Finite Group Databasebeta

← #128046

Subgroups — #128046

Order 512.

609 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 126 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#20723C212yesmaximal
2#20719C211yesmaximal, characteristic
3#7305C214yesmaximal
4C2×C4×#41C2211yescharacteristic
5C2×#432C411yescharacteristic, complemented
6C2×#432C411yescharacteristic, complemented
7#2283—22no—
8#1820—22no—
9#1818C412yescomplemented
10#1799—22no—
11#1799C414yescomplemented
12#1798—22no—
13#1782C2214yes—
14#1780C2212yes—
15C4×#99—22no—
16C4×#99C414yescomplemented
17#1749—22no—
18C22×#41C2311yescharacteristic
19C4×#41—42no—
20C4×#41—42no—
21C2×C4×Q8Q811yescharacteristic
22C2×C4×Q8Q811yescharacteristic
23C2×C4×D4C2×C412yes—
24C2×C4×D4C2×C412yes—
25C2×C4×D4D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27C2×#117C2×C411yescharacteristic
28C2×#117D412yes—
29C22×C42C2×C411yescharacteristic
30C22×C42D412yes—
31#432—42no—
32#432—42no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35C2×#99—24no—
36C2×#99C2×C418yes—
37C2×C22⋊C8—24no—
38C2×C22⋊C8C2×C412yes—
39C2×C22⋊C8C2×C412yes—
40C2×#97—21no—
41C2×#97—21no—
42#383—42no—
43#382—42no—
44#381—24no—
45#380—24no—
46C4×C4⋊C4—42no—
47C4×C22⋊C4—24no—
48#376—42no—
49#342—24no—
50#342—24no—
51#336—24no—
52C2×#41—41no—
53C2×#41—41no—
54C2×#41—44no—
55C22×Q8C2×Q811yescharacteristic
56C22×D4C22×C412yes—
57C22×D4C2×D411yescharacteristic
58C23×C4C4211yescharacteristic
59C23×C4C4211yescharacteristic
60C23×C4C22⋊C412yes—
61C23×C4C22⋊C412yes—
62C23×C4C22×C411yescharacteristic
63C23×C4C2×D412yes—
64C22×C8—22no—
65C22×C8—22no—
66C4×Q8—42no—
67C4×Q8—42no—
68C4×D4—42no—
69C4×D4—42no—
70C4×D4—42no—
71C4×D4—42no—
72C4×D4—42no—
73C4×D4—42no—
74#117—41no—
75#117—41no—
76#117—42no—
77#117—42no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4C4⋊C411yescharacteristic
80C2×C4⋊C4C4⋊C411yescharacteristic
81C2×C4⋊C4C4⋊C412yes—
82C2×C4⋊C4C4⋊C412yes—
83C2×C22⋊C4—28no—
84C2×C22⋊C4C4212yes—
85C2×C22⋊C4C22⋊C412yes—
86C2×C22⋊C4C22⋊C412yes—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—44no—
94C2×C42C22⋊C412yes—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22⋊C411yescharacteristic
97C2×C42C4⋊C412yes—
98C2×C42C4⋊C412yes—
99C2×C42C4⋊C411yescharacteristic
100C2×C42C4⋊C411yescharacteristic
101C4⋊C8—42no—
102C4⋊C8—42no—
103#99—48no—
104#99—48no—
105C22⋊C8—48no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108#97—42no—
109#97—42no—
110#95—44no—
111#95—44no—
112#95—44no—
113#95—44no—
114C24C2×C4211yescharacteristic
115C24C2×C22⋊C412yes—
116#41—48no—
117#41—48no—
118C2×Q8—41no—
119C2×Q8—42no—
120C2×D4—24no—
121C2×D4—24no—
122C2×D4—42no—
123C2×D4—41no—
124C2×D4—42no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—48no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—81no—
142C22×C4—81no—
143C22×C4#9512yes—
144C22×C4#9512yes—
145C22×C4#9511yescharacteristic
146C22×C4#9511yescharacteristic
147C22×C4#9511yescharacteristic
148C22×C4#9511yescharacteristic
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4C2×C4⋊C411yescharacteristic
151C22×C4C2×C4⋊C412yes—
152C2×C8—44no—
153C2×C8—44no—
154C2×C8—42no—
155C2×C8—42no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—88no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C22⋊C4—48no—
165C22⋊C4—48no—
166C42—22no—
167C42—22no—
168C42—22no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42—21no—
173C42—21no—
174C42—22no—
175C42—22no—
176C42—88no—
177C42#9912yes—
178C42#9912yes—
179C23—22no—
180C23—22no—
181C23—21no—
182C23—22no—
183C23—42no—
184C23—42no—
185C23—41no—
186C23—41no—
187C23C2×#9511yescharacteristic
188C23#41014yes—
189Q8—44no—
190D4—44no—
191D4—48no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—88no—
219C2×C4—88no—
220C2×C4—88no—
221C2×C4#34212yes—
222C2×C4#34212yes—
223C2×C4C2×#9911yescharacteristic
224C2×C4C2×#9911yescharacteristic
225C8—82no—
226C8—82no—
227C22—21no—
228C22—21no—
229C22—24no—
230C22—44no—
231C22—42no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—82no—
237C22—82no—
238C22C2×#34211yescharacteristic
239C22#174012yes—
240C4—22no—
241C4—22no—
242C4—21no—
243C4—21no—
244C4—42no—
245C4—42no—
246C4—41no—
247C4—41no—
248C4—42no—
249C4—41no—
250C4—168no—
251C2—22no—
252C2—44no—
253C2—81no—
254C2#810011yescharacteristic
255C2#1257711yescharacteristic
256C2#1257711yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC24C2×C42
Frattini, prime radical, cosocleC22×#41C23
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#432C4semidirect
6. C2×#432C4semidirect
9. #1818C4semidirect
11. #1799C4semidirect
16. C4×#99C4semidirect

Maximal subgroups

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

subgroupindex
#207232
#207192
#73052