Subgroups — #5574

Order 192.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#239	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#893	C₂	1	3	yes	maximal, complemented
5	D₃×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
6	C₄×D₁₂	C₂	1	3	yes	maximal, complemented
7	#872	C₂	1	3	yes	maximal, complemented
8	#520	—	3	1	no	maximal, Sylow
9	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
10	#239	C₄	1	8	yes	complemented
11	C₂×D₁₂	C₂²	1	3	yes	—
12	C₂×C₄×D₃	C₂²	1	6	yes	—
13	C₂×C₄×D₃	C₂²	1	3	yes	—
14	C₃×C₄⋊C₄	C₂²	1	3	yes	—
15	C₃×C₄²	C₂²	1	3	yes	—
16	(C₂×D₃)⋊C₄	C₂²	1	3	yes	—
17	(C₂×D₃)⋊C₄	C₂²	1	3	yes	—
18	C₄⋊Dic₃	C₂²	1	3	yes	—
19	Dic₃⋊C₄	C₂²	1	1	yes	characteristic
20	Dic₃⋊C₄	C₂²	1	3	yes	—
21	C₄×Dic₃	C₂²	1	3	yes	—
22	C₂×#41	—	3	1	no	—
23	C₄×Q₈	—	3	1	no	—
24	C₄×Q₈	D₃	1	1	yes	characteristic, complemented
25	C₄×D₄	—	3	3	no	—
26	C₄×D₄	—	3	3	no	—
27	#117	—	3	3	no	—
28	C₂×C₄⋊C₄	—	3	3	no	—
29	C₂²×D₃	C₂³	1	3	yes	—
30	C₃×Q₈	C₂×C₄	1	4	yes	complemented
31	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₄	C₂³	1	3	yes	—
33	C₂×C₃×C₄	C₂³	1	3	yes	—
34	C₂×Dic₃	C₂³	1	1	yes	characteristic
35	C₂×Dic₃	C₂³	1	3	yes	—
36	C₂×Dic₃	C₂³	1	1	yes	characteristic
37	D₁₂	C₂×C₄	1	12	yes	complemented
38	C₄×D₃	—	4	6	no	—
39	C₄×D₃	C₂×C₄	1	12	yes	complemented
40	#41	—	3	8	no	—
41	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
42	C₂×D₄	—	3	3	no	—
43	C₂²×C₄	—	3	3	no	—
44	C₂²×C₄	—	3	6	no	—
45	C₄⋊C₄	—	3	3	no	—
46	C₄⋊C₄	—	3	3	no	—
47	C₄⋊C₄	—	3	1	no	—
48	C₄⋊C₄	C₂×D₃	1	3	yes	—
49	C₂²⋊C₄	—	3	3	no	—
50	C₂²⋊C₄	—	3	3	no	—
51	C₄²	—	3	3	no	—
52	C₄²	C₂×D₃	1	3	yes	—
53	C₂²×C₃	C₂⁴	1	1	yes	characteristic
54	C₂×D₃	—	2	3	no	—
55	C₂×D₃	—	2	3	no	—
56	C₂×D₃	C₂²×C₄	1	6	yes	—
57	C₃×C₄	—	2	3	no	—
58	C₃×C₄	—	2	1	no	—
59	C₃×C₄	C₂²×C₄	1	6	yes	—
60	Dic₃	—	2	3	no	—
61	Dic₃	—	2	1	no	—
62	Dic₃	C₂²×C₄	1	2	yes	—
63	C₂³	—	3	3	no	—
64	Q₈	C₄×D₃	1	4	yes	complemented
65	D₄	—	3	12	no	—
66	C₂×C₄	—	3	1	no	—
67	C₂×C₄	—	3	12	no	—
68	C₂×C₄	—	3	1	no	—
69	C₂×C₄	—	3	3	no	—
70	C₂×C₄	—	12	6	no	—
71	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₃	1	3	yes	—
73	C₂×C₄	C₂²×D₃	1	3	yes	—
74	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
76	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
77	D₃	—	2	6	no	—
78	C₂²	—	3	6	no	—
79	C₂²	—	6	3	no	—
80	C₂²	—	6	3	no	—
81	C₂²	C₂³×D₃	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	3	no	—
84	C₄	—	3	2	no	—
85	C₄	—	6	1	no	—
86	C₄	—	6	3	no	—
87	C₄	C₂×C₄×D₃	1	6	yes	—
88	C₃	#520	1	1	yes	characteristic, complemented, Sylow
89	C₂	—	6	6	no	—
90	C₂	C₂²×C₄×D₃	1	1	yes	characteristic
91	C₂	#1007	1	1	yes	characteristic
92	C₂	#1009	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄
Fitting	C₃×C₄×Q₈	C₂
socle, cosocle	C₂²×C₃	C₂⁴
prime radical	C₂×#239	C₂
hypercenter	C₄×Q₈	D₃
hypocenter	C₃	#520
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×Q₈	C₂	semidirect
3. Q₈×Dic₃	C₂	semidirect
4. #893	C₂	semidirect
5. D₃×C₄⋊C₄	C₂	semidirect
6. C₄×D₁₂	C₂	semidirect
7. #872	C₂	semidirect
10. #239	C₄	semidirect
24. C₄×Q₈	D₃	semidirect
30. C₃×Q₈	C₂×C₄	semidirect
37. D₁₂	C₂×C₄	semidirect
39. C₄×D₃	C₂×C₄	semidirect
64. Q₈	C₄×D₃	semidirect
88. C₃	#520	semidirect

Maximal subgroups

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

subgroup	index
C₂×#239	2
C₃×C₄×Q₈	2
Q₈×Dic₃	2
#893	2
D₃×C₄⋊C₄	2
C₄×D₁₂	2
#872	2
#520	3