Finite Group Databasebeta

← #6820915

Subgroups — #6820915

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#40024C212yesmaximal
2#33311C211yesmaximal, characteristic
3#32387C212yesmaximal
4#32382C212yesmaximal
5#32381C212yesmaximal
6#32376C212yesmaximal
7#29871C212yesmaximal
8#28832C211yesmaximal, characteristic
9#28832C211yesmaximal, characteristic
10#28828C211yesmaximal, characteristic
11#28827C211yesmaximal, characteristic
12#28535C211yesmaximal, characteristic
13#28535C211yesmaximal, characteristic
14#28015C211yesmaximal, characteristic
15#28015C211yesmaximal, characteristic
16#28012C211yesmaximal, characteristic
17#28012C211yesmaximal, characteristic
18#25460C212yesmaximal
19#25459C212yesmaximal
20#23160C211yesmaximal, characteristic
21#23160C211yesmaximal, characteristic
22C8×#125C211yesmaximal, characteristic
23#21276C211yesmaximal, characteristic
24C2×#499C2211yescharacteristic
25C2×#499C2211yescharacteristic
26C2×#499C2212yes—
27C2×#479C2212yes—
28C2×#479C2212yes—
29C2×C4×Q16C2211yescharacteristic
30#2859C2212yes—
31#2802—22no—
32#2776—22no—
33#2766—22no—
34#2719—22no—
35#2655—22no—
36#2571C2212yes—
37#2570C2212yes—
38#2570C2212yes—
39#2565C2212yes—
40#2565C2212yes—
41#2552C2212yes—
42#2533—22no—
43#2528—22no—
44#2514C2212yes—
45#2514C2212yes—
46#2511C2212yes—
47#2511C2212yes—
48#2370C2212yes—
49#2370C2212yes—
50#2336—22no—
51C4×#125C2211yescharacteristic
52#2295C2212yes—
53#2097C2212yes—
54#2097C2212yes—
55#2096C2212yes—
56#2096C2212yes—
57#2085C2212yes—
58#2085C2212yes—
59#2085C2212yes—
60#2085C2212yes—
61#2084C2212yes—
62#2084C2212yes—
63#2083C2212yes—
64#2083C2212yes—
65#2074C2212yes—
66#2074C2212yes—
67#2072C2212yes—
68#2072C2212yes—
69#2068C2212yes—
70#2068C2212yes—
71#2066C2212yes—
72#2066C2212yes—
73#1989C2212yes—
74#1989C2212yes—
75#1969C2211yescharacteristic
76#1969C2211yescharacteristic
77#1969C2211yescharacteristic
78#1969C2211yescharacteristic
79#1969C2211yescharacteristic
80#1969C2211yescharacteristic
81#1949C2212yes—
82#1949C2212yes—
83#1948C2212yes—
84#1947C2212yes—
85#1942C2211yescharacteristic
86#1942C2211yescharacteristic
87#1942C2211yescharacteristic
88#1942C2211yescharacteristic
89#1942C2211yescharacteristic
90#1942C2211yescharacteristic
91#1939C2211yescharacteristic
92#1939C2211yescharacteristic
93#1933C2212yes—
94#1933C2212yes—
95#1928C2212yes—
96#1928C2212yes—
97#1927C2212yes—
98#1927C2212yes—
99#1920C2212yes—
100#1920C2212yes—
101#1919C2212yes—
102#1919C2212yes—
103#1916C2211yescharacteristic
104#1916C2211yescharacteristic
105#1916C2211yescharacteristic
106#1916C2211yescharacteristic
107#1846C2211yescharacteristic
108#1846C2211yescharacteristic
109#1846C2212yes—
110#1846C2211yescharacteristic
111#1846C2211yescharacteristic
112#1845C2211yescharacteristic
113#1845C2211yescharacteristic
114#1845C2211yescharacteristic
115#1845C2211yescharacteristic
116#1844C2212yes—
117#1831C2211yescharacteristic
118C4×#107C2211yescharacteristic
119C4×#107C2211yescharacteristic
120C4×#107C2211yescharacteristic
121C4×#107C2211yescharacteristic
122C4×#107C2211yescharacteristic
123C8×C4⋊C4C2211yescharacteristic
124C8×C4⋊C4C2211yescharacteristic
125C8×C4⋊C4C2211yescharacteristic
