Finite Group Databasebeta

← #189049

Subgroups — #189049

Order 512.

801 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60268C211yesmaximal, characteristic
2#17817C211yesmaximal, characteristic, complemented
3#17815C211yesmaximal, characteristic, complemented
4#17757C211yesmaximal, characteristic
5#17737C211yesmaximal, characteristic
6#8641C211yesmaximal, characteristic, complemented
7#8641C211yesmaximal, characteristic, complemented
8#8640C211yesmaximal, characteristic, complemented
9#8640C211yesmaximal, characteristic, complemented
10#8639C211yesmaximal, characteristic, complemented
11#8638C211yesmaximal, characteristic, complemented
12#8637C211yesmaximal, characteristic, complemented
13#8636C211yesmaximal, characteristic, complemented
14#8129C211yesmaximal, characteristic, complemented
15#8128C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#2994—21no—
18#2994—21no—
19#2993—21no—
20#2993—21no—
21#2991—21no—
22#2990—21no—
23#2989—21no—
24#2988—21no—
25#2960C2211yescharacteristic
26#2960C2211yescharacteristic
27#2933C412yes—
28#2933C412yes—
29#2931C2211yescharacteristic
30#2931C2211yescharacteristic
31#2925C412yes—
32#2920C2211yescharacteristic
33#2920C2211yescharacteristic
34#2920C2211yescharacteristic
35C2×#431C2211yescharacteristic
36#2906C412yes—
37C2×#423C2211yescharacteristic
38C2×#423C2211yescharacteristic
39#1574—21no—
40#1574—21no—
41#1573—21no—
42#1573—21no—
43#1572—21no—
44#1572—21no—
45#1572—21no—
46#1572—21no—
47#1567C2211yescharacteristic
48#1567C2211yescharacteristic
49#1567C2211yescharacteristic
50#1567C2211yescharacteristic
51#1566C2211yescharacteristic
52#1566C2211yescharacteristic
53#1566C2211yescharacteristic
54#1566C2211yescharacteristic
55#1561C2211yescharacteristic
56#1561C2211yescharacteristic
57#1561C2211yescharacteristic
58#1561C2211yescharacteristic
59#1560C2211yescharacteristic
60#1560C2211yescharacteristic
61#1560C2211yescharacteristic
62#1560C2211yescharacteristic
63#1559—21no—
64#1557—21no—
65#1554—21no—
66#1554—21no—
67#1553—21no—
68#1553—21no—
69#1542—21no—
70#1540—21no—
71#1309C2211yescharacteristic
72#1309C2211yescharacteristic
73#1309C2211yescharacteristic
74#1309C2211yescharacteristic
75#1309C2211yescharacteristic
76#1309C2211yescharacteristic
77#1309C2211yescharacteristic
78#1309C2211yescharacteristic
79C2×#131C2×C412yes—
80C2×#131C2×C412yes—
81C22×M16—21no—
82C22×M16C2×C412yes—
83#533D411yescharacteristic
84#533D411yescharacteristic
85#533C2311yescharacteristic
86#532D411yescharacteristic
87#532D411yescharacteristic
88#532C2311yescharacteristic
89#529—21no—
90#529—21no—
91#529—21no—
92#529—21no—
93#529Q812yes—
94#529Q812yes—
95#525—21no—
96#525—21no—
97#525C2×C412yes—
98C4×#41—21no—
99C4×#41—21no—
100C4×#41C2×C412yes—
101C22×C42C2311yescharacteristic
102#446—21no—
103#446—21no—
104#446—21no—
105#446—21no—
106C8×Q8—21no—
107C8×Q8—21no—
108C8×Q8—21no—
109C8×Q8—21no—
110#436—22no—
111#436—22no—
112#436—21no—
113#436—21no—
114#436—21no—
115#436—21no—
116#435—22no—
117#435—22no—
118#435—22no—
119#435—22no—
120C8×D4—22no—
121C8×D4—22no—
122C8×D4—21no—
123C8×D4—21no—
124C8×D4—21no—
125C8×D4—21no—
126#433—22no—
127#433—22no—
128#432—22no—
129#432—21no—
130#432—21no—
131#432—21no—
132#432—21no—
133#432—21no—
134#432—21no—
135#431—22no—
136#431—21no—
137#431—21no—
138#431—21no—
139#431—21no—
140#431—21no—
141#431—21no—
142#431D411yescharacteristic
143#431D411yescharacteristic
144#431D411yescharacteristic
145#431D411yescharacteristic
146#431D411yescharacteristic
147#431D411yescharacteristic
148#431D411yescharacteristic
149#431D411yescharacteristic
150#424—22no—
151#424—22no—
152#424C2×C412yes—
153#424C2×C412yes—
154#423—21no—
155#423—21no—
156#423—22no—
157#423—21no—
158#423—21no—
159#423—21no—
160#423C2×C412yes—
161#423C2311yescharacteristic
162#423C2311yescharacteristic
163#423C2311yescharacteristic
164#423C2311yescharacteristic
165#423C2311yescharacteristic
166#423C2311yescharacteristic
