Finite Group Databasebeta

← #6499345

Subgroups — #6499345

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#35286C211yesmaximal, characteristic
3#33263C211yesmaximal, characteristic, complemented
4#33263C211yesmaximal, characteristic, complemented
5#33258C211yesmaximal, characteristic, complemented
6#33256C211yesmaximal, characteristic, complemented
7#30307C211yesmaximal, characteristic, complemented
8#30307C211yesmaximal, characteristic, complemented
9#30305C211yesmaximal, characteristic, complemented
10#30304C211yesmaximal, characteristic, complemented
11#29325C214yesmaximal, complemented
12#29324C214yesmaximal, complemented
13#29309C214yesmaximal
14#29308C214yesmaximal
15#21984C211yesmaximal, characteristic
16#21984C211yesmaximal, characteristic
17#21982C211yesmaximal, characteristic
18#21981C211yesmaximal, characteristic
19#20462C211yesmaximal, characteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2647C2212yes—
31#2647C2212yes—
32#2629C2212yes—
33#2628C2212yes—
34#2571—21no—
35#2571—21no—
36#2570—21no—
37#2570—21no—
38#2552—21no—
39#2552—21no—
40#2550—21no—
41#2550—21no—
42#2442—21no—
43#2442—21no—
44#2442—21no—
45#2442—21no—
46#2441—21no—
47#2441—21no—
48#2439—21no—
49#2439—21no—
50#2331C2212yes—
51#2330C2212yes—
52#2317C2212yes—
53#2316C2212yes—
54#2285C2211yescharacteristic
55#2285C2211yescharacteristic
56#2285C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2085C2214yes—
60#2085C2214yes—
61#2084C2214yes—
62#2083C2214yes—
63#2059C2214yes—
64#2059C2214yes—
65#2058C2214yes—
66#2057C2214yes—
67#1949C2214yes—
68#1949C2214yes—
69#1948C2214yes—
70#1947C2214yes—
71#1944C2214yes—
72#1944C2214yes—
73#1943C2214yes—
74#1943C2214yes—
75#1942C2214yes—
76#1942C2214yes—
77#1941C2214yes—
78#1941C2214yes—
79#1854C2214yes—
80#1854C2214yes—
81#1853C2214yes—
82#1852C2214yes—
83#1845C2211yescharacteristic
84#1845C2211yescharacteristic
85#1845C2211yescharacteristic
86#1845C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1844C2211yescharacteristic
90#1844C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1768C2212yes—
100#1768C2212yes—
101#1750C2212yes—
102#1750C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8C2312yes—
110C2×#125—21no—
111C2×#125—21no—
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—21no—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#117—21no—
118C2×#117—21no—
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126#431Q818yescomplemented
127#431Q818yescomplemented
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×#97C2312yes—
141C2×#97C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148#400—22no—
149#400—22no—
150#400—22no—
151#400—22no—
152#400—22no—
153#399—22no—
154#398—22no—
155#398—22no—
156#398—22no—
157#398—22no—
158#398—22no—
159#398—22no—
160#396—22no—
161#396—22no—
162#395—22no—
163#393—22no—
164#391—22no—
165#391—22no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#385—22no—
203#385—22no—
204#384—22no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384—21no—
209#384C2312yes—
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2312yes—
219#382—22no—
220#382—21no—
221#382—21no—
222#382—21no—
223#382—21no—
224#382—21no—
225#382—21no—
226#382—21no—
227#382—21no—
228#382C2312yes—
229#382C2312yes—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—22no—
235#380—22no—
236#380C2312yes—
237#380C2312yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C22⋊C4D411yescharacteristic
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252#376C2311yescharacteristic
253#376C2311yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268C22×Q8—22no—
