Finite Group Databasebeta

Subgroups — #10591035

Order 576.

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

normal subgroups only

This group has 199 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	#64121	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	#63551	C₂	1	1	yes	maximal, characteristic, complemented
6	#63548	C₂	1	1	yes	maximal, characteristic, complemented
7	#63542	C₂	1	1	yes	maximal, characteristic, complemented
8	#4916	—	3	1	no	maximal
9	#4912	—	3	1	no	maximal
10	#3830	—	2	1	no	—
11	#3796	—	2	1	no	—
12	C₃²×D₈	C₂²	1	1	yes	characteristic
13	#3754	—	2	1	no	—
14	#3746	C₂²	1	1	yes	characteristic
15	C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
16	C₃×Dic₁₂	C₂²	1	1	yes	characteristic
17	C₃×D₂₄	C₂²	1	1	yes	characteristic
18	#3715	—	2	1	no	—
19	C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	#919	—	3	1	no	—
22	#912	—	3	1	no	—
23	#912	—	3	1	no	—
24	#912	—	3	1	no	—
25	#912	—	6	1	no	—
26	C₃×SD₃₂	—	3	1	no	—
27	C₃×D₁₆	—	3	1	no	—
28	#829	—	3	1	no	—
29	#827	—	3	1	no	—
30	#827	D₃	1	1	yes	characteristic, complemented
31	#826	D₃	1	1	yes	characteristic, complemented
32	Dic₂₄	—	3	1	no	—
33	#800	—	3	1	no	—
34	D₃×C₁₆	—	3	1	no	—
35	D₃×C₁₆	—	3	1	no	—
36	C₃²×D₄	—	2	1	no	—
37	#638	—	4	1	no	—
38	C₂×C₃⋊Dic₃	—	4	1	no	—
39	C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
40	#634	—	2	1	no	—
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	#508	—	9	1	no	Sylow
48	#239	—	6	1	no	—
49	#237	—	6	1	no	—
50	#237	—	6	1	no	—
51	#237	—	6	1	no	—
52	#237	—	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	#214	—	6	1	no	—
61	#214	—	6	1	no	—
62	#214	—	6	2	no	—
63	Dic₁₂	—	3	1	no	—
64	Dic₁₂	—	3	1	no	—
65	Dic₁₂	—	6	1	no	—
66	Dic₁₂	C₂×D₃	1	1	yes	characteristic
67	D₂₄	C₂×D₃	1	1	yes	characteristic
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
74	C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
75	C₂²×C₃²	—	4	1	no	—
76	C₂×C₃⋊D₃	D₈	1	1	yes	characteristic
77	C₂×C₃×D₃	—	4	1	no	—
78	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₃⋊Dic₃	—	4	1	no	—
80	C₃⋊Dic₃	D₈	1	1	yes	characteristic
81	C₃×Dic₃	—	4	1	no	—
82	#135	—	9	1	no	—
83	#135	—	9	1	no	—
84	Q₃₂	—	9	1	no	—
85	SD₃₂	—	3	1	no	—
86	SD₃₂	—	9	1	no	—
87	D₁₆	—	3	1	no	—
88	C₂×C₁₆	—	9	1	no	—
89	C₃×Q₈	—	6	1	no	—
90	C₃×D₄	—	2	2	no	—
91	C₃×D₄	—	2	1	no	—
92	C₃×D₄	—	2	1	no	—
93	C₃×D₄	—	6	1	no	—
94	#67	—	12	1	no	—
95	#67	—	12	1	no	—
96	#67	—	12	1	no	—
97	#67	—	12	2	no	—
98	C₂×Dic₃	—	12	1	no	—
99	C₂×Dic₃	—	12	1	no	—
100	C₂×Dic₃	—	12	1	no	—
101	C₂×Dic₃	—	12	2	no	—
102	D₁₂	—	2	1	no	—
103	D₁₂	—	12	1	no	—
104	C₄×D₃	—	3	1	no	—
