Finite Group Databasebeta

Subgroups — #134597

Order 512.

290 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 110 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	#21732	C₂	1	1	yes	maximal, characteristic
2	#12900	C₂	1	1	yes	maximal, characteristic
3	#12900	C₂	1	1	yes	maximal, characteristic
4	#11956	C₂	1	2	yes	maximal
5	#7270	C₂	1	2	yes	maximal
6	#3160	C₂²	1	1	yes	characteristic
7	#2960	C₄	1	2	yes	—
8	C₂×#364	—	2	1	no	—
9	C₂×#364	—	2	1	no	—
10	C₂×#364	C₂²	1	2	yes	—
11	C₂×#364	C₂²	1	2	yes	—
12	#2150	—	2	1	no	—
13	#2150	—	2	1	no	—
14	#1994	C₄	1	2	yes	—
15	#1945	—	2	1	no	—
16	#1945	—	2	1	no	—
17	#1848	C₂²	1	1	yes	characteristic
18	#1772	—	2	1	no	—
19	#1772	—	2	1	no	—
20	C₄²×C₈	C₂²	1	1	yes	characteristic
21	#1366	—	2	4	no	—
22	#533	C₂×C₄	1	2	yes	—
23	C₂×M₃₂	—	2	2	no	—
24	C₂×M₃₂	—	2	2	no	—
25	#500	—	2	1	no	—
26	#499	D₄	1	2	yes	—
27	#499	Q₈	1	2	yes	—
28	#498	—	2	1	no	—
29	#446	—	4	2	no	—
30	C₈×Q₈	—	4	2	no	—
31	C₂×#108	—	2	1	no	—
32	C₂×#108	—	2	1	no	—
33	#427	—	2	1	no	—
34	#427	—	2	1	no	—
35	#424	—	2	2	no	—
36	#423	C₂×C₄	1	2	yes	—
37	#405	—	2	2	no	—
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	C₄³	C₂×C₄	1	1	yes	characteristic
45	#364	—	2	2	no	—
46	#364	—	2	2	no	—
47	#364	—	2	2	no	—
48	#364	—	2	2	no	—
49	#364	—	4	1	no	—
50	#364	—	4	1	no	—
51	#364	—	4	1	no	—
52	#364	—	4	1	no	—
53	C₄⋊C₁₆	—	4	2	no	—
54	C₄⋊C₁₆	—	4	2	no	—
55	#339	—	2	2	no	—
56	#339	—	2	2	no	—
57	C₂×M₁₆	—	2	2	no	—
58	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
62	C₄⋊Q₈	—	2	2	no	—
63	#125	C₄⋊C₄	1	4	yes	—
64	C₂²⋊Q₈	—	4	2	no	—
65	C₄×Q₈	—	4	2	no	—
66	#117	—	2	2	no	—
67	C₂×C₄²	—	4	1	no	—
68	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
71	M₃₂	—	4	2	no	—
72	M₃₂	—	4	2	no	—
73	M₃₂	—	4	2	no	—
74	M₃₂	—	4	2	no	—
75	C₂×C₁₆	—	4	2	no	—
76	C₂×C₁₆	—	4	2	no	—
77	#108	—	4	1	no	—
78	#108	—	4	1	no	—
79	#108	—	4	1	no	—
80	#108	—	4	1	no	—
81	#107	—	2	2	no	—
82	#107	—	2	2	no	—
83	#106	—	2	2	no	—
84	#106	—	2	2	no	—
85	C₄⋊C₈	—	4	2	no	—
86	C₄⋊C₈	—	4	4	no	—
87	#97	—	4	2	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	—	4	2	no	—
95	C₄×C₈	SD₁₆	1	1	yes	characteristic
96	C₄×C₈	SD₁₆	1	1	yes	characteristic
97	C₄×C₈	SD₁₆	1	1	yes	characteristic
98	C₄×C₈	SD₁₆	1	1	yes	characteristic
99	C₄×C₈	Q₁₆	1	1	yes	characteristic
100	C₄×C₈	Q₁₆	1	1	yes	characteristic
101	C₄×C₈	Q₁₆	1	1	yes	characteristic
102	C₄×C₈	Q₁₆	1	1	yes	characteristic
103	C₄×C₈	C₂×D₄	1	1	yes	characteristic
104	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
105	C₄×C₈	#41	1	1	yes	characteristic
106	C₄×C₈	#41	1	1	yes	characteristic
107	C₂×Q₈	—	4	2	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	M₁₆	—	4	2	no	—
113	M₁₆	—	4	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	#100	1	1	yes	characteristic
119	C₂×C₈	#100	1	1	yes	characteristic
120	C₂×C₈	#103	1	1	yes	characteristic
121	C₂×C₈	#103	1	1	yes	characteristic
122	C₂×C₈	#103	1	1	yes	characteristic
123	C₂×C₈	#103	1	1	yes	characteristic
124	C₂×C₈	#121	1	1	yes	characteristic
125	C₂×C₈	#121	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
132	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₈	#137	1	1	yes	characteristic
135	C₂×C₈	#137	1	1	yes	characteristic
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₂²⋊C₄	—	4	2	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	4	4	no	—
144	C₄²	—	4	2	no	—
145	C₄²	#103	1	1	yes	characteristic
146	C₄²	#103	1	1	yes	characteristic
147	C₄²	#103	1	1	yes	characteristic
148	C₄²	#103	1	1	yes	characteristic
149	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₄²	#117	1	1	yes	characteristic
151	C₁₆	—	8	2	no	—
152	C₁₆	—	8	2	no	—
153	C₂³	#413	1	1	yes	characteristic
154	Q₈	—	8	4	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	#380	1	1	yes	characteristic
166	C₂×C₄	C₂×#100	1	1	yes	characteristic
167	C₂×C₄	C₂×#103	1	1	yes	characteristic
168	C₂×C₄	C₂×#103	1	1	yes	characteristic
169	C₂×C₄	#417	1	1	yes	characteristic
170	C₂×C₄	#417	1	1	yes	characteristic
171	C₈	—	4	1	no	—
172	C₈	—	8	2	no	—
173	C₈	#333	1	1	yes	characteristic
174	C₈	#333	1	1	yes	characteristic
175	C₈	#333	1	1	yes	characteristic
176	C₈	#333	1	1	yes	characteristic
177	C₈	#460	1	1	yes	characteristic
178	C₈	#460	1	1	yes	characteristic
179	C₈	#462	1	1	yes	characteristic
180	C₈	#462	1	1	yes	characteristic
181	C₈	#475	1	1	yes	characteristic
182	C₈	#475	1	1	yes	characteristic
183	C₈	#477	1	1	yes	characteristic
184	C₈	#477	1	1	yes	characteristic
185	C₂²	—	4	1	no	—
186	C₂²	#1550	1	1	yes	characteristic
187	C₂²	#1550	1	1	yes	characteristic
188	C₂²	#1800	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	1	no	—
193	C₄	—	4	1	no	—
194	C₄	—	4	1	no	—
195	C₄	—	8	4	no	—
196	C₄	C₂×#333	1	1	yes	characteristic
197	C₄	#1560	1	1	yes	characteristic
198	C₄	#1803	1	1	yes	characteristic
199	C₄	#1803	1	1	yes	characteristic
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#8587	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1560
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8587
prime radical	C₂³	#413
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#21732	2
#12900	2
#11956	2
#7270	2