Finite Group Databasebeta

← #134873

Subgroups — #134873

Order 512.

382 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 72 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#22031C211yesmaximal, characteristic
2#11820C212yesmaximal
3#8620C211yesmaximal, characteristic, complemented
4#8608C211yesmaximal, characteristic, complemented
5#7320C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C2×#408C411yescharacteristic, complemented
8#2388C2211yescharacteristic
9#1955—21no—
10#1924—21no—
11#1923—21no—
12#1870C2211yescharacteristic
13#1869—21no—
14#1851C411yescharacteristic, complemented
15#1807—21no—
16#1799C2212yes—
17#1780C411yescharacteristic, complemented
18#1752C411yescharacteristic, complemented
19#1568—21no—
20#1466C2212yes—
21#1351—22no—
22#1343—22no—
23#1290—21no—
24#1289—21no—
25C22×#41C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2311yescharacteristic
31C8×D4—41no—
32C8×D4—41no—
33#432—22no—
34C2×#106—41no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8D411yescharacteristic
37C2×#99—22no—
38#408—41no—
39#408—41no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8C2×C411yescharacteristic
46C2×C22⋊C8C2×C411yescharacteristic
47C2×C22⋊C8D411yescharacteristic
48C2×#97—21no—
49C2×C4×C8D411yescharacteristic
50#395Q811yescharacteristic
51#394—21no—
52#394Q811yescharacteristic
53#393D411yescharacteristic
54#392—21no—
55#392D411yescharacteristic
56#380—22no—
57C4×C22⋊C4—21no—
58C4×C22⋊C4C2×C411yescharacteristic
59#342—22no—
60#340—41no—
61#336—21no—
62#336—21no—
63#336—21no—
64#335—41no—
65#334—41no—
66#323—22no—
67#323—22no—
68C2×#41—41no—
69C2×#41—41no—
70C2×#41—81no—
71C2×#41—81no—
72C22×Q8C4⋊C411yescharacteristic
73C22×D4—21no—
74C22×D4—21no—
75C22×D4C4⋊C411yescharacteristic
76C22×D4#4111yescharacteristic
77C23×C4—21no—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C2×M16—41no—
81C2×M16—81no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—81no—
89C22×C8—81no—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C4×D4—41no—
95C4×D4—41no—
96C4×D4—81no—
97C4×D4—81no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4C2×Q811yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4C2×D411yescharacteristic
108C2×C22⋊C4#4111yescharacteristic
109C2×C42—21no—
110C2×C42C2×D411yescharacteristic
111#106—82no—
112C4⋊C8—41no—
113C4⋊C8—42no—
114C4⋊C8—81no—
115#99—82no—
116C22⋊C8—41no—
117C22⋊C8—41no—
118C22⋊C8—42no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121C22⋊C8—41no—
122C22⋊C8—42no—
123C22⋊C8—81no—
124C22⋊C8—81no—
125C22⋊C8—81no—
126#97—42no—
127#97—81no—
128C4×C8—21no—
129C4×C8—21no—
130#95—21no—
131#95—21no—
132#95—42no—
133#95C4⋊C411yescharacteristic
134#95C4⋊C411yescharacteristic
135C24—21no—
136C24#11711yescharacteristic
137C24#12111yescharacteristic
138#41—84no—
139#41—82no—
140#41—82no—
141C2×Q8—41no—
142C2×Q8—81no—
143C2×D4—22no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—81no—
150C2×D4—81no—
151C2×D4—81no—
152C2×D4—81no—
153C2×D4—81no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—82no—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C2×C4⋊C411yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4C22⋊Q811yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177M16—82no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—82no—
196C2×C8#9911yescharacteristic
197C2×C8#9911yescharacteristic
198C4⋊C4—41no—
199C4⋊C4—81no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—42no—
205C22⋊C4—81no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C42—21no—
212C42—21no—
213C42—41no—
214C42—41no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23#38011yescharacteristic
232C23#47112yes—
233Q8—82no—
234D4—44no—
235D4—82no—
236D4—84no—
237D4—84no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
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—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—162no—
269C2×C4—162no—
270C2×C4—162no—
271C2×C4C2×#9911yescharacteristic
272C2×C4#41011yescharacteristic
273C8—21no—
274C8—21no—
275C8—41no—
276C8—41no—
277C8—41no—
278C8—81no—
279C8—81no—
280C8—162no—
281C22—21no—
282C22—21no—
283C22—22no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22#179911yescharacteristic
301C22#181411yescharacteristic
302C22#181511yescharacteristic
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—322no—
312C4#212111yescharacteristic
313C4#212211yescharacteristic
314C2—21no—
315C2—21no—
316C2—42no—
317C2—81no—
318C2—161no—
319C2#862611yescharacteristic
320C2#1279511yescharacteristic
321C2#1280811yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radical#2388C22
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
3. #8620C2semidirect
4. #8608C2semidirect
5. #7320C2semidirect
7. C2×#408C4semidirect
14. #1851C4semidirect
17. #1780C4semidirect
18. #1752C4semidirect

Maximal subgroups

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

subgroupindex
#220312
#118202
#86202
#86082
#73202