Subgroups — #67767

Order 384.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#443	C₂	1	1	yes	maximal, characteristic
2	#4463	C₂	1	1	yes	maximal, characteristic, complemented
3	#4455	C₂	1	1	yes	maximal, characteristic, complemented
4	#1336	—	3	1	no	maximal, Sylow
5	C₃×#135	C₄	1	1	yes	characteristic
6	C₃×#131	—	2	1	no	—
7	C₃×#108	C₄	1	1	yes	characteristic
8	C₃×#104	—	2	1	no	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	#812	—	2	1	no	—
11	C₄×C₃⋊C₈	—	2	1	no	—
12	#443	D₃	1	1	yes	characteristic
13	#364	—	3	1	no	—
14	#322	—	3	1	no	—
15	C₃×#41	—	2	1	no	—
16	C₃×Q₁₆	C₈	1	1	yes	characteristic, complemented
17	C₃×SD₁₆	—	2	1	no	—
18	C₃×D₈	C₈	1	1	yes	characteristic, complemented
19	C₃×M₁₆	—	2	1	no	—
20	C₃×M₁₆	—	4	1	no	—
21	C₂×C₃×C₈	—	4	1	no	—
22	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₈	D₄	1	1	yes	characteristic
24	C₃×C₄²	D₄	1	1	yes	characteristic
25	C₂×C₃⋊C₈	—	2	2	no	—
26	C₃⋊C₁₆	—	4	1	no	—
27	#135	Dic₃	1	1	yes	characteristic
28	#131	—	2	1	no	—
29	M₃₂	—	6	1	no	—
30	#108	Dic₃	1	1	yes	characteristic
31	#104	—	2	1	no	—
32	C₄×C₈	—	6	1	no	—
33	C₄×C₈	C₂×D₃	1	1	yes	characteristic
34	C₃×Q₈	—	2	1	no	—
35	C₃×D₄	—	2	1	no	—
36	C₃×D₄	—	4	1	no	—
37	C₂×C₃×C₄	—	2	1	no	—
38	C₂×C₃×C₄	—	4	1	no	—
39	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
40	C₃×C₈	—	4	1	no	—
41	C₃×C₈	C₂×C₈	1	1	yes	characteristic
42	C₃×C₈	M₁₆	1	1	yes	characteristic
43	C₃×C₈	D₈	1	1	yes	characteristic
44	C₃×C₈	SD₁₆	1	1	yes	characteristic
45	C₃⋊C₈	—	4	2	no	—
46	#41	—	2	1	no	—
47	Q₁₆	C₃⋊C₈	1	1	yes	characteristic, complemented
48	SD₁₆	—	2	1	no	—
49	D₈	C₃⋊C₈	1	1	yes	characteristic, complemented
50	M₁₆	—	2	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	—	4	1	no	—
53	C₂×C₈	—	6	2	no	—
54	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
55	C₂×C₈	#67	1	1	yes	characteristic
56	C₄²	#67	1	1	yes	characteristic
57	C₁₆	—	12	1	no	—
58	C₂²×C₃	—	4	1	no	—
59	C₂²×C₃	#104	1	1	yes	characteristic
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	—	4	1	no	—
62	C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
63	C₃×C₄	#102	1	1	yes	characteristic
64	Q₈	—	2	1	no	—
65	D₄	—	2	1	no	—
66	D₄	—	4	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	#217	1	1	yes	characteristic
70	C₈	—	4	1	no	—
71	C₈	—	12	2	no	—
72	C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
73	C₈	#208	1	1	yes	characteristic
74	C₈	#213	1	1	yes	characteristic
75	C₈	#214	1	1	yes	characteristic
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	—	8	1	no	—
78	C₂×C₃	D₄⋊C₈	1	1	yes	characteristic
79	C₂²	—	4	1	no	—
80	C₂²	#837	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	4	1	no	—
83	C₄	#830	1	1	yes	characteristic
84	C₄	#832	1	1	yes	characteristic
85	C₃	#1336	1	1	yes	characteristic, complemented, Sylow
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊(C₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#832
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#443	C₂
socle	C₂×C₃	D₄⋊C₈
prime radical	C₃×#135	C₄
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#443	D₃
hypocenter	C₃	#1336
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #4463	C₂	semidirect
3. #4455	C₂	semidirect
16. C₃×Q₁₆	C₈	semidirect
18. C₃×D₈	C₈	semidirect
47. Q₁₆	C₃⋊C₈	semidirect
49. D₈	C₃⋊C₈	semidirect
85. C₃	#1336	semidirect

Maximal subgroups

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

subgroup	index
C₃×#443	2
#4463	2
#4455	2
#1336	3