Subgroups — #69763

Order 384.

216 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	#4936	C₂	1	2	yes	maximal
3	#4551	C₂	1	4	yes	maximal
4	#1835	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	C₂²	1	2	yes	—
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
8	C₃×C₄×C₈	C₄	1	4	yes	complemented
9	C₃⋊#95	—	2	4	no	—
10	C₂×C₄×C₈	D₃	1	1	yes	characteristic
11	#382	—	3	2	no	—
12	#340	—	3	4	no	—
13	C₂²×C₃×C₄	D₄	1	2	yes	—
14	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
15	C₂²×Dic₃	—	2	4	no	—
16	C₂×C₃×C₈	—	2	4	no	—
17	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
18	C₃×C₄²	C₂×C₄	1	2	yes	—
19	C₃×C₄²	Q₈	1	2	yes	—
20	C₄⋊Dic₃	—	2	8	no	—
21	C₂²×C₈	C₂×D₃	1	2	yes	—
22	C₂×C₄⋊C₄	—	3	4	no	—
23	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
24	C₄×C₈	Dic₃	1	4	yes	complemented
25	#95	—	6	4	no	—
26	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₃×C₄	—	2	4	no	—
28	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
29	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₃×C₄	#41	1	4	yes	—
32	C₂×Dic₃	—	4	4	no	—
33	C₂×Dic₃	—	4	8	no	—
34	C₃×C₈	—	2	4	no	—
35	C₂²×C₄	—	6	4	no	—
36	C₂²×C₄	D₁₂	1	2	yes	—
37	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
38	C₂×C₈	—	2	4	no	—
39	C₂×C₈	C₂×Dic₃	1	4	yes	—
40	C₄⋊C₄	—	6	8	no	—
41	C₄²	Dic₆	1	2	yes	—
42	C₄²	C₂×Dic₃	1	2	yes	—
43	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₃	#124	1	2	yes	—
45	C₂²×C₃	#135	1	4	yes	—
46	C₃×C₄	—	2	4	no	—
47	C₃×C₄	#117	1	2	yes	—
48	C₃×C₄	#125	1	2	yes	—
49	Dic₃	—	8	4	no	—
50	C₂³	C₂×D₁₂	1	1	yes	characteristic
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	—	12	4	no	—
53	C₂×C₄	—	12	8	no	—
54	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
55	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
56	C₂×C₄	#235	1	4	yes	—
57	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
58	C₈	—	2	4	no	—
59	C₂×C₃	#382	1	1	yes	characteristic
60	C₂×C₃	#427	1	2	yes	—
61	C₂×C₃	#488	1	4	yes	—
62	C₂²	#875	1	2	yes	—
63	C₂²	#904	1	4	yes	—
64	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	24	4	no	—
67	C₄	#870	1	2	yes	—
68	C₄	#926	1	2	yes	—
69	C₃	#1835	1	1	yes	characteristic, complemented, Sylow
70	C₂	#4697	1	4	yes	—
71	C₂	#4936	1	1	yes	characteristic
72	C₂	#5106	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#1835
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₃×C₄×C₈	C₄	semidirect
24. C₄×C₈	Dic₃	semidirect
69. C₃	#1835	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
#4936	2
#4551	2
#1835	3