Finite Group Databasebeta

← #242906

Subgroups — #242906

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21984C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21870C211yesmaximal, characteristic
4#21819C211yesmaximal, characteristic
5#10626C211yesmaximal, characteristic, complemented
6#10626C211yesmaximal, characteristic, complemented
7#10624C211yesmaximal, characteristic, complemented
8#10623C211yesmaximal, characteristic, complemented
9#10620C211yesmaximal, characteristic, complemented
10#10620C211yesmaximal, characteristic, complemented
11#10618C211yesmaximal, characteristic, complemented
12#10617C211yesmaximal, characteristic, complemented
13#8046C211yesmaximal, characteristic
14#8039C211yesmaximal, characteristic
15#7879C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#1993—21no—
24#1987—21no—
25#1984—21no—
26#1984—21no—
27#1984C2211yescharacteristic
28#1984C2211yescharacteristic
29#1984C2211yescharacteristic
30#1984C2211yescharacteristic
31#1981—21no—
32#1960—21no—
33#1949—21no—
34#1949—21no—
35#1892—21no—
36#1892—21no—
37#1873—21no—
38#1873—21no—
39#1854—21no—
40#1854—21no—
41#1845C2211yescharacteristic
42#1844C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1818—21no—
46#1818—21no—
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1803C2211yescharacteristic
54#1803C2211yescharacteristic
55#1296C2211yescharacteristic
56#1296C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68#533—22no—
69#533—22no—
70#533—22no—
71#533—22no—
72C2×C4⋊Q8D411yescharacteristic
73C2×C4⋊Q8D411yescharacteristic
74C2×#125—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8D411yescharacteristic
77C2×C22⋊Q8D411yescharacteristic
78C2×C4×Q8—21no—
79C2×C4×Q8C2311yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×#117—21no—
82C2×#117—21no—
83C2×#117—21no—
84C22×C42C2311yescharacteristic
85#432—22no—
86#432—22no—
87#431—22no—
88#431—22no—
89#431—22no—
90#431—22no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8D411yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8D411yescharacteristic
117C2×C22⋊C8D411yescharacteristic
118C2×#97—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127#382—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156C22×Q8—21no—
157C22×Q8C2×D411yescharacteristic
158C22×Q8C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4#4111yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—42no—
169C22×C8—81no—
170C22×C8—81no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179#125—42no—
180#125—42no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—42no—
196C4×Q8—42no—
197#117—42no—
198#117—42no—
199#117—81no—
200#117—81no—
201#117—81no—
202#117—81no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42#4111yescharacteristic
239C2×C42C2411yescharacteristic
240#107—81no—
241#107—81no—
242#106—81no—
243#106—81no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—42no—
259#103—81no—
260#103—81no—
261#103—81no—
262#103—81no—
263#103—81no—
264#103—81no—
265#103—81no—
266#103—81no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271#97—42no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—42no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95—41no—
281C24#12111yescharacteristic
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12312yes—
319C22×C4#12312yes—
320C22×C4#12312yes—
321C22×C4#12412yes—
322C22×C4#12612yes—
323C22×C4#12612yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—84no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C4⋊C4—81no—
390C4⋊C4—81no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—82no—
400C42—82no—
401C42#12312yes—
402C42#12312yes—
403C42#12312yes—
404C42#12412yes—
405C42#12612yes—
406C42#12612yes—
407C42C2×#4111yescharacteristic
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23#39712yes—
418C23C2×#12111yescharacteristic
419Q8—41no—
420Q8—41no—
421Q8—41no—
422Q8—41no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426Q8—41no—
427Q8—84no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4#39712yes—
471C2×C4#39712yes—
472C2×C4#39712yes—
473C2×C4C2×#12311yescharacteristic
474C2×C4C2×#12311yescharacteristic
475C2×C4C2×#12311yescharacteristic
476C2×C4C2×#12411yescharacteristic
477C2×C4C2×#12611yescharacteristic
478C2×C4C2×#12611yescharacteristic
479C2×C4#57611yescharacteristic
480C2×C4#57711yescharacteristic
481C2×C4#57711yescharacteristic
482C2×C4#57711yescharacteristic
483C2×C4#57711yescharacteristic
484C2×C4#57811yescharacteristic
485C2×C4#57811yescharacteristic
486C2×C4#57811yescharacteristic
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—81no—
492C8—81no—
493C8—81no—
494C8—81no—
495C8—81no—
496C8—81no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22C2×#39711yescharacteristic
505C22#304311yescharacteristic
506C22#304411yescharacteristic
507C22#306211yescharacteristic
508C22#306311yescharacteristic
509C22#306711yescharacteristic
510C22#306711yescharacteristic
511C4—22no—
512C4—22no—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—81no—
517C4—81no—
518C4—81no—
519C4—81no—
520C4—161no—
521C4—161no—
522C4—161no—
523C4—161no—
524C4#309611yescharacteristic
525C4#309711yescharacteristic
526C4#309711yescharacteristic
527C4#309711yescharacteristic
528C4#309811yescharacteristic
529C4#309911yescharacteristic
530C4#313711yescharacteristic
531C4#313711yescharacteristic
532C2—42no—
533C2#1956811yescharacteristic
534C2#1956911yescharacteristic
535C2#1957411yescharacteristic
536C2#1957511yescharacteristic
537C2#2363311yescharacteristic
538C2#2364311yescharacteristic
539C2#2364611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
5. #10626C2semidirect
6. #10626C2semidirect
7. #10624C2semidirect
8. #10623C2semidirect
9. #10620C2semidirect
10. #10620C2semidirect
11. #10618C2semidirect
12. #10617C2semidirect

Maximal subgroups

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

subgroupindex
#219842
#218852
#218702
#218192
#106262
#106242
#106232
#106202
#106182
#106172
#80462
#80392
#78792