Subgroups — #64105

Order 288.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#634	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#3752	C₂	1	1	yes	maximal, characteristic
5	#3752	C₂	1	1	yes	maximal, characteristic
6	#3750	C₂	1	1	yes	maximal, characteristic, complemented
7	#3750	C₂	1	1	yes	maximal, characteristic, complemented
8	#896	—	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	C₄×C₃⋊D₃	—	2	2	no	—
17	#634	—	2	2	no	—
18	C₂×C₄×D₃	—	3	4	no	—
19	C₂×Dic₆	—	3	4	no	—
20	C₃×C₄⋊C₄	D₃	1	4	yes	complemented
21	(C₂×D₃)⋊C₄	—	3	4	no	—
22	(C₂×D₃)⋊C₄	—	3	4	no	—
23	Dic₃⋊C₄	—	3	4	no	—
24	Dic₃⋊C₄	—	3	4	no	—
25	C₂²×C₃²	C₂³	1	1	yes	characteristic
26	C₂×C₃⋊D₃	—	2	1	no	—
27	C₂×C₃⋊D₃	—	2	1	no	—
28	C₂×C₃⋊D₃	Q₈	1	2	yes	complemented
29	C₃²×C₄	—	2	1	no	—
30	C₃²×C₄	—	2	1	no	—
31	C₃²×C₄	—	2	1	no	—
32	C₃⋊Dic₃	—	2	1	no	—
33	C₃⋊Dic₃	—	2	1	no	—
34	C₃⋊Dic₃	D₄	1	2	yes	—
35	C₂²⋊Q₈	—	9	1	no	Sylow
36	C₂²×D₃	—	3	4	no	—
37	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
38	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
39	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
40	C₂×Dic₃	—	3	4	no	—
41	C₂×Dic₃	—	3	4	no	—
42	C₂×Dic₃	—	3	4	no	—
43	C₄×D₃	—	6	8	no	—
44	Dic₆	—	6	8	no	—
45	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₃²	#41	1	1	yes	characteristic
48	C₃⋊D₃	—	2	2	no	—
49	C₂×Q₈	—	9	1	no	—
50	C₂²×C₄	—	9	1	no	—
51	C₄⋊C₄	—	9	1	no	—
52	C₄⋊C₄	—	9	1	no	—
53	C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic, complemented
54	C₂²⋊C₄	—	9	1	no	—
55	C₂²⋊C₄	—	9	1	no	—
56	C₂²×C₃	C₂²×D₃	1	4	yes	—
57	C₂×D₃	—	3	8	no	—
58	C₂×D₃	—	6	4	no	—
59	C₂×D₃	—	6	4	no	—
60	C₃×C₄	—	2	4	no	—
61	C₃×C₄	—	2	4	no	—
62	C₃×C₄	—	2	4	no	—
63	Dic₃	—	3	8	no	—
64	Dic₃	—	6	4	no	—
65	Dic₃	—	6	4	no	—
66	C₃²	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
67	C₂³	—	9	1	no	—
68	Q₈	—	18	2	no	—
69	C₂×C₄	—	9	1	no	—
70	C₂×C₄	—	9	1	no	—
71	C₂×C₄	—	9	1	no	—
72	C₂×C₄	—	18	2	no	—
73	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
74	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
75	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
76	C₂×C₃	#235	1	4	yes	—
77	C₂×C₃	D₃×D₄	1	4	yes	—
78	C₂×C₃	D₃×Q₈	1	4	yes	—
79	D₃	—	6	8	no	—
80	C₂²	—	9	2	no	—
81	C₂²	—	18	1	no	—
82	C₂²	—	18	1	no	—
83	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	9	2	no	—
88	C₄	—	18	1	no	—
89	C₄	—	18	1	no	—
90	C₃	#896	1	4	yes	complemented
91	C₂	—	18	2	no	—
92	C₂	#3828	1	1	yes	characteristic
93	C₂	D₄×C₃⋊D₃	1	1	yes	characteristic
94	C₂	Q₈×C₃⋊D₃	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	C₂²⋊Q₈
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
2. C₂×#634	C₂	semidirect
3. C₃²×C₄⋊C₄	C₂	semidirect
6. #3750	C₂	semidirect
7. #3750	C₂	semidirect
20. C₃×C₄⋊C₄	D₃	semidirect
28. C₂×C₃⋊D₃	Q₈	semidirect
53. C₄⋊C₄	C₃⋊D₃	semidirect
66. C₃²	C₂²⋊Q₈	semidirect
90. C₃	#896	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊D₃	2
C₂×#634	2
C₃²×C₄⋊C₄	2
#3752	2
#3750	2
#896	3