Subgroups — C₄×#3776 #10593444

Order 576.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3776	C₂	1	1	yes	maximal, characteristic
2	C₄×#627	C₂	1	2	yes	maximal
3	C₄×#622	C₂	1	1	yes	maximal, characteristic
4	#63743	C₂	1	1	yes	maximal, characteristic
5	#63740	C₂	1	2	yes	maximal
6	C₂×#627	C₂²	1	2	yes	—
7	C₂×#622	C₂²	1	1	yes	characteristic
8	C₂×#622	C₂²	1	1	yes	characteristic
9	#3776	C₄	1	4	yes	direct product factor
10	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
11	C₃×C₄×Dic₃	—	2	2	no	—
12	#3724	C₂²	1	2	yes	—
13	Dic₃⋊Dic₃	—	2	2	no	—
14	C₄×Dic₆	—	6	2	no	—
15	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
16	C₂×C₃⋊Dic₃	D₄	1	1	yes	characteristic
17	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
18	C₂×C₃×Dic₃	—	2	2	no	—
19	C₂×C₃×Dic₃	—	2	2	no	—
20	#627	—	2	2	no	—
21	#627	C₂×C₄	1	4	yes	—
22	#622	—	2	1	no	—
23	#622	C₂×C₄	1	2	yes	—
24	C₄×Q₁₆	—	9	1	no	maximal, Sylow
25	C₂×Dic₆	—	6	2	no	—
26	C₃×C₄²	—	6	2	no	—
27	C₄⋊Dic₃	—	6	2	no	—
28	Dic₃⋊C₄	—	6	2	no	—
29	Dic₃⋊C₄	—	6	2	no	—
30	C₄×Dic₃	—	2	2	no	—
31	C₄×Dic₃	—	6	2	no	—
32	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
33	C₃²×C₄	Q₁₆	1	2	yes	complemented
34	C₃⋊Dic₃	—	2	1	no	—
35	C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
36	C₃⋊Dic₃	#41	1	1	yes	characteristic
37	C₃×Dic₃	—	2	4	no	—
38	C₃×Dic₃	—	4	2	no	—
39	C₂×Q₁₆	—	9	1	no	—
40	C₄×Q₈	—	9	2	no	—
41	#107	—	9	1	no	—
42	#103	—	9	2	no	—
43	C₄×C₈	—	9	1	no	—
44	C₂×C₃×C₄	—	2	2	no	—
45	C₂×C₃×C₄	—	6	2	no	—
46	C₂×C₃×C₄	—	6	2	no	—
47	C₂×Dic₃	—	2	2	no	—
48	C₂×Dic₃	—	2	2	no	—
49	C₂×Dic₃	—	6	2	no	—
50	C₂×Dic₃	—	6	2	no	—
51	Dic₆	—	6	4	no	—
52	Dic₆	—	6	4	no	—
53	C₂×C₃²	C₄×D₄	1	1	yes	characteristic
54	C₂×C₃²	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₃²	#135	1	1	yes	characteristic
56	C₂×Q₈	—	9	2	no	—
57	Q₁₆	—	9	4	no	—
58	C₂×C₈	—	9	1	no	—
59	C₂×C₈	—	9	1	no	—
60	C₄⋊C₄	—	9	2	no	—
61	C₄⋊C₄	—	18	2	no	—
62	C₄²	—	6	2	no	—
63	C₄²	—	9	1	no	—
64	C₂²×C₃	—	2	2	no	—
65	C₃×C₄	—	2	4	no	—
66	C₃×C₄	—	6	4	no	—
67	C₃×C₄	—	12	2	no	—
68	Dic₃	—	2	4	no	—
69	Dic₃	—	4	2	no	—
70	Dic₃	—	6	2	no	—
71	Dic₃	—	6	2	no	—
72	Dic₃	—	12	2	no	—
73	C₃²	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
74	Q₈	—	9	4	no	—
75	Q₈	—	18	2	no	—
76	C₂×C₄	—	6	2	no	—
77	C₂×C₄	—	6	2	no	—
78	C₂×C₄	—	9	1	no	—
79	C₂×C₄	—	9	1	no	—
80	C₂×C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic
81	C₈	—	9	2	no	—
82	C₈	—	18	1	no	—
83	C₂×C₃	—	2	2	no	—
84	C₂×C₃	—	2	2	no	—
85	C₂×C₃	—	2	2	no	—
86	C₂²	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic
87	C₄	—	6	4	no	—
88	C₄	—	9	1	no	—
89	C₄	—	9	1	no	—
90	C₄	—	12	2	no	—
91	C₄	—	18	1	no	—
92	C₄	#3776	1	2	yes	direct product factor
93	C₃	—	2	2	no	—
94	C₂	#64223	1	1	yes	characteristic
95	C₂	C₄×GO⁺(4,𝔽₂)	1	1	yes	characteristic
96	C₂	C₂×#3776	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	GO⁺(4,𝔽₂)
commutator	C₃⋊Dic₃	C₂²×C₄
Frattini	C₂²	C₂×GO⁺(4,𝔽₂)
Fitting	C₂×C₃²×C₄	D₄
socle, prime radical	C₂²×C₃²	C₂×D₄
cosocle	C₂×C₃⋊Dic₃	C₂³
hypocenter	C₃²	C₄×Q₁₆
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
9. #3776	C₄	direct (N × H)
33. C₃²×C₄	Q₁₆	semidirect
73. C₃²	C₄×Q₁₆	semidirect
92. C₄	#3776	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#3776	2
C₄×#627	2
C₄×#622	2
#63743	2
#63740	2
C₄×Q₁₆	9

Subgroups — C4×#3776 #10593444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#3776 #10593444