Finite Group Databasebeta

← #7628690

Subgroups — #7628690

Order 512.

5230 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C214yesmaximal
2#35304C2116yesmaximal, complemented
3#35120C218yesmaximal, complemented
4#35052C218yesmaximal
5#34878C2116yesmaximal, complemented
6#34642C212yesmaximal, complemented
7#34638C212yesmaximal, complemented
8#34074C214yesmaximal, complemented
9#34012C212yesmaximal
10C22×#376C211yesmaximal, characteristic
11C2×#533C2214yes—
12C23×C42C2212yes—
13C23×C42C2211yescharacteristic
14#2857C22116yes—
15#2855C22116yes—
16#2851C22116yes—
17#2848C22116yes—
18#2660C22116yescomplemented
19#2659C22116yes—
20#2647C22116yescomplemented
21#2646C22116yes—
22#2612C22164yescomplemented
23#2606C22132yescomplemented
24#2605C22132yescomplemented
25#2569C22132yes—
26#2549C22132yes—
27#2438C22164yes—
28#2402C22116yescomplemented
29#2399C22116yescomplemented
30#2331C22116yescomplemented
31#2330C22116yescomplemented
32#2329C22132yes—
33C4×C22⋊Q8C22132yes—
34#2296C2218yes—
35#2295C22132yescomplemented
36#2285C2214yes—
37#2285C2214yes—
38#2285C2218yes—
39#2283C2218yes—
40#2283C2218yes—
41#2283C2218yes—
42#2280C2214yes—
43#2280C2218yes—
44C4×#117C2214yes—
45C4×#117C2218yes—
46C2×#376C2212yes—
47C2×#376C2214yes—
48C2×#376C2218yes—
49C2×#376C2211yescharacteristic
50C2×#376C2211yescharacteristic
51C2×#376C2218yes—
52C22×#95C2214yes—
53C24×C4C2311yescharacteristic
54C24×C4C2314yes—
55C24×C4C2312yes—
56#533—232no—
57C2×C4⋊Q8C2318yes—
58C2×#125C2318yes—
59C2×C22⋊Q8C23116yes—
60C2×C22⋊Q8C23132yes—
61C2×C4×Q8C23116yes—
62C2×#117C2318yes—
63C2×#117C23116yes—
64C22×C42—216no—
65C22×C42Q8116yescomplemented
66C22×C42C2318yes—
67C22×C42C2314yes—
68C22×C42C2312yes—
69C22×C42C2318yes—
70C22×C42C2311yescharacteristic
71C22×C42C2314yes—
72C22×C42C2312yes—
73C22×C42C2314yes—
74C22×C42C2311yescharacteristic
75C22×C42C2312yes—
76#400C23164yes—
77#398C23164yes—
78#398C23164yes—
79#395C23164yes—
80#391C23164yes—
81#391C23132yes—
82#389C23164yes—
83#389C23132yes—
84#389C23132yes—
85#387C23164yes—
86#387C23164yes—
87#387C23164yes—
88#385C23164yes—
89#385C23132yes—
90#384C2318yes—
91#384C2318yes—
92#383C23116yes—
93#382C2318yes—
94#382C23116yes—
95#382C2318yes—
96#381C2318yes—
97#381C23116yes—
98#380C23132yes—
99#380C23116yes—
100#380C23116yes—
101#380C23116yes—
102#380C23116yes—
103#379C2318yes—
104C4×C4⋊C4C23116yes—
105C4×C4⋊C4C23116yes—
106C4×C4⋊C4C23116yes—
107C4×C4⋊C4C23116yes—
108C4×C4⋊C4C23132yes—
109C4×C22⋊C4C23116yes—
110C4×C22⋊C4C23116yes—
111C4×C22⋊C4C23116yes—
112#376C2318yes—
113#376C23116yes—
114#376C23116yes—
115#376C23116yes—
116#376C2314yes—
117#376C2314yes—
118#376C2314yes—
119#376C2314yes—
120#376C2318yes—
121#376C2318yes—
122C2×#95C23116yes—
123C2×#95C2318yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C23116yes—
127C43C2318yes—
128C25C2411yescharacteristic
129C22×Q8C24116yes—
130C23×C4—28no—
131C23×C4—216no—
132C23×C4—48no—
133C23×C4—416no—
134C23×C4C2×Q818yes—
135C23×C4C2×Q8116yes—
136C23×C4C24116yes—
137C23×C4C2414yes—
138C23×C4C2412yes—
139C23×C4C2411yescharacteristic
140C23×C4C2412yes—
141C23×C4C2414yes—
142C23×C4C2412yes—
143C23×C4C2414yes—
144C23×C4C2412yes—
145C23×C4C2414yes—
146C23×C4C2411yescharacteristic
147C4⋊Q8—232no—
148#125—232no—
149C22⋊Q8—264no—
150C22⋊Q8—464no—
151C4×Q8—264no—
152#117—232no—
153#117—432no—
154C2×C4⋊C4C24116yes—
155C2×C4⋊C4C24132yes—
156C2×C4⋊C4C24116yes—
157C2×C4⋊C4C24132yes—
158C2×C4⋊C4C24132yes—
159C2×C4⋊C4C24164yes—
160C2×C22⋊C4C24116yes—
161C2×C22⋊C4C24132yes—