269C23×C4—22no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2411yescharacteristic
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C4⋊Q8—44no—
278C4⋊Q8—82no—
279C4⋊Q8—82no—
280#125—44no—
281#125—82no—
282#125—82no—
283#125—82no—
284#125—82no—
285#125—82no—
286#125—82no—
287#117—42no—
288#117—42no—
289#117—44no—
290#117—82no—
291#117—82no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4C2×D411yescharacteristic
347C2×C22⋊C4C2×D411yescharacteristic
348C2×C22⋊C4C2×D411yescharacteristic
349C2×C22⋊C4C2×D411yescharacteristic
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×Q814yes—
364C2×C42C2×Q814yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—44no—
372#107—44no—
373#107—44no—
374#107—44no—
375#106—44no—
376#106—44no—
377#106—44no—
378#106—44no—
379C4⋊C8C2×Q818yes—
380C4⋊C8C2×Q818yes—
381C22⋊C8C2×Q818yes—
382C22⋊C8C2×Q818yes—
383#97—42no—
384#97—42no—
385C4×C8—42no—
386C4×C8—42no—
387C4×C8C2×Q814yes—
388C4×C8C2×Q814yes—
389C4×C8C2×Q814yes—
390C4×C8C2×Q814yes—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95C2×D411yescharacteristic
412#95C2×D411yescharacteristic
413#95C2×D411yescharacteristic
414#95C2×D411yescharacteristic
415#95C2×D411yescharacteristic
416#95C2×D411yescharacteristic
417#95C2×D411yescharacteristic
418#95C2×D411yescharacteristic
419C24C22×D411yescharacteristic
420C2×Q8—44no—
421C2×Q8—82no—
422C2×Q8—82no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
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—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C4⋊Q814yes—
441C22×C4C4⋊Q814yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×Q812yes—
456C22×C4C22×Q812yes—
457C22×C4C2511yescharacteristic
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8C22×Q814yes—
467C2×C8C22×Q814yes—
468C2×C8C22×Q814yes—
469C2×C8C22×Q814yes—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—44no—
538C42—82no—
539C42—82no—
540C42C4⋊Q814yes—
541C42C4⋊Q814yes—
542C42C22×Q812yes—
543C42C22×Q812yes—
544C42C22×Q812yes—
545C42C22×Q812yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23C2×C4⋊Q812yes—
553C23C23×D411yescharacteristic
554Q8—164no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—164no—
614C2×C4C2×C4⋊Q812yes—
615C2×C4C2×C4⋊Q812yes—
616C2×C4C2×C4⋊Q812yes—
617C2×C4C2×C4⋊Q812yes—
618C2×C4C2×C4⋊Q812yes—
619C2×C4#57612yes—
620C2×C4#57712yes—
621C2×C4#57712yes—
622C2×C4#57812yes—
623C2×C4C23×Q811yescharacteristic
624C2×C4C23×Q811yescharacteristic
625C2×C4#58512yes—
626C2×C4#58512yes—
627C8—24no—
628C8—24no—
629C8—24no—
630C8—24no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22C22×C4⋊Q811yescharacteristic
639C22#344911yescharacteristic
640C22#344911yescharacteristic
641C22C2×#57611yescharacteristic
642C22C2×#57711yescharacteristic
643C22C2×#57711yescharacteristic
644C22C2×#57811yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#316314yes—
665C4#316414yes—
666C4#350412yes—
667C4#350412yes—
668C2—42no—
669C2#3533511yescharacteristic
670C2C2×#316311yescharacteristic
671C2C2×#316411yescharacteristic
672C2#6100611yescharacteristic
673C2#6100611yescharacteristic
674C2#6100811yescharacteristic
675C2#6100811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33263C2semidirect
4. #33263C2semidirect
5. #33258C2semidirect
6. #33256C2semidirect
7. #30307C2semidirect
8. #30307C2semidirect
9. #30305C2semidirect
10. #30304C2semidirect
11. #29325C2semidirect
12. #29324C2semidirect
126. #431Q8semidirect
127. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#352862
#332632
#332582
#332562
#303072
#303052
#303042
#293252
#293242
#293092
#293082
#219842
#219822
#219812
#204622