Subgroups — #5721

Order 192.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃	C₂	1	2	yes	maximal
3	#944	C₂	1	2	yes	maximal
4	#890	C₂	1	2	yes	maximal
5	#889	C₂	1	4	yes	maximal
6	#887	C₂	1	2	yes	maximal
7	C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
9	#557	—	3	1	no	maximal, Sylow
10	C₂×C₃×Q₈	C₂²	1	2	yes	—
11	C₂×Dic₆	C₂²	1	1	yes	characteristic
12	C₂×Dic₆	C₂²	1	1	yes	characteristic
13	C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₄²	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₃	C₂²	1	4	yes	—
19	Dic₃⋊C₄	C₂²	1	2	yes	—
20	Dic₃⋊C₄	C₂²	1	2	yes	—
21	Dic₃⋊C₄	C₂²	1	4	yes	—
22	Dic₃⋊C₄	C₂²	1	4	yes	—
23	C₄×Dic₃	C₂²	1	2	yes	—
24	C₄×Dic₃	C₂²	1	2	yes	—
25	C₄×Dic₃	C₂²	1	4	yes	—
26	C₄⋊Q₈	—	3	2	no	—
27	C₄⋊Q₈	D₃	1	1	yes	characteristic
28	#125	—	3	4	no	—
29	#125	—	3	2	no	—
30	C₄×Q₈	—	3	2	no	—
31	C₄×Q₈	—	3	2	no	—
32	C₄×Q₈	—	3	1	no	—
33	C₄×Q₈	—	3	1	no	—
34	C₃×Q₈	—	2	4	no	—
35	C₂×C₃×C₄	C₂³	1	4	yes	—
36	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂×Dic₃	C₂³	1	2	yes	—
40	C₂×Dic₃	C₂³	1	2	yes	—
41	C₂×Dic₃	C₂³	1	2	yes	—
42	C₂×Dic₃	C₂³	1	2	yes	—
43	Dic₆	—	2	2	no	—
44	Dic₆	Q₈	1	4	yes	complemented
45	C₂×Q₈	—	3	1	no	—
46	C₂×Q₈	—	3	1	no	—
47	C₂×Q₈	C₂×D₃	1	2	yes	—
48	C₄⋊C₄	—	3	1	no	—
49	C₄⋊C₄	—	3	2	no	—
50	C₄⋊C₄	—	3	2	no	—
51	C₄⋊C₄	—	3	4	no	—
52	C₄⋊C₄	—	3	1	no	—
53	C₄⋊C₄	—	3	4	no	—
54	C₄⋊C₄	—	3	4	no	—
55	C₄⋊C₄	C₂×D₃	1	2	yes	—
56	C₄⋊C₄	C₂×D₃	1	2	yes	—
57	C₄²	—	3	4	no	—
58	C₄²	—	3	2	no	—
59	C₄²	—	3	2	no	—
60	C₄²	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃	C₂⁴	1	1	yes	characteristic
62	C₃×C₄	—	2	4	no	—
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	C₂×Q₈	1	2	yes	—
65	C₃×C₄	#41	1	2	yes	—
66	Dic₃	—	2	2	no	—
67	Dic₃	—	2	2	no	—
68	Dic₃	—	2	2	no	—
69	Dic₃	C₂×Q₈	1	4	yes	—
70	Q₈	—	2	4	no	—
71	Q₈	—	3	4	no	—
72	Q₈	—	6	2	no	—
73	C₂×C₄	—	3	2	no	—
74	C₂×C₄	—	3	2	no	—
75	C₂×C₄	—	3	2	no	—
76	C₂×C₄	—	3	2	no	—
77	C₂×C₄	C₂²×D₃	1	4	yes	—
78	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
79	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
82	C₂×C₃	C₂×#41	1	1	yes	characteristic
83	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
84	C₂²	C₂³×D₃	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	2	1	no	—
87	C₄	—	3	4	no	—
88	C₄	—	6	2	no	—
89	C₄	—	6	2	no	—
90	C₄	—	6	2	no	—
91	C₄	#237	1	2	yes	—
92	C₄	D₃×Q₈	1	2	yes	—
93	C₃	#557	1	1	yes	characteristic, complemented, Sylow
94	C₂	C₂×#237	1	1	yes	characteristic
95	C₂	C₂×D₃×Q₈	1	1	yes	characteristic
96	C₂	#1007	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₃
hypocenter	C₃	#557
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
44. Dic₆	Q₈	semidirect
93. C₃	#557	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄⋊Q₈	2
Q₈×Dic₃	2
#944	2
#890	2
#889	2
#887	2
C₄×Dic₆	2
#557	3