Subgroups — #67570

Order 384.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4693	C₂	1	1	yes	maximal, characteristic
2	C₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	#4463	C₂	1	1	yes	maximal, characteristic, complemented
4	#1359	—	3	1	no	maximal, Sylow
5	C₂×#204	—	2	1	no	—
6	C₂×#204	—	2	1	no	—
7	#874	C₄	1	1	yes	characteristic
8	#869	C₄	1	1	yes	characteristic
9	C₃×M₃₂	—	2	1	no	—
10	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	#812	—	2	1	no	—
12	#492	—	3	1	no	—
13	#364	—	3	1	no	—
14	#364	D₃	1	1	yes	characteristic, complemented
15	C₂×D₁₂	—	2	1	no	—
16	C₂×D₁₂	—	4	1	no	—
17	C₂×Dic₆	—	2	1	no	—
18	C₂×C₃×C₈	D₄	1	1	yes	characteristic
19	C₂×C₃×C₈	D₄	1	1	yes	characteristic
20	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₄⋊Dic₃	—	4	1	no	—
22	#204	—	4	1	no	—
23	#204	—	4	1	no	—
24	#204	—	4	1	no	—
25	#204	—	4	1	no	—
26	C₃×C₁₆	—	4	1	no	—
27	C₃⋊C₁₆	—	4	1	no	—
28	C₂×SD₁₆	—	6	1	no	—
29	C₂×SD₁₆	—	6	1	no	—
30	C₄⋊Q₈	—	3	1	no	—
31	#127	—	3	1	no	—
32	M₃₂	—	2	1	no	—
33	M₃₂	—	6	1	no	—
34	C₄×C₈	C₂×D₃	1	1	yes	characteristic
35	C₂²×D₃	—	4	1	no	—
36	C₂×C₃×C₄	—	2	1	no	—
37	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×Dic₃	—	4	1	no	—
39	D₁₂	—	4	1	no	—
40	D₁₂	—	4	1	no	—
41	D₁₂	—	8	2	no	—
42	Dic₆	—	4	1	no	—
43	Dic₆	—	4	1	no	—
44	C₃×C₈	D₈	1	1	yes	characteristic
45	C₃×C₈	D₈	1	1	yes	characteristic
46	C₃×C₈	SD₁₆	1	1	yes	characteristic
47	C₃×C₈	SD₁₆	1	1	yes	characteristic
48	C₂×Q₈	—	6	1	no	—
49	C₂×D₄	—	6	1	no	—
50	C₂×D₄	—	12	1	no	—
51	SD₁₆	—	12	1	no	—
52	SD₁₆	—	12	1	no	—
53	SD₁₆	—	12	1	no	—
54	SD₁₆	—	12	1	no	—
55	C₂×C₈	D₁₂	1	1	yes	characteristic
56	C₂×C₈	#67	1	1	yes	characteristic
57	C₄⋊C₄	—	12	1	no	—
58	C₄²	C₄×D₃	1	1	yes	characteristic
59	C₁₆	—	4	1	no	—
60	C₁₆	—	12	1	no	—
61	C₂²×C₃	#100	1	1	yes	characteristic
62	C₂×D₃	—	8	2	no	—
63	C₂×D₃	—	8	1	no	—
64	C₃×C₄	—	2	2	no	—
65	C₃×C₄	#102	1	1	yes	characteristic
66	C₃×C₄	#102	1	1	yes	characteristic
67	Dic₃	—	8	1	no	—
68	C₂³	—	12	1	no	—
69	Q₈	—	12	1	no	—
70	Q₈	—	12	1	no	—
71	D₄	—	12	1	no	—
72	D₄	—	12	1	no	—
73	D₄	—	24	2	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	12	1	no	—
76	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
77	C₈	#204	1	1	yes	characteristic
78	C₈	D₂₄	1	1	yes	characteristic
79	C₈	#213	1	1	yes	characteristic
80	C₈	#215	1	1	yes	characteristic
81	C₂×C₃	—	2	1	no	—
82	C₂×C₃	#331	1	1	yes	characteristic
83	D₃	—	16	1	no	—
84	C₂²	—	24	2	no	—
85	C₂²	—	24	1	no	—
86	C₂²	#823	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	24	1	no	—
89	C₄	#809	1	1	yes	characteristic
90	C₄	#821	1	1	yes	characteristic
91	C₃	#1359	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	2	1	no	—
93	C₂	—	48	1	no	—
94	C₂	#4485	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4485
commutator	C₃×C₄²	C₂×C₄
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#364	C₂
socle	C₂×C₃	#331
prime radical	#874	C₄
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#364	D₃
hypocenter	C₃	#1359
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. C₃×#364	C₂	semidirect
3. #4463	C₂	semidirect
14. #364	D₃	semidirect
91. C₃	#1359	semidirect

Maximal subgroups

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

subgroup	index
#4693	2
C₃×#364	2
#4463	2
#1359	3