Subgroups — C₄×#815 #69748

Order 384.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#815	C₂	1	4	yes	maximal
3	C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄×#97	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₃	C₄	1	4	yes	—
9	C₂×C₄×Dic₃	C₂²	1	2	yes	—
10	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
11	C₃×C₄×C₈	C₄	1	4	yes	complemented
12	#815	C₄	1	16	yes	direct product factor
13	C₄×C₃⋊C₈	C₄	1	4	yes	complemented
14	C₂×#97	—	3	4	no	—
15	C₂×C₄×C₈	—	3	1	no	—
16	C₂×C₄×C₈	D₃	1	1	yes	characteristic
17	C₄³	—	3	1	no	—
18	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
19	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₈	C₂×C₄	1	8	yes	—
22	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
23	C₃×C₄²	C₂×C₄	1	2	yes	—
24	C₃×C₄²	C₂×C₄	1	2	yes	—
25	C₄×Dic₃	—	2	8	no	—
26	C₄×Dic₃	C₂×C₄	1	8	yes	—
27	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
28	C₂×C₃⋊C₈	C₂×C₄	1	8	yes	—
29	C₂²×C₈	—	3	2	no	—
30	C₂²×C₈	C₂×D₃	1	2	yes	—
31	C₂×C₄²	—	3	2	no	—
32	C₂×C₄²	—	3	4	no	—
33	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
34	#97	—	3	16	no	—
35	C₄×C₈	—	3	4	no	—
36	C₄×C₈	Dic₃	1	4	yes	complemented
37	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄	C₄²	1	4	yes	—
39	C₂×C₃×C₄	M₁₆	1	8	yes	—
40	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
42	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₃	—	2	8	no	—
44	C₂×Dic₃	C₄²	1	8	yes	—
45	C₃×C₈	C₄²	1	8	yes	complemented
46	C₃⋊C₈	C₄²	1	8	yes	complemented
47	C₂²×C₄	—	3	4	no	—
48	C₂²×C₄	C₄×D₃	1	2	yes	—
49	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
50	C₂×C₈	—	3	4	no	—
51	C₂×C₈	—	3	8	no	—
52	C₂×C₈	C₄×D₃	1	8	yes	—
53	C₂×C₈	C₂×Dic₃	1	4	yes	—
54	C₄²	—	3	8	no	—
55	C₄²	—	6	8	no	—
56	C₄²	C₄×D₃	1	2	yes	—
57	C₄²	C₂×Dic₃	1	2	yes	—
58	C₂²×C₃	C₂×C₄²	1	2	yes	—
59	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
60	C₂²×C₃	C₂×M₁₆	1	4	yes	—
61	C₃×C₄	#97	1	8	yes	complemented
62	C₃×C₄	C₂×C₄²	1	2	yes	—
63	C₃×C₄	C₂×C₄²	1	2	yes	—
64	Dic₃	—	2	8	no	—
65	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
66	C₂×C₄	—	3	8	no	—
67	C₂×C₄	—	6	8	no	—
68	C₂×C₄	ΓL(1,𝔽₂₅)	1	8	yes	—
69	C₂×C₄	C₄×Dic₃	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
71	C₂×C₄	C₂×C₄×D₃	1	4	yes	—
72	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
73	C₈	—	3	8	no	—
74	C₈	C₄×Dic₃	1	8	yes	complemented
75	C₂×C₃	C₄³	1	1	yes	characteristic
76	C₂×C₃	C₂×#97	1	2	yes	—
77	C₂×C₃	C₄×M₁₆	1	4	yes	—
78	C₂²	C₄²×D₃	1	2	yes	—
79	C₂²	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
80	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
81	C₄	—	6	8	no	—
82	C₄	#815	1	8	yes	direct product factor
83	C₄	C₄²×D₃	1	2	yes	—
84	C₄	C₂×C₄×Dic₃	1	2	yes	—
85	C₃	C₄×#97	1	1	yes	characteristic, complemented, Sylow
86	C₂	C₄×ΓL(1,𝔽₂₅)	1	4	yes	—
87	C₂	C₄²×Dic₃	1	1	yes	characteristic
88	C₂	C₂×#815	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₃
commutator	C₂×C₃	C₄³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	C₄×#97
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
11. C₃×C₄×C₈	C₄	semidirect
12. #815	C₄	direct (N × H)
13. C₄×C₃⋊C₈	C₄	semidirect
36. C₄×C₈	Dic₃	semidirect
45. C₃×C₈	C₄²	semidirect
46. C₃⋊C₈	C₄²	semidirect
61. C₃×C₄	#97	semidirect
74. C₈	C₄×Dic₃	semidirect
82. C₄	#815	direct (N × H)
85. C₃	C₄×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₈	2
C₂×#815	2
C₄²×Dic₃	2
C₂×C₄×C₃⋊C₈	2
C₄×#97	3

Subgroups — C4×#815 #69748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#815 #69748