Subgroups — #67552

Order 384.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5085	C₂	1	1	yes	maximal, characteristic
2	C₃×#356	C₂	1	1	yes	maximal, characteristic, complemented
3	#4548	C₂	1	1	yes	maximal, characteristic, complemented
4	#1410	—	3	1	no	maximal, Sylow
5	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
6	C₂×#215	—	2	1	no	—
7	C₂×#215	—	2	1	no	—
8	#874	C₄	1	1	yes	characteristic
9	C₃×#101	—	2	1	no	—
10	#836	—	2	1	no	—
11	C₄×C₃⋊C₈	C₄	1	1	yes	characteristic
12	#492	—	3	1	no	—
13	#356	—	3	1	no	—
14	#356	D₃	1	1	yes	characteristic, complemented
15	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
16	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
17	C₂×D₁₂	—	2	1	no	—
18	C₂×D₁₂	—	4	1	no	—
19	C₃×M₁₆	—	4	1	no	—
20	C₃×C₄⋊C₄	—	4	1	no	—
21	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
22	#215	—	4	2	no	—
23	#215	—	4	2	no	—
24	#208	—	4	1	no	—
25	C₂×C₃⋊C₈	—	2	1	no	—
26	C₂×SD₁₆	—	6	1	no	—
27	C₂×SD₁₆	—	6	1	no	—
28	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
29	#127	—	3	1	no	—
30	#101	—	2	1	no	—
31	#101	—	6	1	no	—
32	C₄×C₈	—	3	1	no	—
33	C₂²×D₃	—	4	1	no	—
34	C₃×Q₈	—	4	1	no	—
35	C₃×Q₈	—	4	1	no	—
36	C₂×C₃×C₄	—	2	1	no	—
37	C₂×C₃×C₄	—	2	1	no	—
38	C₂×C₃×C₄	—	2	1	no	—
39	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
40	D₁₂	—	4	1	no	—
41	D₁₂	—	4	1	no	—
42	D₁₂	—	8	2	no	—
43	C₃×C₈	—	8	1	no	—
44	C₃⋊C₈	—	2	2	no	—
45	C₃⋊C₈	—	8	1	no	—
46	C₂×Q₈	D₁₂	1	1	yes	characteristic
47	C₂×Q₈	#67	1	1	yes	characteristic
48	C₂×D₄	—	6	1	no	—
49	C₂×D₄	—	12	1	no	—
50	SD₁₆	—	12	2	no	—
51	SD₁₆	—	12	2	no	—
52	M₁₆	—	4	1	no	—
53	M₁₆	—	12	1	no	—
54	C₂×C₈	—	6	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄²	C₄×D₃	1	1	yes	characteristic
57	C₂²×C₃	#99	1	1	yes	characteristic
58	C₂×D₃	—	8	2	no	—
59	C₂×D₃	—	8	1	no	—
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	—	4	1	no	—
63	C₃×C₄	—	4	1	no	—
64	C₂³	—	12	1	no	—
65	Q₈	—	4	1	no	—
66	Q₈	—	4	1	no	—
67	D₄	—	12	1	no	—
68	D₄	—	12	1	no	—
69	D₄	—	24	2	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
74	C₈	—	6	2	no	—
75	C₈	—	8	1	no	—
76	C₈	—	24	1	no	—
77	C₂×C₃	—	2	1	no	—
78	C₂×C₃	#353	1	1	yes	characteristic
79	D₃	—	16	1	no	—
80	C₂²	—	24	2	no	—
81	C₂²	—	24	1	no	—
82	C₂²	#806	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₃	#1410	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	1	no	—
89	C₂	—	48	1	no	—
90	C₂	#4474	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4474
commutator	C₃×C₄²	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₃
Fitting	C₃×#356	C₂
socle	C₂×C₃	#353
prime radical	#874	C₄
cosocle	C₃×C₄⋊Q₈	C₂²
hypercenter	#356	D₃
hypocenter	C₃	#1410
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
2. C₃×#356	C₂	semidirect
3. #4548	C₂	semidirect
14. #356	D₃	semidirect
87. C₃	#1410	semidirect

Maximal subgroups

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

subgroup	index
#5085	2
C₃×#356	2
#4548	2
#1410	3