Subgroups — C₂²×#869 #85670

Order 384.

1624 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₆	C₂	1	2	yes	maximal
2	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊Dic₃	C₂	1	4	yes	maximal
4	C₂×#869	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄⋊Q₈	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂³×C₃×C₄	C₂²	1	2	yes	—
8	C₂³×Dic₃	C₂²	1	4	yes	—
9	C₂²×Dic₆	—	2	16	no	—
10	C₂²×Dic₆	C₂²	1	24	yes	—
11	C₂×C₃×C₄²	C₂²	1	12	yes	—
12	C₂×C₄⋊Dic₃	C₂²	1	48	yes	—
13	#869	C₂²	1	64	yes	direct product factor
14	C₂³×Q₈	—	3	2	no	—
15	C₂×C₄⋊Q₈	—	3	24	no	—
16	C₂²×C₄⋊C₄	—	3	4	no	—
17	C₂²×C₄²	D₃	1	1	yes	characteristic
18	C₂⁴×C₃	C₂³	1	1	yes	characteristic
19	C₂²×C₃×C₄	D₄	1	8	yes	—
20	C₂²×C₃×C₄	Q₈	1	16	yes	complemented
21	C₂²×C₃×C₄	C₂³	1	6	yes	—
22	C₂²×C₃×C₄	C₂³	1	12	yes	—
23	C₂²×Dic₃	—	2	16	no	—
24	C₂²×Dic₃	C₂³	1	24	yes	—
25	C₂×Dic₆	—	2	96	no	—
26	C₂×Dic₆	C₂³	1	32	yes	—
27	C₃×C₄²	C₂³	1	16	yes	—
28	C₄⋊Dic₃	C₂³	1	64	yes	—
29	C₂²×Q₈	—	3	24	no	—
30	C₂²×Q₈	—	6	16	no	—
31	C₂³×C₄	—	3	4	no	—
32	C₂³×C₄	C₂×D₃	1	2	yes	—
33	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
34	C₄⋊Q₈	—	3	64	no	—
35	C₂×C₄⋊C₄	—	3	48	no	—
36	C₂×C₄²	C₂×D₃	1	12	yes	—
37	C₂³×C₃	C₂×D₄	1	4	yes	—
38	C₂³×C₃	C₂×Q₈	1	8	yes	—
39	C₂³×C₃	C₂⁴	1	3	yes	—
40	C₂×C₃×C₄	C₂×D₄	1	24	yes	—
41	C₂×C₃×C₄	C₂×Q₈	1	48	yes	complemented
42	C₂×C₃×C₄	C₂⁴	1	4	yes	—
43	C₂×C₃×C₄	C₂⁴	1	8	yes	—
44	C₂×Dic₃	—	2	48	no	—
45	C₂×Dic₃	C₂⁴	1	16	yes	—
46	Dic₆	—	2	64	no	—
47	C₂⁴	C₂²×D₃	1	1	yes	characteristic
48	C₂×Q₈	—	3	32	no	—
49	C₂×Q₈	—	6	96	no	—
50	C₂²×C₄	—	3	24	no	—
51	C₂²×C₄	—	6	16	no	—
52	C₂²×C₄	Dic₆	1	16	yes	complemented
53	C₂²×C₄	D₁₂	1	8	yes	—
54	C₂²×C₄	C₂²×D₃	1	12	yes	—
55	C₂²×C₄	C₂²×D₃	1	6	yes	—
56	C₄⋊C₄	—	3	64	no	—
57	C₄²	C₂²×D₃	1	16	yes	—
58	C₂²×C₃	C₄⋊Q₈	1	16	yes	complemented
59	C₂²×C₃	C₂²×D₄	1	6	yes	—
60	C₂²×C₃	C₂²×Q₈	1	12	yes	—
61	C₂²×C₃	C₂⁵	1	1	yes	characteristic
62	C₃×C₄	C₂²×D₄	1	8	yes	—
63	C₃×C₄	C₂²×Q₈	1	16	yes	complemented
64	Dic₃	—	2	16	no	—
65	C₂³	C₂×Dic₆	1	8	yes	—
66	C₂³	C₂×D₁₂	1	4	yes	—
67	C₂³	C₂³×D₃	1	3	yes	—
68	Q₈	—	6	64	no	—
69	C₂×C₄	—	3	16	no	—
70	C₂×C₄	—	6	48	no	—
71	C₂×C₄	C₂×Dic₆	1	48	yes	complemented
72	C₂×C₄	C₂×D₁₂	1	24	yes	—
73	C₂×C₄	C₂³×D₃	1	8	yes	—
74	C₂×C₄	C₂³×D₃	1	4	yes	—
75	C₂×C₃	C₂×C₄⋊Q₈	1	12	yes	complemented
76	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
77	C₂×C₃	C₂³×Q₈	1	2	yes	—
78	C₂²	#869	1	16	yes	direct product factor
79	C₂²	C₂²×Dic₆	1	12	yes	—
80	C₂²	C₂²×D₁₂	1	6	yes	—
81	C₂²	C₂⁴×D₃	1	1	yes	characteristic
82	C₄	—	6	16	no	—
83	C₄	C₂²×Dic₆	1	16	yes	complemented
84	C₄	C₂²×D₁₂	1	8	yes	—
85	C₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
86	C₂	C₂×#869	1	12	yes	direct product factor
87	C₂	C₂³×Dic₆	1	2	yes	—
88	C₂	C₂³×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle, prime radical	C₂⁴×C₃	C₂³
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
4. C₂×#869	C₂	direct (N × H)
13. #869	C₂²	direct (N × H)
20. C₂²×C₃×C₄	Q₈	semidirect
41. C₂×C₃×C₄	C₂×Q₈	semidirect
52. C₂²×C₄	Dic₆	semidirect
58. C₂²×C₃	C₄⋊Q₈	semidirect
63. C₃×C₄	C₂²×Q₈	semidirect
71. C₂×C₄	C₂×Dic₆	semidirect
75. C₂×C₃	C₂×C₄⋊Q₈	semidirect
78. C₂²	#869	direct (N × H)
83. C₄	C₂²×Dic₆	semidirect
85. C₃	C₂²×C₄⋊Q₈	semidirect
86. C₂	C₂×#869	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₆	2
C₂²×C₃×C₄²	2
C₂²×C₄⋊Dic₃	2
C₂×#869	2
C₂²×C₄⋊Q₈	3

Subgroups — C22×#869 #85670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#869 #85670