Finite Group Databasebeta

← #355925

Subgroups — #355925

Order 512.

2345 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60070C212yesmaximal
2#60065C211yesmaximal, characteristic
3#10766C212yesmaximal, complemented
4#10759C212yesmaximal, complemented
5#10752C214yesmaximal, complemented
6#10743C214yesmaximal, complemented
7C2×#585C2211yescharacteristic
8C2×#568C2211yescharacteristic
9C22×#131C2211yescharacteristic
10#3428C2214yes—
11#2884C418yescomplemented
12#2883—28no—
13#2883C418yescomplemented
14C2×#410—28no—
15C2×#410C2214yes—
16C2×#410C2214yes—
17#2880—28no—
18#2879C414yescomplemented
19#2879C2212yes—
20C2×#408C414yescomplemented
21C2×#408C2212yes—
22#1735C2214yescomplemented
23#1733C2218yescomplemented
24C2×H(3,ℤ/4)C2214yescomplemented
25#585—44no—
26#585—44no—
27C2×2−1+4C2×C414yes—
28C2×2−1+4Q812yes—
29C2×H(5,𝔽2)C2×C414yes—
30C2×H(5,𝔽2)D414yes—
31C2×H(5,𝔽2)Q812yes—
32C22×#41C2×C414yes—
33C22×#41D418yes—
34C22×#41C2311yescharacteristic
35C22×#41C2312yes—
36#568—44no—
37C2×#131—24no—
38C2×#131—24no—
39C2×#131—44no—
40C22×M16C2×C414yescomplemented
41C22×M16C2×C412yescomplemented
42C22×M16C2312yes—
43C22×M16C2311yescharacteristic
44C23×C8C2×C412yescomplemented
45C23×C8C2311yescharacteristic
46#520—48no—
47#519C2×C418yescomplemented
48#518—28no—
49#518C2×C418yescomplemented
50C4×#41—48no—
51C2×#117—28no—
52C2×#117C2318yes—
53#410—232no—
54#410—28no—
55#410—28no—
56#410C2×C4116yes—
57#410C2×C4116yescomplemented
58C2×#99—232no—
59C2×#99—28no—
60C2×#99—28no—
61#408—28no—
62#408—28no—
63#408—24no—
64#408—24no—
65#408C2×C414yescomplemented
66#408C2×C418yescomplemented
67#408C2×C414yescomplemented
68#407—28no—
69#407—24no—
70C2×C22⋊C8—28no—
71C2×C22⋊C8—24no—
72#340—44no—
73#339—48no—
74H(3,ℤ/4)—44no—
752−1+4—416no—
762−1+4—48no—
77H(5,𝔽2)—416no—
78H(5,𝔽2)—416no—
79H(5,𝔽2)—48no—
80C2×#41—28no—
81C2×#41—28no—
82C2×#41—24no—
83C2×#41—24no—
84C2×#41—48no—
85C2×#41—48no—
86C2×#41—416no—
87C2×#41—416no—
88C2×#41C4218yes—
89C2×#41C22⋊C4116yes—
90C2×#41C22⋊C418yes—
91C2×#41C4⋊C418yes—
92C2×#41C4⋊C418yes—
93C2×#41C4⋊C418yes—
94C2×#41C22×C414yes—
95C2×#41C22×C414yes—
96C2×#41C22×C414yes—
97C2×#41C2×D414yes—
98C2×#41C2×D414yes—
99C2×#41C2×Q814yes—
100C22×Q8C4214yes—
101C22×Q8C4⋊C418yes—
102C22×Q8C22×C412yes—
103C22×Q8C2×Q811yescharacteristic
104C22×D4C4214yes—
105C22×D4C22⋊C4116yes—
106C22×D4C4⋊C418yes—
107C22×D4C4⋊C418yes—
108C22×D4C22×C412yes—
109C22×D4C2×D414yes—
110C22×D4C2×D412yes—
111C22×D4C2×Q811yescharacteristic
112C23×C4C22⋊C418yes—
113C23×C4C22×C412yes—
114C23×C4C2×D414yes—
115C23×C4C2411yescharacteristic
116#131—416no—
117#131—416no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—24no—
123C2×M16—28no—
124C2×M16—44no—
125C2×M16—48no—
126C2×M16C22×C414yes—
127C22×C8—24no—
128C22×C8—24no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—24no—
132C22×C8—44no—
133C22×C8C22×C411yescharacteristic
134C22×C8C22×C412yes—
135C22×C8C22×C411yescharacteristic
136C4×Q8—416no—
137C4×D4—416no—
138C4×D4—432no—
139#117—28no—
140#117—216no—
141#117—28no—
142#117—28no—
143#117—48no—
144#117—48no—
145#117C22×C4116yes—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—48no—
148C2×C22⋊C4—216no—
149C2×C22⋊C4—28no—
150C2×C42—28no—
151C2×C42—48no—
152#99—232no—
153#99—232no—
154C22⋊C8—28no—
155C22⋊C8—216no—
156C22⋊C8—28no—
157C24#9518yes—
158C24C2×C22⋊C414yes—
159C24C2×C4⋊C412yes—
160C24C22×D411yescharacteristic
161#41—28no—
162#41—28no—
163#41—216no—
164#41—216no—
165#41—28no—
166#41—28no—
167#41—28no—
168#41—28no—
