Finite Group Databasebeta

← #214726

Subgroups — #214726

Order 512.

1002 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36010C211yesmaximal, characteristic
2#20633C211yesmaximal, characteristic
3#17350C211yesmaximal, characteristic
4#9470C212yesmaximal, complemented
5#9383C212yesmaximal, complemented
6#9199C212yesmaximal, complemented
7#9025C212yesmaximal, complemented
8#8763C212yesmaximal, complemented
9#8382C212yesmaximal, complemented
10C2×#552C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2972C2212yes—
13C22×#99—21no—
14#2638C411yescharacteristic, complemented
15#2633C411yescharacteristic, complemented
16#2626C2211yescharacteristic
17#2611C411yescharacteristic, complemented
18#2599C411yescharacteristic, complemented
19#2564C2211yescharacteristic
20#2561C411yescharacteristic, complemented
21#2486C2211yescharacteristic
22#2482C411yescharacteristic, complemented
23#2480—21no—
24#2475—21no—
25#2466—21no—
26#2460—21no—
27#2419—21no—
28#2405—21no—
29#2398C411yescharacteristic, complemented
30#2395C411yescharacteristic, complemented
31C2×#397C2211yescharacteristic
32#2359—21no—
33#2353C2211yescharacteristic
34#2342—21no—
35#2315C2212yes—
36#1823C2214yescomplemented
37#1799—21no—
38#1799—21no—
39#1799C2212yes—
40#1799C2212yes—
41C2×#342—21no—
42#1520C2214yescomplemented
43#1460—21no—
44#1459—21no—
45#1459C2212yes—
46#1458—21no—
47C2×#323—21no—
48C2×#323C2212yes—
49#1294C2214yescomplemented
50#1289C2214yescomplemented
51C23×D4D411yescharacteristic
52#552—44no—
53C2×#126—21no—
54C2×#125—21no—
55C2×#125C2×C411yescharacteristic, complemented
56C2×#123—21no—
57C2×#123—21no—
58C2×C22⋊Q8C2×C411yescharacteristic, complemented
59C2×C22⋊Q8C2311yescharacteristic
60#518—44no—
61C2×C4×Q8C2×C411yescharacteristic, complemented
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic, complemented
73C2×#117D411yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4C2311yescharacteristic
81C8×D4—44no—
82#431—22no—
83C2×#99—22no—
84C2×#99—22no—
85C2×#99—44no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8C2314yes—
88#400—21no—
89#400C2×C411yescharacteristic, complemented
90#399C2×C411yescharacteristic
91#399C2×C411yescharacteristic
92#399C2×C411yescharacteristic, complemented
93#398—21no—
94#398C2×C411yescharacteristic, complemented
95#398C2×C411yescharacteristic, complemented
96#397—21no—
97#397—21no—
98#397—21no—
99#397—21no—
100#397C2×C411yescharacteristic
101#397C2×C411yescharacteristic
102#397C2×C411yescharacteristic
103#397C2×C411yescharacteristic
104#397C2×C411yescharacteristic, complemented
105#397C2×C411yescharacteristic, complemented
106#396—21no—
107#396—21no—
108#395C2×C411yescharacteristic, complemented
109#394—21no—
110#394—21no—
111#394—21no—
112#394—21no—
113#393C2×C411yescharacteristic
114#393C2×C411yescharacteristic
115#393C2×C411yescharacteristic, complemented
116#389—21no—
117#389—21no—
118#388—21no—
119#388—21no—
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#387—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#386—21no—
132#386—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—44no—
142#383C2×C411yescharacteristic, complemented
143#381—21no—
144#381—21no—
145#381C2×C411yescharacteristic, complemented
146#380—21no—
147#380—21no—
148#380—22no—
149#380C2314yes—
150#380C2311yescharacteristic
151#379—21no—
152#379—21no—
153#379—21no—
154C4×C4⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4C2×C411yescharacteristic, complemented
160#376C2×C411yescharacteristic, complemented
161C2×#95—21no—
162C2×#95C2311yescharacteristic
163#342—22no—
164#342—22no—
165#340—44no—
166#328—44no—
167#324—22no—
168#323—22no—
169#323—22no—
170#323—22no—
171C25C22⋊C411yescharacteristic
172C25C2×D411yescharacteristic
173C22×Q8C22×C411yescharacteristic
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—42no—
178C22×D4C22⋊C411yescharacteristic
179C22×D4C22⋊C411yescharacteristic
180C22×D4C2×D411yescharacteristic
181C22×D4C2×D411yescharacteristic
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—42no—
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2411yescharacteristic
194C22×C8—24no—
195C22×C8—42no—
196#126—44no—
197#125—42no—
198#125—44no—
199#123—44no—
200#123—42no—
201#123—42no—
202C22⋊Q8—24no—
203C22⋊Q8—42no—
204C4×Q8—42no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213#117—42no—
214#117—42no—
215#117—44no—
216#117—42no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C22⋊C4—24no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22×C411yescharacteristic
