Finite Group Databasebeta

← #204937

Subgroups — #204937

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic
3#20546C211yesmaximal, characteristic
4C2×#1556C211yesmaximal, characteristic
5C2×#1552C211yesmaximal, characteristic
6C2×#1552C211yesmaximal, characteristic
7C22×#332C211yesmaximal, characteristic
8#9708C212yesmaximal, complemented
9#9703C212yesmaximal, complemented
10#9700C212yesmaximal, complemented
11#9692C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18#2868C2212yes—
19#2862C2212yes—
20#2660C414yes—
21#2567—22no—
22#2546—22no—
23#2502—21no—
24#2502—21no—
25#2434—22no—
26#2317C414yes—
27C4×#117C2211yescharacteristic
28#1940—22no—
29#1850—22no—
30#1842C2212yes—
31#1831C2212yes—
32#1771—22no—
33#1767C2212yes—
34C4×C4⋊C8C2212yes—
35#1752—21no—
36C8×C22⋊C4—21no—
37#1556—24no—
38#1552—24no—
39#1552—24no—
40C2×#332—24no—
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2211yescharacteristic
44C2×#332C2212yes—
45C2×#332C2211yescharacteristic
46C2×#332C2212yes—
47C2×#332C2211yescharacteristic
48C2×#332C2211yescharacteristic
49C2×#332C2212yes—
50C2×#332C2212yes—
51C23×Q8—21no—
52C23×C8—21no—
53C23×C8—21no—
54#533—22no—
55#533—22no—
56#533—22no—
57#533—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8C2×C412yes—
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2311yescharacteristic
69C2×#125C2×C412yes—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×#117C2×C412yes—
75C2×#117C2×C412yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×C412yes—
78C22×C42C2311yescharacteristic
79#432—22no—
80#432—22no—
81#432—22no—
82#432D414yes—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8D414yes—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2312yes—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×#97—21no—
99C2×#97—21no—
100C2×#97—22no—
101C2×C4×C8—22no—
102#398—24no—
103#398—24no—
104#395—28no—
105#391—22no—
106#391—22no—
107#391—24no—
108#389—28no—
109#385—24no—
110#385—24no—
111#384—22no—
112#384—22no—
113#384C2×C418yes—
114#382C2×C418yes—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4D412yes—
117C4×C22⋊C4D412yes—
118C4×C22⋊C4D411yescharacteristic
119C4×C22⋊C4D411yescharacteristic
120#376C2×C412yes—
121C43C2312yes—
122#336—24no—
123#336—22no—
124#336—22no—
125#332—24no—
126#332—24no—
127#332—24no—
128#332—24no—
129#332—22no—
130#332—22no—
131#332—22no—
132#332—22no—
133#332—24no—
134#332—24no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—82no—
142C22×Q8—82no—
143C23×C4—22no—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—24no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—42no—
155C22×C8—82no—
156C22×C8—82no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—84no—
176C4⋊Q8—84no—
177#125—24no—
178#125—24no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187#117—28no—
188#117—24no—
189#117—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C22⋊C414yes—
204C2×C4⋊C4C22×C414yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C42—22no—
211C2×C42C22×C414yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D414yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42C2411yescharacteristic
223C4⋊C8—24no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—22no—
233C4⋊C8—42no—
234C4⋊C8—84no—
235C4⋊C8C2×D412yes—
236C4⋊C8C2×D412yes—
237C4⋊C8C2×D414yes—
238C22⋊C8—24no—
239C22⋊C8—42no—
240#97—24no—
241#97—42no—
242#97—82no—
243C4×C8—82no—
244#95—28no—
245#95—28no—
246#95C22⋊C414yes—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249C24C2×C22⋊C411yescharacteristic
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—44no—
258C2×Q8—42no—
259C2×Q8—44no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—44no—
263C2×Q8—84no—
264C2×Q8—88no—
265C2×Q8—84no—
266C2×Q8—84no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—44no—
276C22×C4—84no—
277C22×C4C2×C22⋊C414yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C414yes—
282C22×C4C4×D412yes—
283C22×C4C4×D412yes—
284C22×C4#12112yes—
285C22×C4#12112yes—
286C22×C4#12412yes—
287C22×C4#12712yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—44no—
303C2×C8—42no—
304C2×C8—84no—
305C2×C8—84no—
306C4⋊C4—28no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—88no—
324C4⋊C4—88no—
325C22⋊C4—24no—
326C22⋊C4—44no—
327C22⋊C4#10118yes—
328C42—24no—
329C42—22no—
330C42—22no—
331C42—24no—
332C42—22no—
333C42—22no—
334C42—28no—
335C42—44no—
336C42C4×D412yes—
337C42C4×D412yes—
338C42#12112yes—
339C42#12112yes—
340C42#12412yes—
341C42#12712yes—
342C42C22×D411yescharacteristic
343C42C22×D411yescharacteristic
344C42C2×#4111yescharacteristic
345C42C2×#4111yescharacteristic
346C23—22no—
347C23—22no—
348C23#39012yes—
349C23C2×#10114yes—
350C23C22×C22⋊C411yescharacteristic
351Q8—88no—
352Q8—82no—
353Q8—84no—
354Q8—81no—
355Q8—82no—
356Q8—81no—
357Q8—82no—
358Q8—81no—
359Q8—81no—
360Q8—82no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—44no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—88no—
385C2×C4#39012yes—
386C2×C4#39012yes—
387C2×C4#39012yes—
388C2×C4C2×#10118yes—
389C2×C4C2×C4×D411yescharacteristic
390C2×C4C2×C4×D411yescharacteristic
391C2×C4C2×#12111yescharacteristic
392C2×C4C2×#12111yescharacteristic
393C2×C4C2×#12411yescharacteristic
394C2×C4C2×#12711yescharacteristic
395C2×C4#57612yes—
396C2×C4#57712yes—
397C2×C4#57712yes—
398C2×C4#57812yes—
399C8—42no—
400C8—42no—
401C8—44no—
402C8—81no—
403C8—81no—
404C8—82no—
405C22—22no—
406C22—28no—
407C22—22no—
408C22#196614yes—
409C22C2×#39011yescharacteristic
410C22C22×#10112yes—
411C22#289712yes—
412C22#289812yes—
413C4—22no—
414C4—28no—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—22no—
419C4—44no—
420C4—82no—
421C4—84no—
422C4—82no—
423C4#304211yescharacteristic
424C4#304411yescharacteristic
425C4#304612yes—
426C4#313612yes—
427C4#315411yescharacteristic
428C4#315511yescharacteristic
429C2—24no—
430C2#1777414yes—
431C2#2120611yescharacteristic
432C2C2×#196611yescharacteristic
433C2#2126211yescharacteristic
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
8. #9708C2semidirect
9. #9703C2semidirect
10. #9700C2semidirect
11. #9692C2semidirect

Maximal subgroups

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

subgroupindex
#352812
#219762
#205462
C2×#15562
C2×#15522
C22×#3322
#97082
#97032
#97002
#96922