Finite Group Databasebeta

← #208321

Subgroups — #208321

Order 512.

795 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#35284C211yesmaximal, characteristic
2#20812C211yesmaximal, characteristic
3#20549C211yesmaximal, characteristic
4#9900C212yesmaximal, complemented
5#9694C212yesmaximal, complemented
6#9140C212yesmaximal
7#9139C212yesmaximal
8#8236C212yesmaximal, complemented
9#8233C212yesmaximal, complemented
10C2×#433—21no—
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15#2857C2212yes—
16#2647—21no—
17#2629—21no—
18#2507C414yes—
19#2497C414yes—
20#2446—22no—
21#2429—22no—
22#2330—21no—
23#2314—21no—
24C4×C4⋊Q8C2212yes—
25C4×#117C2211yescharacteristic
26#1842C2212yes—
27#1832—21no—
28#1798—21no—
29#1797C2212yes—
30#1771—22no—
31#1768—21no—
32#1767C2211yescharacteristic
33#1767C2212yes—
34C4×C4⋊C8C2211yescharacteristic
35C4×C4⋊C8C2212yes—
36#1753—22no—
37C2×#332C2212yes—
38C2×#332C2212yes—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×M16—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2312yes—
48C2×#125—21no—
49C2×C22⋊Q8—22no—
50C2×C4×Q8—22no—
51C2×#117D411yescharacteristic
52C2×#117D411yescharacteristic
53C22×C42C2311yescharacteristic
54#433—44no—
55#432—24no—
56#423—24no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—22no—
66C2×#97—21no—
67C2×#97—21no—
68C2×#97—22no—
69C2×C4×C8—21no—
70C2×C4×C8—22no—
71#400—24no—
72#398—24no—
73#398—24no—
74#396—24no—
75#391—22no—
76#391—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—24no—
81#389C2×C414yes—
82#389C2×C414yes—
83#389C2×C414yes—
84#388—24no—
85#387—24no—
86#387—24no—
87#384—21no—
88#384—21no—
89#384C2×C414yes—
90#384C2312yes—
91#382—21no—
92#380—22no—
93#380C2×C414yes—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2×C414yes—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4D412yes—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4D412yes—
100#376D411yescharacteristic
101#376D411yescharacteristic
102C43C2311yescharacteristic
103C43C2311yescharacteristic
104#336—22no—
105#336—21no—
106#336—21no—
107#336—22no—
108#336—22no—
109#332—24no—
110#332—24no—
111#332—24no—
112#332—24no—
113C22×Q8—22no—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C2×M16—24no—
118C2×M16—82no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—82no—
128C4⋊Q8—82no—
129#125—42no—
130#125—42no—
131C22⋊Q8—82no—
132C22⋊Q8—82no—
133C4×Q8—82no—
134C4×Q8—82no—
135#117—22no—
136#117—22no—
137#117—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C22×C414yes—
150C2×C22⋊C4—24no—
151C2×C22⋊C4C22⋊C412yes—
152C2×C22⋊C4C22⋊C412yes—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—24no—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C412yes—
158C2×C42C22×C412yes—
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2411yescharacteristic
168C4⋊C8—24no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—44no—
176C4⋊C8—44no—
177C4⋊C8#4114yes—
178C4⋊C8#4114yes—
179C22⋊C8—24no—
180C22⋊C8—44no—
181#97—24no—
182#97—42no—
183#97—82no—
184C4×C8—42no—
185C4×C8—82no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95C22⋊C412yes—
192#95C22⋊C412yes—
193#95C22⋊C412yes—
194#95C22⋊C412yes—
195C24C2×C22⋊C411yescharacteristic
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—42no—
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C23×C411yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224M16—84no—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—48no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C4⋊C4—84no—
252C4⋊C4—84no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—84no—
256C42—22no—
257C42—21no—
258C42—21no—
259C42—22no—
260C42—21no—
261C42—22no—
262C42—21no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—42no—
268C42—42no—
269C42—84no—
270C42AGL(1,ℤ/8)12yes—
271C42#13712yes—
272C42#13714yes—
273C42C2×#4112yes—
274C42C2×#4112yes—
275C23—22no—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—21no—
280C23C22×C22⋊C411yescharacteristic
281Q8—82no—
282Q8—82no—
283Q8—82no—
284Q8—82no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—82no—
315C2×C4#41714yes—
316C2×C4#41914yes—
317C2×C4C4×#4112yes—
318C2×C4#52511yescharacteristic
319C2×C4#52511yescharacteristic
320C2×C4#53212yes—
321C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
322C2×C4C2×#13712yes—
323C2×C4C2×#13711yescharacteristic
324C2×C4#57612yes—
325C2×C4#57712yes—
326C8—44no—
327C8—44no—
328C8—82no—
329C8—82no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22#229711yescharacteristic
336C22#288912yes—
337C22C2×#41911yescharacteristic
338C22C2×#41711yescharacteristic
339C22#289712yes—
340C4—24no—
341C4—24no—
342C4—22no—
343C4—42no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—84no—
349C4—84no—
350C4#311912yes—
351C4#312211yescharacteristic
352C4#312611yescharacteristic
353C4#318212yes—
354C4#318512yes—
355C2—81no—
356C2#1776112yes—
357C2#1776512yes—
358C2#2139911yescharacteristic
359C2#2140911yescharacteristic
360C2#2142911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9900C2semidirect
5. #9694C2semidirect
8. #8236C2semidirect
9. #8233C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#208122
#205492
#99002
#96942
#91402
#91392
#82362
#82332