Finite Group Databasebeta

Subgroups — #152815

Order 512.

204 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 84 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	#13627	C₂	1	1	yes	maximal, characteristic, complemented
2	#13558	C₂	1	1	yes	maximal, characteristic
3	#13556	C₂	1	1	yes	maximal, characteristic
4	#13551	C₂	1	1	yes	maximal, characteristic, complemented
5	#12614	C₂	1	1	yes	maximal, characteristic
6	#7469	C₂	1	1	yes	maximal, characteristic, complemented
7	#7467	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#504	C₂²	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂×M₆₄	C₂²	1	1	yes	characteristic
12	C₂×M₆₄	C₂²	1	1	yes	characteristic
13	C₂×#350	C₄	1	1	yes	characteristic
14	#2108	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	#1764	C₄	1	1	yes	characteristic
17	C₄⋊C₃₂	—	2	2	no	—
18	#1401	—	2	2	no	—
19	#1400	—	2	2	no	—
20	#1398	—	2	2	no	—
21	#1379	C₄	1	1	yes	characteristic
22	#1376	C₄	1	1	yes	characteristic
23	C₂×#131	C₂×C₄	1	1	yes	characteristic
24	#504	—	2	1	no	—
25	#504	—	2	1	no	—
26	#504	—	2	1	no	—
27	#504	—	2	1	no	—
28	C₂×M₃₂	—	2	1	no	—
29	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
30	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₁₆	—	2	1	no	—
32	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₆	C₂³	1	1	yes	characteristic
34	C₂×#104	C₈	1	1	yes	characteristic
35	#416	C₈	1	1	yes	characteristic
36	#405	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	M₆₄	—	2	1	no	—
39	M₆₄	—	2	1	no	—
40	M₆₄	—	2	1	no	—
41	M₆₄	—	2	1	no	—
42	M₆₄	—	2	1	no	—
43	M₆₄	—	2	1	no	—
44	M₆₄	—	2	1	no	—
45	M₆₄	D₄	1	2	yes	—
46	C₂×C₃₂	—	2	1	no	—
47	C₂×C₃₂	—	2	1	no	—
48	C₂×C₃₂	—	2	1	no	—
49	C₂×C₃₂	D₄	1	2	yes	—
50	#350	—	2	2	no	—
51	#346	—	2	1	no	—
52	C₄×C₁₆	—	2	1	no	—
53	C₄×C₁₆	—	2	1	no	—
54	C₄×C₁₆	—	2	1	no	—
55	#341	C₈	1	1	yes	characteristic
56	#339	C₈	1	1	yes	characteristic
57	C₂×#41	C₂×C₈	1	1	yes	characteristic
58	#131	—	2	1	no	—
59	#131	—	2	1	no	—
60	#131	—	2	1	no	—
61	#131	—	2	1	no	—
62	C₂×M₁₆	—	2	1	no	—
63	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
64	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
68	#117	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
70	M₃₂	—	2	1	no	—
71	M₃₂	—	2	1	no	—
72	M₃₂	—	2	1	no	—
73	M₃₂	—	4	1	no	—
74	M₃₂	—	4	1	no	—
75	M₃₂	M₁₆	1	2	yes	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	4	1	no	—
82	C₂×C₁₆	—	4	1	no	—
83	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
84	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
85	C₂×C₁₆	#41	1	1	yes	characteristic
86	C₂×C₁₆	#41	1	1	yes	characteristic
87	C₂×C₁₆	#41	1	1	yes	characteristic
88	C₂×C₁₆	#41	1	1	yes	characteristic
89	#104	—	2	1	no	—
90	#104	—	2	1	no	—
91	#103	—	2	1	no	—
92	#102	—	2	1	no	—
93	#97	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₃₂	—	2	2	no	—
98	C₃₂	—	2	2	no	—
99	C₃₂	—	4	1	no	—
100	C₃₂	—	4	1	no	—
101	#41	—	2	1	no	—
102	#41	—	2	1	no	—
103	#41	—	2	1	no	—
104	#41	—	2	1	no	—
105	C₂×Q₈	M₃₂	1	1	yes	characteristic
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	M₃₂	1	1	yes	characteristic
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
111	M₁₆	—	2	1	no	—
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	4	1	no	—
116	M₁₆	—	4	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	#117	1	1	yes	characteristic
125	C₂×C₈	C₄×D₄	1	1	yes	characteristic
126	C₂×C₈	C₄×D₄	1	1	yes	characteristic
127	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₈	#131	1	1	yes	characteristic
129	C₂×C₈	#131	1	1	yes	characteristic
130	C₄⋊C₄	—	2	1	no	—
131	C₂²⋊C₄	—	2	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	2	1	no	—
134	C₄²	M₃₂	1	1	yes	characteristic
135	C₄²	M₃₂	1	1	yes	characteristic
136	C₁₆	—	2	2	no	—
137	C₁₆	—	4	1	no	—
138	C₁₆	—	4	1	no	—
139	C₁₆	#121	1	1	yes	characteristic
140	C₁₆	#123	1	1	yes	characteristic
141	C₁₆	#124	1	1	yes	characteristic
142	C₁₆	#126	1	1	yes	characteristic
143	C₂³	—	2	1	no	—
144	C₂³	#504	1	1	yes	characteristic
145	Q₈	—	2	1	no	—
146	Q₈	—	2	1	no	—
147	D₄	—	2	1	no	—
148	D₄	—	2	1	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	#431	1	1	yes	characteristic
159	C₂×C₄	C₈×D₄	1	1	yes	characteristic
160	C₂×C₄	C₈×D₄	1	1	yes	characteristic
161	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
162	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
163	C₂×C₄	#504	1	1	yes	characteristic
164	C₈	—	4	1	no	—
165	C₈	—	4	1	no	—
166	C₈	—	4	1	no	—
167	C₈	#388	1	1	yes	characteristic
168	C₈	#433	1	1	yes	characteristic
169	C₈	#435	1	1	yes	characteristic
170	C₈	#436	1	1	yes	characteristic
171	C₂²	—	4	1	no	—
172	C₂²	—	4	1	no	—
173	C₂²	—	8	1	no	—
174	C₂²	#2164	1	1	yes	characteristic
175	C₂²	#2168	1	1	yes	characteristic
176	C₂²	#2168	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	4	1	no	—
179	C₄	—	4	1	no	—
180	C₄	—	8	1	no	—
181	C₄	#1923	1	1	yes	characteristic
182	C₄	#2163	1	1	yes	characteristic
183	C₄	#2168	1	1	yes	characteristic
184	C₄	#2169	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	—	8	1	no	—
189	C₂	#13121	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13121
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #13627	C₂	semidirect
4. #13551	C₂	semidirect
6. #7469	C₂	semidirect
7. #7467	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13627	2
#13558	2
#13556	2
#13551	2
#12614	2
#7469	2
#7467	2