Finite Group Databasebeta

← #7624451

Subgroups — #7624451

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62360C211yesmaximal, characteristic, complemented
2C2×#3396C211yesmaximal, characteristic
3#33967C218yesmaximal, complemented
4#33942C218yesmaximal, complemented
5#33781C214yesmaximal, complemented
6#33774C212yesmaximal
7#33670C214yesmaximal, complemented
8#33664C212yesmaximal, complemented
9#33497C211yesmaximal, characteristic, complemented
10#3502—21no—
11C2×#559—21no—
12#3396—24no—
13#3396C2218yes—
14#3395C2214yescomplemented
15Q8×Q16C2214yescomplemented
16#3363—24no—
17#3249—22no—
18#3249—21no—
19#3170C2212yes—
20#3170C2214yes—
21C2×#495C2212yes—
22C2×#493C2211yescharacteristic
23C2×#460—21no—
24C2×#463—22no—
25C2×C8×Q8C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×C4×D8C2212yes—
28#2251C2218yescomplemented
29#2248C2214yes—
30#2243C2218yescomplemented
31#2240C2214yes—
32#2236C2214yes—
33#2227C2214yescomplemented
34#2224C2218yescomplemented
35#2211C22116yescomplemented
36#2209C2214yescomplemented
37#2208C2218yescomplemented
38#2207C2218yes—
39#2188C2214yes—
40#2187C2218yes—
41#2183C2218yescomplemented
42#2175C22116yescomplemented
43C4×Q32C2218yescomplemented
44#2163C2214yes—
45#2162C2212yes—
46C22×Q16C2311yescharacteristic
47C22×SD16—22no—
48C22×D8—21no—
49C22×D8C2312yes—
50#559—28no—
51#559—44no—
52Q82—24no—
53#557—24no—
54#557—44no—
55#544—41no—
56#544—42no—
57#533—24no—
58#533—22no—
59C2×#127—21no—
60C2×#124—22no—
61C2×#124—41no—
62C2×#123—42no—
63#519—41no—
64C2×C4×Q8—21no—
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4—22no—
67C2×C4×D4—41no—
68C2×Q32C2318yes—
69C2×SD32C23116yes—
70#500C2314yes—
71#500C2312yes—
72#495—24no—
73#495C2318yes—
74#494C2314yes—
75#494C2312yes—
76#493—22no—
77#493C2314yes—
78#487—24no—
79#487—22no—
80#477—24no—
81#477—28no—
82#467—22no—
83#467—24no—
84#467C2314yes—
85#463—216no—
86#462—24no—
87#462—28no—
88#460—28no—
89C8×Q8D414yes—
90C8×Q8C2314yes—
91C4×Q16—24no—
92C4×Q16—22no—
93C4×Q16—22no—
94C4×Q16C2314yes—
95C4×Q16C2318yes—
96C4×Q16C2314yes—
97C4×D8—24no—
98C4×D8C2318yes—
99C2×#107C2311yescharacteristic
100C2×C4⋊C8D412yes—
101C2×C4⋊C8D411yescharacteristic
102C2×#103—21no—
103C2×#102—22no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2312yes—
106#366C2318yes—
107C4⋊C16C2314yes—
108C4⋊C16C2318yes—
109#358C2318yes—
110#358C2318yes—
111#357C23116yes—
112C22⋊C16C2314yes—
113#346C2318yes—
114C4×C16C2314yes—
115C22×Q8—21no—
116C22×Q8D811yescharacteristic
117C22×D4—22no—
118C22×D4—41no—
119C22×D4—41no—
120C23×C4—42no—
121C2×Q16—21no—
122C2×Q16—24no—
123C2×Q16—22no—
124C2×Q16—42no—
125C2×Q16C2412yes—
126C2×Q16C2414yes—
127C2×SD16—28no—
128C2×SD16—84no—
129C2×D8—22no—
130C2×D8—24no—
131C2×D8—44no—
132C2×D8—82no—
133C2×D8C2414yes—
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D412yes—
136C22×C8C2411yescharacteristic
137C4⋊Q8—24no—
138C4⋊Q8—22no—
139C4⋊Q8—24no—
140C4⋊Q8—22no—
141C4⋊Q8—44no—
142C4⋊Q8—48no—
143#127—24no—
144#127—42no—
145#125—44no—
146#125—48no—
147#125—44no—
148#125—42no—
149#124—28no—
150#124—48no—
151#124—44no—
152#123—416no—
153C22⋊Q8—24no—
154C22⋊Q8—44no—
155C22⋊Q8—42no—
156C4×Q8—24no—
157C4×Q8—28no—
158C4×Q8—24no—
159C4×Q8—44no—
160C4×Q8—44no—
161C4×Q8—42no—
162C4×Q8—44no—
163C4×Q8—42no—
164C4×Q8D814yes—
165C4×Q8C2×D414yes—
166C4×D4—28no—
167C4×D4—48no—
168C4×D4—44no—
