Finite Group Databasebeta

← #364217

Subgroups — #364217

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic
2#14349C211yesmaximal, characteristic
3#13796C211yesmaximal, characteristic
4#11381C212yesmaximal
5#11353C212yesmaximal
6#10912C214yesmaximal
7#10801C214yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9#2329—24no—
10#2329—24no—
11C2×#391C412yescomplemented
12C2×#389C412yescomplemented
13C2×#387C412yescomplemented
14C2×#387C412yescomplemented
15C2×#385C2212yes—
16C2×#380C2211yescharacteristic
17C2×C4×C22⋊C4C2212yes—
18C2×C4×C22⋊C4C2212yes—
19C22×#95C2211yescharacteristic
20C2×#342C2218yes—
21C2×#342C2218yes—
22#1443—24no—
23#1434—24no—
24C22⋊#95C2212yes—
25C22⋊#95C2212yes—
26C22⋊#95C2212yes—
27C22⋊#95C2212yes—
28C22⋊#95C2212yes—
29C23×Q8C2×C411yescharacteristic
30C24×C4C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C24×C4C2311yescharacteristic
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—28no—
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C412yes—
42C22×C22⋊C4C2314yes—
43C22×C22⋊C4C2312yes—
44C22×C22⋊C4C2314yes—
45C22×C22⋊C4C2312yes—
46C22×C42—22no—
47C22×C42C2×C412yes—
48C22×C42C2×C412yes—
49#391—28no—
50#389—28no—
51#387—28no—
52#387—28no—
53#385—24no—
54#385—24no—
55#380—22no—
56#380—22no—
57C4×C22⋊C4—24no—
58C4×C22⋊C4—24no—
59C4×C22⋊C4—28no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95C2×C412yes—
72C2×#95C2×C412yes—
73C2×#95C2×C412yes—
74C2×#95C2×C412yes—
75C2×#95C2×C412yes—
76C2×#95C2×C412yes—
77C2×#95C2×C412yes—
78C2×#95D414yes—
79C2×#95D414yes—
80#342—216no—
81#342—216no—
82C25C2411yescharacteristic
83C22×Q8—22no—
84C22×Q8—24no—
85C22×Q8—81no—
86C22×Q8—81no—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—81no—
93C23×C4—81no—
94C23×C4—81no—
95C23×C4—81no—
96C23×C4—81no—
97C23×C4—81no—
98C23×C4C22×C412yes—
99C23×C4C22×C412yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C412yes—
104C23×C4C22×C412yes—
105C23×C4C22×C412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C22⋊Q8—216no—
113C22⋊Q8—216no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—81no—
124C2×C4⋊C4—81no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4—82no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4#4118yes—
141C2×C22⋊C4#4118yes—
142C2×C42—28no—
143C2×C42—24no—
144C2×C42—24no—
145C2×C42—82no—
146C2×C42—82no—
147C2×C42—82no—
148C2×C42—82no—
149#95—28no—
150#95—24no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—82no—
156#95—82no—
157#95—82no—
158#95—82no—
159#95—82no—
160#95—82no—
161#95—82no—
162#95—82no—
163#95—82no—
164#95—82no—
165#95C22⋊C418yes—
166#95C22⋊C418yes—
167C24—41no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24C23×C411yescharacteristic
172C24C22×D411yescharacteristic
173C24C22×D411yescharacteristic
174C24C2×#4114yes—
175C24C2×#4114yes—
176C242−1+414yes—
177C2×Q8—28no—
178C2×Q8—82no—
179C2×Q8—82no—
180C2×Q8—88no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—28no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
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—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—88no—
243C22×C4—88no—
244C22×C4—88no—
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C414yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4#11714yes—
250C22×C4#11714yes—
251C4⋊C4—28no—
252C4⋊C4—216no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C22⋊C4—28no—
258C22⋊C4—28no—
259C22⋊C4—216no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—48no—
263C22⋊C4—48no—
264C22⋊C4—48no—
265C42—84no—
266C42—84no—
267C42—88no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—44no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—44no—
285C23—44no—
286C23C22×C22⋊C411yescharacteristic
287C23C2×#11712yes—
288C23C4×#4112yes—
289C23#51912yes—
290C23#52512yes—
291C23#52512yes—
292C23#52512yes—
293C23#53212yes—
294C23#53612yes—
295C23#53612yes—
296C23#55218yes—
297C23#55618yes—
298Q8—88no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—84no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—48no—
373C22—48no—
374C22#228011yescharacteristic
375C22#229711yescharacteristic
376C22#231011yescharacteristic
377C22#235614yes—
378C22#259914yes—
379C22#260414yes—
380C22#288311yescharacteristic
381C22#288311yescharacteristic
382C22#288311yescharacteristic
383C22#288411yescharacteristic
384C4—44no—
385C4—44no—
386C4—42no—
387C4—42no—
388C4—44no—
389C4—44no—
390C4—88no—
391C4—82no—
392C4—82no—
393C4—164no—
394C2—24no—
395C2—24no—
396C2—24no—
397C2#1396111yescharacteristic
398C2#2072411yescharacteristic
399C2#2072511yescharacteristic
400C2#2142211yescharacteristic
401C2#2142311yescharacteristic
402C2#2182711yescharacteristic
403C2#2182811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. C2×#391C4semidirect
12. C2×#389C4semidirect
13. C2×#387C4semidirect
14. C2×#387C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
#143492
#137962
#113812
#113532
#109122
#108012