Finite Group Databasebeta

← #239758

Subgroups — #239758

Order 512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2#21959C211yesmaximal, characteristic
3#21400C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6C22×Q8⋊C8C211yesmaximal, characteristic
7#10423C214yesmaximal, complemented
8#10415C214yesmaximal, complemented
9#7897C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×#419—21no—
15C22×C4×C8C2211yescharacteristic
16#2855—22no—
17#2627—22no—
18#2582—22no—
19#2581—22no—
20#2568C2212yes—
21#2567C2212yes—
22#2565—22no—
23#2317—21no—
24#2310—21no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#1990C2212yes—
28#1956C2212yes—
29#1948—22no—
30#1948C2214yes—
31#1892—22no—
32#1807—22no—
33#1805—22no—
34#1803C2214yes—
35C4×#103—22no—
36#1736—21no—
37#1550—24no—
38C2×#333C2212yes—
39C2×#332C2212yes—
40C2×Q8⋊C8—24no—
41C2×Q8⋊C8C2214yes—
42#1279C2214yes—
43C23×Q8D411yescharacteristic
44C22×M16—21no—
45C23×C8—21no—
46C23×C8—21no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8D412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8D412yes—
64C2×C4×Q8D412yes—
65C2×C4×Q8C2312yes—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C2×#106—24no—
74#423—22no—
75#423—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8D414yes—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82#419—44no—
83C2×#103—24no—
84C2×#103—24no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8—21no—
88C2×C4×C8C2312yes—
89#400—24no—
90#398—28no—
91#395—24no—
92#391—24no—
93#391—22no—
94#391—22no—
95#391—24no—
96#389—24no—
97#389—24no—
98#387—28no—
99#385—24no—
100#384—22no—
101#384D412yes—
102#384C2312yes—
103#382—22no—
104#382—21no—
105#382—21no—
106#381—24no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380D412yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2312yes—
115#340—24no—
116#340—24no—
117#340—24no—
118#333—24no—
119#332—24no—
120Q8⋊C8—28no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8SD1614yes—
129C22×Q8C2×D412yes—
130C22×Q8C2×D412yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—42no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—24no—
142C22×C8—22no—
143C22×C8—44no—
144C22×C8—42no—
145C22×C8—82no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154#125—42no—
155#125—42no—
156C22⋊Q8—24no—
157C22⋊Q8—24no—
158C22⋊Q8—42no—
159C22⋊Q8—42no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174#117—42no—
175#117—42no—
176#117—42no—
177#117—42no—
178#117—42no—
179#117—42no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D414yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—41no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—41no—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42C2411yescharacteristic
219#106—84no—
220C4⋊C8—24no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—42no—
225C4⋊C8C2×D414yes—
226#103—48no—
227#103—48no—
228C4×C8—22no—
229C4×C8—22no—
230C4×C8—42no—
231#95—28no—
232#95—28no—
233#95—28no—
234#95—22no—
235#95—22no—
236#95—41no—
237#95—41no—
238C24#12011yescharacteristic
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—28no—
248C2×Q8—28no—
249C2×Q8—44no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8C2×SD1614yes—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—82no—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12112yes—
277C22×C4#12112yes—
278C22×C4#12112yes—
279C22×C4#12312yes—
280C22×C4#12312yes—
281C22×C4#12412yes—
282C22×C4C2×SD1612yes—
283C22×C4#13512yes—
284C22×C4#13712yes—
285C22×C4#13712yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289M16—84no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—48no—
294C2×C8—42no—
295C2×C8—44no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—84no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—82no—
325C4⋊C4—88no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—82no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42—44no—
335C42—82no—
336C42#12112yes—
337C42#12112yes—
338C42#12112yes—
339C42#12312yes—
340C42#12312yes—
341C42#12412yes—
342C42C22×D411yescharacteristic
343C42C2×#4111yescharacteristic
344C42C2×#4111yescharacteristic
345C42C2×#4111yescharacteristic
346C23—22no—
347C23—22no—
348C23#39412yes—
349C23#44812yes—
350C23#45212yes—
351C23C2×#12011yescharacteristic
352Q8—24no—
353Q8—24no—
354Q8—44no—
355Q8—42no—
356Q8—42no—
357Q8—48no—
358Q8—82no—
359Q8—82no—
360Q8—81no—
361Q8—81no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—48no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—84no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4#39412yes—
388C2×C4#39412yes—
389C2×C4#39412yes—
390C2×C4#44812yes—
391C2×C4#45212yes—
392C2×C4#46514yes—
393C2×C4#47014yes—
394C2×C4#48314yes—
395C2×C4#48514yes—
396C2×C4C2×#12111yescharacteristic
397C2×C4C2×#12111yescharacteristic
398C2×C4C2×#12111yescharacteristic
399C2×C4C2×#12311yescharacteristic
400C2×C4C2×#12311yescharacteristic
401C2×C4C2×#12411yescharacteristic
402C2×C4C22×SD1611yescharacteristic
403C2×C4C2×#13511yescharacteristic
404C2×C4C2×#13711yescharacteristic
405C2×C4C2×#13711yescharacteristic
406C2×C4#57712yes—
407C2×C4#57812yes—
408C8—44no—
409C8—44no—
410C8—82no—
411C22—22no—
412C22—22no—
413C22—24no—
414C22—24no—
415C22#203514yes—
416C22C2×#39411yescharacteristic
417C22C2×#44811yescharacteristic
418C22C2×#45211yescharacteristic
419C22#301312yes—
420C22#301812yes—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—22no—
425C4—44no—
426C4—82no—
427C4—82no—
428C4—82no—
429C4—84no—
430C4—162no—
431C4#203514yes—
432C4#304211yescharacteristic
433C4#304411yescharacteristic
434C4C2×#46511yescharacteristic
435C4C2×#47011yescharacteristic
436C4C2×#48311yescharacteristic
437C4C2×#48511yescharacteristic
438C4#313712yes—
439C2—24no—
440C2#1866912yes—
441C2#1867612yes—
442C2C2×#203511yescharacteristic
443C2#2349211yescharacteristic
444C2#2351211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10423C2semidirect
8. #10415C2semidirect

Maximal subgroups

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

subgroupindex
#351792
#219592
#214002
#206472
C2×#15502
C22×Q8⋊C82
#104232
#104152
#78972