167#423C2311yescharacteristic
168#423C2311yescharacteristic
169#423C2311yescharacteristic
170#423C2311yescharacteristic
171C2×C4⋊C8—21no—
172C2×C4⋊C8—21no—
173C2×C4⋊C8—22no—
174C2×C4⋊C8—21no—
175C2×C4⋊C8—21no—
176C2×C4⋊C8C2×C411yescharacteristic
177C2×C4⋊C8C2×C411yescharacteristic
178#408C2×C412yes—
179#408C2×C412yes—
180#407—21no—
181#407—21no—
182#407—21no—
183C2×C22⋊C8C2×C411yescharacteristic
184C2×C22⋊C8C2×C411yescharacteristic
185#405C2×C412yes—
186#405C2×C412yes—
187C4×M16—21no—
188C4×M16—21no—
189C4×M16—21no—
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2311yescharacteristic
192#335—21no—
193#335—21no—
194#335—21no—
195#335—21no—
196#335—21no—
197#335—21no—
198#335—21no—
199#335—21no—
200#334—21no—
201#334—21no—
202#334—21no—
203#334—21no—
204#334—21no—
205#334—21no—
206#334—21no—
207#334—21no—
208#333—21no—
209#333—21no—
210#333—21no—
211#333—21no—
212#332—21no—
213#332—21no—
214#332—21no—
215#332—21no—
216#332—21no—
217#332—21no—
218#332—21no—
219#332—21no—
220#331—21no—
221#331—21no—
222#331—21no—
223#331—21no—
224C2×#41—21no—
225C2×#41—21no—
226C2×#41C22×C412yes—
227C23×C4C22×C411yescharacteristic
228C23×C4#4111yescharacteristic
229C23×C4#4111yescharacteristic
230#131—82no—
231#131—82no—
232C2×M16—22no—
233C2×M16—22no—
234C2×M16—22no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—41no—
244C2×M16—82no—
245C2×M16—82no—
246C2×M16C22×C412yes—
247C2×M16C22×C412yes—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—41no—
252C22×C8C22×C411yescharacteristic
253C22×C8C22×C411yescharacteristic
254C22×C8C22×C411yescharacteristic
255C22×C8C22×C411yescharacteristic
256C4⋊Q8C2×D411yescharacteristic
257C4⋊Q8C2×D411yescharacteristic
258C4⋊Q8C2×D411yescharacteristic
259#127C2×D411yescharacteristic
260#126—22no—
261#126—22no—
262#125—21no—
263#125—21no—
264#125C2×Q812yes—
265#124—21no—
266#124—21no—
267#124C2×Q812yes—
268#123—22no—
269#123—22no—
270#123—22no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275C22⋊Q8C4⋊C412yes—
276C22⋊Q8C4⋊C412yes—
277#121—21no—
278#121—21no—
279#121—21no—
280#121—21no—
281#121C4⋊C412yes—
282#121C4⋊C412yes—
283#120—22no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×Q8—21no—
287C4×Q8—21no—
288C4×Q8C4⋊C412yes—
289C4×Q8C4⋊C412yes—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4C4⋊C412yes—
301C4×D4C4⋊C412yes—
302#117—21no—
303#117—21no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117C22×C412yes—
308C2×C42—22no—
309C2×C42C22×C411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×Q811yescharacteristic
317C2×C42C2×Q811yescharacteristic
318C2×C42#4111yescharacteristic
319C2×C42#4111yescharacteristic
320C2×C42C2411yescharacteristic
321C4⋊C8—22no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—41no—
348C4⋊C8—41no—
349C4⋊C8C22⋊C411yescharacteristic
350C4⋊C8C22⋊C411yescharacteristic
351C4⋊C8C22⋊C411yescharacteristic
352C4⋊C8C22⋊C411yescharacteristic
353C4⋊C8C22⋊C411yescharacteristic
354C4⋊C8C22⋊C411yescharacteristic
355C4⋊C8C22⋊C411yescharacteristic
356C4⋊C8C22⋊C411yescharacteristic
357C22⋊C8—22no—
358C22⋊C8—22no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—41no—
362C22⋊C8—41no—
363C22⋊C8—41no—
364C22⋊C8—41no—
365C22⋊C8—41no—
366C22⋊C8—41no—
367C22⋊C8C22⋊C411yescharacteristic
368C22⋊C8C22⋊C411yescharacteristic
369C22⋊C8C22⋊C411yescharacteristic
370C22⋊C8C22⋊C411yescharacteristic
371C22⋊C8C22⋊C411yescharacteristic
372C22⋊C8C22⋊C411yescharacteristic
373C22⋊C8C22⋊C411yescharacteristic
374C22⋊C8C22⋊C411yescharacteristic
375#97—22no—
376#97—22no—
377#97—41no—
378#97—41no—
379#97—41no—
380C4×C8—22no—
381C4×C8—22no—
382C4×C8—41no—
383C4×C8—41no—
384C4×C8—41no—
385C4×C8C2×D411yescharacteristic
386C4×C8C2×D411yescharacteristic
387C4×C8C2×D411yescharacteristic
388C4×C8C2×D411yescharacteristic
389C4×C8C2×D411yescharacteristic
390C4×C8C2×D411yescharacteristic
391C4×C8C2×D411yescharacteristic
392C4×C8C2×D411yescharacteristic
393C24#11711yescharacteristic
