Finite Group Databasebeta

Subgroups — #10596221

Order 576.

774 conjugacy classes of subgroups; only the 259 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 259 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	#64157	C₂	1	1	yes	maximal, characteristic, complemented
4	#64116	C₂	1	3	yes	maximal, complemented
5	M₁₆×C₃⋊D₃	C₂	1	3	yes	maximal, complemented
6	#64110	C₂	1	3	yes	maximal, complemented
7	C₂×#3743	C₂	1	3	yes	maximal, complemented
8	#5750	—	3	4	no	maximal
9	C₃²×#41	C₂²	1	1	yes	characteristic
10	#3832	C₄	1	1	yes	characteristic
11	Q₈×C₃⋊D₃	C₄	1	1	yes	characteristic
12	#3830	C₄	1	3	yes	—
13	D₄×C₃⋊D₃	C₄	1	3	yes	—
14	#3828	C₂²	1	3	yes	—
15	C₂×C₄×C₃⋊D₃	C₂²	1	3	yes	—
16	C₃²×M₁₆	C₂²	1	3	yes	complemented
17	C₂×C₃²×C₈	C₂²	1	3	yes	complemented
18	#3748	C₂²	1	3	yes	complemented
19	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	3	yes	complemented
20	#3743	C₂²	1	1	yes	characteristic
21	#3743	C₂²	1	3	yes	—
22	#3743	C₂²	1	3	yes	complemented
23	#3743	C₂²	1	3	yes	complemented
24	C₈×C₃⋊D₃	C₂²	1	6	yes	complemented
25	D₃×#41	—	3	4	no	—
26	C₃×#131	D₃	1	4	yes	complemented
27	#948	—	3	4	no	—
28	#907	—	3	12	no	—
29	D₃×M₁₆	—	3	12	no	—
30	#901	—	3	12	no	—
31	C₂×ΓL(1,𝔽₂₅)	—	3	12	no	—
32	C₂²×C₃⋊D₃	C₂×C₄	1	3	yes	—
33	C₃²×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₃²×D₄	C₂×C₄	1	3	yes	—
35	C₂×C₃²×C₄	C₂³	1	3	yes	—
36	#638	C₂×C₄	1	6	yes	—
37	C₂×C₃⋊Dic₃	C₂×C₄	1	3	yes	—
38	C₄⋊(C₃⋊D₃)	C₂×C₄	1	3	yes	—
39	C₄×C₃⋊D₃	C₂×C₄	1	3	yes	—
40	C₄×C₃⋊D₃	C₂×C₄	1	3	yes	—
41	C₄×C₃⋊D₃	C₂³	1	3	yes	—
42	C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
43	#634	C₂×C₄	1	3	yes	—
44	C₃²×C₈	C₂³	1	3	yes	—
45	C₃²×C₈	C₂³	1	1	yes	characteristic
46	C₃⋊(C₃⋊C₈)	C₂³	1	3	yes	—
47	C₃⋊(C₃⋊C₈)	C₂³	1	1	yes	characteristic
48	#568	—	9	1	no	Sylow
49	C₃×#41	C₂×D₃	1	4	yes	—
50	#239	—	3	4	no	—
51	D₃×Q₈	—	3	4	no	—
52	#237	—	3	12	no	—
53	D₃×D₄	—	3	12	no	—
54	#235	—	3	12	no	—
55	C₂×C₄×D₃	—	3	12	no	—
56	C₃×M₁₆	C₂×D₃	1	12	yes	complemented
57	C₂×C₃×C₈	C₂×D₃	1	12	yes	complemented
58	#208	—	3	12	no	—
59	C₂×C₃⋊C₈	—	3	12	no	—
60	ΓL(1,𝔽₂₅)	—	3	4	no	—
61	ΓL(1,𝔽₂₅)	—	3	12	no	—
62	ΓL(1,𝔽₂₅)	—	3	12	no	—
63	ΓL(1,𝔽₂₅)	—	3	12	no	—
64	D₃×C₈	—	3	24	no	—
65	C₂²×C₃²	C₂²×C₄	1	3	yes	—
66	C₂×C₃⋊D₃	—	4	3	no	—
67	C₂×C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃⋊D₃	C₂²×C₄	1	3	yes	—
69	C₃²×C₄	C₂²×C₄	1	3	yes	—
70	C₃²×C₄	C₂⁴	1	1	yes	characteristic
71	C₃⋊Dic₃	C₂²×C₄	1	3	yes	—
72	C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×#41	—	9	1	no	—
74	#131	—	9	1	no	—
75	#131	—	9	3	no	—
76	#131	—	9	3	no	—
77	#131	C₃⋊D₃	1	1	yes	characteristic, complemented
78	C₂×M₁₆	—	9	3	no	—
79	C₂×M₁₆	—	9	3	no	—
80	C₂²×D₃	—	3	12	no	—
81	C₃×Q₈	C₄×D₃	1	4	yes	—
82	C₃×D₄	C₄×D₃	1	12	yes	—
83	C₂×C₃×C₄	C₂²×D₃	1	12	yes	—
84	#67	—	3	24	no	—
85	C₂×Dic₃	—	3	12	no	—
86	D₁₂	—	3	12	no	—
87	C₄×D₃	—	3	12	no	—
88	C₄×D₃	—	3	12	no	—
89	C₄×D₃	—	3	12	no	—
90	C₄×D₃	—	3	4	no	—
91	Dic₆	—	3	12	no	—
92	C₃×C₈	C₂²×D₃	1	12	yes	—
93	C₃×C₈	C₂²×D₃	1	4	yes	—
94	C₃⋊C₈	—	3	12	no	—
95	C₃⋊C₈	—	3	4	no	—
96	C₂×C₃²	—	2	3	no	—
97	C₂×C₃²	C₂³×C₄	1	1	yes	characteristic
98	C₃⋊D₃	—	2	1	no	—
99	C₃⋊D₃	—	2	3	no	—
100	#41	—	9	1	no	—
101	#41	—	9	3	no	—
102	#41	—	9	3	no	—
103	#41	C₂×C₃⋊D₃	1	1	yes	characteristic
104	C₂×Q₈	—	9	1	no	—
105	C₂×D₄	—	9	3	no	—
106	C₂²×C₄	—	9	3	no	—
107	M₁₆	—	9	3	no	—
108	M₁₆	—	9	3	no	—
109	M₁₆	—	9	1	no	—
110	M₁₆	—	9	3	no	—
111	M₁₆	—	9	3	no	—
112	M₁₆	C₂×C₃⋊D₃	1	3	yes	complemented
113	C₂×C₈	—	9	3	no	—
114	C₂×C₈	—	9	6	no	—
115	C₂×C₈	C₂×C₃⋊D₃	1	3	yes	complemented
116	C₂²×C₃	C₂×C₄×D₃	1	12	yes	—
117	C₂×D₃	—	3	4	no	—
118	C₂×D₃	—	3	12	no	—
119	C₂×D₃	—	12	12	no	—
120	C₃×C₄	C₂×C₄×D₃	1	12	yes	—
121	C₃×C₄	C₂³×D₃	1	4	yes	—
122	Dic₃	—	3	12	no	—
123	Dic₃	—	3	4	no	—
124	C₃²	#568	1	1	yes	characteristic, complemented, Sylow
125	C₂³	—	9	3	no	—
126	Q₈	—	9	3	no	—
127	Q₈	C₄×C₃⋊D₃	1	1	yes	characteristic
128	D₄	—	9	6	no	—
129	D₄	—	9	3	no	—
130	D₄	C₄×C₃⋊D₃	1	3	yes	—
131	C₂×C₄	—	9	3	no	—
132	C₂×C₄	—	9	3	no	—
133	C₂×C₄	—	9	1	no	—
134	C₂×C₄	—	9	3	no	—
135	C₂×C₄	—	9	3	no	—
136	C₂×C₄	C₂²×C₃⋊D₃	1	3	yes	—
137	C₈	—	9	1	no	—
138	C₈	—	9	3	no	—
139	C₈	C₂²×C₃⋊D₃	1	3	yes	—
140	C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
141	C₂×C₃	—	2	12	no	—
142	C₂×C₃	C₂²×C₄×D₃	1	4	yes	—
143	D₃	—	6	4	no	—
144	D₃	—	6	12	no	—
145	C₂²	—	9	1	no	—
146	C₂²	—	9	3	no	—
147	C₂²	—	36	3	no	—
148	C₂²	C₂×C₄×C₃⋊D₃	1	3	yes	—
149	C₄	—	9	1	no	—
150	C₄	—	9	3	no	—
151	C₄	C₂×C₄×C₃⋊D₃	1	3	yes	—
152	C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
153	C₃	#5750	1	4	yes	complemented
154	C₂	—	2	3	no	—
155	C₂	—	18	1	no	—
156	C₂	—	18	3	no	—
157	C₂	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×C₃⋊D₃
commutator, socle	C₂×C₃²	C₂³×C₄
Fitting	C₃²×#131	C₂
prime radical	#41×C₃⋊D₃	C₂
cosocle	C₃²×C₄	C₂⁴
hypercenter	#131	C₃⋊D₃
hypocenter	C₃²	#568
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ #131

