Subgroups — #68929

Order 384.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5118	C₂	1	2	yes	maximal
2	C₂×#815	C₂	1	1	yes	maximal, characteristic
3	C₃×#341	C₂	1	1	yes	maximal, characteristic
4	#4558	C₂	1	2	yes	maximal
5	#4552	C₂	1	1	yes	maximal, characteristic
6	#1838	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
10	C₂×#208	C₂²	1	2	yes	—
11	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	#815	C₄	1	4	yes	—
13	Dic₃⋊C₈	—	2	4	no	—
14	#423	—	3	2	no	—
15	C₂×#97	—	3	1	no	—
16	#341	—	3	2	no	—
17	#341	—	3	1	no	—
18	#341	D₃	1	1	yes	characteristic
19	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃	Q₈	1	2	yes	—
21	C₃×M₁₆	—	2	2	no	—
22	C₃×M₁₆	—	2	2	no	—
23	C₂×C₃×C₈	—	2	2	no	—
24	C₂×C₃×C₈	—	2	2	no	—
25	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
26	C₄×Dic₃	C₂×C₄	1	2	yes	—
27	C₄×Dic₃	D₄	1	2	yes	—
28	#208	—	2	2	no	—
29	#208	—	2	2	no	—
30	C₂×C₃⋊C₈	—	2	2	no	—
31	C₂×C₃⋊C₈	—	2	2	no	—
32	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×M₁₆	—	3	2	no	—
34	C₂×M₁₆	C₂×D₃	1	2	yes	—
35	C₂²×C₈	—	3	1	no	—
36	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
37	C₂×C₄²	—	3	1	no	—
38	C₄⋊C₈	—	6	4	no	—
39	#97	—	3	4	no	—
40	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
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	C₂×C₃×C₄	#41	1	2	yes	—
45	C₂×Dic₃	—	4	2	no	—
46	C₂×Dic₃	C₄⋊C₄	1	4	yes	—
47	C₃×C₈	—	2	2	no	—
48	C₃×C₈	—	4	2	no	—
49	C₃⋊C₈	—	2	2	no	—
50	C₃⋊C₈	—	4	2	no	—
51	C₂²×C₄	—	3	2	no	—
52	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
53	M₁₆	—	2	2	no	—
54	M₁₆	—	2	2	no	—
55	M₁₆	—	6	2	no	—
56	M₁₆	—	6	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	3	2	no	—
60	C₂×C₈	—	6	2	no	—
61	C₂×C₈	—	6	2	no	—
62	C₂×C₈	C₄×D₃	1	2	yes	—
63	C₄²	—	3	2	no	—
64	C₄²	—	3	2	no	—
65	C₂²×C₃	—	2	2	no	—
66	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₃	#125	1	2	yes	—
68	C₃×C₄	#117	1	1	yes	characteristic
69	C₃×C₄	#117	1	1	yes	characteristic
70	C₃×C₄	#124	1	2	yes	—
71	Dic₃	—	2	4	no	—
72	C₂³	D₃×Q₈	1	1	yes	characteristic
73	C₂×C₄	—	3	4	no	—
74	C₂×C₄	—	12	2	no	—
75	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
76	C₂×C₄	#235	1	2	yes	—
77	C₂×C₄	D₃×D₄	1	1	yes	characteristic
78	C₂×C₄	#237	1	2	yes	—
79	C₈	—	2	2	no	—
80	C₈	—	4	2	no	—
81	C₈	—	6	2	no	—
82	C₈	—	12	2	no	—
83	C₂×C₃	—	2	2	no	—
84	C₂×C₃	#382	1	1	yes	characteristic
85	C₂×C₃	#430	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	#889	1	2	yes	—
88	C₂²	D₃×C₄⋊C₄	1	1	yes	characteristic
89	C₄	—	6	4	no	—
90	C₄	#872	1	1	yes	characteristic
91	C₄	#877	1	1	yes	characteristic
92	C₄	#885	1	2	yes	—
93	C₃	#1838	1	1	yes	characteristic, complemented, Sylow
94	C₂	—	2	2	no	—
95	C₂	#4650	1	1	yes	characteristic
96	C₂	#4893	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#341	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂³×C₃	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#341	D₃
hypocenter	C₃	#1838
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

Complements

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

N	H ≅ G/N	type
93. C₃	#1838	semidirect

Maximal subgroups

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

subgroup	index
#5118	2
C₂×#815	2
C₃×#341	2
#4558	2
#4552	2
#1838	3