Finite Group Databasebeta

Subgroups — #10591036

Order 576.

218 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 44 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	D₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#63810	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#827	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#826	C₂	1	1	yes	maximal, characteristic, complemented
5	#63549	C₂	1	1	yes	maximal, characteristic, complemented
6	#63546	C₂	1	1	yes	maximal, characteristic, complemented
7	#63544	C₂	1	1	yes	maximal, characteristic, complemented
8	#4914	—	3	1	no	maximal
9	#4911	—	3	1	no	maximal
10	D₄×C₃⋊D₃	—	2	1	no	—
11	#3796	—	2	1	no	—
12	C₃²×D₈	C₂²	1	1	yes	characteristic
13	#3753	—	2	1	no	—
14	#3745	C₂²	1	1	yes	characteristic
15	C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
16	C₃×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
17	C₃×D₂₄	C₂²	1	1	yes	characteristic, complemented
18	#3715	—	2	1	no	—
19	C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
20	C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
21	#919	—	3	1	no	—
22	#912	—	3	1	no	—
23	D₃×D₈	—	3	1	no	—
24	D₃×D₈	—	3	1	no	—
25	D₃×D₈	—	6	1	no	—
26	C₃×SD₃₂	—	3	1	no	—
27	C₃×D₁₆	—	3	1	no	—
28	#828	—	3	1	no	—
29	#827	D₃	1	1	yes	characteristic, complemented
30	#826	—	3	1	no	—
31	#826	D₃	1	1	yes	characteristic, complemented
32	#800	—	3	1	no	—
33	D₄₈	—	3	1	no	—
34	#798	—	3	1	no	—
35	#798	—	3	1	no	—
36	C₂²×C₃⋊D₃	—	4	1	no	—
37	C₃²×D₄	—	2	1	no	—
38	#638	—	4	1	no	—
39	C₄⋊(C₃⋊D₃)	—	2	1	no	—
40	C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
41	C₃×D₁₂	—	2	1	no	—
42	C₃×Dic₆	—	2	1	no	—
43	#626	—	4	1	no	—
44	D₃×Dic₃	—	4	1	no	—
45	C₃²×C₈	C₂³	1	1	yes	characteristic
46	C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
47	#509	—	9	1	no	Sylow
48	#239	—	6	1	no	—
49	#237	—	6	1	no	—
50	D₃×D₄	—	6	1	no	—
51	D₃×D₄	—	6	1	no	—
52	D₃×D₄	—	6	2	no	—
53	C₃×Q₁₆	—	3	1	no	—
54	C₃×D₈	—	2	1	no	—
55	C₃×D₈	—	3	1	no	—
56	C₃×D₈	C₂×D₃	1	1	yes	characteristic
57	C₃×D₈	C₂×D₃	1	1	yes	characteristic
58	#215	—	6	1	no	—
59	#214	—	6	1	no	—
60	#213	—	6	1	no	—
61	#213	—	6	1	no	—
62	#213	—	6	2	no	—
63	Dic₁₂	C₂×D₃	1	1	yes	characteristic, complemented
64	D₂₄	—	3	1	no	—
65	D₂₄	—	3	1	no	—
66	D₂₄	—	6	1	no	—
67	D₂₄	C₂×D₃	1	1	yes	characteristic, complemented
68	D₃×C₈	—	3	1	no	—
69	D₃×C₈	—	3	1	no	—
70	D₃×C₈	—	6	1	no	—
71	C₃×C₁₆	—	3	1	no	—
72	C₃×C₁₆	—	3	1	no	—
73	C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
74	C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
75	C₂²×C₃²	—	4	1	no	—
76	C₂×C₃⋊D₃	—	4	1	no	—
77	C₂×C₃⋊D₃	—	8	2	no	—
78	C₂×C₃⋊D₃	D₈	1	1	yes	characteristic
79	C₂×C₃×D₃	—	4	1	no	—
80	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₃⋊Dic₃	D₈	1	1	yes	characteristic
82	C₃×Dic₃	—	4	1	no	—
83	#135	—	9	1	no	—
84	C₂×D₈	—	9	1	no	—
85	SD₃₂	—	3	1	no	—
86	SD₃₂	—	9	1	no	—
87	D₁₆	—	3	1	no	—
88	D₁₆	—	9	1	no	—
89	M₃₂	—	9	1	no	—
90	C₂²×D₃	—	12	1	no	—
91	C₂²×D₃	—	12	1	no	—
92	C₂²×D₃	—	12	2	no	—
93	C₃×Q₈	—	6	1	no	—
94	C₃×D₄	—	2	2	no	—
95	C₃×D₄	—	2	1	no	—
96	C₃×D₄	—	2	1	no	—
97	C₃×D₄	—	6	1	no	—
98	#67	—	12	1	no	—
99	#67	—	12	1	no	—
100	#67	—	12	1	no	—
101	#67	—	12	2	no	—
102	C₂×Dic₃	—	12	1	no	—
