Subgroups — #69840

Order 384.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#819	C₂	1	2	yes	maximal
3	C₄×#208	C₂	1	2	yes	maximal
4	#4552	C₂	1	2	yes	maximal
5	#1778	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×#208	C₂²	1	4	yes	—
10	C₃×#97	C₄	1	4	yes	—
11	#819	C₄	1	8	yes	complemented
12	#803	—	2	2	no	—
13	C₄×C₃⋊C₈	—	2	2	no	—
14	C₂×#108	—	3	2	no	—
15	C₄×M₁₆	—	3	2	no	—
16	C₂×#97	D₃	1	1	yes	characteristic
17	#341	—	3	2	no	—
18	C₂²×C₃×C₄	Q₈	1	2	yes	—
19	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₂×C₃×C₈	—	2	2	no	—
21	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
22	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
24	C₃×C₄²	C₂×C₄	1	2	yes	—
25	C₃×C₄²	D₄	1	2	yes	—
26	#208	—	2	4	no	—
27	#208	C₂×C₄	1	8	yes	—
28	C₂×C₃⋊C₈	—	2	4	no	—
29	C₂×M₁₆	—	3	4	no	—
30	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
31	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
32	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
33	#108	—	3	8	no	—
34	#97	—	6	2	no	—
35	#97	Dic₃	1	4	yes	—
36	C₄×C₈	—	6	2	no	—
37	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₃×C₄	—	4	2	no	—
39	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
40	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
41	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃×C₄	#41	1	2	yes	—
44	C₃×C₈	—	2	2	no	—
45	C₃×C₈	C₄²	1	4	yes	—
46	C₃⋊C₈	—	2	8	no	—
47	C₂²×C₄	Dic₆	1	2	yes	—
48	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
49	M₁₆	—	3	8	no	—
50	M₁₆	—	6	4	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	6	4	no	—
53	C₂×C₈	C₄×D₃	1	4	yes	—
54	C₂×C₈	C₂×Dic₃	1	2	yes	—
55	C₂×C₈	C₂×Dic₃	1	2	yes	—
56	C₄²	D₁₂	1	2	yes	—
57	C₄²	C₂×Dic₃	1	2	yes	—
58	C₂²×C₃	—	2	2	no	—
59	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₃	C₄×Q₈	1	2	yes	—
61	C₃×C₄	—	2	4	no	—
62	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
63	C₃×C₄	#117	1	1	yes	characteristic
64	C₃×C₄	C₄×D₄	1	2	yes	—
65	C₂³	C₂×Dic₆	1	1	yes	characteristic
66	C₂×C₄	—	4	2	no	—
67	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
68	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
69	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
70	C₂×C₄	#235	1	2	yes	—
71	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	6	8	no	—
74	C₈	C₄×Dic₃	1	4	yes	—
75	C₂×C₃	—	2	2	no	—
76	C₂×C₃	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₃	#430	1	2	yes	—
78	C₂²	—	2	2	no	—
79	C₂²	C₄×Dic₆	1	2	yes	—
80	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	C₄×D₁₂	1	2	yes	—
83	C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
84	C₄	#926	1	1	yes	characteristic
85	C₃	#1778	1	1	yes	characteristic, complemented, Sylow
86	C₂	—	2	2	no	—
87	C₂	C₄×C₄⋊Dic₃	1	1	yes	characteristic
88	C₂	#5125	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator	C₃×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#97	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂³×C₃	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#97	D₃
hypocenter	C₃	#1778
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
11. #819	C₄	semidirect
85. C₃	#1778	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#97	2
C₂×#819	2
C₄×#208	2
#4552	2
#1778	3