162C2×C42—216no—
163C2×C42—28no—
164C2×C42—232no—
165C2×C42—232no—
166C2×C42C2×Q8132yes—
167C2×C42C2×Q8132yes—
168C2×C42C2×Q8116yes—
169C2×C42C2×Q818yes—
170C2×C42C2×Q818yes—
171C2×C42#41116yes—
172C2×C42C24116yes—
173C2×C42C2414yes—
174C2×C42C24116yes—
175C2×C42C2414yes—
176C2×C42C24132yes—
177C2×C42C2414yes—
178C2×C42C2412yes—
179C2×C42C2414yes—
180C2×C42C2412yes—
181C2×C42C2412yes—
182C2×C42C2412yes—
183C2×C42C2414yes—
184C2×C42C2414yes—
185#95C24132yes—
186#95C24116yes—
187#95C24116yes—
188#95C24132yes—
189#95C24132yes—
190#95C24132yes—
191#95C24132yes—
192#95C24164yes—
193C24—24no—
194C24—28no—
195C24C22×Q814yes—
196C24C2511yescharacteristic
197C24C2512yes—
198C2×Q8—232no—
199C2×Q8—432no—
200C22×C4—24no—
201C22×C4—216no—
202C22×C4—232no—
203C22×C4—28no—
204C22×C4—28no—
205C22×C4—28no—
206C22×C4—232no—
207C22×C4—232no—
208C22×C4—432no—
209C22×C4—44no—
210C22×C4—48no—
211C22×C4—44no—
212C22×C4—48no—
213C22×C4—416no—
214C22×C4—48no—
215C22×C4—464no—
216C22×C4C22×Q814yes—
217C22×C4C22×Q8116yes—
218C22×C4C22×Q818yes—
219C22×C4C22×Q818yes—
220C22×C4C22×Q818yes—
221C22×C4C22×Q818yes—
222C22×C4C22×Q8116yes—
223C22×C4C22×Q814yes—
224C22×C4C2×#41116yes—
225C22×C4C2×#4118yes—
226C22×C4C2518yes—
227C22×C4C25132yes—
228C22×C4C2512yes—
229C22×C4C2514yes—
230C22×C4C2514yes—
231C22×C4C2518yes—
232C22×C4C2511yescharacteristic
233C22×C4C2511yescharacteristic
234C4⋊C4—232no—
235C4⋊C4—264no—
236C4⋊C4—232no—
237C4⋊C4—432no—
238C4⋊C4—432no—
239C4⋊C4—464no—
240C22⋊C4—232no—
241C22⋊C4—264no—
242C42—232no—
243C42—28no—
244C42—28no—
245C42—232no—
246C42—432no—
247C42C22×Q8116yes—
248C42C22×Q8116yes—
249C42C22×Q818yes—
250C42C22×Q818yes—
251C42C22×Q818yes—
252C42C22×Q818yes—
253C42C2×#41116yes—
254C42C2×#41116yes—
255C23—232no—
256C23—24no—
257C23—24no—
258C23—24no—
259C23—28no—
260C23—216no—
261C23C23×Q812yes—
262C23C23×Q814yes—
263C23C22×#4114yes—
264C23#58518yes—
265C23C2611yescharacteristic
266Q8—832no—
267C2×C4—24no—
268C2×C4—28no—
269C2×C4—28no—
270C2×C4—216no—
271C2×C4—28no—
272C2×C4—232no—
273C2×C4—216no—
274C2×C4—216no—
275C2×C4—28no—
276C2×C4—232no—
277C2×C4—264no—
278C2×C4—216no—
279C2×C4—48no—
280C2×C4—416no—
281C2×C4—48no—
282C2×C4—416no—
283C2×C4—464no—
284C2×C4—416no—
285C2×C4#533116yes—
286C2×C4C23×Q814yes—
287C2×C4C23×Q818yes—
288C2×C4C23×Q818yes—
289C2×C4C23×Q812yes—
290C2×C4C23×Q812yes—
291C2×C4C22×#41116yes—
292C2×C4C22×#4114yes—
293C2×C4C22×#4114yes—
294C22—232no—
295C22—28no—
296C22—216no—
297C22—24no—
298C22—24no—
299C22—24no—
300C22C2×#53314yes—
301C22#345818yes—
302C22C24×Q811yescharacteristic
303C22C23×#4112yes—
304C22C2×#58514yes—
305C4—216no—
306C4—24no—
307C4—28no—
308C4—28no—
309C4—48no—
310C4—432no—
311C4C2×#53314yes—
312C4C2×#53314yes—
313C2—24no—
314C2—28no—
315C2C22×#53311yescharacteristic
316C2C2×#345812yes—
317C2#6281414yes—
subgroupG/H
centerC22×C4C25
commutator, Frattini, socle, cosocleC23C26
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35304C2semidirect
3. #35120C2semidirect
5. #34878C2semidirect
6. #34642C2semidirect
7. #34638C2semidirect
8. #34074C2semidirect
18. #2660C22semidirect
20. #2647C22semidirect
22. #2612C22semidirect
23. #2606C22semidirect
24. #2605C22semidirect
28. #2402C22semidirect
29. #2399C22semidirect
30. #2331C22semidirect
31. #2330C22semidirect
35. #2295C22semidirect
65. C22×C42Q8semidirect

Maximal subgroups

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

subgroupindex
#366252
#353042
#351202
#350522
#348782
#346422
#346382
#340742
#340122
C22×#3762