272C2×C22⋊C4C22×C411yescharacteristic
273C2×C22⋊C4C22×C411yescharacteristic
274C2×C22⋊C4C2×D411yescharacteristic
275C2×C22⋊C4C2×D411yescharacteristic
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C22⋊C4C2×D411yescharacteristic
278C2×C22⋊C4C2×D411yescharacteristic
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C4⋊C8—42no—
288#99—44no—
289#99—44no—
290C22⋊C8—24no—
291C22⋊C8—24no—
292C22⋊C8—42no—
293C4×C8—42no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—24no—
314#95—42no—
315#95C22×C411yescharacteristic
316#95C22×C411yescharacteristic
317#95C22×C411yescharacteristic
318#95C22×C411yescharacteristic
319C24—22no—
320C24—21no—
321C24—21no—
322C24—22no—
323C24—22no—
324C24—41no—
325C24—81no—
326C24—81no—
327C24—81no—
328C24—81no—
329C24C2×C22⋊C411yescharacteristic
330C24C2×C22⋊C411yescharacteristic
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C24#13514yes—
334C24C22×D411yescharacteristic
335C2×Q8—22no—
336C2×Q8—81no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4AGL(1,ℤ/8)12yes—
352C2×D4#13712yes—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—84no—
388C22×C4—162no—
389C22×C4—162no—
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×C4C2×C22⋊C411yescharacteristic
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4H(5,𝔽2)14yes—
401C2×C8—44no—
402C2×C8—84no—
403C2×C8—162no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—81no—
420C4⋊C4—81no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4—84no—
425C4⋊C4—162no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—81no—
452C22⋊C4—81no—
453C22⋊C4—81no—
454C22⋊C4—81no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C22⋊C4—81no—
458C22⋊C4—81no—
459C22⋊C4—84no—
460C42—22no—
461C42—22no—
462C42—41no—
463C42—41no—
464C42—42no—
465C42—42no—
466C42—81no—
467C42—81no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—24no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—41no—
480C23—41no—
481C23—44no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—82no—
491C23—82no—
492C23—84no—
493C23—161no—
494C23—161no—
495C23—161no—
496C23—161no—
497C23—162no—
498C23—162no—
499C23#41612yes—
500C23#41612yes—
501C23#41712yes—
502C23C22×C22⋊C411yescharacteristic
503C23#53412yes—
504C23C2×#13512yes—
505C23C2×AGL(1,ℤ/8)11yescharacteristic
506C23C2×#13711yescharacteristic
507Q8—82no—
508D4—24no—
509D4—24no—
510D4—44no—
511D4—44no—
512D4—42no—
513D4—42no—
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—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
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—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—161no—
568C2×C4—161no—
569C2×C4—161no—
570C2×C4—161no—
571C2×C4—162no—
572C2×C4—168no—
573C2×C4#41712yes—
574C2×C4#51812yes—
575C2×C4#53412yes—
576C8—84no—
577C22—21no—
578C22—22no—
579C22—21no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—42no—
585C22—41no—
586C22—41no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—84no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—84no—
604C22—81no—
605C22—81no—
606C22—164no—
607C22#230811yescharacteristic
608C22C2×#41611yescharacteristic
609C22C2×#41711yescharacteristic
610C22#318812yes—
611C22#319012yes—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—164no—
631C2—21no—
632C2—21no—
633C2—22no—
634C2—22no—
635C2—42no—
636C2—42no—
637C2—42no—
638C2—81no—
639C2#2176311yescharacteristic
640C2#2177711yescharacteristic
641C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9470C2semidirect
5. #9383C2semidirect
6. #9199C2semidirect
7. #9025C2semidirect
8. #8763C2semidirect
9. #8382C2semidirect
14. #2638C4semidirect
15. #2633C4semidirect
17. #2611C4semidirect
18. #2599C4semidirect
20. #2561C4semidirect
22. #2482C4semidirect
29. #2398C4semidirect
30. #2395C4semidirect
36. #1823C22semidirect
42. #1520C22semidirect
49. #1294C22semidirect
50. #1289C22semidirect
55. C2×#125C2×C4semidirect
58. C2×C22⋊Q8C2×C4semidirect
61. C2×C4×Q8C2×C4semidirect
72. C2×#117C2×C4semidirect
89. #400C2×C4semidirect
92. #399C2×C4semidirect
94. #398C2×C4semidirect
95. #398C2×C4semidirect
104. #397C2×C4semidirect
105. #397C2×C4semidirect
108. #395C2×C4semidirect
115. #393C2×C4semidirect
142. #383C2×C4semidirect
145. #381C2×C4semidirect
159. C4×C22⋊C4C2×C4semidirect
160. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
#360102
#206332
#173502
#94702
#93832
#91992
#90252
#87632
#83822