Finite Group Databasebeta

← #134784

Subgroups — #134784

Order 512.

380 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 82 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#21561C211yesmaximal, characteristic
2#12794C212yesmaximal
3#8614C211yesmaximal, characteristic
4#8609C211yesmaximal, characteristic
5#7236C212yesmaximal
6C2×C8×D4C2211yescharacteristic
7C22×#97C411yescharacteristic, complemented
8C4×#123C2211yescharacteristic
9C2×#352—22no—
10#1924—21no—
11#1917—21no—
12#1869C2211yescharacteristic
13#1850C411yescharacteristic, complemented
14#1817C411yescharacteristic, complemented
15#1808—21no—
16#1800C2212yes—
17#1799C2212yes—
18#1771—21no—
19#1752—21no—
20#1752C411yescharacteristic, complemented
21#1566—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×#97—21no—
44C2×#97—21no—
45C2×#97—21no—
46C2×#97—41no—
47C2×#97—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#334—41no—
68#334—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—22no—
90#123—22no—
91C4×D4—41no—
92C4×D4—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4#4111yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4C4⋊C411yescharacteristic
101C2×C22⋊C4C4⋊C411yescharacteristic
102C2×C22⋊C4C2×Q811yescharacteristic
103C2×C22⋊C4#4111yescharacteristic
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—41no—
108C2×C42—41no—
109C2×C42C2×D411yescharacteristic
110#107—82no—
111C4⋊C8—41no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115#100—42no—
116#100—42no—
117#99—42no—
118#99—42no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121C22⋊C8—42no—
122C22⋊C8—41no—
123#97—42no—
124#97—42no—
125#97—41no—
126#97—41no—
127C4×C8—21no—
128C4×C8—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—42no—
135C24C2×C4⋊C411yescharacteristic
136C24#12111yescharacteristic
137C2×D4—41no—
138C2×D4—41no—
139C2×D4AGL(1,ℤ/8)12yes—
140C2×D4#13712yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—81no—
158C22×C4—82no—
159C22×C4C2×C22⋊C411yescharacteristic
160C22×C4#11711yescharacteristic
161C22×C4#12111yescharacteristic
162C22×C4C22⋊Q811yescharacteristic
163C22×C4C22⋊Q811yescharacteristic
164M16—82no—
165M16—82no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—42no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—41no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—42no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—81no—
185C2×C8—82no—
186C2×C8#9911yescharacteristic
187C2×C8#9911yescharacteristic
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C22⋊C4—22no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—81no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C42—21no—
203C42—21no—
204C42—41no—
205C42—41no—
206C42—42no—
207C42—41no—
208C42—41no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23#38011yescharacteristic
218C23#42812yes—
219C23#46911yescharacteristic
220C23#47011yescharacteristic
221D4—42no—
222D4—42no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—162no—
254C2×C4—162no—
255C2×C4—162no—
256C2×C4C2×#9911yescharacteristic
257C2×C4#41011yescharacteristic
258C2×C4#46812yes—
259C8—21no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8—42no—
264C8—81no—
265C8—162no—
266C8#35212yescomplemented
267C22—21no—
268C22—21no—
269C22—22no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22#179911yescharacteristic
282C22#180911yescharacteristic
283C22#181211yescharacteristic
284C4—22no—
285C4—21no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4—322no—
294C4C2×#35211yescharacteristic
295C4#212011yescharacteristic
296C2—21no—
297C2—21no—
298C2—41no—
299C2—42no—
300C2—41no—
301C2#861011yescharacteristic
302C2#1279411yescharacteristic
303C2#1280611yescharacteristic
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×#97C4semidirect
13. #1850C4semidirect
14. #1817C4semidirect
20. #1752C4semidirect
266. C8#352semidirect

Maximal subgroups

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

subgroupindex
#215612
#127942
#86142
#86092
#72362