169#41—216no—
170#41—48no—
171#41—48no—
172#41—432no—
173#41—416no—
174#41—416no—
175#41—416no—
176#41—416no—
177C2×Q8—28no—
178C2×Q8—24no—
179C2×Q8—48no—
180C2×Q8—416no—
181C2×Q8#9518yes—
182C2×Q8C2×C4214yes—
183C2×Q8C2×C4⋊C414yes—
184C2×Q8C2×C4⋊C414yes—
185C2×D4—216no—
186C2×D4—28no—
187C2×D4—28no—
188C2×D4—24no—
189C2×D4—48no—
190C2×D4—416no—
191C2×D4—432no—
192C2×D4—416no—
193C2×D4#9518yes—
194C2×D4#95116yes—
195C2×D4C2×C4214yes—
196C2×D4C2×C22⋊C418yes—
197C2×D4C2×C22⋊C418yes—
198C2×D4C2×C4⋊C414yes—
199C2×D4C2×C4⋊C414yes—
200C2×D4C2×C4⋊C418yes—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—28no—
204C22×C4—28no—
205C22×C4—48no—
206C22×C4—48no—
207C22×C4—416no—
208C22×C4—48no—
209C22×C4#95116yes—
210C22×C4#9518yes—
211C22×C4C2×C4214yes—
212C22×C4C2×C22⋊C418yes—
213C22×C4C2×C22⋊C414yes—
214C22×C4C2×C22⋊C414yes—
215C22×C4C2×C4⋊C418yes—
216C22×C4C2×C4⋊C412yes—
217C22×C4C23×C411yescharacteristic
218C22×C4C23×C412yes—
219C22×C4C22×D412yes—
220C22×C4C22×Q811yescharacteristic
221C22×C4H(5,𝔽2)14yes—
222C22×C4H(5,𝔽2)14yes—
223M16—28no—
224M16—48no—
225M16—48no—
226M16—44no—
227M16—44no—
228C2×C8—21no—
229C2×C8—24no—
230C2×C8—22no—
231C2×C8—21no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—24no—
235C2×C8—48no—
236C2×C8—48no—
237C2×C8—44no—
238C2×C8—44no—
239C4⋊C4—216no—
240C4⋊C4—416no—
241C4⋊C4—48no—
242C22⋊C4—216no—
243C22⋊C4—416no—
244C22⋊C4—48no—
245C42—216no—
246C42—416no—
247C42—48no—
248C23—28no—
249C23—24no—
250C23—48no—
251C23—81no—
252C23—84no—
253C23—82no—
254C23—88no—
255C23C2×#9514yes—
256C23C2×#9518yes—
257C23C22×C22⋊C412yes—
258C23C22×C4⋊C411yescharacteristic
259C23#53414yes—
260Q8—28no—
261Q8—28no—
262Q8—28no—
263Q8—48no—
264D4—216no—
265D4—216no—
266D4—216no—
267D4—28no—
268D4—28no—
269D4—28no—
270D4—48no—
271D4—416no—
272C2×C4—28no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—28no—
276C2×C4—24no—
277C2×C4—28no—
278C2×C4—216no—
279C2×C4—28no—
280C2×C4—416no—
281C2×C4—48no—
282C2×C4—416no—
283C2×C4—168no—
284C2×C4—1616no—
285C2×C4C2×#9514yes—
286C2×C4C2×#9518yes—
287C2×C4C22×C4211yescharacteristic
288C2×C4C22×C22⋊C411yescharacteristic
289C2×C4C22×C4⋊C412yes—
290C2×C4#51812yes—
291C2×C4#51812yes—
292C2×C4#51818yes—
293C2×C4#53414yes—
294C2×C4#53414yes—
295C2×C4#54314yes—
296C8—41no—
297C8—41no—
298C8—42no—
299C8—44no—
300C22—24no—
301C22—28no—
302C22—44no—
303C22—42no—
304C22—41no—
305C22—48no—
306C22—44no—
307C22—88no—
308C22—84no—
309C22—168no—
310C22C22×#9511yescharacteristic
311C22#230814yes—
312C22#231512yes—
313C4—28no—
314C4—24no—
315C4—816no—
316C4#227312yes—
317C4#230811yescharacteristic
318C4#230811yescharacteristic
319C4#231514yes—
320C2—24no—
321C2—21no—
322C2—22no—
323C2—48no—
324C2—44no—
325C2#1379411yescharacteristic
subgroupG/H
centerC4#2308
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC2#13794
prime radicalC2×#585C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10766C2semidirect
4. #10759C2semidirect
5. #10752C2semidirect
6. #10743C2semidirect
11. #2884C4semidirect
13. #2883C4semidirect
18. #2879C4semidirect
20. C2×#408C4semidirect
22. #1735C22semidirect
23. #1733C22semidirect
24. C2×H(3,ℤ/4)C22semidirect
40. C22×M16C2×C4semidirect
41. C22×M16C2×C4semidirect
44. C23×C8C2×C4semidirect
47. #519C2×C4semidirect
49. #518C2×C4semidirect
57. #410C2×C4semidirect
65. #408C2×C4semidirect
66. #408C2×C4semidirect
67. #408C2×C4semidirect

Maximal subgroups

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

subgroupindex
#600702
#600652
#107662
#107592
#107522
#107432