Finite Group Databasebeta

← #163719

Subgroups — #163719

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35039C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic
3#22014C211yesmaximal, characteristic
4#7725C211yesmaximal, characteristic, complemented
5#7721C211yesmaximal, characteristic, complemented
6#7709C211yesmaximal, characteristic, complemented
7#7705C211yesmaximal, characteristic, complemented
8#7701C211yesmaximal, characteristic, complemented
9#7697C211yesmaximal, characteristic, complemented
10#7689C211yesmaximal, characteristic, complemented
11#7677C211yesmaximal, characteristic, complemented
12#7648C211yesmaximal, characteristic, complemented
13#7646C211yesmaximal, characteristic, complemented
14#7623C211yesmaximal, characteristic, complemented
15#7613C211yesmaximal, characteristic, complemented
16#2864C2211yescharacteristic
17#2863C2211yescharacteristic
18#2857C2211yescharacteristic
19#2855C2211yescharacteristic
20#2652—21no—
21#2638—21no—
22#2588—21no—
23#2577—21no—
24#2523—21no—
25#2506C411yescharacteristic
26#2499C411yescharacteristic
27#2495C411yescharacteristic
28#2485C411yescharacteristic
29#2460—21no—
30#2451C411yescharacteristic
31#2439C411yescharacteristic
32#2421C411yescharacteristic
33#2415C411yescharacteristic
34#2336—21no—
35#2332—21no—
36C4×C22⋊Q8C2211yescharacteristic
37#2296C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1955—21no—
40#1954—21no—
41#1940—21no—
42#1924—21no—
43#1924—21no—
44#1924—21no—
45#1924—21no—
46#1831C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1817C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1770—21no—
54#1767C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56#1749C2211yescharacteristic
57#1749C2211yescharacteristic
58#1745C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#125—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8C2×C411yescharacteristic
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×#117C2×C411yescharacteristic
85C2×#117C2×C411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106#400—21no—
107#400—21no—
108#400—21no—
109#398—21no—
110#398—21no—
111#397—21no—
112#397—21no—
113#397—21no—
114#397—21no—
115#397—21no—
116#397—21no—
117#396—21no—
118#395—21no—
119#394—21no—
120#394—21no—
121#393—21no—
122#391—21no—
123#391—21no—
124#391—21no—
125#391C2×C411yescharacteristic
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#391C2311yescharacteristic
129#389—21no—
130#389—21no—
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#389C2×C411yescharacteristic
138#389C2311yescharacteristic
139#389C2311yescharacteristic
140#389C2311yescharacteristic
141#388—21no—
142#388—21no—
143#388—21no—
144#388—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#387—21no—
161#387—21no—
162#385—21no—
163#385—21no—
164#385—21no—
165#385—21no—
166#384C2×C411yescharacteristic
167#383—21no—
168#383—21no—
169#382C2×C411yescharacteristic
170#382C2×C411yescharacteristic
171#381—21no—
172#380—21no—
173#380C2×C411yescharacteristic
174#380C2×C411yescharacteristic
175#380C2×C411yescharacteristic
176#380C2×C411yescharacteristic
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4C2×C411yescharacteristic
181C4×C4⋊C4C2×C411yescharacteristic
182C4×C4⋊C4D411yescharacteristic
183C4×C4⋊C4D411yescharacteristic
184C4×C22⋊C4—21no—
185C4×C22⋊C4C2×C411yescharacteristic
186C4×C22⋊C4C2×C411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191C43C2311yescharacteristic
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C22×Q8—21no—
207C22×Q8C22×C411yescharacteristic
208C22×D4C22⋊C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C4⋊Q8—81no—
224#125—44no—
225C22⋊Q8—42no—
226C22⋊Q8—42no—
227C22⋊Q8—42no—
228C22⋊Q8—81no—
229C4×Q8—81no—
230C4×D4—42no—
231C4×D4—42no—
232C4×D4—41no—
233C4×D4—41no—
234#117—42no—
235#117—81no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—81no—
281C2×C42—81no—
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2411yescharacteristic
295C4⋊C8—42no—
296C4⋊C8—42no—
297C4⋊C8—42no—
298C4⋊C8—81no—
299C4⋊C8—81no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—81no—
303C22⋊C8—81no—
304C22⋊C8—81no—
305C22⋊C8—81no—
306C22⋊C8—81no—
307C22⋊C8—81no—
308#97—42no—
309#97—81no—
310#97—81no—
311#97—81no—
312C4×C8—42no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346C24C2×C22⋊C411yescharacteristic
347C24C2×C22⋊C411yescharacteristic
348C2×Q8—22no—
349C2×Q8—42no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—81no—
353C2×Q8—81no—
354C2×D4—22no—
355C2×D4—41no—
356C2×D4—41no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C2×C8—22no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C42—22no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—22no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—42no—
483C42—42no—
484C42—81no—
485C42—81no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C22×C22⋊C411yescharacteristic
501Q8—82no—
502Q8—162no—
503D4—44no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4#42112yes—
574C2×C4#56812yes—
575C2×C4#57511yescharacteristic
576C2×C4#57511yescharacteristic
577C2×C4#57711yescharacteristic
578C2×C4#57811yescharacteristic
579C8—42no—
580C8—81no—
581C8—81no—
582C8—81no—
583C8—81no—
584C8—81no—
585C8—81no—
586C8—81no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22#287911yescharacteristic
602C22#288311yescharacteristic
603C22#288911yescharacteristic
604C22#289611yescharacteristic
605C22#289811yescharacteristic
606C22C2×#42111yescharacteristic
607C22#290111yescharacteristic
608C4—22no—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—81no—
631C2—81no—
632C2—81no—
633C2#1740611yescharacteristic
634C2#1749411yescharacteristic
635C2#1750611yescharacteristic
636C2#1760011yescharacteristic
637C2#1762511yescharacteristic
638C2#1768311yescharacteristic
639C2#1776711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7725C2semidirect
5. #7721C2semidirect
6. #7709C2semidirect
7. #7705C2semidirect
8. #7701C2semidirect
9. #7697C2semidirect
10. #7689C2semidirect
11. #7677C2semidirect
12. #7648C2semidirect
13. #7646C2semidirect
14. #7623C2semidirect
15. #7613C2semidirect

Maximal subgroups

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

subgroupindex
#350392
#228392
#220142
#77252
#77212
#77092
#77052
#77012
#76972
#76892
#76772
#76482
#76462
#76232
#76132