Finite Group Databasebeta

← #6477780

Subgroups — #6477780

Order 512.

1876 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3#33148C214yesmaximal, complemented
4#29968C214yesmaximal, complemented
5#25990C2116yesmaximal, complemented
6#21938C212yesmaximal
7#21930C211yesmaximal, characteristic
8C2×#1823C212yesmaximal
9C2×#543—21no—
10C22×#117—21no—
11C23×C4⋊C4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13#2857C2214yes—
14#2659—22no—
15#2549—24no—
16#2440C2218yes—
17#2344—24no—
18#2344C2218yes—
19#2315—21no—
20C2×#389C2212yes—
21C2×#387—22no—
22C2×#381—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2212yes—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#2100C2218yescomplemented
28#2088C22116yescomplemented
29#2058C2218yescomplemented
30#1987C2218yescomplemented
31#1948C2218yescomplemented
32#1920C22116yescomplemented
33#1853C2218yes—
34#1853C2218yes—
35#1850C2218yes—
36#1823C2214yes—
37#1823C2214yes—
38#1823C2218yes—
39#1807C2218yes—
40#1797C2214yes—
41C2×#340C2218yes—
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2314yes—
45#543—48no—
46C2×C4⋊Q8C2314yes—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—28no—
49C2×#117—28no—
50C2×#117—48no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4Q818yescomplemented
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2314yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2314yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42—22no—
66C22×C42C2312yes—
67C2×#106C23116yes—
68C2×C4⋊C8C2318yes—
69C2×C22⋊C8—28no—
70C2×C22⋊C8C2314yes—
71C2×C22⋊C8C2314yes—
72#398—28no—
73#395—28no—
74#391—24no—
75#389—24no—
76#389—24no—
77#389C2318yes—
78#389C2318yes—
79#387—216no—
80#385—28no—
81#384C2314yes—
82#381—28no—
83#380—24no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380C2314yes—
88#380C2314yes—
89#380C2314yes—
90C4×C4⋊C4—24no—
91C4×C4⋊C4C2318yes—
92C4×C22⋊C4D414yes—
93C4×C22⋊C4C2314yes—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D412yes—
99#340C23116yes—
100#340C23116yes—
101#340C23116yes—
102#336C2318yes—
103C25C2×D411yescharacteristic
104C22×Q8—24no—
105C23×C4—28no—
106C23×C4—21no—
107C23×C4—24no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—44no—
112C23×C4—44no—
113C23×C4—44no—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×Q814yes—
119C23×C4#4114yes—
120C23×C4C2412yes—
121C23×C4C2411yescharacteristic
122C22×C8—28no—
123C22×C8—48no—
124C22×C8C2418yes—
125C4⋊Q8—48no—
126C22⋊Q8—432no—
127C22⋊Q8—416no—
128#117—432no—
129C2×C4⋊C4—216no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—48no—
146C2×C4⋊C4—48no—
147C2×C4⋊C4C2×Q814yes—
148C2×C4⋊C4C2×Q814yes—
149C2×C4⋊C4C2×Q8116yes—
150C2×C4⋊C4C2418yes—
151C2×C4⋊C4C2414yes—
152C2×C4⋊C4C2412yes—
153C2×C4⋊C4C2412yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—48no—
160C2×C22⋊C4SD1618yes—
161C2×C22⋊C4C2×D414yes—
162C2×C22⋊C4C2×D414yes—
163C2×C42—28no—
164C2×C42—44no—
165C2×C42—48no—
166C2×C42C2×D414yes—
167C2×C42C2414yes—
168#106—416no—
169C4⋊C8—48no—
170C22⋊C8—28no—
171C22⋊C8—28no—
