Finite Group Databasebeta

← #381312

Subgroups — #381312

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2#14684C211yesmaximal, characteristic
3C4×#381C211yesmaximal, characteristic
4#11381C212yesmaximal
5#11353C212yesmaximal
6#11197C214yesmaximal
7#10912C214yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C23×C42C2211yescharacteristic
10#2855—22no—
11#2855—22no—
12#2855—22no—
13#2855—22no—
14C2×#391C412yescomplemented
15C2×#389C412yescomplemented
16C2×#387C414yescomplemented
17C2×#385C2212yes—
18C2×#381C2212yes—
19C2×#381C2212yes—
20C2×#381C2212yes—
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23C2×#342C2218yes—
24C2×#342C2218yes—
25#1434—22no—
26#1434—22no—
27#1434—22no—
28C22⋊#95C2212yes—
29C22⋊#95C2212yes—
30C22⋊#95C2212yes—
31C4×#95—22no—
32C23×Q8C2×C411yescharacteristic
33C24×C4C2311yescharacteristic
34C24×C4C2311yescharacteristic
35C24×C4C2311yescharacteristic
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—44no—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2×C414yes—
43C22×C22⋊C4C2314yes—
44C22×C22⋊C4C2312yes—
45C22×C22⋊C4C2312yes—
46C22×C22⋊C4C2314yes—
47C22×C42—22no—
48C22×C42—44no—
49C22×C42C2×C412yes—
50C22×C42C2×C412yes—
51C22×C42D412yes—
52C22×C42D412yes—
53C22×C42D412yes—
54C22×C42D412yes—
55#391—22no—
56#391—22no—
57#391—22no—
58#391—22no—
59#389—22no—
60#389—22no—
61#389—22no—
62#389—22no—
63#387—24no—
64#387—24no—
65#387—24no—
66#387—24no—
67#385—22no—
68#385—24no—
69#385—22no—
70#381—24no—
71#381—22no—
72#381—22no—
73#381—28no—
74#381—44no—
75#380—22no—
76#380—22no—
77C4×C22⋊C4—44no—
78C2×#95—24no—
79C2×#95—24no—
80C2×#95—24no—
81C2×#95—24no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95C2×C412yes—
86C2×#95C2×C414yes—
87C2×#95C2×C414yes—
88C2×#95C2×C414yes—
89#342—216no—
90#342—48no—
91C25C2411yescharacteristic
92C22×Q8—22no—
93C22×Q8—42no—
94C22×Q8—44no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—44no—
103C23×C4—44no—
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C412yes—
106C23×C4C22×C412yes—
107C23×C4C22×C412yes—
108C23×C4C22×C414yes—
109C23×C4C22×C412yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C22⋊Q8—48no—
120C22⋊Q8—48no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—48no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4#4118yes—
145C2×C42—44no—
146C2×C42—44no—
147C2×C42—44no—
148C2×C42—44no—
149C2×C42—44no—
150C2×C42—44no—
151C2×C42—44no—
152C2×C42—48no—
153C2×C42C22⋊C412yes—
154C2×C42C22⋊C412yes—
155C2×C42C22⋊C412yes—
156C2×C42C22⋊C412yes—
157C2×C42C22⋊C412yes—
158C2×C42C22⋊C412yes—
159C2×C42C22⋊C412yes—
160C2×C42C22⋊C412yes—
161#95—216no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—48no—
170#95—48no—
171#95—48no—
172#95—44no—
173#95—44no—
174#95—44no—
175C24—42no—
176C24—42no—
177C24C23×C411yescharacteristic
178C24C22×D411yescharacteristic
179C24C22×D411yescharacteristic
180C24C2×#4114yes—
181C24H(5,𝔽2)14yes—
182C242−1+414yes—
183C2×Q8—44no—
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8—88no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—416no—
227C22×C4—416no—
228C22×C4—48no—
229C22×C4—48no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—48no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C4⋊C4—44no—
250C4⋊C4—48no—
251C4⋊C4—44no—
252C4⋊C4—48no—
253C4⋊C4—48no—
254C4⋊C4—44no—
255C22⋊C4—216no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—48no—
261C22⋊C4—48no—
262C22⋊C4—48no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—88no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—44no—
279C23—44no—
280C23—48no—
281C23—48no—
282C23C22×C22⋊C411yescharacteristic
283C23C4×#4112yes—
284C23#51812yes—
285C23#51912yes—
286C23#52512yes—
287C23#53212yes—
288C23#53412yes—
289C23#53412yes—
290C23#53612yes—
291C23#53612yes—
292C23#53918yes—
293C23#54218yes—
294Q8—88no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—48no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—48no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—416no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—84no—
333C2×C4—88no—
334C2×C4—84no—
335C2×C4—88no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—88no—
339C2×C4—88no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4#41014yes—
344C2×C4#41014yes—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—24no—
349C22—24no—
350C22—24no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—48no—
358C22—48no—
359C22#229711yescharacteristic
360C22#230811yescharacteristic
361C22#231011yescharacteristic
362C22#233214yes—
363C22#263814yes—
364C22#266314yes—
365C22C2×#41012yes—
366C22C2×#41012yes—
367C4—24no—
368C4—24no—
369C4—42no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—88no—
374C4—84no—
375C4—82no—
376C4—82no—
377C4—164no—
378C2—24no—
379C2—24no—
380C2—24no—
381C2#1429211yescharacteristic
382C2#2142112yes—
383C2#2174711yescharacteristic
384C2#2174811yescharacteristic
385C2#2182612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. C2×#391C4semidirect
15. C2×#389C4semidirect
16. C2×#387C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
#146842
C4×#3812
#113812
#113532
#111972
#109122