Finite Group Databasebeta

← #6528593

Subgroups — #6528593

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic
3#29335C214yesmaximal
4#27983C214yesmaximal
5#27757C218yesmaximal, complemented
6#27738C218yesmaximal, complemented
7#21860C212yesmaximal
8#21385C212yesmaximal
9#20536C211yesmaximal, characteristic
10C2×#533C2212yes—
11C2×#431C2212yes—
12#2857C2214yes—
13#2659—22no—
14#2584—24no—
15#2571—24no—
16#2571C2218yes—
17#2567C2214yes—
18#2546C2214yes—
19C2×#400—22no—
20#2316—21no—
21#2316—21no—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#382—21no—
26C2×#382C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2099C2218yes—
30#2074C22116yes—
31#2043C2218yes—
32#1990C2214yes—
33#1987C2214yes—
34#1981C2214yes—
35#1960C2218yes—
36#1956C2214yes—
37#1935C22116yes—
38#1872C2214yes—
39#1872C2218yes—
40#1872C2214yes—
41#1836C2218yes—
42#1803C2218yes—
43#1770C2218yes—
44C4×#103C2218yes—
45#1749C2214yes—
46#533—28no—
47#533—28no—
48C2×C4⋊Q8C2312yes—
49C2×C4⋊Q8C2314yes—
50C2×C4⋊Q8C2312yes—
51C2×#125—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8C2314yes—
54C2×C4×Q8—22no—
55C2×C4×Q8C2314yes—
56C2×#117—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C42—21no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68#431—28no—
69C2×C4⋊C8C2314yes—
70C2×#103C23116yes—
71C2×#103C2318yes—
72C2×#103C2318yes—
73C2×C22⋊C8C2314yes—
74C2×#97C2318yes—
75C2×C4×C8C2314yes—
76#400—216no—
77#398—28no—
78#395—28no—
79#391C2318yes—
80#389—24no—
81#389—24no—
82#389—28no—
83#389—24no—
84#389—24no—
85#389—24no—
86#389—24no—
87#389—24no—
88#389C2318yes—
89#384—22no—
90#384C2314yes—
91#384C2314yes—
92#382—28no—
93#382—24no—
94#382C2318yes—
95#380—24no—
96#380—24no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4D414yes—
99C4×C4⋊C4C2314yes—
100C4×C4⋊C4C2314yes—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D412yes—
106#340C23116yes—
107#340C2318yes—
108#340C2318yes—
109#336C2318yes—
110C22×Q8—22no—
111C22×Q8C2414yes—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8C2418yes—
121C4⋊Q8—24no—
122C4⋊Q8—24no—
123C4⋊Q8—24no—
124C4⋊Q8—24no—
125C4⋊Q8—48no—
126#125—48no—
127C22⋊Q8—28no—
128C22⋊Q8—28no—
129C22⋊Q8—48no—
130C4×Q8—28no—
131C4×Q8—28no—
132C4×Q8—48no—
133#117—48no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—84no—
157C2×C4⋊C4C2×D414yes—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2412yes—
160C2×C4⋊C4C2412yes—
161C2×C4⋊C4C2414yes—
162C2×C4⋊C4C2414yes—
163C2×C22⋊C4—24no—
164C2×C42—24no—
165C2×C42—24no—
166C2×C42—44no—
167C2×C42—44no—
168C2×C42C2×D414yes—
169C2×C42C2×D412yes—
170C2×C42#4118yes—
171C2×C42C2414yes—
172C2×C42C2412yes—
173C4⋊C8—28no—
174#103—48no—
175#103—416no—
176#103—48no—
177C22⋊C8—28no—
178#97—48no—
179C4×C8—28no—
180#95—24no—
181#95—28no—
182#95—28no—
183#95—28no—
184#95—28no—
185#95—28no—
186#95C2×D418yes—
187C24C22×D411yescharacteristic
188C2×Q8—24no—
189C2×Q8—24no—
190C2×Q8—28no—
191C2×Q8—28no—
192C2×Q8—44no—
193C2×Q8—82no—
194C2×Q8—82no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4#12414yes—
220C22×C4#12414yes—
221C22×C4#13714yes—
222C22×C4#13714yes—
223C22×C4C22×D414yes—
224C22×C4C22×D414yes—
225C22×C4C22×D412yes—
226C22×C4C22×D411yescharacteristic
227C22×C4C22×D412yes—
228C22×C4C2×#4114yes—
229C22×C4C2511yescharacteristic
230C2×C8—28no—
231C2×C8—28no—
232C2×C8—28no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—28no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—48no—
246C4⋊C4—48no—
247C4⋊C4—44no—
248C4⋊C4—48no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C4⋊C4—84no—
252C4⋊C4—88no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C22⋊C4—48no—
256C42—28no—
257C42—44no—
258C42—48no—
259C42—48no—
260C42#12418yes—
261C42C2×#4114yes—
262C42C2×#4114yes—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23C2×#12412yes—
268C23C2×#13712yes—
269C23C2×#13712yes—
270C23C23×D411yescharacteristic
271Q8—48no—
272Q8—48no—
273Q8—88no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—28no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—84no—
304C2×C4—84no—
305C2×C4—88no—
306C2×C4#491116yes—
307C2×C4C2×#12412yes—
308C2×C4C2×#12414yes—
309C2×C4C2×#12414yes—
310C2×C4C2×#13714yes—
311C2×C4C2×#13714yes—
312C2×C4C22×#4112yes—
313C2×C4#58512yes—
314C2×C4#58512yes—
315C8—48no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22C2×#49114yes—
323C22C22×#12411yescharacteristic
324C22#348511yescharacteristic
325C22#348711yescharacteristic
326C22C22×#13711yescharacteristic
327C22C22×#13711yescharacteristic
328C22#358511yescharacteristic
329C22#358611yescharacteristic
330C4—22no—
331C4—24no—
332C4—44no—
333C4—44no—
334C4—82no—
335C4—84no—
336C4—82no—
337C4—84no—
338C4—84no—
339C4C2×#49118yes—
340C4#351214yes—
341C2—24no—
342C2#3564911yescharacteristic
343C2C22×#49111yescharacteristic
344C2#6062111yescharacteristic
345C2#6172111yescharacteristic
346C2#6172311yescharacteristic
347C2#6188911yescharacteristic
348C2#6189211yescharacteristic
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
5. #27757C2semidirect
6. #27738C2semidirect

Maximal subgroups

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

subgroupindex
#351732
C2×#25712
#293352
#279832
#277572
#277382
#218602
#213852
#205362