Finite Group Databasebeta

← #511576

Subgroups — #511576

Order 512.

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

normal subgroups only

This group has 706 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#32931C212yesmaximal
2#13494C212yesmaximal, complemented
3#13492C211yesmaximal, characteristic, complemented
4#13383C212yesmaximal, complemented
5#13382C212yesmaximal, complemented
6#13382C212yesmaximal, complemented
7#13306C211yesmaximal, characteristic, complemented
8#13306C212yesmaximal, complemented
9#12679C211yesmaximal, characteristic
10C24×C8C2211yescharacteristic
11C22×Q32C2211yescharacteristic, complemented
12C2×#467C2212yes—
13C2×#467C2212yes—
14C2×#467C2212yes—
15#2848—22no—
16C2×#367C2212yescomplemented
17C2×#366C2211yescharacteristic, complemented
18C2×#358C2212yescomplemented
19C2×#358C2212yescomplemented
20C2×#358C2212yescomplemented
21C2×C22⋊C16C2211yescharacteristic
22C2×C22⋊C16C2212yes—
23C2×C22⋊C16C2211yescharacteristic
24C2×C22⋊C16C2212yes—
25#1892—22no—
26#1892—22no—
27#1892—22no—
28#1808C2212yes—
29#1808C2212yes—
30#1808C2212yes—
31#1380C2211yescharacteristic, complemented
32#1380C2212yescomplemented
33#1380C2212yescomplemented
34#1380C2211yescharacteristic, complemented
35C24×C4D411yescharacteristic
36C22×Q16C2312yes—
37C23×C8—41no—
38C23×C8—42no—
39C23×C8—41no—
40C23×C8—42no—
41C23×C8D411yescharacteristic
42C23×C8D412yes—
43C23×C8C2311yescharacteristic
44C23×C8C2312yes—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×Q32—41no—
49C2×Q32—42no—
50C22×C16C2311yescharacteristic
51C22×C16C2311yescharacteristic
52C22×C16C2312yes—
53#467—42no—
54#467—42no—
55#467—42no—
56#467—42no—
57#467—42no—
58#467—42no—
59C2×#107C2312yes—
60C2×#107C2312yes—
61C2×#107C2312yes—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—22no—
65#385—42no—
66#385—42no—
67#385—42no—
68#380—22no—
69#380—22no—
70#380—22no—
71#379—42no—
72#367—42no—
73#366—41no—
74#358—42no—
75#358—42no—
76#358—42no—
77C22⋊C16—41no—
78C22⋊C16—42no—
79C22⋊C16—41no—
80C22⋊C16—42no—
81#340—22no—
82#340—22no—
83#340—22no—
84C25D811yescharacteristic
85C22×Q8—22no—
86C23×C4—42no—
87C23×C4—41no—
88C23×C4—42no—
89C23×C4—41no—
90C23×C4D811yescharacteristic
91C23×C4D812yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C2×Q16—22no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—22no—
99C2×Q16—22no—
100C22×C8—21no—
101C22×C8—22no—
102C22×C8—21no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—42no—
111C22×C8—41no—
112C22×C8—42no—
113C22×C8—42no—
114C22×C8—42no—
115C22×C8—44no—
116C22×C8—48no—
117C22×C8—44no—
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D412yes—
120C22×C8C2411yescharacteristic
121C22⋊Q8—44no—
122C22⋊Q8—44no—
123C22⋊Q8—44no—
124C22⋊Q8—44no—
125C22⋊Q8—44no—
126C22⋊Q8—44no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139Q32—44no—
140C2×C16—22no—
141C2×C16—22no—
142C2×C16—22no—
143C2×C16—21no—
144C2×C16—22no—
145C2×C16—21no—
146C2×C16—22no—
147#107—22no—
148#107—22no—
149#107—22no—
150#107—22no—
151#107—22no—
152#107—22no—
153#103—44no—
154#103—44no—
155#103—44no—
156#95—42no—
157#95—42no—
158#95—42no—
159C24—22no—
160C24—24no—
161C24—42no—
162C24—41no—
163C24—41no—
164C24C2×D811yescharacteristic
165C24C2×D812yes—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—48no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4C2×D811yescharacteristic
195C22×C4C2×D812yes—
196C22×C4C22×D411yescharacteristic
197Q16—42no—
198Q16—42no—
199Q16—42no—
200Q16—42no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—21no—
206C2×C8—48no—
207C2×C8—44no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8H(5,𝔽2)12yes—
216C2×C8H(5,𝔽2)11yescharacteristic
217C2×C8H(5,𝔽2)11yescharacteristic
218C2×C8H(5,𝔽2)12yes—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—84no—
226C4⋊C4—84no—
227C4⋊C4—84no—
228C22⋊C4—84no—
229C22⋊C4—84no—
230C22⋊C4—84no—
231C22⋊C4—84no—
232C22⋊C4—84no—
233C22⋊C4—84no—
234C16—41no—
235C16—42no—
236C16—41no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—21no—
247C23—48no—
248C23—44no—
249C23—44no—
250C23—48no—
251C23—44no—
252C23#50812yes—
253C23#51014yes—
254C23C22×D811yescharacteristic
255Q8—82no—
256Q8—82no—
257Q8—82no—
258Q8—82no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—48no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4#53412yes—
286C2×C4#53411yescharacteristic
287C2×C4#57512yes—
288C2×C4#57511yescharacteristic
289C8—41no—
290C8—42no—
291C8#56011yescharacteristic
292C8#56012yes—
293C8H(3,𝔽4)11yescharacteristic
294C22—24no—
295C22—24no—
296C22—22no—
297C22—24no—
298C22—24no—
299C22—416no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—42no—
306C22—42no—
307C22#318612yes—
308C22#318611yescharacteristic
309C22C2×#50811yescharacteristic
310C22C2×#51012yes—
311C22#341411yescharacteristic
312C4—41no—
313C4—42no—
314C4—162no—
315C4—162no—
316C4—162no—
317C4—162no—
318C4#281211yescharacteristic
319C4#319712yes—
320C4#319711yescharacteristic
321C2—42no—
322C2—44no—
323C2#3270611yescharacteristic
324C2#3375612yes—
325C2#3375711yescharacteristic
326C2#3376111yescharacteristic
327C2#3376212yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13494C2semidirect
3. #13492C2semidirect
4. #13383C2semidirect
5. #13382C2semidirect
6. #13382C2semidirect
7. #13306C2semidirect
8. #13306C2semidirect
11. C22×Q32C22semidirect
16. C2×#367C22semidirect
17. C2×#366C22semidirect
18. C2×#358C22semidirect
19. C2×#358C22semidirect
20. C2×#358C22semidirect
31. #1380C22semidirect
32. #1380C22semidirect
33. #1380C22semidirect
34. #1380C22semidirect

Maximal subgroups

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

subgroupindex
#329312
#134942
#134922
#133832
#133822
#133062
#126792