Finite Group Databasebeta

← #128045

Subgroups — #128045

Order 512.

623 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 130 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#20718C211yesmaximal, characteristic
3#7305C214yesmaximal
4C2×C4×#41C2211yescharacteristic
5C2×#431C411yescharacteristic, complemented
6C2×#431C411yescharacteristic, 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
17C4×C22⋊C8—22no—
18C22×#41C2311yescharacteristic
19C4×#41—41no—
20C4×#41—41no—
21C4×#41—41no—
22C4×#41—41no—
23C2×C4×Q8Q811yescharacteristic
24C2×C4×Q8Q811yescharacteristic
25C2×C4×D4C2×C412yes—
26C2×C4×D4C2×C412yes—
27C2×C4×D4D411yescharacteristic
28C2×C4×D4D411yescharacteristic
29C2×#117C2×C411yescharacteristic
30C2×#117D412yes—
31C22×C42C2×C411yescharacteristic
32C22×C42D412yes—
33#431—41no—
34#431—41no—
35#431—41no—
36#431—41no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×#99—24no—
40C2×#99C2×C418yes—
41C2×C22⋊C8—24no—
42C2×C22⋊C8C2×C412yes—
43C2×C22⋊C8C2×C412yes—
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46#383—42no—
47#382—42no—
48#381—24no—
49#380—24no—
50C4×C4⋊C4—42no—
51C4×C22⋊C4—24no—
52#376—42no—
53#342—24no—
54#342—24no—
55#336—24no—
56C2×#41—41no—
57C2×#41—41no—
58C2×#41—44no—
59C22×Q8C2×Q811yescharacteristic
60C22×D4C22×C412yes—
61C22×D4C2×D411yescharacteristic
62C23×C4C4211yescharacteristic
63C23×C4C4211yescharacteristic
64C23×C4C22⋊C412yes—
65C23×C4C22⋊C412yes—
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D412yes—
68C22×C8—22no—
69C22×C8—22no—
70C4×Q8—41no—
71C4×Q8—41no—
72C4×Q8—41no—
73C4×Q8—41no—
74C4×D4—42no—
75C4×D4—42no—
76C4×D4—42no—
77C4×D4—42no—
78C4×D4—41no—
79C4×D4—41no—
80C4×D4—41no—
81C4×D4—41no—
82#117—41no—
83#117—41no—
84#117—42no—
85#117—42no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4C4⋊C411yescharacteristic
88C2×C4⋊C4C4⋊C411yescharacteristic
89C2×C4⋊C4C4⋊C412yes—
90C2×C4⋊C4C4⋊C412yes—
91C2×C22⋊C4—28no—
92C2×C22⋊C4C4212yes—
93C2×C22⋊C4C22⋊C412yes—
94C2×C22⋊C4C22⋊C412yes—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—44no—
104C2×C42C22⋊C412yes—
105C2×C42C22⋊C411yescharacteristic
106C2×C42C22⋊C411yescharacteristic
107C2×C42C4⋊C412yes—
108C2×C42C4⋊C412yes—
109C2×C42C4⋊C411yescharacteristic
110C2×C42C4⋊C411yescharacteristic
111C4⋊C8—42no—
112C4⋊C8—42no—
113#99—48no—
114#99—48no—
115C22⋊C8—48no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C4×C8—41no—
119C4×C8—41no—
120C4×C8—41no—
121C4×C8—41no—
122#95—44no—
123#95—44no—
124#95—44no—
125#95—44no—
126C24C2×C4211yescharacteristic
127C24C2×C22⋊C412yes—
128#41—48no—
129#41—48no—
130C2×Q8—41no—
131C2×Q8—42no—
132C2×D4—24no—
133C2×D4—24no—
134C2×D4—42no—
135C2×D4—41no—
136C2×D4—42no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—48no—
157C22×C4#9511yescharacteristic
158C22×C4#9511yescharacteristic
159C22×C4#9512yes—
160C22×C4#9512yes—
161C22×C4#9511yescharacteristic
162C22×C4#9511yescharacteristic
163C22×C4C2×C22⋊C411yescharacteristic
164C22×C4C2×C4⋊C411yescharacteristic
165C22×C4C2×C4⋊C412yes—
166C2×C8—44no—
167C2×C8—44no—
168C2×C8—42no—
169C2×C8—42no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—88no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—48no—
179C22⋊C4—48no—
180C42—22no—
181C42—22no—
182C42—21no—
183C42—21no—
184C42—22no—
185C42—21no—
186C42—22no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—22no—
191C42—22no—
192C42—88no—
193C42#9911yescharacteristic
194C42#9911yescharacteristic
195C42#9911yescharacteristic
196C42#9911yescharacteristic
197C23—22no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23—42no—
202C23—42no—
203C23—41no—
204C23—41no—
205C23C2×#9511yescharacteristic
206C23#41014yes—
207Q8—44no—
208D4—44no—
209D4—48no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—88no—
239C2×C4—88no—
240C2×C4—88no—
241C2×C4#34211yescharacteristic
242C2×C4#34211yescharacteristic
243C2×C4#34211yescharacteristic
244C2×C4#34211yescharacteristic
245C2×C4C2×#9911yescharacteristic
246C2×C4C2×#9911yescharacteristic
247C8—82no—
248C8—82no—
249C22—21no—
250C22—21no—
251C22—24no—
252C22—44no—
253C22—42no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—82no—
259C22—82no—
260C22C2×#34211yescharacteristic
261C22#174012yes—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—21no—
266C4—41no—
267C4—41no—
268C4—42no—
269C4—42no—
270C4—42no—
271C4—41no—
272C4—168no—
273C4#139112yes—
274C4#139112yes—
275C2—22no—
276C2—44no—
277C2—81no—
278C2#810011yescharacteristic
279C2C2×#139111yescharacteristic
280C2C2×#139111yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC24C2×C42
Frattini, prime radical, cosocleC22×#41C23
socleC22C2×#342
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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×#431C4semidirect
6. C2×#431C4semidirect
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
#207182
#73052