Subgroups — #68059

Order 384.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4694	C₂	1	2	yes	maximal
2	#4693	C₂	1	1	yes	maximal, characteristic
3	#4681	C₂	1	2	yes	maximal, complemented
4	#4680	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	#1708	—	3	1	no	maximal, Sylow
7	C₂×#204	—	2	2	no	—
8	#874	C₂²	1	1	yes	characteristic
9	#869	C₂²	1	2	yes	—
10	#869	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₈	C₂²	1	2	yes	—
13	#821	—	2	4	no	—
14	#817	—	2	2	no	—
15	#816	—	2	4	no	—
16	#498	—	3	1	no	—
17	#492	—	3	1	no	—
18	#487	—	3	2	no	—
19	#486	—	3	2	no	—
20	C₈²	D₃	1	1	yes	characteristic, complemented
21	C₂×D₁₂	—	2	2	no	—
22	C₂×D₁₂	—	2	1	no	—
23	C₂×Dic₆	—	2	2	no	—
24	C₂×Dic₆	—	2	1	no	—
25	C₂×C₃×C₈	D₄	1	4	yes	—
26	C₂×C₃×C₈	D₄	1	2	yes	—
27	C₃×C₄²	C₂³	1	1	yes	characteristic
28	C₄⋊Dic₃	—	2	2	no	—
29	C₄⋊Dic₃	—	2	2	no	—
30	C₄⋊Dic₃	—	2	2	no	—
31	#204	—	4	4	no	—
32	C₂×SD₁₆	—	6	2	no	—
33	C₄⋊Q₈	—	3	2	no	—
34	C₄⋊Q₈	—	3	1	no	—
35	#127	—	3	1	no	—
36	#106	—	6	2	no	—
37	#103	—	6	4	no	—
38	#102	—	6	4	no	—
39	C₄×C₈	C₂×D₃	1	2	yes	—
40	C₄×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂²×D₃	—	4	1	no	—
42	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
44	C₂×Dic₃	—	4	1	no	—
45	C₂×Dic₃	—	4	2	no	—
46	D₁₂	—	4	2	no	—
47	D₁₂	—	4	4	no	—
48	Dic₆	—	4	2	no	—
49	Dic₆	—	4	4	no	—
50	C₃×C₈	—	2	4	no	—
51	C₃×C₈	SD₁₆	1	4	yes	—
52	C₂×Q₈	—	6	2	no	—
53	C₂×Q₈	—	6	1	no	—
54	C₂×D₄	—	6	2	no	—
55	C₂×D₄	—	6	1	no	—
56	SD₁₆	—	12	4	no	—
57	C₂×C₈	D₁₂	1	4	yes	—
58	C₂×C₈	D₁₂	1	2	yes	—
59	C₄⋊C₄	—	6	2	no	—
60	C₄⋊C₄	—	6	2	no	—
61	C₄⋊C₄	—	6	2	no	—
62	C₄²	C₂²×D₃	1	1	yes	characteristic
63	C₂²×C₃	#127	1	1	yes	characteristic
64	C₂×D₃	—	8	1	no	—
65	C₂×D₃	—	8	2	no	—
66	C₃×C₄	C₂×SD₁₆	1	2	yes	—
67	C₃×C₄	#135	1	4	yes	—
68	Dic₃	—	8	1	no	—
69	Dic₃	—	8	2	no	—
70	C₂³	—	12	1	no	—
71	Q₈	—	12	2	no	—
72	Q₈	—	12	4	no	—
73	D₄	—	12	2	no	—
74	D₄	—	12	4	no	—
75	C₂×C₄	—	12	1	no	—
76	C₂×C₄	—	12	2	no	—
77	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
78	C₂×C₄	C₂×D₁₂	1	2	yes	—
79	C₈	—	2	4	no	—
80	C₈	#204	1	4	yes	—
81	C₂×C₃	#492	1	1	yes	characteristic
82	C₂×C₃	#495	1	2	yes	—
83	D₃	—	16	1	no	—
84	C₂²	—	24	1	no	—
85	C₂²	—	24	2	no	—
86	C₂²	#874	1	1	yes	characteristic
87	C₄	—	24	1	no	—
88	C₄	—	24	2	no	—
89	C₄	C₂×#204	1	2	yes	—
90	C₄	#904	1	4	yes	—
91	C₃	#1708	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	48	1	no	—
93	C₂	#4693	1	1	yes	characteristic
94	C₂	#4696	1	2	yes	—

	subgroup	G/H
center	C₂²	#874
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×C₈²	C₂
socle	C₂²×C₃	#127
prime radical	#874	C₂²
hypercenter	C₈²	D₃
hypocenter	C₃	#1708
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
3. #4681	C₂	semidirect
4. #4680	C₂	semidirect
5. C₃×C₈²	C₂	semidirect
20. C₈²	D₃	semidirect
91. C₃	#1708	semidirect

Maximal subgroups

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

subgroup	index
#4694	2
#4693	2
#4681	2
#4680	2
C₃×C₈²	2
#1708	3