Finite Group Databasebeta

← #6528444

Subgroups — #6528444

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic
3#29301C214yesmaximal
4#28835C218yesmaximal, complemented
5#28799C218yesmaximal, complemented
6#27956C214yesmaximal
7#21982C212yesmaximal
8#21359C212yesmaximal
9#20536C211yesmaximal, characteristic
10C2×#544—21no—
11C2×#431C2212yes—
12#2857C2214yes—
13#2659—22no—
14#2571—24no—
15#2571C2218yes—
16#2552C2214yes—
17#2551C2214yes—
18#2550—24no—
19C2×#400—22no—
20#2316—21no—
21C2×#389—21no—
22C2×#389C2212yes—
23#2285C2212yes—
24C2×#384C2211yescharacteristic
25C2×#382—21no—
26C2×#382C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2085C22116yes—
30#2083C2218yes—
31#2058C2218yes—
32#1993C2218yes—
33#1948C2214yes—
34#1948C2214yes—
35#1947C2218yes—
36#1943C22116yes—
37#1853C2214yes—
38#1853C2214yes—
39#1852C2218yes—
40#1847C2218yes—
41#1845C2214yes—
42#1834C2214yes—
43C4×#106C2218yes—
44#1770C2218yes—
45#1749C2214yes—
46#544—48no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8C2314yes—
49C2×#125—24no—
50C2×C22⋊Q8—24no—
51C2×#117—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C42—21no—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63#431Q8116yescomplemented
64C2×#107C23116yes—
65C2×#106C2318yes—
66C2×#106C2318yes—
67C2×C4⋊C8C2314yes—
68C2×C22⋊C8C2314yes—
69C2×#97C2318yes—
70C2×C4×C8C2314yes—
71#400—216no—
72#398—28no—
73#395—28no—
74#391—24no—
75#389—28no—
76#389—24no—
77#389—24no—
78#389—24no—
79#389—24no—
80#389C2318yes—
81#389C2318yes—
82#384—22no—
83#384C2314yes—
84#384C2314yes—
85#384C2312yes—
86#384C2312yes—
87#382—28no—
88#382—24no—
89#382—22no—
90#382C2318yes—
91#380—24no—
92#380—22no—
93#380C2314yes—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4D414yes—
97C4×C4⋊C4C2314yes—
98C4×C4⋊C4C2314yes—
99C4×C4⋊C4C2314yes—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95D412yes—
105#340C23116yes—
106#340C2318yes—
107#340C2318yes—
108#336C2318yes—
109C22×Q8—24no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C22×C8C2418yes—
119C4⋊Q8—48no—
120C4⋊Q8—416no—
121#125—416no—
122C22⋊Q8—416no—
123#117—48no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—82no—
145C2×C4⋊C4—84no—
146C2×C4⋊C4C2×D414yes—
147C2×C4⋊C4C2×D414yes—
148C2×C4⋊C4C2412yes—
149C2×C4⋊C4C2412yes—
150C2×C4⋊C4C2414yes—
151C2×C4⋊C4C2414yes—
152C2×C4⋊C4C2414yes—
153C2×C22⋊C4—24no—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—44no—
157C2×C42—44no—
158C2×C42C2×D414yes—
159C2×C42C2×D412yes—
160C2×C42C2×Q818yes—
161C2×C42C2414yes—
162C2×C42C2412yes—
163#107—416no—
164#106—48no—
165#106—48no—
166C4⋊C8C2×Q8116yes—
167C22⋊C8C2×Q8116yes—
168#97—48no—
169C4×C8C2×Q8116yes—
170#95—24no—
171#95—28no—
172#95—28no—
173#95—28no—
174#95—24no—
175#95—24no—
176#95—28no—
177#95C2×D418yes—
178C24C22×D411yescharacteristic
179C2×Q8—48no—
180C2×Q8—84no—
181C2×Q8—84no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4C4⋊Q814yes—
207C22×C4C4⋊Q814yes—
208C22×C4AGL(1,ℤ/8)14yes—
209C22×C4#13714yes—
210C22×C4C22×D414yes—
211C22×C4C22×D414yes—
212C22×C4C22×D412yes—
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D412yes—
215C22×C4C22×Q814yes—
216C22×C4C2511yescharacteristic
217C2×C8—28no—
218C2×C8—28no—
219C2×C8C22×Q8116yes—
220C4⋊C4—28no—
221C4⋊C4—28no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—28no—
225C4⋊C4—28no—
226C4⋊C4—28no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—48no—
231C4⋊C4—416no—
232C4⋊C4—48no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C4⋊C4—84no—
238C4⋊C4—84no—
239C4⋊C4—88no—
240C4⋊C4—84no—
241C22⋊C4—48no—
242C42—28no—
243C42—44no—
244C42—48no—
245C42—48no—
246C42C4⋊Q818yes—
247C42C22×Q814yes—
248C42C22×Q814yes—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23C2×C4⋊Q812yes—
254C23C2×AGL(1,ℤ/8)12yes—
255C23C2×#13712yes—
256C23C23×D411yescharacteristic
257Q8—168no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—28no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—88no—
290C2×C4#501116yes—
291C2×C4C2×C4⋊Q812yes—
292C2×C4C2×C4⋊Q814yes—
293C2×C4C2×C4⋊Q814yes—
294C2×C4C2×AGL(1,ℤ/8)14yes—
295C2×C4C2×#13714yes—
296C2×C4C23×Q812yes—
297C2×C4#58512yes—
298C2×C4#58512yes—
299C8—216no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22C2×#50114yes—
307C22C22×C4⋊Q811yescharacteristic
308C22#348511yescharacteristic
309C22#348711yescharacteristic
310C22C22×AGL(1,ℤ/8)11yescharacteristic
311C22C22×#13711yescharacteristic
312C22#358511yescharacteristic
313C22#358611yescharacteristic
314C4—22no—
315C4—24no—
316C4—44no—
317C4—44no—
318C4—82no—
319C4—84no—
320C4—82no—
321C4—84no—
322C4—84no—
323C4C2×#50118yes—
324C4#350414yes—
325C2—24no—
326C2#3564011yescharacteristic
327C2C22×#50111yescharacteristic
328C2#6066411yescharacteristic
329C2#6172111yescharacteristic
330C2#6172211yescharacteristic
331C2#6188911yescharacteristic
332C2#6189111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #28835C2semidirect
5. #28799C2semidirect
63. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#25712
#293012
#288352
#287992
#279562
#219822
#213592
#205362