Subgroups — #67798

Order 384.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#497	C₂	1	1	yes	maximal, characteristic
2	#4542	C₂	1	1	yes	maximal, characteristic
3	#4541	C₂	1	1	yes	maximal, characteristic
4	#1407	—	3	1	no	maximal, Sylow
5	C₂×C₃×Q₁₆	—	2	1	no	—
6	C₂×C₃×SD₁₆	—	2	1	no	—
7	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
8	C₃×#124	C₂²	1	1	yes	characteristic
9	C₃×#97	C₄	1	1	yes	characteristic, complemented
10	#836	—	2	1	no	—
11	#834	—	2	1	no	—
12	#497	D₃	1	1	yes	characteristic
13	#355	—	3	1	no	—
14	#354	—	3	1	no	—
15	C₂×C₃×Q₈	—	2	1	no	—
16	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
17	C₂×C₃×D₄	D₄	1	1	yes	characteristic
18	C₃×Q₁₆	—	4	2	no	—
19	C₃×SD₁₆	—	4	2	no	—
20	C₂×C₃×C₈	—	2	1	no	—
21	C₃×C₄⋊C₄	—	4	1	no	—
22	C₃×C₂²⋊C₄	—	4	1	no	—
23	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
24	#217	—	4	1	no	—
25	#208	—	4	1	no	—
26	C₂×Q₁₆	—	2	1	no	—
27	C₂×SD₁₆	—	2	1	no	—
28	C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
29	#124	C₂×D₃	1	1	yes	characteristic
30	#101	—	6	1	no	—
31	#99	—	6	1	no	—
32	#97	Dic₃	1	1	yes	characteristic, complemented
33	C₂³×C₃	—	2	1	no	—
34	C₃×Q₈	—	4	1	no	—
35	C₃×Q₈	—	4	1	no	—
36	C₃×Q₈	—	4	1	no	—
37	C₃×D₄	—	4	1	no	—
38	C₂×C₃×C₄	—	2	1	no	—
39	C₂×C₃×C₄	—	2	1	no	—
40	C₂×C₃×C₄	—	4	1	no	—
41	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×Dic₃	—	8	1	no	—
43	C₃×C₈	—	2	2	no	—
44	C₃⋊C₈	—	8	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	#67	1	1	yes	characteristic
47	C₂×D₄	#67	1	1	yes	characteristic
48	Q₁₆	—	4	2	no	—
49	SD₁₆	—	4	2	no	—
50	M₁₆	—	12	1	no	—
51	C₂×C₈	—	2	1	no	—
52	C₄⋊C₄	—	4	1	no	—
53	C₂²⋊C₄	—	4	1	no	—
54	C₂²⋊C₄	—	12	1	no	—
55	C₄²	C₂×Dic₃	1	1	yes	characteristic
56	C₂²×C₃	—	4	1	no	—
57	C₂²×C₃	—	8	1	no	—
58	C₂²×C₃	#99	1	1	yes	characteristic
59	C₃×C₄	—	2	1	no	—
60	C₃×C₄	—	4	1	no	—
61	C₃×C₄	—	4	1	no	—
62	C₃×C₄	—	8	1	no	—
63	Dic₃	—	16	1	no	—
64	C₂³	—	2	1	no	—
65	Q₈	—	4	1	no	—
66	Q₈	—	4	1	no	—
67	Q₈	—	4	1	no	—
68	D₄	—	4	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	24	1	no	—
73	C₂×C₄	#217	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	24	1	no	—
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	—	8	1	no	—
78	C₂×C₃	#353	1	1	yes	characteristic
79	C₂²	—	4	1	no	—
80	C₂²	—	8	1	no	—
81	C₂²	#834	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	48	1	no	—
87	C₃	#1407	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	1	no	—
89	C₂	—	8	1	no	—
90	C₂	#4545	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4545
commutator	C₃×C₄²	C₂×C₄
Frattini	#124	C₂×D₃
Fitting	C₃×#497	C₂
socle	C₂×C₃	#353
prime radical	C₂×C₃×D₄	D₄
cosocle	C₃×#124	C₂²
hypercenter	#497	D₃
hypocenter	C₃	#1407
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #497

Complements

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

N	H ≅ G/N	type
7. C₃×C₄⋊Q₈	C₄	semidirect
9. C₃×#97	C₄	semidirect
28. C₄⋊Q₈	Dic₃	semidirect
32. #97	Dic₃	semidirect
87. C₃	#1407	semidirect

Maximal subgroups

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

subgroup	index
C₃×#497	2
#4542	2
#4541	2
#1407	3