Subgroups — C₂×#3828 #64358

Order 288.

492 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	#3828	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
6	C₂×#634	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#235	—	3	4	no	maximal
8	C₂³×C₃²	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊D₃	C₂²	1	2	yes	—
10	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₄	C₂²	1	4	yes	complemented
13	#638	C₂²	1	8	yes	complemented
14	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
15	C₄⋊(C₃⋊D₃)	C₂²	1	4	yes	complemented
16	C₄×C₃⋊D₃	C₂²	1	8	yes	complemented
17	#634	C₂²	1	4	yes	complemented
18	C₂²×C₃×C₄	D₃	1	4	yes	complemented
19	C₂×#67	—	3	8	no	—
20	#235	—	3	32	no	—
21	C₂×D₁₂	—	3	4	no	—
22	C₂×C₄×D₃	—	3	8	no	—
23	C₂×Dic₆	—	3	4	no	—
24	C₂²×C₃²	—	2	2	no	—
25	C₂²×C₃²	C₂³	1	2	yes	—
26	C₂²×C₃²	C₂³	1	1	yes	characteristic
27	C₂×C₃⋊D₃	—	2	4	no	—
28	C₂×C₃⋊D₃	C₂³	1	4	yes	—
29	C₃²×C₄	C₂³	1	2	yes	—
30	C₃²×C₄	C₂³	1	2	yes	—
31	C₃⋊Dic₃	C₂³	1	4	yes	—
32	C₂×#41	—	9	1	no	Sylow
33	C₂³×C₃	C₂×D₃	1	4	yes	—
34	C₂²×D₃	—	3	8	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	16	yes	complemented
38	#67	—	3	32	no	—
39	C₂×Dic₃	—	3	8	no	—
40	D₁₂	—	3	16	no	—
41	C₄×D₃	—	3	32	no	—
42	Dic₆	—	3	16	no	—
43	C₂×C₃²	—	2	2	no	—
44	C₂×C₃²	#41	1	2	yes	complemented
45	C₂×C₃²	C₂⁴	1	1	yes	characteristic
46	C₃⋊D₃	—	2	4	no	—
47	#41	—	9	8	no	—
48	C₂×Q₈	—	9	1	no	—
49	C₂×D₄	—	9	2	no	—
50	C₂×D₄	—	9	1	no	—
51	C₂²×C₄	—	9	2	no	—
52	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic, complemented
53	C₂²×C₃	—	2	8	no	—
54	C₂²×C₃	C₂²×D₃	1	8	yes	—
55	C₂²×C₃	C₂²×D₃	1	4	yes	—
56	C₂×D₃	—	3	16	no	—
57	C₂×D₃	—	6	16	no	—
58	C₃×C₄	C₂²×D₃	1	8	yes	—
59	C₃×C₄	C₂²×D₃	1	8	yes	—
60	Dic₃	—	3	16	no	—
61	C₃²	C₂×#41	1	1	yes	characteristic, complemented, Sylow
62	C₂³	—	9	2	no	—
63	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
64	Q₈	—	9	4	no	—
65	D₄	—	9	8	no	—
66	D₄	—	9	4	no	—
67	C₂×C₄	—	9	2	no	—
68	C₂×C₄	—	9	8	no	—
69	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
70	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
71	C₂×C₄	C₂×C₃⋊D₃	1	4	yes	complemented
72	C₂×C₃	—	2	8	no	—
73	C₂×C₃	#235	1	8	yes	complemented
74	C₂×C₃	C₂³×D₃	1	4	yes	—
75	D₃	—	6	16	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	9	4	no	—
78	C₂²	—	18	4	no	—
79	C₂²	C₂²×C₃⋊D₃	1	2	yes	—
80	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
81	C₄	—	9	4	no	—
82	C₄	C₂²×C₃⋊D₃	1	2	yes	—
83	C₄	C₂²×C₃⋊D₃	1	2	yes	—
84	C₃	C₂×#235	1	4	yes	complemented
85	C₂	—	2	2	no	—
86	C₂	—	18	4	no	—
87	C₂	#3828	1	2	yes	direct product factor
88	C₂	C₂³×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂⁴
Frattini	C₂	C₂³×C₃⋊D₃
Fitting	C₂²×C₃²×C₄	C₂
socle	C₂²×C₃²	C₂³
hypercenter	C₂²×C₄	C₃⋊D₃
hypocenter	C₃²	C₂×#41
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ 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. #3828	C₂	direct (N × H)
4. C₂×C₄⋊(C₃⋊D₃)	C₂	semidirect
5. C₂×C₄×C₃⋊D₃	C₂	semidirect
6. C₂×#634	C₂	semidirect
12. C₂×C₃²×C₄	C₂²	semidirect
13. #638	C₂²	semidirect
15. C₄⋊(C₃⋊D₃)	C₂²	semidirect
16. C₄×C₃⋊D₃	C₂²	semidirect
17. #634	C₂²	semidirect
18. C₂²×C₃×C₄	D₃	semidirect
37. C₂×C₃×C₄	C₂×D₃	semidirect
44. C₂×C₃²	#41	semidirect
52. C₂²×C₄	C₃⋊D₃	semidirect
61. C₃²	C₂×#41	semidirect
71. C₂×C₄	C₂×C₃⋊D₃	semidirect
73. C₂×C₃	#235	semidirect
84. C₃	C₂×#235	semidirect
87. C₂	#3828	direct (N × H)

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
#3828	2
C₂×C₄⋊(C₃⋊D₃)	2
C₂×C₄×C₃⋊D₃	2
C₂×#634	2
C₂×#235	3

Subgroups — C2×#3828 #64358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3828 #64358