Subgroups — #68095

Order 384.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4699	C₂	1	1	yes	maximal, characteristic
2	#4681	C₂	1	2	yes	maximal
3	#4680	C₂	1	2	yes	maximal
4	#4680	C₂	1	1	yes	maximal, characteristic
5	C₃×#322	C₂	1	1	yes	maximal, characteristic
6	#1716	—	3	1	no	maximal, Sylow
7	C₂×Dic₁₂	—	2	2	no	—
8	#869	C₂²	1	2	yes	—
9	#869	C₂²	1	1	yes	characteristic
10	#869	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₈	C₂²	1	2	yes	—
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#817	—	2	4	no	—
14	#817	—	2	2	no	—
15	#816	—	2	4	no	—
16	#498	—	3	2	no	—
17	#498	—	3	1	no	—
18	#494	—	3	1	no	—
19	#487	—	3	2	no	—
20	#322	D₃	1	1	yes	characteristic
21	C₂×Dic₆	—	2	2	no	—
22	C₂×Dic₆	—	2	1	no	—
23	C₂×Dic₆	—	2	1	no	—
24	C₂×C₃×C₈	D₄	1	4	yes	—
25	C₂×C₃×C₈	D₄	1	2	yes	—
26	C₃×C₄²	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₃	—	2	2	no	—
28	C₄⋊Dic₃	—	2	2	no	—
29	C₄⋊Dic₃	—	2	2	no	—
30	C₄⋊Dic₃	—	2	2	no	—
31	Dic₁₂	—	4	4	no	—
32	C₂×Q₁₆	—	6	2	no	—
33	C₄⋊Q₈	—	3	1	no	—
34	C₄⋊Q₈	—	3	2	no	—
35	C₄⋊Q₈	—	3	1	no	—
36	#106	—	6	4	no	—
37	#106	—	6	2	no	—
38	#103	—	6	4	no	—
39	C₄×C₈	C₂×D₃	1	2	yes	—
40	C₄×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
43	C₂×Dic₃	—	4	1	no	—
44	C₂×Dic₃	—	4	1	no	—
45	C₂×Dic₃	—	4	2	no	—
46	Dic₆	—	4	2	no	—
47	Dic₆	—	4	2	no	—
48	Dic₆	—	4	4	no	—
49	C₃×C₈	—	2	4	no	—
50	C₃×C₈	SD₁₆	1	4	yes	—
51	C₂×Q₈	—	6	2	no	—
52	C₂×Q₈	—	6	1	no	—
53	C₂×Q₈	—	6	1	no	—
54	Q₁₆	—	12	4	no	—
55	C₂×C₈	D₁₂	1	4	yes	—
56	C₂×C₈	D₁₂	1	2	yes	—
57	C₄⋊C₄	—	6	2	no	—
58	C₄⋊C₄	—	6	2	no	—
59	C₄⋊C₄	—	6	2	no	—
60	C₄⋊C₄	—	6	2	no	—
61	C₄²	C₂²×D₃	1	1	yes	characteristic
62	C₂²×C₃	#127	1	1	yes	characteristic
63	C₃×C₄	C₂×SD₁₆	1	2	yes	—
64	C₃×C₄	#137	1	4	yes	—
65	Dic₃	—	8	1	no	—
66	Dic₃	—	8	1	no	—
67	Dic₃	—	8	2	no	—
68	Q₈	—	12	2	no	—
69	Q₈	—	12	2	no	—
70	Q₈	—	12	4	no	—
71	C₂×C₄	—	12	1	no	—
72	C₂×C₄	—	12	1	no	—
73	C₂×C₄	—	12	2	no	—
74	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
75	C₂×C₄	C₂×D₁₂	1	2	yes	—
76	C₈	—	2	4	no	—
77	C₈	#204	1	4	yes	—
78	C₂×C₃	#492	1	1	yes	characteristic
79	C₂×C₃	#497	1	2	yes	—
80	C₂²	#874	1	1	yes	characteristic
81	C₄	—	24	1	no	—
82	C₄	—	24	1	no	—
83	C₄	—	24	2	no	—
84	C₄	C₂×#204	1	2	yes	—
85	C₄	#909	1	4	yes	—
86	C₃	#1716	1	1	yes	characteristic, complemented, Sylow
87	C₂	#4693	1	1	yes	characteristic
88	C₂	#4715	1	2	yes	—

	subgroup	G/H
center	C₂²	#874
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#322	C₂
socle, prime radical	C₂²×C₃	#127
hypercenter	#322	D₃
hypocenter	C₃	#1716
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
86. C₃	#1716	semidirect

Maximal subgroups

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

subgroup	index
#4699	2
#4681	2
#4680	2
C₃×#322	2
#1716	3