Subgroups — #67797

Order 384.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#496	C₂	1	1	yes	maximal, characteristic
2	#4542	C₂	1	1	yes	maximal, characteristic, complemented
3	#4541	C₂	1	1	yes	maximal, characteristic, complemented
4	#1406	—	3	1	no	maximal, Sylow
5	C₂×C₃×SD₁₆	—	2	1	no	—
6	C₂×C₃×D₈	—	2	1	no	—
7	C₃×#127	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	#496	D₃	1	1	yes	characteristic
13	#355	—	3	1	no	—
14	#354	—	3	1	no	—
15	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
16	C₂×C₃×D₄	—	2	1	no	—
17	C₂×C₃×D₄	—	4	1	no	—
18	C₂×C₃×D₄	D₄	1	1	yes	characteristic
19	C₃×SD₁₆	—	4	2	no	—
20	C₃×D₈	—	4	2	no	—
21	C₂×C₃×C₈	—	2	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×SD₁₆	—	2	1	no	—
27	C₂×D₈	—	2	1	no	—
28	#127	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	C₂³×C₃	—	4	1	no	—
35	C₃×Q₈	—	4	1	no	—
36	C₃×D₄	—	4	1	no	—
37	C₃×D₄	—	4	1	no	—
38	C₃×D₄	—	4	1	no	—
39	C₃×D₄	—	8	2	no	—
40	C₂×C₃×C₄	—	2	1	no	—
41	C₂×C₃×C₄	—	2	1	no	—
42	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×Dic₃	—	8	1	no	—
44	C₃×C₈	—	2	2	no	—
45	C₃⋊C₈	—	8	1	no	—
46	C₂×Q₈	#67	1	1	yes	characteristic
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	#67	1	1	yes	characteristic
50	SD₁₆	—	4	2	no	—
51	D₈	—	4	2	no	—
52	M₁₆	—	12	1	no	—
53	C₂×C₈	—	2	1	no	—
54	C₂²⋊C₄	—	4	1	no	—
55	C₂²⋊C₄	—	12	1	no	—
56	C₄²	C₂×Dic₃	1	1	yes	characteristic
57	C₂²×C₃	—	4	1	no	—
58	C₂²×C₃	—	8	1	no	—
59	C₂²×C₃	—	8	2	no	—
60	C₂²×C₃	—	8	1	no	—
61	C₂²×C₃	#99	1	1	yes	characteristic
62	C₃×C₄	—	2	1	no	—
63	C₃×C₄	—	4	1	no	—
64	C₃×C₄	—	4	1	no	—
65	Dic₃	—	16	1	no	—
66	C₂³	—	2	1	no	—
67	C₂³	—	4	1	no	—
68	Q₈	—	4	1	no	—
69	D₄	—	4	1	no	—
70	D₄	—	4	1	no	—
71	D₄	—	4	1	no	—
72	D₄	—	8	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	24	1	no	—
76	C₂×C₄	#217	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	24	1	no	—
79	C₂×C₃	—	2	1	no	—
80	C₂×C₃	—	8	1	no	—
81	C₂×C₃	—	16	1	no	—
82	C₂×C₃	#353	1	1	yes	characteristic
83	C₂²	—	4	1	no	—
84	C₂²	—	8	1	no	—
85	C₂²	—	8	2	no	—
86	C₂²	—	8	1	no	—
87	C₂²	#834	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	4	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	48	1	no	—
92	C₃	#1406	1	1	yes	characteristic, complemented, Sylow
93	C₂	—	2	1	no	—
94	C₂	—	8	1	no	—
95	C₂	—	16	1	no	—
96	C₂	#4545	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4545
commutator	C₃×C₄²	C₂×C₄
Frattini	#124	C₂×D₃
Fitting, prime radical	C₃×#496	C₂
socle	C₂×C₃	#353
cosocle	C₃×#124	C₂²
hypercenter	#496	D₃
hypocenter	C₃	#1406
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #496

Complements

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

N	H ≅ G/N	type
2. #4542	C₂	semidirect
3. #4541	C₂	semidirect
7. C₃×#127	C₄	semidirect
9. C₃×#97	C₄	semidirect
28. #127	Dic₃	semidirect
32. #97	Dic₃	semidirect
92. C₃	#1406	semidirect

Maximal subgroups

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

subgroup	index
C₃×#496	2
#4542	2
#4541	2
#1406	3