Finite Group Databasebeta

← #127808

Subgroups — #127808

Order 512.

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

normal subgroups only

This group has 639 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#21374C211yesmaximal, characteristic
2#20723C212yesmaximal
3#7283C214yesmaximal, complemented
4C2×C4×#41C2211yescharacteristic
5C2×#431C411yescharacteristic, complemented
6C2×C4×M16C411yescharacteristic, complemented
7#2283—22no—
8#1985C412yescomplemented
9#1955C2212yes—
10#1870—22no—
11#1820C414yescomplemented
12#1799—22no—
13#1799C414yescomplemented
14#1782—22no—
15C4×#99—22no—
16C4×#99C2214yes—
17C4×C22⋊C8—22no—
18C22×#41D411yescharacteristic
19C22×M16—21no—
20C4×#41—41no—
21C4×#41—41no—
22C4×#41—41no—
23C4×#41—41no—
24C2×C4×Q8C2×C411yescharacteristic
25C2×C4×Q8Q811yescharacteristic
26C2×C4×D4C2×C412yes—
27C2×C4×D4D412yes—
28C2×C4×D4D411yescharacteristic
29C2×C4×D4C2311yescharacteristic
30C2×#117C2×C412yes—
31C2×#117Q811yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D412yes—
34#431—41no—
35#431—41no—
36C2×C4⋊C8C2×C412yes—
37C2×#99—24no—
38C2×#99—24no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—24no—
41C4×M16—41no—
42C4×M16—41no—
43C2×#97C2×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#342C2×C418yes—
55#336—24no—
56C2×#41—41no—
57C2×#41—41no—
58C2×#41—44no—
59C22×Q8C4⋊C411yescharacteristic
60C22×D4C22⋊C412yes—
61C22×D4C2×D411yescharacteristic
62C23×C4C4211yescharacteristic
63C23×C4C22⋊C412yes—
64C23×C4C22⋊C412yes—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C4⋊C411yescharacteristic
67C23×C4C2×D412yes—
68C2×M16—44no—
69C2×M16—42no—
70C22×C8—22no—
71C22×C8—22no—
72C4×Q8—41no—
73C4×Q8—41no—
74C4×Q8—41no—
75C4×Q8—41no—
76C4×D4—42no—
77C4×D4—42no—
78C4×D4—41no—
79C4×D4—41no—
80C4×D4—42no—
81C4×D4—42no—
82C4×D4—41no—
83C4×D4—41no—
84#117—42no—
85#117—42no—
86#117—41no—
87#117—41no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4C4212yes—
90C2×C4⋊C4C4⋊C412yes—
91C2×C4⋊C4C4⋊C411yescharacteristic
92C2×C4⋊C4C2×Q811yescharacteristic
93C2×C22⋊C4—28no—
94C2×C22⋊C4C22⋊C412yes—
95C2×C22⋊C4C4⋊C412yes—
96C2×C22⋊C4C22×C412yes—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—44no—
106C2×C42C4211yescharacteristic
107C2×C42C22⋊C412yes—
108C2×C42C22⋊C411yescharacteristic
109C2×C42C4⋊C412yes—
110C2×C42C4⋊C412yes—
111C2×C42C4⋊C411yescharacteristic
112C2×C42C22×C411yescharacteristic
113C4⋊C8—42no—
114#99—48no—
115#99—48no—
116C22⋊C8—48no—
117C22⋊C8—42no—
118#97—42no—
119C4×C8—41no—
120C4×C8—41no—
121C4×C8—41no—
122C4×C8—41no—
123#95—44no—
124#95—44no—
125#95—44no—
126#95—44no—
127C24#9511yescharacteristic
128C24C2×C22⋊C412yes—
129#41—48no—
130#41—48no—
131C2×Q8—41no—
132C2×Q8—42no—
133C2×D4—24no—
134C2×D4—24no—
135C2×D4—42no—
136C2×D4—41no—
137C2×D4—42no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—48no—
158C22×C4#9512yes—
159C22×C4#9511yescharacteristic
160C22×C4#9511yescharacteristic
161C22×C4#9511yescharacteristic
162C22×C4#9512yes—
163C22×C4C2×C4211yescharacteristic
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4C2×C4⋊C411yescharacteristic
166C22×C4C2×C4⋊C412yes—
167M16—48no—
168C2×C8—24no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—44no—
172C2×C8—42no—
173C4⋊C4—42no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—88no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C22⋊C4—48no—
182C22⋊C4—48no—
183C42—21no—
184C42—21no—
185C42—22no—
186C42—22no—
187C42—21no—
188C42—21no—
189C42—22no—
190C42—22no—
191C42—21no—
192C42—22no—
193C42—21no—
194C42—22no—
195C42—88no—
196C42#9911yescharacteristic
197C42#9911yescharacteristic
198C42#9911yescharacteristic
199C42#9911yescharacteristic
200C23—22no—
201C23—22no—
202C23—21no—
203C23—22no—
204C23—42no—
205C23—42no—
206C23—41no—
207C23—41no—
208C23C2×#9511yescharacteristic
209C23#42114yes—
210Q8—44no—
211D4—44no—
212D4—48no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—88no—
242C2×C4—88no—
243C2×C4—88no—
244C2×C4#34211yescharacteristic
245C2×C4#34211yescharacteristic
246C2×C4#34211yescharacteristic
247C2×C4#34211yescharacteristic
248C2×C4C2×#9911yescharacteristic
249C2×C4C2×#9911yescharacteristic
250C8—44no—
251C8—82no—
252C22—21no—
253C22—21no—
254C22—24no—
255C22—44no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—82no—
262C22—82no—
263C22#173312yes—
264C22C2×#34211yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—41no—
272C4—42no—
273C4—42no—
274C4—41no—
275C4—168no—
276C4#139112yes—
277C4#212612yes—
278C2—22no—
279C2—44no—
280C2—81no—
281C2#804911yescharacteristic
282C2C2×#139111yescharacteristic
283C2#1258111yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7283C2semidirect
5. C2×#431C4semidirect
6. C2×C4×M16C4semidirect
8. #1985C4semidirect
11. #1820C4semidirect
13. #1799C4semidirect

Maximal subgroups

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

subgroupindex
#213742
#207232
#72832