Subgroups — #68068

Order 384.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4685	C₂	1	2	yes	maximal
2	C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	#4455	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
6	#1450	—	3	1	no	maximal, Sylow
7	C₂×D₃×C₈	C₄	1	2	yes	—
8	C₄²×D₃	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₈	C₂²	1	2	yes	—
11	(C₂×D₃)⋊C₈	C₄	1	4	yes	—
12	Dic₃⋊C₈	C₄	1	4	yes	—
13	C₈×Dic₃	C₄	1	2	yes	—
14	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊C₈	C₂²	1	2	yes	—
16	#431	—	3	2	no	—
17	C₂×C₄×C₈	—	3	1	no	—
18	#322	—	3	1	no	—
19	#322	D₃	1	1	yes	characteristic, complemented
20	C₈²	—	3	2	no	—
21	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
25	C₃×C₄²	C₂³	1	1	yes	characteristic
26	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₃	C₂×C₄	1	2	yes	—
28	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
30	D₃×C₈	C₈	1	8	yes	complemented
31	C₂²×C₈	—	3	2	no	—
32	C₂×C₄²	—	3	1	no	—
33	C₄⋊C₈	—	3	4	no	—
34	C₂²⋊C₈	—	3	4	no	—
35	C₄×C₈	—	3	2	no	—
36	C₄×C₈	—	3	1	no	—
37	C₄×C₈	—	3	2	no	—
38	C₄×C₈	C₂×D₃	1	2	yes	—
39	C₄×C₈	C₂×D₃	1	1	yes	characteristic
40	C₂²×D₃	C₄²	1	1	yes	characteristic
41	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
43	C₂×Dic₃	C₄²	1	2	yes	—
44	C₂×Dic₃	C₄²	1	1	yes	characteristic
45	C₄×D₃	—	2	4	no	—
46	C₄×D₃	C₂×C₈	1	4	yes	—
47	C₃×C₈	—	2	4	no	—
48	C₃×C₈	C₂×C₈	1	4	yes	—
49	C₃⋊C₈	—	2	4	no	—
50	C₃⋊C₈	C₂×C₈	1	4	yes	—
51	C₂²×C₄	—	3	1	no	—
52	C₂²×C₄	—	3	2	no	—
53	C₂×C₈	—	3	2	no	—
54	C₂×C₈	—	3	8	no	—
55	C₂×C₈	—	3	4	no	—
56	C₂×C₈	C₄×D₃	1	4	yes	—
57	C₂×C₈	C₄×D₃	1	2	yes	—
58	C₄²	—	3	1	no	—
59	C₄²	—	3	2	no	—
60	C₄²	C₂²×D₃	1	1	yes	characteristic
61	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
62	C₂×D₃	—	2	2	no	—
63	C₂×D₃	C₄×C₈	1	2	yes	—
64	C₃×C₄	C₂²×C₈	1	2	yes	—
65	C₃×C₄	#131	1	4	yes	—
66	Dic₃	—	2	2	no	—
67	Dic₃	C₄×C₈	1	2	yes	—
68	C₂³	—	3	1	no	—
69	C₂×C₄	—	3	1	no	—
70	C₂×C₄	—	3	4	no	—
71	C₂×C₄	—	3	2	no	—
72	C₂×C₄	—	6	4	no	—
73	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
74	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
75	C₈	—	2	4	no	—
76	C₈	—	3	4	no	—
77	C₈	—	6	4	no	—
78	C₈	D₃×C₈	1	4	yes	—
79	C₂×C₃	C₂×C₄×C₈	1	1	yes	characteristic
80	C₂×C₃	#405	1	2	yes	—
81	D₃	—	2	2	no	—
82	C₂²	—	3	2	no	—
83	C₂²	—	6	2	no	—
84	C₂²	C₄²×D₃	1	1	yes	characteristic
85	C₄	—	3	2	no	—
86	C₄	—	6	2	no	—
87	C₄	C₂×D₃×C₈	1	2	yes	—
88	C₄	#907	1	4	yes	—
89	C₃	#1450	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	6	2	no	—
91	C₂	C₄×D₃×C₈	1	1	yes	characteristic
92	C₂	#4708	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₃
commutator	C₂×C₃	C₂×C₄×C₈
Fitting	C₃×#322	C₂
socle	C₂²×C₃	C₂×C₄²
prime radical	C₂²×D₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	#322	D₃
hypocenter	C₃	#1450
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. C₃×#322	C₂	semidirect
4. #4455	C₂	semidirect
5. C₈×C₃⋊C₈	C₂	semidirect
19. #322	D₃	semidirect
30. D₃×C₈	C₈	semidirect
89. C₃	#1450	semidirect

Maximal subgroups

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

subgroup	index
#4685	2
C₄×D₃×C₈	2
C₃×#322	2
#4455	2
C₈×C₃⋊C₈	2
#1450	3