103	D₁₂	—	2	1	no	—
104	D₁₂	—	6	2	no	—
105	D₁₂	—	6	1	no	—
106	D₁₂	—	6	1	no	—
107	D₁₂	—	12	1	no	—
108	C₄×D₃	—	3	1	no	—
109	C₄×D₃	—	3	1	no	—
110	C₄×D₃	—	6	1	no	—
111	C₄×D₃	—	12	1	no	—
112	Dic₆	—	2	1	no	—
113	C₃×C₈	—	2	1	no	—
114	C₃×C₈	C₂²×D₃	1	1	yes	characteristic
115	C₃×C₈	C₂²×D₃	1	1	yes	characteristic
116	C₃⋊C₈	—	3	1	no	—
117	C₃⋊C₈	—	3	1	no	—
118	C₃⋊C₈	—	6	1	no	—
119	C₂×C₃²	—	8	1	no	—
120	C₂×C₃²	C₂×D₈	1	1	yes	characteristic
121	C₃⋊D₃	—	2	1	no	—
122	C₃⋊D₃	—	8	1	no	—
123	C₃×D₃	—	8	1	no	—
124	#41	—	18	1	no	—
125	C₂×D₄	—	18	1	no	—
126	Q₁₆	—	3	1	no	—
127	SD₁₆	—	18	1	no	—
128	D₈	—	3	1	no	—
129	D₈	—	9	1	no	—
130	D₈	—	18	1	no	—
131	D₈	D₃²	1	1	yes	characteristic
132	C₂×C₈	—	9	1	no	—
133	C₁₆	—	3	1	no	—
134	C₁₆	—	3	1	no	—
135	C₂²×C₃	—	4	2	no	—
136	C₂²×C₃	—	4	1	no	—
137	C₂²×C₃	—	4	1	no	—
138	C₂²×C₃	—	12	1	no	—
139	C₂×D₃	—	3	1	no	—
140	C₂×D₃	—	3	1	no	—
141	C₂×D₃	—	4	1	no	—
142	C₂×D₃	—	6	1	no	—
143	C₂×D₃	—	12	2	no	—
144	C₂×D₃	—	12	1	no	—
145	C₂×D₃	—	12	1	no	—
146	C₂×D₃	—	24	4	no	—
147	C₂×D₃	—	24	2	no	—
148	C₂×D₃	—	24	2	no	—
149	C₃×C₄	—	2	1	no	—
150	C₃×C₄	—	12	1	no	—
151	C₃×C₄	D₃×D₄	1	1	yes	characteristic
152	C₃×C₄	D₃×D₄	1	1	yes	characteristic
153	Dic₃	—	3	1	no	—
154	Dic₃	—	3	1	no	—
155	Dic₃	—	4	1	no	—
156	Dic₃	—	6	1	no	—
157	C₃²	#509	1	1	yes	characteristic, complemented, Sylow
158	C₂³	—	36	1	no	—
159	Q₈	—	6	1	no	—
160	D₄	—	2	1	no	—
161	D₄	—	6	1	no	—
162	D₄	—	18	1	no	—
163	D₄	—	36	1	no	—
164	D₄	—	36	1	no	—
165	C₂×C₄	—	9	1	no	—
166	C₂×C₄	—	36	1	no	—
167	C₈	—	9	1	no	—
168	C₈	C₂×D₃²	1	1	yes	characteristic
169	C₂×C₃	—	2	1	no	—
170	C₂×C₃	—	8	2	no	—
171	C₂×C₃	—	8	1	no	—
172	C₂×C₃	—	8	1	no	—
173	C₂×C₃	—	24	1	no	—
174	C₂×C₃	D₃×D₈	1	1	yes	characteristic
175	C₂×C₃	D₃×D₈	1	1	yes	characteristic
176	D₃	—	6	2	no	—
177	D₃	—	6	1	no	—
178	D₃	—	6	1	no	—
179	D₃	—	8	1	no	—
180	D₃	—	24	2	no	—
181	D₃	—	24	1	no	—
182	D₃	—	24	1	no	—
183	C₂²	—	4	1	no	—
184	C₂²	—	9	1	no	—
185	C₂²	—	12	1	no	—
186	C₂²	—	36	1	no	—
187	C₂²	—	72	2	no	—
188	C₄	—	9	1	no	—
189	C₄	—	12	1	no	—
190	C₄	#3811	1	1	yes	characteristic
191	C₃	—	2	1	no	—
192	C₃	#4911	1	1	yes	characteristic, complemented
193	C₃	#4914	1	1	yes	characteristic, complemented
194	C₂	—	8	1	no	—
195	C₂	—	18	1	no	—
196	C₂	—	24	1	no	—
197	C₂	—	72	1	no	—
198	C₂	#63926	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#63926
commutator, cosocle	C₃²×C₈	C₂³
Frattini	C₈	C₂×D₃²
Fitting	C₃²×D₈	C₂²
socle	C₂×C₃²	C₂×D₈
hypercenter	D₈	D₃²
hypocenter	C₃²	#509
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₈×C₃⋊D₃	C₂	semidirect
2. #63810	C₂	semidirect
3. C₃×#827	C₂	semidirect
4. C₃×#826	C₂	semidirect
5. #63549	C₂	semidirect
6. #63546	C₂	semidirect
7. #63544	C₂	semidirect
16. C₃×Dic₁₂	C₂²	semidirect
17. C₃×D₂₄	C₂²	semidirect
19. C₃×C₃⋊C₁₆	C₂²	semidirect
20. C₃×C₃⋊C₁₆	C₂²	semidirect
29. #827	D₃	semidirect
31. #826	D₃	semidirect
63. Dic₁₂	C₂×D₃	semidirect
67. D₂₄	C₂×D₃	semidirect
73. C₃⋊C₁₆	C₂×D₃	semidirect
74. C₃⋊C₁₆	C₂×D₃	semidirect
157. C₃²	#509	semidirect
192. C₃	#4911	semidirect
193. C₃	#4914	semidirect

Maximal subgroups

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

subgroup	index
D₈×C₃⋊D₃	2
#63810	2
C₃×#827	2
C₃×#826	2
#63549	2
#63546	2
#63544	2
#4914	3
#4911	3