	maximal
6	C₂²×C₃×D₃	—	2	1	no	—
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×Dic₃	—	2	1	no	—
9	C₃×D₁₂	—	2	1	no	—
10	C₃×D₁₂	C₄	1	2	yes	complemented
11	C₃⋊(C₃⋊C₈)	—	2	1	no	—
12	C₂×C₃×D₄	—	3	1	no	—
13	C₂×D₁₂	D₃	1	1	yes	characteristic
14	C₃×C₄⋊C₄	—	3	1	no	—
15	C₄⋊Dic₃	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃⋊C₈	—	3	1	no	—
17	C₂×C₃⋊C₈	—	3	1	no	—
18	C₂×C₃⋊C₈	—	6	1	no	—
19	C₂²×C₃²	D₄	1	1	yes	characteristic
20	C₂×C₃×D₃	—	2	2	no	—
21	C₂×C₃×D₃	—	4	1	no	—
22	C₂×C₃×D₃	—	4	1	no	—
23	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₃²×C₄	D₄	1	1	yes	characteristic
25	C₃×Dic₃	—	4	1	no	—
26	#102	—	9	1	no	Sylow
27	C₂³×C₃	—	6	1	no	—
28	C₂²×D₃	—	2	1	no	—
29	C₃×D₄	—	3	2	no	—
30	C₃×D₄	—	6	1	no	—
31	C₂×C₃×C₄	—	2	1	no	—
32	C₂×C₃×C₄	—	6	1	no	—
33	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
34	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
35	C₂×Dic₃	—	2	1	no	—
36	D₁₂	—	2	1	no	—
37	D₁₂	Dic₃	1	2	yes	complemented
38	C₃⋊C₈	—	6	2	no	—
39	C₃⋊C₈	—	6	1	no	—
40	C₃⋊C₈	—	6	1	no	—
41	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₃²	D₈	1	1	yes	characteristic
43	C₂×C₃²	SD₁₆	1	1	yes	characteristic
44	C₃×D₃	—	4	2	no	—
45	C₂×D₄	—	3	1	no	—
46	C₂×C₈	—	9	1	no	—
47	C₄⋊C₄	—	3	1	no	—
48	C₂²×C₃	—	2	1	no	—
49	C₂²×C₃	—	6	2	no	—
50	C₂²×C₃	—	12	1	no	—
51	C₂²×C₃	—	12	1	no	—
52	C₂²×C₃	#67	1	1	yes	characteristic
53	C₂²×C₃	#67	1	1	yes	characteristic
54	C₂×D₃	—	2	2	no	—
55	C₂×D₃	—	4	1	no	—
56	C₂×D₃	—	4	1	no	—
57	C₃×C₄	—	2	1	no	—
58	C₃×C₄	—	2	1	no	—
59	C₃×C₄	—	12	1	no	—
60	C₃×C₄	C₄×D₃	1	1	yes	characteristic
61	C₃×C₄	D₁₂	1	1	yes	characteristic
62	C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
63	C₃×C₄	#67	1	1	yes	characteristic
64	Dic₃	—	4	1	no	—
65	C₃²	#102	1	1	yes	characteristic, complemented, Sylow
66	C₂³	—	6	1	no	—
67	D₄	—	3	2	no	—
68	D₄	—	6	1	no	—
69	C₂×C₄	—	6	1	no	—
70	C₂×C₄	D₃²	1	1	yes	characteristic
71	C₈	—	18	1	no	—
72	C₂×C₃	—	2	1	no	—
73	C₂×C₃	—	2	1	no	—
74	C₂×C₃	—	2	1	no	—
75	C₂×C₃	—	12	2	no	—
76	C₂×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
77	C₂×C₃	#213	1	1	yes	characteristic
78	C₂×C₃	#213	1	1	yes	characteristic
79	C₂×C₃	#214	1	1	yes	characteristic
80	C₂×C₃	#215	1	1	yes	characteristic
81	C₂×C₃	#217	1	1	yes	characteristic
82	D₃	—	4	2	no	—
83	C₂²	—	6	2	no	—
84	C₂²	—	12	1	no	—
85	C₂²	—	12	1	no	—
86	C₂²	#625	1	1	yes	characteristic
87	C₄	—	12	1	no	—
88	C₄	D₃×Dic₃	1	1	yes	characteristic
89	C₄	#626	1	1	yes	characteristic
90	C₃	—	2	1	no	—
91	C₃	#809	1	1	yes	characteristic, complemented
92	C₃	#832	1	1	yes	characteristic, complemented
93	C₂	—	12	2	no	—
94	C₂	#3713	1	1	yes	characteristic
95	C₂	#3715	1	1	yes	characteristic
96	C₂	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#625
commutator	C₃²×C₄	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃²
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle	C₂²×C₃²	D₄
prime radical	C₂×C₃×D₁₂	C₂
hypocenter	C₃²	#102
LCS	G ▷ C₃²×C₄ ▷ 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. C₂×C₃⋊(C₃⋊C₈)	C₂	semidirect
3. C₃×C₄⋊Dic₃	C₂	semidirect
10. C₃×D₁₂	C₄	semidirect
15. C₄⋊Dic₃	D₃	semidirect
37. D₁₂	Dic₃	semidirect
65. C₃²	#102	semidirect
91. C₃	#809	semidirect
92. C₃	#832	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₁₂	2
C₂×C₃⋊(C₃⋊C₈)	2
C₃×C₄⋊Dic₃	2
#832	3
#809	3