394#41—42no—
395#41—42no—
396#41—42no—
397#41—42no—
398#41—81no—
399#41—81no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8C2×C4⋊C412yes—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4C2×C4⋊C412yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C4⋊C411yescharacteristic
421C22×C4C2×C4⋊C411yescharacteristic
422C22×C4#11711yescharacteristic
423C22×C4#11711yescharacteristic
424C22×C4#11711yescharacteristic
425C22×C4#12111yescharacteristic
426C22×C4#12111yescharacteristic
427C22×C4#12111yescharacteristic
428C22×C4#12111yescharacteristic
429C22×C4C22⋊Q811yescharacteristic
430C22×C4C22⋊Q811yescharacteristic
431C22×C4C22⋊Q811yescharacteristic
432C22×C4C22⋊Q811yescharacteristic
433C22×C4C23×C411yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436M16—42no—
437M16—42no—
438M16—82no—
439M16—82no—
440M16—82no—
441M16—82no—
442M16—81no—
443M16—81no—
444M16—81no—
445M16—81no—
446M16—81no—
447M16—81no—
448M16—81no—
449M16—81no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—82no—
465C2×C8—82no—
466C2×C8—82no—
467C2×C8—82no—
468C2×C8C2×C22⋊C411yescharacteristic
469C2×C8C2×C22⋊C411yescharacteristic
470C2×C8C2×C22⋊C411yescharacteristic
471C2×C8C2×C22⋊C411yescharacteristic
472C2×C8C2×C22⋊C411yescharacteristic
473C2×C8C2×C22⋊C411yescharacteristic
474C2×C8C2×C22⋊C411yescharacteristic
475C2×C8C2×C22⋊C411yescharacteristic
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4C2×C4⋊C412yes—
487C4⋊C4C2×C4⋊C412yes—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42C2×C22⋊C411yescharacteristic
501C42C2×C22⋊C411yescharacteristic
502C42C2×C4⋊C411yescharacteristic
503C42C2×C4⋊C411yescharacteristic
504C42#12111yescharacteristic
505C42#12111yescharacteristic
506C42#12111yescharacteristic
507C42#12111yescharacteristic
508C42C22⋊Q811yescharacteristic
509C42C22⋊Q811yescharacteristic
510C42C22⋊Q811yescharacteristic
511C42C22⋊Q811yescharacteristic
512C42C22×D411yescharacteristic
513C42C22×D411yescharacteristic
514C42C22×D411yescharacteristic
515C42C22×Q811yescharacteristic
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23#38011yescharacteristic
521C23#38011yescharacteristic
522C23C2×#11711yescharacteristic
523Q8—42no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—41no—
528D4—42no—
529D4—44no—
530D4—44no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—82no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4#38011yescharacteristic
571C2×C4#38011yescharacteristic
572C2×C4#38011yescharacteristic
573C2×C4#38011yescharacteristic
574C2×C4#38011yescharacteristic
575C2×C4#38011yescharacteristic
576C2×C4C22×C22⋊C411yescharacteristic
577C2×C4C22×C4⋊C411yescharacteristic
578C2×C4C2×#12111yescharacteristic
579C2×C4C2×#12111yescharacteristic
580C2×C4C2×C22⋊Q811yescharacteristic
581C2×C4C2×C22⋊Q811yescharacteristic
582C8—21no—
583C8—21no—
584C8—21no—
585C8—21no—
586C8—21no—
587C8—21no—
588C8—21no—
589C8—21no—
590C8—42no—
591C8—42no—
592C8—81no—
593C8—81no—
594C8—81no—
595C8—81no—
596C8—81no—
597C8—81no—
598C8—81no—
599C8—81no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—82no—
613C22C2×#38011yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4#288811yescharacteristic
627C4#288911yescharacteristic
628C4#306711yescharacteristic
629C4#306711yescharacteristic
630C4#306811yescharacteristic
631C4#306911yescharacteristic
632C2—41no—
633C2—41no—
634C2—41no—
635C2—82no—
636C2#2064011yescharacteristic
637C2#2067811yescharacteristic
638C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17817C2semidirect
3. #17815C2semidirect
6. #8641C2semidirect
7. #8641C2semidirect
8. #8640C2semidirect
9. #8640C2semidirect
10. #8639C2semidirect
11. #8638C2semidirect
12. #8637C2semidirect
13. #8636C2semidirect
14. #8129C2semidirect
15. #8128C2semidirect

Maximal subgroups

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

subgroupindex
#602682
#178172
#178152
#177572
#177372
#86412
#86402
#86392
#86382
#86372
#86362
#81292
#81282