105	C₄×D₃	—	3	1	no	—
106	C₄×D₃	—	6	1	no	—
107	C₄×D₃	—	12	1	no	—
108	Dic₆	—	2	1	no	—
109	Dic₆	—	6	2	no	—
110	Dic₆	—	6	1	no	—
111	Dic₆	—	6	1	no	—
112	C₃×C₈	—	2	1	no	—
113	C₃×C₈	C₂²×D₃	1	1	yes	characteristic
114	C₃×C₈	C₂²×D₃	1	1	yes	characteristic
115	C₃⋊C₈	—	3	1	no	—
116	C₃⋊C₈	—	3	1	no	—
117	C₃⋊C₈	—	6	1	no	—
118	C₂×C₃²	—	8	1	no	—
119	C₂×C₃²	C₂×D₈	1	1	yes	characteristic
120	C₃⋊D₃	—	2	1	no	—
121	C₃×D₃	—	8	1	no	—
122	#41	—	18	1	no	—
123	#41	—	18	1	no	—
124	Q₁₆	—	3	1	no	—
125	Q₁₆	—	9	1	no	—
126	SD₁₆	—	18	1	no	—
127	SD₁₆	—	18	1	no	—
128	D₈	—	3	1	no	—
129	D₈	D₃²	1	1	yes	characteristic
130	C₂×C₈	—	9	1	no	—
131	C₁₆	—	3	1	no	—
132	C₁₆	—	3	1	no	—
133	C₂²×C₃	—	4	2	no	—
134	C₂²×C₃	—	4	1	no	—
135	C₂²×C₃	—	4	1	no	—
136	C₂²×C₃	—	12	1	no	—
137	C₂×D₃	—	3	1	no	—
138	C₂×D₃	—	3	1	no	—
139	C₂×D₃	—	4	1	no	—
140	C₂×D₃	—	6	1	no	—
141	C₃×C₄	—	2	1	no	—
142	C₃×C₄	—	12	1	no	—
143	C₃×C₄	D₃×D₄	1	1	yes	characteristic
144	C₃×C₄	D₃×D₄	1	1	yes	characteristic
145	Dic₃	—	3	1	no	—
146	Dic₃	—	3	1	no	—
147	Dic₃	—	4	1	no	—
148	Dic₃	—	6	1	no	—
149	Dic₃	—	12	2	no	—
150	Dic₃	—	12	1	no	—
151	Dic₃	—	12	1	no	—
152	C₃²	#508	1	1	yes	characteristic, complemented, Sylow
153	Q₈	—	6	1	no	—
154	Q₈	—	18	1	no	—
155	D₄	—	2	1	no	—
156	D₄	—	6	1	no	—
157	D₄	—	36	1	no	—
158	D₄	—	36	1	no	—
159	C₂×C₄	—	9	1	no	—
160	C₂×C₄	—	36	1	no	—
161	C₂×C₄	—	36	1	no	—
162	C₈	—	9	1	no	—
163	C₈	C₂×D₃²	1	1	yes	characteristic
164	C₂×C₃	—	2	1	no	—
165	C₂×C₃	—	8	2	no	—
166	C₂×C₃	—	8	1	no	—
167	C₂×C₃	—	8	1	no	—
168	C₂×C₃	—	24	1	no	—
169	C₂×C₃	D₃×D₈	1	1	yes	characteristic
170	C₂×C₃	D₃×D₈	1	1	yes	characteristic
171	D₃	—	6	1	no	—
172	D₃	—	6	1	no	—
173	D₃	—	8	1	no	—
174	D₃	—	12	1	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	9	1	no	—
177	C₂²	—	12	1	no	—
178	C₄	—	9	1	no	—
179	C₄	—	12	1	no	—
180	C₄	—	36	1	no	—
181	C₄	#3811	1	1	yes	characteristic
182	C₃	—	2	1	no	—
183	C₃	#4912	1	1	yes	characteristic, complemented
184	C₃	#4916	1	1	yes	characteristic, complemented
185	C₂	—	8	1	no	—
186	C₂	—	18	1	no	—
187	C₂	—	24	1	no	—
188	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	#508
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. #64121	C₂	semidirect
2. #63810	C₂	semidirect
3. C₃×#827	C₂	semidirect
4. C₃×#826	C₂	semidirect
5. #63551	C₂	semidirect
6. #63548	C₂	semidirect
7. #63542	C₂	semidirect
30. #827	D₃	semidirect
31. #826	D₃	semidirect
152. C₃²	#508	semidirect
183. C₃	#4912	semidirect
184. C₃	#4916	semidirect

Maximal subgroups

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

subgroup	index
#64121	2
#63810	2
C₃×#827	2
C₃×#826	2
#63551	2
#63548	2
#63542	2
#4916	3
#4912	3