Subgroups — C₂×#3797 #64299

Order 288.

339 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×C₃×Dic₆	C₂	1	2	yes	maximal, complemented
3	C₂×#627	C₂	1	2	yes	maximal, complemented
4	C₂×#624	C₂	1	2	yes	maximal
5	#3797	C₂	1	8	yes	maximal, direct product factor
6	C₂×D₃×Q₈	—	3	2	no	maximal
7	C₂²×C₃⋊D₃	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
10	C₄×C₃⋊D₃	C₂²	1	4	yes	—
11	C₂×C₃×Dic₃	C₂²	1	4	yes	—
12	C₃×Dic₆	C₂²	1	8	yes	complemented
13	#627	C₂²	1	8	yes	complemented
14	#624	C₂²	1	8	yes	—
15	C₂×C₃×Q₈	—	3	2	no	—
16	D₃×Q₈	—	3	16	no	—
17	C₂×C₄×D₃	—	3	2	no	—
18	C₂×C₄×D₃	—	3	4	no	—
19	C₂×C₄×D₃	—	6	1	no	—
20	C₂×Dic₆	—	3	4	no	—
21	C₂×Dic₆	D₃	1	2	yes	complemented
22	C₂²×C₃²	C₂³	1	1	yes	characteristic
23	C₂×C₃⋊D₃	Q₈	1	4	yes	complemented
24	C₂×C₃⋊D₃	C₂³	1	2	yes	—
25	C₃²×C₄	C₂³	1	2	yes	—
26	C₃⋊Dic₃	C₂³	1	2	yes	—
27	C₃×Dic₃	C₂³	1	8	yes	—
28	C₂²×Q₈	—	9	1	no	Sylow
29	C₂²×D₃	—	3	2	no	—
30	C₂²×D₃	—	6	1	no	—
31	C₃×Q₈	—	3	8	no	—
32	C₂×C₃×C₄	—	2	1	no	—
33	C₂×C₃×C₄	—	3	4	no	—
34	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
35	C₂×Dic₃	—	3	2	no	—
36	C₂×Dic₃	—	6	1	no	—
37	C₂×Dic₃	C₂×D₃	1	4	yes	—
38	C₄×D₃	—	3	8	no	—
39	C₄×D₃	—	3	16	no	—
40	C₄×D₃	—	6	4	no	—
41	Dic₆	—	3	16	no	—
42	Dic₆	C₂×D₃	1	8	yes	complemented
43	C₂×C₃²	C₂×Q₈	1	2	yes	complemented
44	C₂×C₃²	C₂⁴	1	1	yes	characteristic
45	C₃⋊D₃	C₂×Q₈	1	4	yes	complemented
46	C₂×Q₈	—	3	2	no	—
47	C₂×Q₈	—	9	2	no	—
48	C₂×Q₈	—	9	8	no	—
49	C₂²×C₄	—	9	2	no	—
50	C₂²×C₄	—	9	1	no	—
51	C₂²×C₃	—	2	1	no	—
52	C₂²×C₃	C₂²×D₃	1	2	yes	—
53	C₂×D₃	—	3	8	no	—
54	C₂×D₃	—	3	4	no	—
55	C₂×D₃	—	6	2	no	—
56	C₂×D₃	—	6	4	no	—
57	C₃×C₄	—	2	2	no	—
58	C₃×C₄	—	3	8	no	—
59	C₃×C₄	C₂²×D₃	1	4	yes	—
60	Dic₃	—	3	4	no	—
61	Dic₃	—	6	2	no	—
62	Dic₃	C₂²×D₃	1	8	yes	—
63	C₃²	C₂²×Q₈	1	1	yes	characteristic, complemented, Sylow
64	C₂³	—	9	1	no	—
65	Q₈	—	3	8	no	—
66	Q₈	—	9	8	no	—
67	C₂×C₄	—	3	4	no	—
68	C₂×C₄	—	9	8	no	—
69	C₂×C₄	—	9	1	no	—
70	C₂×C₄	—	9	4	no	—
71	C₂×C₄	D₃²	1	1	yes	characteristic
72	C₂×C₃	—	2	1	no	—
73	C₂×C₃	—	2	2	no	—
74	C₂×C₃	D₃×Q₈	1	4	yes	complemented
75	C₂×C₃	C₂³×D₃	1	2	yes	—
76	D₃	—	3	8	no	—
77	D₃	—	6	4	no	—
78	C₂²	—	9	2	no	—
79	C₂²	—	9	4	no	—
80	C₂²	C₂×D₃²	1	1	yes	characteristic
81	C₄	—	3	8	no	—
82	C₄	—	9	2	no	—
83	C₄	C₂×D₃²	1	2	yes	—
84	C₃	—	2	1	no	—
85	C₃	C₂×D₃×Q₈	1	2	yes	complemented
86	C₂	—	9	4	no	—
87	C₂	#3797	1	2	yes	direct product factor
88	C₂	C₂²×D₃²	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₃²
commutator, cosocle	C₂×C₃²	C₂⁴
Frattini	C₂	C₂²×D₃²
Fitting	C₂×C₃²×C₄	C₂²
socle	C₂²×C₃²	C₂³
prime radical	C₂²×C₃⋊D₃	C₂²
hypercenter	C₂×C₄	D₃²
hypocenter	C₃²	C₂²×Q₈
LCS	G ▷ 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×Dic₆	C₂	semidirect
3. C₂×#627	C₂	semidirect
5. #3797	C₂	direct (N × H)
12. C₃×Dic₆	C₂²	semidirect
13. #627	C₂²	semidirect
21. C₂×Dic₆	D₃	semidirect
23. C₂×C₃⋊D₃	Q₈	semidirect
42. Dic₆	C₂×D₃	semidirect
43. C₂×C₃²	C₂×Q₈	semidirect
45. C₃⋊D₃	C₂×Q₈	semidirect
63. C₃²	C₂²×Q₈	semidirect
74. C₂×C₃	D₃×Q₈	semidirect
85. C₃	C₂×D₃×Q₈	semidirect
87. C₂	#3797	direct (N × H)

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×C₃×Dic₆	2
C₂×#627	2
C₂×#624	2
#3797	2
C₂×D₃×Q₈	3

Subgroups — C2×#3797 #64299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3797 #64299