Finite Group Databasebeta

← #325463

Subgroups — #325463

Order 512.

634 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33256C211yesmaximal, characteristic, complemented
2#29149C211yesmaximal, characteristic
3#28724C211yesmaximal, characteristic
4#21681C211yesmaximal, characteristic
5#10619C214yesmaximal, complemented
6#9904C211yesmaximal, characteristic, complemented
7#9902C211yesmaximal, characteristic, complemented
8#8654C214yesmaximal, complemented
9#8274C211yesmaximal, characteristic, complemented
10C2×#490—21no—
11C2×#436—22no—
12C2×#436C2212yes—
13#2859C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×#127C2211yescharacteristic
16#2073—22no—
17#2025—22no—
18#1992—21no—
19#1961—21no—
20#1954—22no—
21#1954C2212yes—
22#1941—22no—
23#1941—22no—
24#1844C2212yes—
25#1842C2211yescharacteristic
26#1836—21no—
27#1834C2212yes—
28#1802C2214yes—
29#1802C2214yes—
30#1767C2211yescharacteristic
31#1767C2211yescharacteristic
32C42×C8C2211yescharacteristic
33C2×#335C2212yes—
34#1296C2214yes—
35#1296C2214yes—
36#1277C2212yes—
37C22×M16—22no—
38C2×#127D411yescharacteristic
39C2×#125—21no—
40C2×#125—21no—
41C2×C4×D4—22no—
42C2×C4×D4C2312yes—
43#490—44no—
44#436—28no—
45#436—48no—
46C2×#106—22no—
47C2×#106—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×C4⋊C8C2312yes—
52C2×#102—24no—
53C2×#102—24no—
54C2×C22⋊C8—24no—
55C2×C22⋊C8—22no—
56C2×#97—21no—
57C2×#97—21no—
58C2×C4×C8D412yes—
59C2×C4×C8D412yes—
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62#390—22no—
63#390D412yes—
64#389—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#384D411yescharacteristic
69#384C2312yes—
70#384C2312yes—
71#382—21no—
72#382—21no—
73C43C2311yescharacteristic
74#340—22no—
75#340—22no—
76#340—24no—
77#340—24no—
78#336—21no—
79#336—21no—
80#335—24no—
81C22×D4—22no—
82C22×D4C2×D412yes—
83C23×C4—22no—
84C2×M16—48no—
85C2×M16—44no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D412yes—
92#127—22no—
93#127—22no—
94#125—42no—
95#125—42no—
96#125—82no—
97C4×D4—24no—
98C4×D4—24no—
99C4×D4—44no—
100C4×D4—44no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4C2×D412yes—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—22no—
113C2×C42C2×D412yes—
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42#4112yes—
117C2×C42C2411yescharacteristic
118#106—44no—
119#106—44no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—24no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—44no—
126#102—48no—
127#102—84no—
128C22⋊C8—44no—
129C22⋊C8—48no—
130#97—42no—
131#97—42no—
132C4×C8—24no—
133C4×C8—22no—
134C4×C8—42no—
135C4×C8#4114yes—
136#95—42no—
137#95—42no—
138C24—22no—
139C2×D4—24no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—24no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4—44no—
146C22×C4—22no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4#12112yes—
154C22×C4#12112yes—
155C22×C4C22×D411yescharacteristic
156C22×C4C22×D411yescharacteristic
157C22×C4C2×#4111yescharacteristic
158M16—88no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—44no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8#12412yes—
172C2×C8#12412yes—
173C2×C8AGL(1,ℤ/8)14yes—
174C4⋊C4—24no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—44no—
183C4⋊C4—82no—
184C4⋊C4—82no—
185C4⋊C4—84no—
186C4⋊C4—84no—
187C22⋊C4—48no—
188C22⋊C4—44no—
189C42—22no—
190C42—22no—
191C42—22no—
192C42—42no—
193C42AGL(1,ℤ/8)12yes—
194C42#13712yes—
195C42C2×#4112yes—
196C422−1+412yes—
197C23—44no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—44no—
202C23C2×#12111yescharacteristic
203D4—44no—
204D4—44no—
205D4—88no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—84no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4#46814yes—
224C2×C4C2×#12411yescharacteristic
225C2×C4#53611yescharacteristic
226C2×C4#54712yes—
227C2×C4#55612yes—
228C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
229C2×C4C2×AGL(1,ℤ/8)12yes—
230C2×C4C2×#13711yescharacteristic
231C2×C4#57512yes—
232C2×C4#57712yes—
233C8—22no—
234C8—22no—
235C8—24no—
236C8—82no—
237C8—82no—
238C8—82no—
239C22—82no—
240C22—82no—
241C22—82no—
242C22—84no—
243C22—84no—
244C22#254711yescharacteristic
245C22#303612yes—
246C22C2×#46811yescharacteristic
247C22#305711yescharacteristic
248C22#306712yes—
249C4—22no—
250C4—24no—
251C4—42no—
252C4—82no—
253C4—162no—
254C4—162no—
255C4#313511yescharacteristic
256C4#313811yescharacteristic
257C4#323412yes—
258C4#335412yes—
259C4#335612yes—
260C2—162no—
261C2#1927912yes—
262C2#1932812yes—
263C2#2871311yescharacteristic
264C2#2872211yescharacteristic
265C2#2873411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33256C2semidirect
5. #10619C2semidirect
6. #9904C2semidirect
7. #9902C2semidirect
8. #8654C2semidirect
9. #8274C2semidirect

Maximal subgroups

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

subgroupindex
#332562
#291492
#287242
#216812
#106192
#99042
#99022
#86542
#82742