126C8×C4⋊C4C2211yescharacteristic
127C8×C4⋊C4C2211yescharacteristic
128C8×C4⋊C4C2211yescharacteristic
129C4×#103C2212yes—
130C42×C8C2211yescharacteristic
131C22×Q16C2311yescharacteristic
132C2×#125—22no—
133C2×#125D411yescharacteristic
134C2×#125C2312yes—
135C2×#125C2312yes—
136C2×C4×Q8C2312yes—
137#499—24no—
138#499—24no—
139#499—44no—
140#479—44no—
141#479—44no—
142C4×Q16—24no—
143C2×#107C2311yescharacteristic
144C2×#107C2311yescharacteristic
145C2×#107C2311yescharacteristic
146C2×#107C2311yescharacteristic
147C2×#107C2311yescharacteristic
148C2×#107C2311yescharacteristic
149C2×#107C2311yescharacteristic
150C2×#107C2312yes—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#106C2312yes—
155C2×#106C2312yes—
156C2×#106C2312yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2311yescharacteristic
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177C2×C4×C8C2311yescharacteristic
178C2×C4×C8C2311yescharacteristic
179C2×C4×C8C2311yescharacteristic
180C2×C4×C8C2311yescharacteristic
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2311yescharacteristic
183#401—22no—
184#401—22no—
185#400—22no—
186#400—22no—
187#400—22no—
188#400—22no—
189#400—22no—
190#400—22no—
191#400—22no—
192#400—22no—
193#400—22no—
194#400—22no—
195#398—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#398—22no—
200#398—22no—
201#398—22no—
202#398—22no—
203#391C2312yes—
204#391C2312yes—
205#391C2312yes—
206#391C2312yes—
207#391C2312yes—
208#391C2312yes—
209#389—22no—
210#389—22no—
211#389—22no—
212#389—22no—
213#389—22no—
214#389—22no—
215#389—22no—
216#389—22no—
217#389D411yescharacteristic
218#389D411yescharacteristic
219#389C2312yes—
220#389C2312yes—
221#389C2312yes—
222#389C2312yes—
223#389C2312yes—
224#389C2312yes—
225#387—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#387—22no—
230#387—22no—
231#387—22no—
232#387—22no—
233#387—22no—
234#387—22no—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387D411yescharacteristic
246#387D411yescharacteristic
247#387D411yescharacteristic
248#387D411yescharacteristic
249#384C2312yes—
250#384C2312yes—
251#384C2312yes—
252#384C2312yes—
253#383—22no—
254#383—22no—
255#382—22no—
256#382D411yescharacteristic
257#382C2312yes—
258#382C2312yes—
259#382C2312yes—
260#382C2312yes—
261C4×C4⋊C4—22no—
262C4×C4⋊C4—22no—
263C4×C4⋊C4—22no—
264C4×C4⋊C4C2312yes—
265C4×C4⋊C4C2312yes—
266C4×C4⋊C4C2312yes—
267C4×C4⋊C4C2312yes—
268C4×C4⋊C4C2312yes—
269C4×C4⋊C4C2311yescharacteristic
270C4×C4⋊C4C2311yescharacteristic
271C4×C4⋊C4C2311yescharacteristic
272C4×C4⋊C4C2311yescharacteristic
273C4×C4⋊C4C2311yescharacteristic
274C4×C4⋊C4C2311yescharacteristic
275#376—22no—
276C43C2311yescharacteristic
277#340C2312yes—
278#340C2312yes—
279#340C2312yes—
280#340C2312yes—
281#340C2312yes—
282#340C2312yes—
283#340C2312yes—
284#340C2312yes—
285#340C2312yes—
286#340C2312yes—
287#340C2312yes—
288#336C2311yescharacteristic
289#336C2311yescharacteristic
290#336C2311yescharacteristic
291#336C2311yescharacteristic
292#336C2311yescharacteristic
293#336C2311yescharacteristic
294#336C2311yescharacteristic
295#336C2311yescharacteristic
296#336C2311yescharacteristic
297#336C2311yescharacteristic