172#95—28no—
173#95—24no—
174#95—24no—
175#95—24no—
176#95—28no—
177#95—28no—
178#95C2×D418yes—
179C24—42no—
180C24—42no—
181C24C22⋊Q814yes—
182C24C2×SD1614yes—
183C24AGL(1,ℤ/8)14yes—
184C24C22×D412yes—
185C24C22×D411yescharacteristic
186C2×Q8—48no—
187C2×Q8—84no—
188C2×Q8—84no—
189C22×C4—24no—
190C22×C4—28no—
191C22×C4—24no—
192C22×C4—216no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—48no—
200C22×C4—48no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—48no—
210C22×C4—48no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—432no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—48no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4C22⋊Q814yes—
221C22×C4C22⋊Q818yes—
222C22×C4C2×SD1618yes—
223C22×C4C22×D414yes—
224C22×C4C22×D412yes—
225C22×C4C22×D411yescharacteristic
226C22×C4C22×D414yes—
227C22×C4C22×Q814yes—
228C22×C4C22×Q812yes—
229C22×C4C2×#4112yes—
230C22×C4C2×#4114yes—
231C22×C4C2511yescharacteristic
232C2×C8—28no—
233C2×C8—216no—
234C2×C8—88no—
235C4⋊C4—24no—
236C4⋊C4—216no—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—28no—
241C4⋊C4—216no—
242C4⋊C4—44no—
243C4⋊C4—48no—
244C4⋊C4—416no—
245C4⋊C4—416no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C4⋊C4—84no—
249C4⋊C4C22×Q814yes—
250C4⋊C4C22×Q814yes—
251C22⋊C4—28no—
252C22⋊C4—48no—
253C22⋊C4—48no—
254C22⋊C4—48no—
255C22⋊C4C2×SD16116yes—
256C42—28no—
257C42—44no—
258C42—416no—
259C23—22no—
260C23—24no—
261C23—22no—
262C23—24no—
263C23—44no—
264C23—44no—
265C23—48no—
266C23—48no—
267C23#476116yes—
268C23C2×C22⋊Q812yes—
269C23C2×C22⋊Q814yes—
270C23C22×SD1614yes—
271C23C22×SD1612yes—
272C23C2×AGL(1,ℤ/8)14yes—
273C23C2×AGL(1,ℤ/8)12yes—
274C23C23×D411yescharacteristic
275Q8—168no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—432no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—48no—
303C2×C4—44no—
304C2×C4—48no—
305C2×C4—84no—
306C2×C4—88no—
307C2×C4—168no—
308C2×C4C2×C22⋊Q814yes—
309C2×C4C2×C22⋊Q812yes—
310C2×C4C22×SD1618yes—
311C2×C4C23×Q811yescharacteristic
312C2×C4C22×#4111yescharacteristic
313C2×C4#58514yes—
314C8—48no—
315C22—24no—
316C22—28no—
317C22—24no—
318C22—22no—
319C22—22no—
320C22—24no—
321C22—24no—
322C22—28no—
323C22—416no—
324C22C2×#47618yes—
325C22C2×#47614yes—
326C22C22×C22⋊Q811yescharacteristic
327C22#348512yes—
328C22C23×SD1611yescharacteristic
329C22C22×AGL(1,ℤ/8)11yescharacteristic
330C22#358612yes—
331C4—22no—
332C4—24no—
333C4—28no—
334C4—44no—
335C4—44no—
336C4—44no—
337C4—88no—
338C4—84no—
339C4#345812yes—
340C4#349112yes—
341C2—24no—
342C2—28no—
343C2#3509011yescharacteristic
344C2C22×#47611yescharacteristic
345C2#6049111yescharacteristic
346C2#6169512yes—
347C2#6170412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33148C2semidirect
4. #29968C2semidirect
5. #25990C2semidirect
27. #2100C22semidirect
28. #2088C22semidirect
29. #2058C22semidirect
30. #1987C22semidirect
31. #1948C22semidirect
32. #1920C22semidirect
55. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#23442
#331482
#299682
#259902
#219382
#219302
C2×#18232