169#117—44no—
170#117—42no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4D811yescharacteristic
174C2×C4⋊C4D812yes—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C42—41no—
179C2×C42C2×D412yes—
180C2×C42C2×D411yescharacteristic
181Q32—48no—
182SD32—416no—
183C2×C16C2418yes—
184#107—21no—
185#107—24no—
186#107—22no—
187#107C2414yes—
188#107C2412yes—
189C4⋊C8C2×D414yes—
190C4⋊C8C2×D418yes—
191#103—24no—
192#103—24no—
193#103—28no—
194#103—24no—
195#102—28no—
196C4×C8C2×D414yes—
197C4×C8C2×D412yes—
198C4×C8C2412yes—
199C4×C8C2414yes—
200C24—42no—
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—22no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—41no—
207C2×Q8—44no—
208C2×Q8—42no—
209C2×Q8C2×D814yes—
210C2×D4—24no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—84no—
217C2×D4—162no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—48no—
221C22×C4—164no—
222C22×C4C2×D812yes—
223C22×C4C2×D812yes—
224C22×C4C2×D811yescharacteristic
225C22×C4C2×D811yescharacteristic
226C22×C4C22×D411yescharacteristic
227Q16—28no—
228Q16—24no—
229Q16—42no—
230Q16—82no—
231Q16—84no—
232SD16—88no—
233D8—28no—
234D8—44no—
235D8—84no—
236C2×C8—22no—
237C2×C8—41no—
238C2×C8—42no—
239C2×C8C22×D411yescharacteristic
240C2×C8C22×D414yes—
241C2×C8C22×D412yes—
242C2×C8C2511yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—24no—
245C4⋊C4—41no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—48no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4C2×D814yes—
257C4⋊C4C2×D818yes—
258C22⋊C4—44no—
259C22⋊C4—48no—
260C22⋊C4—48no—
261C22⋊C4—48no—
262C42—44no—
263C42—48no—
264C42—44no—
265C42C2×D814yes—
266C42C2×D812yes—
267C42C22×D414yes—
268C42C22×D412yes—
269C16—28no—
270C23—44no—
271C23—84no—
272C23—84no—
273C23—84no—
274C23C22×D811yescharacteristic
275Q8—28no—
276Q8—42no—
277Q8—44no—
278Q8—42no—
279Q8—44no—
280Q8—82no—
281Q8—164no—
282Q8—162no—
283D4—44no—
284D4—44no—
285D4—84no—
286D4—88no—
287D4—164no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—168no—
300C2×C4#51014yes—
301C2×C4C22×D814yes—
302C2×C4C22×D811yescharacteristic
303C2×C4C22×D814yes—
304C2×C4C22×D812yes—
305C2×C4C22×D812yes—
306C2×C4C23×D411yescharacteristic
307C8—22no—
308C8—24no—
309C8—21no—
310C8#58512yes—
311C22—21no—
312C22—21no—
313C22—84no—
314C22—84no—
315C22—84no—
316C22—168no—
317C22C2×#51012yes—
318C22C23×D811yescharacteristic
319C4—24no—
320C4—24no—
321C4—22no—
322C4—21no—
323C4—82no—
324C4—84no—
325C4—82no—
326C4—84no—
327C4C2×#51014yes—
328C4#348711yescharacteristic
329C4#358511yescharacteristic
330C2—22no—
331C2—164no—
332C2#6188011yescharacteristic
333C2C22×#51011yescharacteristic
334C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC2×#493C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62360C2semidirect
3. #33967C2semidirect
4. #33942C2semidirect
5. #33781C2semidirect
7. #33670C2semidirect
8. #33664C2semidirect
9. #33497C2semidirect
14. #3395C22semidirect
15. Q8×Q16C22semidirect
28. #2251C22semidirect
30. #2243C22semidirect
33. #2227C22semidirect
34. #2224C22semidirect
35. #2211C22semidirect
36. #2209C22semidirect
37. #2208C22semidirect
41. #2183C22semidirect
42. #2175C22semidirect
43. C4×Q32C22semidirect

Maximal subgroups

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

subgroupindex
#623602
C2×#33962
#339672
#339422
#337812
#337742
#336702
#336642
#334972