Complements

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

N	H ≅ G/N	type
2. C₃²×#131	C₂	semidirect
3. #64157	C₂	semidirect
4. #64116	C₂	semidirect
5. M₁₆×C₃⋊D₃	C₂	semidirect
6. #64110	C₂	semidirect
7. C₂×#3743	C₂	semidirect
16. C₃²×M₁₆	C₂²	semidirect
17. C₂×C₃²×C₈	C₂²	semidirect
18. #3748	C₂²	semidirect
19. C₂×C₃⋊(C₃⋊C₈)	C₂²	semidirect
22. #3743	C₂²	semidirect
23. #3743	C₂²	semidirect
24. C₈×C₃⋊D₃	C₂²	semidirect
26. C₃×#131	D₃	semidirect
56. C₃×M₁₆	C₂×D₃	semidirect
57. C₂×C₃×C₈	C₂×D₃	semidirect
77. #131	C₃⋊D₃	semidirect
112. M₁₆	C₂×C₃⋊D₃	semidirect
115. C₂×C₈	C₂×C₃⋊D₃	semidirect
124. C₃²	#568	semidirect
153. C₃	#5750	semidirect

Maximal subgroups

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

subgroup	index
#41×C₃⋊D₃	2
C₃²×#131	2
#64157	2
#64116	2
M₁₆×C₃⋊D₃	2
#64110	2
C₂×#3743	2
#5750	3