Finite Group Databasebeta

← #6528519

Subgroups — #6528519

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3#29111C214yesmaximal
4#27969C214yesmaximal
5#27823C218yesmaximal, complemented
6#27737C218yesmaximal, complemented
7#21833C212yesmaximal
8#20734C212yesmaximal
9#20536C211yesmaximal, characteristic
10C2×#544—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#431C2212yes—
14#2857C2214yes—
15#2659—22no—
16#2568—24no—
17#2568C2218yes—
18#2552C2214yes—
19#2551C2214yes—
20#2550—24no—
21C2×#400—22no—
22#2316—21no—
23C2×#391—21no—
24C2×#391—21no—
25C2×#391C2212yes—
26#2285C2212yes—
27C2×#382C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2101C22116yes—
30#2072C2218yes—
31#2024C2218yes—
32#1993C2218yes—
33#1990C2214yes—
34#1990C2214yes—
35#1988C2218yes—
36#1935C22116yes—
37#1836C2218yes—
38#1824C2214yes—
39#1823C2218yes—
40#1803C2214yes—
41#1803C2214yes—
42#1787C2214yes—
43#1770C2218yes—
44C4×#103C2218yes—
45#1749C2214yes—
46C23×Q8—21no—
47C23×Q8C2311yescharacteristic
48#544—48no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—28no—
52C2×C4⋊Q8C2314yes—
53C2×C4⋊Q8C2314yes—
54C2×#125—24no—
55C2×C22⋊Q8—24no—
56C2×C4×Q8—22no—
57C2×C4×Q8—28no—
58C2×C4×Q8C2314yes—
59C2×#117—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#431—28no—
70C2×C4⋊C8C2314yes—
71C2×#103C2318yes—
72C2×#103C2318yes—
73C2×C22⋊C8C2314yes—
74C2×#97C2318yes—
75C2×C4×C8C2314yes—
76#400—216no—
77#398—28no—
78#395—28no—
79#391—28no—
80#391—28no—
81#391—24no—
82#391—24no—
83#391C2318yes—
84#389—24no—
85#389—24no—
86#389—24no—
87#389C2318yes—
88#384C2312yes—
89#384C2312yes—
90#384C2314yes—
91#382—24no—
92#382—22no—
93#382C2318yes—
94#380—24no—
95#380—22no—
96#380C2314yes—
97C4×C4⋊C4—22no—
98C4×C4⋊C4D414yes—
99C4×C4⋊C4C2314yes—
100C4×C4⋊C4C2314yes—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D412yes—
106#340C2318yes—
107#340C2318yes—
108#340C23116yes—
109#340C23116yes—
110#336C2318yes—
111C22×Q8—24no—
112C22×Q8—21no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—42no—
117C22×Q8—44no—
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8C2412yes—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8C2418yes—
130C4⋊Q8—28no—
131C4⋊Q8—28no—
132C4⋊Q8—48no—
133C4⋊Q8—416no—
134C4⋊Q8—48no—
135#125—416no—
136C22⋊Q8—416no—
137C4×Q8—28no—
138C4×Q8—28no—
139C4×Q8—48no—
140#117—48no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×D414yes—
161C2×C4⋊C4C2412yes—
162C2×C4⋊C4C2414yes—
163C2×C4⋊C4C2414yes—
164C2×C4⋊C4C2414yes—
165C2×C22⋊C4—24no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—44no—
169C2×C42—44no—
170C2×C42C2×D414yes—
171C2×C42C2×D412yes—
172C2×C42#4118yes—
173C2×C42C2414yes—
174C2×C42C2412yes—
175C4⋊C8—28no—
176#103—216no—
177#103—48no—
178C22⋊C8—28no—
179#97—48no—
180C4×C8—28no—
181#95—28no—
182#95—28no—
183#95—28no—
184#95—28no—
185#95—24no—
186#95—24no—
187#95—28no—
188#95C2×D418yes—
189C24C22×D411yescharacteristic
190C2×Q8—24no—
191C2×Q8—22no—
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8—416no—
195C2×Q8—42no—
196C2×Q8—48no—
197C2×Q8—42no—
198C2×Q8—48no—
199C2×Q8—48no—
200C2×Q8—48no—
201C2×Q8—84no—
202C2×Q8—84no—
203C2×Q8—816no—
204C2×Q82−1+414yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4#12414yes—
230C22×C4#12414yes—
231C22×C4AGL(1,ℤ/8)14yes—
232C22×C4#13714yes—
233C22×C4C22×D414yes—
234C22×C4C22×D414yes—
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D412yes—
237C22×C4C22×D412yes—
238C22×C4C2×#4114yes—
239C22×C4C2511yescharacteristic
240C2×C8—28no—
241C2×C8—28no—
242C2×C8—28no—
243C4⋊C4—28no—
244C4⋊C4—28no—
245C4⋊C4—24no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—28no—
249C4⋊C4—48no—
250C4⋊C4—416no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—48no—
254C4⋊C4—48no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—84no—
260C4⋊C4—84no—
261C22⋊C4—48no—
262C42—28no—
263C42—44no—
264C42—48no—
265C42—48no—
266C42#12418yes—
267C42C2×#4114yes—
268C42C2×#4114yes—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23C2×#12412yes—
274C23C2×AGL(1,ℤ/8)12yes—
275C23C2×#13712yes—
276C23C23×D411yescharacteristic
277Q8—28no—
278Q8—48no—
279Q8—44no—
280Q8—816no—
281Q8—168no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—28no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—44no—
295C2×C4—48no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—88no—
314C2×C4#489116yes—
315C2×C4C2×#12412yes—
316C2×C4C2×#12414yes—
317C2×C4C2×#12414yes—
318C2×C4C2×AGL(1,ℤ/8)14yes—
319C2×C4C2×#13714yes—
320C2×C4C22×#4112yes—
321C2×C4C2×2−1+412yes—
322C2×C4#58512yes—
323C8—48no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22C2×#48914yes—
331C22C22×#12411yescharacteristic
332C22#348511yescharacteristic
333C22#348611yescharacteristic
334C22C22×AGL(1,ℤ/8)11yescharacteristic
335C22C22×#13711yescharacteristic
336C22#358611yescharacteristic
337C22#358611yescharacteristic
338C4—22no—
339C4—24no—
340C4—44no—
341C4—44no—
342C4—44no—
343C4—84no—
344C4—84no—
345C4—84no—
346C4—82no—
347C4C2×#48918yes—
348C4#349514yes—
349C2—24no—
350C2#3564411yescharacteristic
351C2C22×#48911yescharacteristic
352C2#6062011yescharacteristic
353C2#6172211yescharacteristic
354C2#6172311yescharacteristic
355C2#6182811yescharacteristic
356C2#6182911yescharacteristic
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. #27823C2semidirect
6. #27737C2semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#25682
#291112
#279692
#278232
#277372
#218332
#207342
#205362