Subgroups — #67780

Order 384.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#470	C₂	1	1	yes	maximal, characteristic
2	#4557	C₂	1	1	yes	maximal, characteristic
3	#4553	C₂	1	1	yes	maximal, characteristic
4	#1340	—	3	1	no	maximal, Sylow
5	C₂×C₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
6	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₃×C₂²⋊Q₈	—	2	1	no	—
8	#926	—	2	1	no	—
9	C₃×#106	C₄	1	1	yes	characteristic, complemented
10	C₃×#103	—	2	1	no	—
11	#812	—	2	1	no	—
12	#470	D₃	1	1	yes	characteristic
13	#349	—	3	1	no	—
14	H(3,ℤ/4)	—	3	1	no	—
15	C₂×C₃×Q₈	—	2	1	no	—
16	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
17	C₃×Q₁₆	—	4	1	no	—
18	C₃×M₁₆	—	4	1	no	—
19	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₈	D₄	1	1	yes	characteristic
21	C₃×C₄⋊C₄	—	2	1	no	—
22	C₃×C₄⋊C₄	—	4	1	no	—
23	C₃×C₂²⋊C₄	—	4	1	no	—
24	#217	—	4	1	no	—
25	C₄⋊Dic₃	—	4	1	no	—
26	C₄×Dic₃	—	2	2	no	—
27	C₃⋊C₁₆	—	4	1	no	—
28	C₂×Q₁₆	Dic₃	1	1	yes	characteristic, complemented
29	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
30	C₂²⋊Q₈	—	2	1	no	—
31	#117	—	6	1	no	—
32	M₃₂	—	6	1	no	—
33	#106	Dic₃	1	1	yes	characteristic, complemented
34	#103	—	2	1	no	—
35	C₂³×C₃	D₈	1	1	yes	characteristic
36	C₃×Q₈	—	4	1	no	—
37	C₃×Q₈	—	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	—	4	1	no	—
42	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
44	C₂×Dic₃	—	4	2	no	—
45	C₃×C₈	—	2	1	no	—
46	C₃×C₈	—	2	1	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂²×C₄	#67	1	1	yes	characteristic
49	Q₁₆	—	4	1	no	—
50	M₁₆	—	4	1	no	—
51	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
52	C₂×C₈	#67	1	1	yes	characteristic
53	C₄⋊C₄	—	2	1	no	—
54	C₄⋊C₄	—	4	1	no	—
55	C₄⋊C₄	—	12	1	no	—
56	C₂²⋊C₄	—	4	1	no	—
57	C₂²⋊C₄	—	12	1	no	—
58	C₄²	—	6	2	no	—
59	C₁₆	—	12	1	no	—
60	C₂²×C₃	—	2	1	no	—
61	C₂²×C₃	—	4	1	no	—
62	C₂²×C₃	#102	1	1	yes	characteristic
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	—	8	1	no	—
65	C₃×C₄	—	8	1	no	—
66	C₃×C₄	#99	1	1	yes	characteristic
67	C₃×C₄	#104	1	1	yes	characteristic
68	Dic₃	—	8	2	no	—
69	C₂³	#213	1	1	yes	characteristic
70	Q₈	—	4	1	no	—
71	Q₈	—	4	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	12	2	no	—
77	C₂×C₄	#214	1	1	yes	characteristic
78	C₂×C₄	#217	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	—	2	1	no	—
81	C₂×C₃	—	2	1	no	—
82	C₂×C₃	—	4	1	no	—
83	C₂×C₃	#327	1	1	yes	characteristic
84	C₂²	—	2	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#832	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	8	1	no	—
89	C₄	—	8	1	no	—
90	C₄	—	24	2	no	—
91	C₄	#834	1	1	yes	characteristic
92	C₄	#837	1	1	yes	characteristic
93	C₃	#1340	1	1	yes	characteristic, complemented, Sylow
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#4537	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4537
commutator	C₂×C₃×C₈	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₃
Fitting	C₃×#470	C₂
socle	C₂×C₃	#327
prime radical	C₂³×C₃	D₈
cosocle	C₂×C₃×M₁₆	C₂²
hypercenter	#470	D₃
hypocenter	C₃	#1340
LCS	G ▷ C₂×C₃×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #470

Complements

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

N	H ≅ G/N	type
5. C₂×C₃×Q₁₆	C₄	semidirect
9. C₃×#106	C₄	semidirect
28. C₂×Q₁₆	Dic₃	semidirect
33. #106	Dic₃	semidirect
93. C₃	#1340	semidirect

Maximal subgroups

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

subgroup	index
C₃×#470	2
#4557	2
#4553	2
#1340	3