298#336C2311yescharacteristic
299#336C2311yescharacteristic
300C22×Q8C2412yes—
301C2×Q16—22no—
302C2×Q16—22no—
303C2×Q16—22no—
304C22×C8C2411yescharacteristic
305C22×C8C2411yescharacteristic
306C22×C8C2411yescharacteristic
307C22×C8C2411yescharacteristic
308C22×C8C2411yescharacteristic
309C22×C8C2411yescharacteristic
310C22×C8C2411yescharacteristic
311C22×C8C2411yescharacteristic
312#125—24no—
313#125—24no—
314#125—24no—
315#125—24no—
316#125—24no—
317#125—48no—
318C4×Q8—24no—
319C4×Q8—24no—
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—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—22no—
376#107—22no—
377#107—22no—
378#107—22no—
379#107—22no—
380#107—22no—
381#107—22no—
382#107—42no—
383#107—42no—
384#107—42no—
385#107—42no—
386#106—24no—
387#106—24no—
388#106—42no—
389#106—42no—
390#106—42no—
391#106—42no—
392C4⋊C8—22no—
393C4⋊C8—22no—
394C4⋊C8—22no—
395C4⋊C8—22no—
396C4⋊C8—22no—
397C4⋊C8—22no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—42no—
402#103—42no—
403#103—42no—
404#103—42no—
405C4×C8—22no—
406C4×C8—22no—
407C4×C8—22no—
408C4×C8—22no—
409C4×C8—22no—
410C4×C8—22no—
411C4×C8—22no—
412C4×C8—22no—
413C4×C8—22no—
414C4×C8#4114yes—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454#95C2×D411yescharacteristic
455#95C2×D411yescharacteristic
456#95C2×D411yescharacteristic
457#95C2×D411yescharacteristic
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C2511yescharacteristic
487Q16—44no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8C2×#4112yes—
507C2×C8C2×#4112yes—
508C2×C82−1+412yes—
509C2×C82−1+412yes—
510C2×C82−1+412yes—
511C2×C82−1+412yes—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4#13514yes—
557C42—22no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—41no—
572C42—41no—
573C42—44no—
574C42—44no—
575C42—82no—
576C42—82no—
577C42—82no—
578C42—82no—
579C42C2×#4112yes—
580C42H(5,𝔽2)12yes—
581C23C23×D411yescharacteristic
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4#55012yes—
647C2×C4#55012yes—
648C2×C4C2×#13512yes—
649C2×C4C2×#13512yes—
650C2×C4C2×#13512yes—
651C2×C4#57812yes—
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×2−1+411yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C8—22no—
659C8—22no—
660C8—22no—
661C8—41no—
662C8—41no—
663C8—41no—
664C8—41no—
665C8#54114yes—
666C22C2×#55011yescharacteristic
667C22#349911yescharacteristic
668C22#349911yescharacteristic
669C22C22×#13511yescharacteristic
670C22C2×#57811yescharacteristic
671C22#358411yescharacteristic
672C22#358411yescharacteristic
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4C2×#54111yescharacteristic
696C4#350911yescharacteristic
697C4#355211yescharacteristic
698C4#355211yescharacteristic
699C2#3938511yescharacteristic
700C2#6139311yescharacteristic
701C2#6140811yescharacteristic
702C2#6247311yescharacteristic
703C2#6247511yescharacteristic
704C2#6249011yescharacteristic
705C2#6249311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#125 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#400242
#333112
#323872
#323822
#323812
#323762
#298712
#288322
#288282
#288272
#285352
#280152
#280122
#254602
#254592
#231602
C8×#1252
#212762