Subgroups — #10589824

Order 576.

318 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#63506	C₂	1	4	yes	maximal
3	#63450	C₂	1	4	yes	maximal
4	#63449	C₂	1	4	yes	maximal
5	C₄×Dic₁₈	C₂	1	2	yes	maximal
6	#5722	—	3	1	no	maximal
7	C₂×Q₈×C₉	C₂²	1	2	yes	—
8	C₂×Dic₁₈	C₂²	1	4	yes	—
9	C₂×Dic₁₈	C₂²	1	2	yes	—
10	C₉×C₄⋊C₄	C₂²	1	4	yes	—
11	C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₉	C₂²	1	2	yes	—
13	Dic₉⋊C₄	C₂²	1	4	yes	—
14	Dic₉⋊C₄	C₂²	1	8	yes	—
15	C₄×Dic₉	C₂²	1	4	yes	—
16	C₄×Dic₉	C₂²	1	4	yes	—
17	C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
18	#944	—	3	4	no	—
19	#888	—	3	4	no	—
20	#887	—	3	4	no	—
21	C₄×Dic₆	—	3	2	no	—
22	Q₈×C₉	—	4	2	no	—
23	C₂×C₄×C₉	C₂³	1	4	yes	—
24	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₉	C₂³	1	2	yes	—
26	C₂×Dic₉	C₂³	1	4	yes	—
27	C₂×Dic₉	C₂³	1	4	yes	—
28	Dic₁₈	—	4	4	no	—
29	Dic₁₈	Q₈	1	8	yes	complemented
30	Q₈²	—	9	1	no	Sylow
31	C₂×C₃×Q₈	C₂×D₃	1	2	yes	—
32	C₂×Dic₆	—	3	2	no	—
33	C₂×Dic₆	—	3	4	no	—
34	C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
35	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
36	C₄⋊Dic₃	—	3	2	no	—
37	Dic₃⋊C₄	—	3	4	no	—
38	Dic₃⋊C₄	—	3	8	no	—
39	C₄×Dic₃	—	3	4	no	—
40	C₄×Dic₃	—	3	4	no	—
41	C₂²×C₉	C₂⁴	1	1	yes	characteristic
42	C₄×C₉	—	2	4	no	—
43	C₄×C₉	—	2	1	no	—
44	C₄×C₉	C₂×Q₈	1	4	yes	—
45	Dic₉	—	2	4	no	—
46	Dic₉	C₂×Q₈	1	8	yes	—
47	C₄⋊Q₈	—	9	4	no	—
48	C₄⋊Q₈	—	9	4	no	—
49	C₄⋊Q₈	D₉	1	1	yes	characteristic
50	C₄×Q₈	—	9	4	no	—
51	C₄×Q₈	—	9	2	no	—
52	C₃×Q₈	—	4	2	no	—
53	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
54	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
56	C₂×Dic₃	—	3	4	no	—
57	C₂×Dic₃	—	3	4	no	—
58	Dic₆	—	3	8	no	—
59	Dic₆	—	12	4	no	—
60	C₂×C₉	C₂²×Q₈	1	2	yes	—
61	C₂×C₉	H(5,𝔽₂)	1	1	yes	characteristic
62	C₂×Q₈	—	9	2	no	—
63	C₂×Q₈	—	9	4	no	—
64	C₂×Q₈	C₂×D₉	1	2	yes	—
65	C₄⋊C₄	—	9	8	no	—
66	C₄⋊C₄	—	9	4	no	—
67	C₄⋊C₄	—	9	2	no	—
68	C₄⋊C₄	C₂×D₉	1	4	yes	—
69	C₄²	—	9	4	no	—
70	C₄²	—	9	4	no	—
71	C₄²	C₂×D₉	1	1	yes	characteristic
72	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
73	C₃×C₄	—	2	4	no	—
74	C₃×C₄	—	2	1	no	—
75	C₃×C₄	D₃×Q₈	1	4	yes	—
76	Dic₃	—	3	8	no	—
77	Dic₃	—	6	4	no	—
78	C₉	Q₈²	1	1	yes	characteristic, complemented, Sylow
79	Q₈	—	4	2	no	—
80	Q₈	—	9	8	no	—
81	Q₈	—	36	4	no	—
82	C₂×C₄	—	9	4	no	—
83	C₂×C₄	—	9	4	no	—
84	C₂×C₄	C₂²×D₉	1	4	yes	—
85	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
86	C₂×C₄	C₂²×D₉	1	2	yes	—
87	C₂×C₃	#1004	1	1	yes	characteristic
88	C₂×C₃	C₂×D₃×Q₈	1	2	yes	—
89	C₂²	C₂³×D₉	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	2	4	no	—
92	C₄	—	9	8	no	—
93	C₄	—	18	4	no	—
94	C₄	Q₈×D₉	1	4	yes	—
95	C₃	#5722	1	1	yes	characteristic
96	C₂	#63710	1	1	yes	characteristic
97	C₂	C₂×Q₈×D₉	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴
Frattini, socle, prime radical	C₂²×C₃	C₂³×D₃
Fitting	C₉×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₉
hypocenter	C₉	Q₈²
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
29. Dic₁₈	Q₈	semidirect
78. C₉	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
C₉×C₄⋊Q₈	2
#63506	2
#63450	2
#63449	2
C₄×Dic₁₈	2
#5722	3