Finite Group Databasebeta

← #204932

Subgroups — #204932

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2#20815C211yesmaximal, characteristic
3#20549C211yesmaximal, characteristic
4C2×#1559C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6C2×#1554C211yesmaximal, characteristic
7C2×#1546C211yesmaximal, characteristic
8#9709C212yesmaximal, complemented
9#9705C212yesmaximal, complemented
10#9697C212yesmaximal, complemented
11#9694C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18#2868C2211yescharacteristic
19#2868C2211yescharacteristic
20#2862C2211yescharacteristic
21#2862C2211yescharacteristic
22#2660—22no—
23#2567C414yes—
24#2546—22no—
25#2502C414yes—
26#2434—22no—
27#2317—22no—
28C4×#117C2211yescharacteristic
29#1842C2212yes—
30#1831C2212yes—
31#1798—22no—
32#1771—22no—
33#1768—22no—
34#1767C2212yes—
35C4×C4⋊C8C2212yes—
36#1753—22no—
37#1559—24no—
38#1556—24no—
39#1554—24no—
40#1546—24no—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48C2×#332C2212yes—
49C23×Q8—21no—
50C22×M16—21no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#125—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8C2×C412yes—
71C2×C4×Q8—21no—
72C2×C4×Q8C2×C412yes—
73C2×#117—21no—
74C2×#117D412yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2×C412yes—
77C22×C42C2311yescharacteristic
78#432—22no—
79#432—22no—
80#432—22no—
81#432D414yes—
82#431—22no—
83#431—22no—
84#423—22no—
85#423D414yes—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×#97—21no—
96C2×#97—21no—
97C2×#97—22no—
98C2×C4×C8—21no—
99C2×C4×C8—22no—
100#398—28no—
101#395—28no—
102#391—22no—
103#391—22no—
104#391C2×C414yes—
105#391C2×C414yes—
106#389—28no—
107#385—28no—
108#384—22no—
109#384—22no—
110#384C2×C414yes—
111#384C2×C414yes—
112#382—22no—
113#382—22no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4D412yes—
116C4×C22⋊C4C2×C412yes—
117C4×C22⋊C4D412yes—
118#376D412yes—
119C43C2311yescharacteristic
120C43C2311yescharacteristic
121#336—24no—
122#336—22no—
123#336—22no—
124#333—24no—
125#333—24no—
126#332—24no—
127#332—24no—
128#332—24no—
129#332—24no—
130#332—24no—
131#332—24no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—82no—
139C22×Q8—82no—
140C22×Q8C22×C412yes—
141C23×C4—22no—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C2×M16—24no—
146C2×M16—82no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—82no—
174C4⋊Q8—82no—
175C4⋊Q8—82no—
176C4⋊Q8—82no—
177#125—42no—
178#125—42no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—42no—
186C4×Q8—42no—
187#117—28no—
188#117—42no—
189#117—42no—
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—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C22⋊C414yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4C22⋊C414yes—
211C2×C42—22no—
212C2×C42C22×C412yes—
213C2×C42C22×C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226C4⋊C8—24no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—48no—
234C4⋊C8C2×D414yes—
235C4⋊C8C2×D414yes—
236C22⋊C8—24no—
237C22⋊C8—42no—
238C22⋊C8—42no—
239#97—24no—
240#97—42no—
241#97—82no—
242C4×C8—42no—
243C4×C8—82no—
244#95—28no—
245#95—28no—
246#95C22⋊C414yes—
247#95C22⋊C414yes—
248C24C2×C22⋊C411yescharacteristic
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—44no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—84no—
266C2×Q8—84no—
267C2×Q8—84no—
268C2×Q8—88no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—84no—
279C22×C4C2×C22⋊C414yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C414yes—
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4#12412yes—
289C22×C4#12712yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293M16—84no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—44no—
303C2×C8—44no—
304C4⋊C4—28no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—88no—
326C4⋊C4—88no—
327C22⋊C4—28no—
328C22⋊C4—44no—
329C42—24no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—42no—
335C42—42no—
336C42—44no—
337C42C4×D412yes—
338C42C4×D412yes—
339C42#12112yes—
340C42#12112yes—
341C42#12412yes—
342C42#12712yes—
343C42#13718yes—
344C42C22×D411yescharacteristic
345C42C22×D411yescharacteristic
346C42C2×#4111yescharacteristic
347C42C2×#4111yescharacteristic
348C23—22no—
349C23—22no—
350C23—22no—
351C23#39012yes—
352C23C22×C22⋊C411yescharacteristic
353Q8—82no—
354Q8—82no—
355Q8—88no—
356Q8—81no—
357Q8—81no—
358Q8—81no—
359Q8—81no—
360Q8—81no—
361Q8—81no—
362Q8—82no—
363Q8—82no—
364Q8—82no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—88no—
390C2×C4#39012yes—
391C2×C4#39012yes—
392C2×C4#39012yes—
393C2×C4#41918yes—
394C2×C4C2×C4×D411yescharacteristic
395C2×C4C2×C4×D411yescharacteristic
396C2×C4C2×#12111yescharacteristic
397C2×C4C2×#12111yescharacteristic
398C2×C4C2×#12411yescharacteristic
399C2×C4C2×#12711yescharacteristic
400C2×C4C2×#13714yes—
401C2×C4#57612yes—
402C2×C4#57712yes—
403C8—44no—
404C8—44no—
405C8—82no—
406C8—82no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—44no—
411C22C2×#39011yescharacteristic
412C22#288912yes—
413C22C2×#41912yes—
414C22#289712yes—
415C4—22no—
416C4—28no—
417C4—22no—
418C4—21no—
419C4—21no—
420C4—22no—
421C4—44no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—82no—
426C4#304111yescharacteristic
427C4#304311yescharacteristic
428C4#304512yes—
429C4#313912yes—
430C4#315212yes—
431C2—42no—
432C2#1776114yes—
433C2#2120211yescharacteristic
434C2#2122011yescharacteristic
435C2#2126311yescharacteristic
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. #9709C2semidirect
9. #9705C2semidirect
10. #9697C2semidirect
11. #9694C2semidirect

Maximal subgroups

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

subgroupindex
#352812
#208152
#205492
C2×#15592
C2×#15562
C2×#15542
C2×#15462
#97092
#97052
#96972
#96942