Finite Group Databasebeta

← #6487859

Subgroups — #6487859

Order 512.

1640 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic
3#32960C214yesmaximal
4#29972C214yesmaximal
5#26123C218yesmaximal, complemented
6#26057C218yesmaximal, complemented
7#21986C212yesmaximal
8#21955C211yesmaximal, characteristic
9#20740C212yesmaximal
10C22×C4×Q8C2211yescharacteristic
11C22×C4×Q8C2212yes—
12C2×#479C2218yes—
13C2×#432C2212yes—
14#2851C2214yes—
15#2646—21no—
16#2646—21no—
17#2571C2214yes—
18#2570—24no—
19#2552C2214yes—
20#2351—24no—
21#2351C2218yes—
22#2317—22no—
23C2×#389—22no—
24C2×#389C2212yes—
25#2285C2212yes—
26C2×#382—21no—
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2101C2218yes—
31#2088C2218yes—
32#2087C2218yes—
33#2059C2218yes—
34#1949C2218yes—
35#1940C2218yes—
36#1865C2218yes—
37#1863C2218yes—
38#1853C2214yes—
39#1852C2214yes—
40#1847C2214yes—
41#1836C2214yes—
42#1823C2214yes—
43#1805C2218yes—
44#1803C2214yes—
45#1803C2214yes—
46#1797C2214yes—
47#1787C2214yes—
48#1763C2218yes—
49C23×Q8C2311yescharacteristic
50C2×#125C2314yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C2×C4×Q8—48no—
55C2×C4×Q8Q818yes—
56C2×C4×Q8C2318yes—
57C2×C4×Q8C2314yes—
58C2×#117—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C42—21no—
66C22×C42—22no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#479—416no—
70#432—28no—
71C2×#107C2318yes—
72C2×#106C2318yes—
73C2×C4⋊C8C2314yes—
74C2×C4⋊C8C2318yes—
75C2×#103C2318yes—
76C2×#103C2318yes—
77C2×C22⋊C8C2314yes—
78C2×#97C2314yes—
79#400—24no—
80#400—24no—
81#398—24no—
82#398—24no—
83#389—24no—
84#389—24no—
85#389—28no—
86#389—28no—
87#389—24no—
88#389C2318yes—
89#389C2314yes—
90#389C2314yes—
91#387—24no—
92#387—24no—
93#385—24no—
94#385—24no—
95#384—24no—
96#384C2312yes—
97#384C2312yes—
98#382—24no—
99#382—22no—
100#382—22no—
101#382—28no—
102#382C2314yes—
103#382C2314yes—
104#380—22no—
105#380C2314yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—28no—
108C4×C4⋊C4D414yes—
109C4×C4⋊C4C2314yes—
110C4×C4⋊C4C2314yes—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340C2318yes—
117#340C2318yes—
118#340C2318yes—
119#340C2318yes—
120#336C2318yes—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8C2×Q814yes—
126C22×Q8C2412yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8C2418yes—
136#125—48no—
137C4×Q8—216no—
138C4×Q8—28no—
139C4×Q8—216no—
140C4×Q8—48no—
141C4×Q8—416no—
142C4×Q8C2×Q8116yes—
143#117—48no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4C2×Q814yes—
166C2×C4⋊C4C2414yes—
167C2×C4⋊C4C2414yes—
168C2×C4⋊C4C2414yes—
169C2×C4⋊C4C2412yes—
170C2×C22⋊C4—24no—
171C2×C42—28no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—28no—
175C2×C42—44no—
176C2×C42—48no—
177C2×C42C2×D414yes—
178C2×C42C2×D412yes—
179C2×C42C2×Q814yes—
180C2×C42#4114yes—
181C2×C42C2414yes—
182C2×C42C2412yes—
183#107—48no—
184#106—48no—
185C4⋊C8—28no—
186C4⋊C8—48no—
187#103—216no—
188#103—216no—
189C22⋊C8—28no—
190#97—28no—
191#95—24no—
192#95—28no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95C2×D414yes—
201#95C2×D414yes—
202C24C22×D411yescharacteristic
203C2×Q8—24no—
204C2×Q8—216no—
205C2×Q8—24no—
206C2×Q8—216no—
207C2×Q8—24no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8C22×Q814yes—
212C2×Q8H(5,𝔽2)14yes—
213C2×Q82−1+414yes—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4C22⋊Q814yes—
238C22×C4C22⋊Q814yes—
239C22×C4C22×D414yes—
240C22×C4C22×D414yes—
241C22×C4C22×D412yes—
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D412yes—
244C22×C4C22×Q812yes—
245C22×C4C2×#4112yes—
246C22×C4C2511yescharacteristic
247C2×C8—28no—
248C2×C8—28no—
249C2×C8—28no—
250C4⋊C4—216no—
251C4⋊C4—24no—
252C4⋊C4—22no—
253C4⋊C4—28no—
254C4⋊C4—28no—
255C4⋊C4—28no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—416no—
259C4⋊C4—416no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—84no—
266C4⋊C4—84no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C4⋊C4C22×Q814yes—
270C22⋊C4—48no—
271C42—216no—
272C42—28no—
273C42—44no—
274C42—416no—
275C42—48no—
276C42C22⋊Q818yes—
277C42C22×Q814yes—
278C42C2×#4114yes—
279C23—22no—
280C23—21no—
281C23—21no—
282C23C2×C22⋊Q812yes—
283C23#57514yes—
284C23C23×D411yescharacteristic
285Q8—216no—
286Q8—24no—
287Q8—24no—
288Q8—24no—
289Q8#54418yes—
290C2×C4—216no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—28no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—21no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—84no—
324C2×C4—88no—
325C2×C4C2×C22⋊Q814yes—
326C2×C4C2×C22⋊Q812yes—
327C2×C4C2×C22⋊Q814yes—
328C2×C4C23×Q811yescharacteristic
329C2×C4C22×#4111yescharacteristic
330C2×C4C2×H(5,𝔽2)12yes—
331C2×C4C2×2−1+412yes—
332C8—48no—
333C22—22no—
334C22—24no—
335C22—24no—
336C22—22no—
337C22#308014yes—
338C22C22×C22⋊Q811yescharacteristic
339C22#348411yescharacteristic
340C22#348611yescharacteristic
341C22C2×#57512yes—
342C22#358612yes—
343C4—24no—
344C4—24no—
345C4—28no—
346C4—22no—
347C4—44no—
348C4—44no—
349C4—44no—
350C4—84no—
351C4—84no—
352C4—84no—
353C4C2×#54412yes—
354C4#349212yes—
355C2—24no—
356C2#3522411yescharacteristic
357C2C2×#308011yescharacteristic
358C2#6049111yescharacteristic
359C2#6157812yes—
360C2#6183212yes—
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. #26123C2semidirect
6. #26057C2semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×#23512
#329602
#299722
#261232
#260572
#219862
#219552
#207402