Finite Group Databasebeta

← #134752

Subgroups — #134752

Order 512.

401 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C8×#123C211yesmaximal, characteristic
2#12794C212yesmaximal
3#8616C211yesmaximal, characteristic
4#8609C211yesmaximal, characteristic
5#7236C212yesmaximal
6C2×C8×D4C2211yescharacteristic
7C22×C4×C8C411yescharacteristic, complemented
8C4×#123C2211yescharacteristic
9C2×#352—22no—
10#1923—21no—
11#1916—21no—
12#1869C2211yescharacteristic
13#1850C411yescharacteristic, complemented
14#1817C411yescharacteristic, complemented
15#1808—21no—
16#1800C2212yes—
17#1799C2212yes—
18C8×C4⋊C4—21no—
19C8×C22⋊C4—21no—
20C8×C22⋊C4C411yescharacteristic, complemented
21#1567—21no—
22#1390—22no—
23#1292—21no—
24#1289—21no—
25C23×C8C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#123D411yescharacteristic
29C2×C4×D4C2311yescharacteristic
30C22×C42C2×C411yescharacteristic
31C8×D4—41no—
32C8×D4—41no—
33C2×#107—41no—
34#423—22no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8D411yescharacteristic
37C2×#100—22no—
38C2×#99—22no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8C2×C411yescharacteristic
41C2×C22⋊C8C2×C411yescharacteristic
42C2×C22⋊C8D411yescharacteristic
43C2×C4×C8—21no—
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8—41no—
47C2×C4×C8—41no—
48C2×C4×C8D411yescharacteristic
49#388—21no—
50#387—21no—
51#386D411yescharacteristic
52#385Q811yescharacteristic
53#381Q811yescharacteristic
54#380—22no—
55C4×C4⋊C4—21no—
56C4×C22⋊C4—21no—
57C4×C22⋊C4C2×C411yescharacteristic
58#352—44no—
59#343—22no—
60#342—22no—
61#340—41no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#335—41no—
68#335—41no—
69C22×D4C2×D411yescharacteristic
70C23×C4C4⋊C411yescharacteristic
71C23×C4C4⋊C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C2×M16—42no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—42no—
82C22×C8—42no—
83C22×C8—42no—
84C22×C8—41no—
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89#123—21no—
90#123—21no—
91#123D812yes—
92#123Q1612yes—
93C4×D4—41no—
94C4×D4—41no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4#4111yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4C4⋊C411yescharacteristic
103C2×C22⋊C4C4⋊C411yescharacteristic
104C2×C22⋊C4C2×Q811yescharacteristic
105C2×C22⋊C4#4111yescharacteristic
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42C2×D411yescharacteristic
112#107—82no—
113C4⋊C8—41no—
114C4⋊C8—42no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117#100—42no—
118#100—42no—
119#99—42no—
120#99—42no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—42no—
124C22⋊C8—41no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8—42no—
128C4×C8—42no—
129C4×C8—41no—
130C4×C8—41no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—42no—
137C24C2×C4⋊C411yescharacteristic
138C24#12111yescharacteristic
139C2×D4—41no—
140C2×D4—41no—
141C2×D4C2×D811yescharacteristic
142C2×D4C2×Q1611yescharacteristic
143C2×D4#13511yescharacteristic
144C2×D4#13511yescharacteristic
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—82no—
163C22×C4#10712yescomplemented
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4#11711yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4C22⋊Q811yescharacteristic
168C22×C4C22⋊Q811yescharacteristic
169M16—82no—
170M16—82no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—42no—
177C2×C8—41no—
178C2×C8—42no—
179C2×C8—41no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—81no—
190C2×C8—82no—
191C2×C8#9911yescharacteristic
192C2×C8#9911yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C22⋊C4—21no—
200C22⋊C4—22no—
201C22⋊C4—22no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—42no—
205C22⋊C4—41no—
206C22⋊C4—82no—
207C22⋊C4—82no—
208C22⋊C4#10712yescomplemented
209C42—21no—
210C42—21no—
211C42—41no—
212C42—41no—
213C42—42no—
214C42—41no—
215C42—41no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23#38011yescharacteristic
225C23C2×#10711yescharacteristic
226C23#42711yescharacteristic
227C23#46611yescharacteristic
228C23#46711yescharacteristic
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—81no—
265C2×C4—162no—
266C2×C4—162no—
267C2×C4—162no—
268C2×C4C2×#9911yescharacteristic
269C2×C4#41011yescharacteristic
270C2×C4#46611yescharacteristic
271C2×C4#46711yescharacteristic
272C8—21no—
273C8—41no—
274C8—41no—
275C8—41no—
276C8—42no—
277C8—81no—
278C8—162no—
279C8#35212yescomplemented
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22#179911yescharacteristic
297C22#180811yescharacteristic
298C22#181211yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—41no—
304C4—42no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—322no—
310C4C2×#35211yescharacteristic
311C4#212011yescharacteristic
312C2—21no—
313C2—21no—
314C2—41no—
315C2—41no—
316C2—41no—
317C2—41no—
318C2#860911yescharacteristic
319C2#1279411yescharacteristic
320C2#1280511yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C22×C4×C8C4semidirect
13. #1850C4semidirect
14. #1817C4semidirect
20. C8×C22⋊C4C4semidirect
163. C22×C4#107semidirect
208. C22⋊C4#107semidirect
279. C8#352semidirect

Maximal subgroups

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

subgroupindex
C8×#1232
#127942
#86162
#86092
#72362