Finite Group Databasebeta

← #511572

Subgroups — #511572

Order 512.

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

normal subgroups only

This group has 784 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#32930C212yesmaximal, complemented
2#13494C212yesmaximal, complemented
3#13493C211yesmaximal, characteristic, complemented
4#13381C212yesmaximal, complemented
5#13381C212yesmaximal, complemented
6#13380C212yesmaximal, complemented
7#13304C211yesmaximal, characteristic, complemented
8#13304C212yesmaximal, complemented
9#12679C211yesmaximal, characteristic, complemented
10C24×C8C2211yescharacteristic
11C22×D16C2211yescharacteristic, complemented
12C2×#466C2212yes—
13C2×#466C2212yes—
14C2×#466C2212yes—
15#2847—22no—
16C2×#367C2212yescomplemented
17C2×#366C2211yescharacteristic, complemented
18C2×#357C2212yescomplemented
19C2×#357C2212yescomplemented
20C2×#357C2212yescomplemented
21C2×C22⋊C16C2212yes—
22C2×C22⋊C16C2211yescharacteristic
23C2×C22⋊C16C2211yescharacteristic
24C2×C22⋊C16C2212yes—
25#1893—22no—
26#1893—22no—
27#1893—22no—
28#1808C2212yes—
29#1808C2212yes—
30#1808C2212yes—
31#1380C2212yescomplemented
32#1380C2211yescharacteristic, complemented
33#1380C2212yescomplemented
34#1380C2211yescharacteristic, complemented
35C24×C4D411yescharacteristic
36C22×D8C2312yes—
37C23×C8—42no—
38C23×C8—41no—
39C23×C8—41no—
40C23×C8—42no—
41C23×C8D411yescharacteristic
42C23×C8D412yes—
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45C2×#121—22no—
46C2×#121—22no—
47C2×#121—22no—
48C2×#120—42no—
49C2×D16—41no—
50C2×D16—42no—
51C22×C16C2311yescharacteristic
52C22×C16C2311yescharacteristic
53C22×C16C2312yes—
54#466—42no—
55#466—42no—
56#466—42no—
57#466—42no—
58#466—42no—
59#466—42no—
60C2×#107C2312yes—
61C2×#107C2312yes—
62C2×#107C2312yes—
63C2×#102—22no—
64C2×#102—22no—
65C2×#102—22no—
66#386—42no—
67#386—42no—
68#386—42no—
69#380—22no—
70#380—22no—
71#380—22no—
72#367—42no—
73#366—41no—
74#357—42no—
75#357—42no—
76#357—42no—
77C22⋊C16—42no—
78C22⋊C16—41no—
79C22⋊C16—41no—
80C22⋊C16—42no—
81#340—22no—
82#340—22no—
83#340—22no—
84C25D811yescharacteristic
85C22×D4—22no—
86C22×D4—42no—
87C22×D4—42no—
88C22×D4—42no—
89C23×C4—42no—
90C23×C4—41no—
91C23×C4—42no—
92C23×C4—41no—
93C23×C4D811yescharacteristic
94C23×C4D812yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C2×D8—22no—
98C2×D8—22no—
99C2×D8—22no—
100C2×D8—22no—
101C2×D8—22no—
102C2×D8—22no—
103C22×C8—22no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—42no—
114C22×C8—41no—
115C22×C8—42no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—44no—
119C22×C8—48no—
120C22×C8—44no—
121C22×C8C2×D412yes—
122C22×C8C2×D411yescharacteristic
123C22×C8C2411yescharacteristic
124#121—44no—
125#121—44no—
126#121—44no—
127#121—44no—
128#121—44no—
129#121—44no—
130#120—88no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140D16—44no—
141C2×C16—22no—
142C2×C16—22no—
143C2×C16—22no—
144C2×C16—21no—
145C2×C16—22no—
146C2×C16—21no—
147C2×C16—22no—
148#107—22no—
149#107—22no—
150#107—22no—
151#107—22no—
152#107—22no—
153#107—22no—
154#102—44no—
155#102—44no—
156#102—44no—
157#95—42no—
158#95—42no—
159#95—42no—
160C24—22no—
161C24—24no—
162C24—42no—
163C24—41no—
164C24—41no—
165C24—42no—
166C24C2×D811yescharacteristic
167C24C2×D812yes—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4—84no—
177C2×D4—84no—
178C2×D4—84no—
179C2×D4—84no—
180C2×D4—84no—
181C2×D4—84no—
182C2×D4—84no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4C2×D811yescharacteristic
205C22×C4C2×D812yes—
206C22×C4C22×D411yescharacteristic
207D8—42no—
208D8—42no—
209D8—42no—
210D8—42no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—21no—
216C2×C8—48no—
217C2×C8—44no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—41no—
222C2×C8—42no—
223C2×C8—41no—
224C2×C8—42no—
225C2×C8H(5,𝔽2)12yes—
226C2×C8H(5,𝔽2)11yescharacteristic
227C2×C8H(5,𝔽2)11yescharacteristic
228C2×C8H(5,𝔽2)12yes—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C22⋊C4—84no—
236C22⋊C4—84no—
237C22⋊C4—84no—
238C22⋊C4—84no—
239C22⋊C4—84no—
240C22⋊C4—84no—
241C16—41no—
242C16—42no—
243C16—41no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—21no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—48no—
255C23—44no—
256C23—44no—
257C23—48no—
258C23—44no—
259C23—82no—
260C23—82no—
261C23—82no—
262C23—82no—
263C23—82no—
264C23—82no—
265C23—82no—
266C23#50812yes—
267C23#50914yes—
268C23C22×D811yescharacteristic
269D4—82no—
270D4—82no—
271D4—82no—
272D4—82no—
273D4—168no—
274D4—168no—
275D4—168no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—48no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4#53412yes—
300C2×C4#53411yescharacteristic
301C2×C4#57512yes—
302C2×C4#57511yescharacteristic
303C8—41no—
304C8—42no—
305C8#56011yescharacteristic
306C8#56012yes—
307C8H(3,𝔽4)11yescharacteristic
308C22—24no—
309C22—24no—
310C22—22no—
311C22—24no—
312C22—24no—
313C22—416no—
314C22—41no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—42no—
320C22—42no—
321C22—162no—
322C22—162no—
323C22—162no—
324C22—162no—
325C22—162no—
326C22—162no—
327C22—162no—
328C22#318612yes—
329C22#318611yescharacteristic
330C22C2×#50811yescharacteristic
331C22C2×#50912yes—
332C22#341411yescharacteristic
333C4—41no—
334C4—42no—
335C4—162no—
336C4—162no—
337C4—162no—
338C4#281211yescharacteristic
339C4#319712yes—
340C4#319711yescharacteristic
341C2—42no—
342C2—44no—
343C2—322no—
344C2#3270611yescharacteristic
345C2#3375412yes—
346C2#3375711yescharacteristic
347C2#3375811yescharacteristic
348C2#3376012yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
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
1. #32930C2semidirect
2. #13494C2semidirect
3. #13493C2semidirect
4. #13381C2semidirect
5. #13381C2semidirect
6. #13380C2semidirect
7. #13304C2semidirect
8. #13304C2semidirect
9. #12679C2semidirect
11. C22×D16C22semidirect
16. C2×#367C22semidirect
17. C2×#366C22semidirect
18. C2×#357C22semidirect
19. C2×#357C22semidirect
20. C2×#357C22semidirect
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
#329302
#134942
#134932
#133812
#133802
#133042
#126792