Finite Group Databasebeta

← #208288

Subgroups — #208288

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#20812C211yesmaximal, characteristic
3#20543C211yesmaximal, characteristic
4#9900C212yesmaximal, complemented
5#9691C212yesmaximal, complemented
6#9128C212yesmaximal
7#9127C212yesmaximal
8#8235C212yesmaximal, complemented
9C4×#333C212yesmaximal, complemented
10C2×#433—21no—
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15#2857C2212yes—
16#2647—21no—
17#2629—21no—
18#2507—22no—
19#2497—22no—
20#2446C414yes—
21#2429C414yes—
22#2330—21no—
23#2314—21no—
24C4×C4⋊Q8C2212yes—
25C4×#117C2211yescharacteristic
26#1842C2212yes—
27#1832—21no—
28#1798—21no—
29#1797C2212yes—
30C8×C4⋊C4—22no—
31#1768—21no—
32#1767C2211yescharacteristic
33C4×C4⋊C8C2211yescharacteristic
34C4×C4⋊C8C2212yes—
35C4×C4⋊C8C2212yes—
36C8×C22⋊C4—22no—
37C2×#333C2212yes—
38C2×#333C2212yes—
39C2×#333C2212yes—
40C2×#333C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C23×C8—21no—
46C2×C4⋊Q8C2×C414yes—
47C2×C4⋊Q8C2312yes—
48C2×#125—21no—
49C2×C22⋊Q8C2×C414yes—
50C2×C4×Q8C2×C414yes—
51C2×#117D411yescharacteristic
52C2×#117D411yescharacteristic
53C22×C42C2311yescharacteristic
54#433—44no—
55#431—24no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—24no—
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×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71#400—24no—
72#398—24no—
73#398—24no—
74#396—24no—
75#391C2×C414yes—
76#391C2×C414yes—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—24no—
83#389C2×C414yes—
84#388—24no—
85#387—24no—
86#387—24no—
87#384—21no—
88#384—21no—
89#384—22no—
90#384C2312yes—
91#382—21no—
92#380—22no—
93#380—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
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#333—24no—
110#333—24no—
111#333—24no—
112#333—24no—
113C22×Q8C22×C414yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—24no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—82no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—84no—
128#125—44no—
129C22⋊Q8—84no—
130C4×Q8—84no—
131#117—24no—
132#117SD1614yes—
133#117Q1614yes—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22×C414yes—
146C2×C22⋊C4—24no—
147C2×C22⋊C4C22⋊C412yes—
148C2×C22⋊C4C22⋊C412yes—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—24no—
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C412yes—
154C2×C42C22×C412yes—
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2411yescharacteristic
164C4⋊C8—24no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—44no—
172C4⋊C8—44no—
173C4⋊C8#4114yes—
174C4⋊C8#4114yes—
175C22⋊C8—24no—
176C22⋊C8—44no—
177#97—42no—
178C4×C8—24no—
179C4×C8—42no—
180C4×C8—82no—
181C4×C8—82no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95C22⋊C412yes—
188#95C22⋊C412yes—
189#95C22⋊C412yes—
190#95C22⋊C412yes—
191C24C2×C22⋊C411yescharacteristic
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8—44no—
195C2×Q8—44no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—42no—
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×SD1612yes—
214C22×C4C2×Q1612yes—
215C22×C4C23×C411yescharacteristic
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—84no—
231C4⋊C4—24no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—48no—
241C4⋊C4—84no—
242C4⋊C4—84no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C4⋊C4#10318yes—
246C22⋊C4—24no—
247C22⋊C4—84no—
248C22⋊C4#10318yes—
249C42—22no—
250C42—21no—
251C42—21no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—42no—
259C42—42no—
260C42—84no—
261C42C2×SD1612yes—
262C42C2×SD1612yes—
263C42C2×Q1612yes—
264C42C2×Q1612yes—
265C42#13514yes—
266C42C2×#4112yes—
267C42C2×#4112yes—
268C23—22no—
269C23—21no—
270C23—22no—
271C23—21no—
272C23C2×#10312yes—
273C23C22×C22⋊C411yescharacteristic
274Q8—84no—
275Q8—84no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—82no—
304C2×C4C2×#10314yes—
305C2×C4C2×#10314yes—
306C2×C4C2×#10312yes—
307C2×C4#41614yes—
308C2×C4C4×#4112yes—
309C2×C4#52511yescharacteristic
310C2×C4#52511yescharacteristic
311C2×C4#53212yes—
312C2×C4C22×SD1611yescharacteristic
313C2×C4C22×Q1611yescharacteristic
314C2×C4C2×#13512yes—
315C2×C4#57712yes—
316C2×C4#57812yes—
317C8—44no—
318C8—44no—
319C8—82no—
320C8—82no—
321C22—24no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22#229711yescharacteristic
327C22#288812yes—
328C22C22×#10311yescharacteristic
329C22C2×#41611yescharacteristic
330C22#289812yes—
331C4—24no—
332C4—24no—
333C4—24no—
334C4—22no—
335C4—22no—
336C4—42no—
337C4—41no—
338C4—41no—
339C4—84no—
340C4—84no—
341C4#312012yes—
342C4#312311yescharacteristic
343C4#312711yescharacteristic
344C4#316712yes—
345C4#317012yes—
346C2—42no—
347C2#1772212yes—
348C2#1772612yes—
349C2#2139811yescharacteristic
350C2#2140011yescharacteristic
351C2#2142811yescharacteristic
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. #9691C2semidirect
8. #8235C2semidirect
9. C4×#333C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#208122
#205432
#99002
#96912
#91282
#91272
#82352
C4×#3332