Finite Group Databasebeta

← #127377

Subgroups — #127377

Order 512.

658 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 138 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#21373C211yesmaximal, characteristic
2#20723C212yesmaximal
3#7237C214yesmaximal, complemented
4C2×C4×#41C2211yescharacteristic
5C2×#431C411yescharacteristic, complemented
6C22×C4×C8C411yescharacteristic, complemented
7#2283—22no—
8#1984C412yescomplemented
9#1954C2212yes—
10#1869—22no—
11#1820C414yescomplemented
12#1799—22no—
13#1799C2214yes—
14#1782—22no—
15C4×#99—22no—
16C4×#99C414yescomplemented
17C4×C22⋊C8—22no—
18C22×#41D411yescharacteristic
19C23×C8—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—
41C2×C4×C8—21no—
42C2×C4×C8—21no—
43C2×C4×C8—41no—
44C2×C4×C8—41no—
45C2×C4×C8C2×C412yes—
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—24no—
57C2×#41—41no—
58C2×#41—41no—
59C2×#41—42no—
60C22×Q8C4⋊C411yescharacteristic
61C22×D4C22⋊C412yes—
62C22×D4C2×D411yescharacteristic
63C23×C4C4211yescharacteristic
64C23×C4C22⋊C412yes—
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C4⋊C411yescharacteristic
68C23×C4C2×D412yes—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—44no—
72C22×C8—42no—
73C4×Q8—41no—
74C4×Q8—41no—
75C4×Q8—41no—
76C4×Q8—41no—
77C4×D4—42no—
78C4×D4—42no—
79C4×D4—41no—
80C4×D4—41no—
81C4×D4—42no—
82C4×D4—42no—
83C4×D4—41no—
84C4×D4—41no—
85#117—42no—
86#117—42no—
87#117—41no—
88#117—41no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4C4212yes—
91C2×C4⋊C4C4⋊C412yes—
92C2×C4⋊C4C4⋊C411yescharacteristic
93C2×C4⋊C4C2×Q811yescharacteristic
94C2×C22⋊C4—28no—
95C2×C22⋊C4C22⋊C412yes—
96C2×C22⋊C4C4⋊C412yes—
97C2×C22⋊C4C22×C412yes—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—44no—
107C2×C42C4211yescharacteristic
108C2×C42C22⋊C412yes—
109C2×C42C22⋊C411yescharacteristic
110C2×C42C4⋊C412yes—
111C2×C42C4⋊C412yes—
112C2×C42C4⋊C411yescharacteristic
113C2×C42C22×C411yescharacteristic
114C4⋊C8—42no—
115#99—48no—
116#99—48no—
117C22⋊C8—48no—
118C22⋊C8—42no—
119C4×C8—42no—
120C4×C8—41no—
121C4×C8—41no—
122C4×C8—41no—
123C4×C8—41no—
124#95—44no—
125#95—44no—
126#95—44no—
127#95—44no—
128C24#9511yescharacteristic
129C24C2×C22⋊C412yes—
130#41—48no—
131#41—48no—
132C2×Q8—22no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—42no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4#13514yes—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—48no—
162C22×C4#9512yes—
163C22×C4#9511yescharacteristic
164C22×C4#9511yescharacteristic
165C22×C4#9511yescharacteristic
166C22×C4#9512yes—
167C22×C4C2×C4211yescharacteristic
168C22×C4C2×C22⋊C411yescharacteristic
169C22×C4C2×C4⋊C411yescharacteristic
170C22×C4C2×C4⋊C412yes—
171C2×C8—24no—
172C2×C8—42no—
173C2×C8—48no—
174C2×C8—42no—
175C2×C8—44no—
176C2×C8—42no—
177C4⋊C4—42no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—41no—
181C4⋊C4—88no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—48no—
186C22⋊C4—48no—
187C42—21no—
188C42—21no—
189C42—22no—
190C42—22no—
191C42—21no—
192C42—22no—
193C42—21no—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—22no—
198C42—22no—
199C42—88no—
200C42#9911yescharacteristic
201C42#9911yescharacteristic
202C42#9911yescharacteristic
203C42#9911yescharacteristic
204C23—21no—
205C23—22no—
206C23—21no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23H(3,ℤ/4)12yes—
213C23C2×#9511yescharacteristic
214C23#41614yes—
215Q8—44no—
216D4—44no—
217D4—48no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—88no—
248C2×C4—88no—
249C2×C4—88no—
250C2×C4#34211yescharacteristic
251C2×C4#34211yescharacteristic
252C2×C4#34211yescharacteristic
253C2×C4#34211yescharacteristic
254C2×C4C2×#9911yescharacteristic
255C2×C4C2×#9911yescharacteristic
256C2×C4#42712yes—
257C8—44no—
258C8—82no—
259C22—21no—
260C22—21no—
261C22—22no—
262C22—24no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—41no—
267C22—42no—
268C22—41no—
269C22—42no—
270C22—82no—
271C22C2×H(3,ℤ/4)11yescharacteristic
272C22#173511yescharacteristic
273C22C2×#34211yescharacteristic
274C4—21no—
275C4—21no—
276C4—21no—
277C4—21no—
278C4—22no—
279C4—41no—
280C4—42no—
281C4—41no—
282C4—42no—
283C4—42no—
284C4—168no—
285C4#139112yes—
286C4#212612yes—
287C2—21no—
288C2—21no—
289C2—44no—
290C2—42no—
291C2#798211yescharacteristic
292C2C2×#139111yescharacteristic
293C2#1258111yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #7237C2semidirect
5. C2×#431C4semidirect
6. C22×C4×C8C4semidirect
8. #1984C4semidirect
11. #1820C4semidirect
16. C4×#99C4semidirect

Maximal subgroups

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

subgroupindex
#213732
#207232
#72372