Finite Group Databasebeta

← #6806011

Subgroups — #6806011

Order 512.

1192 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39754C211yesmaximal, characteristic
2#39641C211yesmaximal, characteristic
3#33294C211yesmaximal, characteristic
4#32341C212yesmaximal
5#32338C212yesmaximal
6#32337C212yesmaximal
7#32334C212yesmaximal
8#30232C212yesmaximal
9#29219C211yesmaximal, characteristic
10#29188C212yesmaximal
11#28698C212yesmaximal
12#28691C211yesmaximal, characteristic
13#26513C211yesmaximal, characteristic
14#26510C211yesmaximal, characteristic
15#26509C211yesmaximal, characteristic
16#26506C211yesmaximal, characteristic
17#26283C211yesmaximal, characteristic
18#26281C211yesmaximal, characteristic
19#26278C211yesmaximal, characteristic
20#26276C211yesmaximal, characteristic
21#22252C212yesmaximal
22C4×#501C211yesmaximal, characteristic
23#21343C211yesmaximal, characteristic
24C2×#501C2211yescharacteristic
25C2×#499C2212yes—
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2802—22no—
29#2791—22no—
30#2791—22no—
31#2774—22no—
32#2660C2212yes—
33#2647C2212yes—
34#2584—21no—
35#2581—21no—
36#2571C2212yes—
37#2571C2211yescharacteristic
38#2570C2212yes—
39#2552C2211yescharacteristic
40#2546—21no—
41#2546—21no—
42#2546C2211yescharacteristic
43#2546C2211yescharacteristic
44#2508—21no—
45#2507C2212yes—
46#2507C2212yes—
47#2453—21no—
48#2446—21no—
49#2446—21no—
50#2365C2211yescharacteristic
51#2365C2211yescharacteristic
52#2355C2211yescharacteristic
53Q8×C4⋊C4C2211yescharacteristic
54#2330C2212yes—
55#2330C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×#125C2211yescharacteristic
58#2295C2211yescharacteristic
59#2101C2212yes—
60#2101C2212yes—
61#2101C2212yes—
62#2101C2212yes—
63#2100C2212yes—
64#2100C2212yes—
65#2099C2212yes—
66#2099C2212yes—
67#2099C2212yes—
68#2099C2212yes—
69#2088C2212yes—
70#2088C2212yes—
71#2087C2212yes—
72#2086C2212yes—
73#2085C2212yes—
74#2085C2212yes—
75#2085C2212yes—
76#2085C2212yes—
77#2084C2212yes—
78#2083C2212yes—
79#1993C2211yescharacteristic
80#1992C2212yes—
81#1992C2212yes—
82#1992C2211yescharacteristic
83#1991C2212yes—
84#1990C2211yescharacteristic
85#1990C2211yescharacteristic
86#1990C2211yescharacteristic
87#1990C2211yescharacteristic
88#1990C2212yes—
89#1988C2211yescharacteristic
90#1988C2211yescharacteristic
91#1988C2211yescharacteristic
92#1988C2211yescharacteristic
93#1984C2211yescharacteristic
94#1984C2211yescharacteristic
95#1949C2212yes—
96#1949C2212yes—
97#1948C2211yescharacteristic
98#1948C2211yescharacteristic
99#1948C2212yes—
100#1947C2211yescharacteristic
101#1947C2211yescharacteristic
102#1947C2212yes—
103#1944C2211yescharacteristic
104#1944C2211yescharacteristic
105#1943C2211yescharacteristic
106#1943C2211yescharacteristic
107#1939C2211yescharacteristic
108#1939C2211yescharacteristic
109#1921C2212yes—
110#1921C2212yes—
111#1921C2212yes—
112#1920C2212yes—
113#1920C2212yes—
114#1920C2212yes—
115#1919C2212yes—
116#1919C2212yes—
117#1919C2212yes—
118#1918C2212yes—
119#1918C2212yes—
120#1918C2212yes—
121#1917C2212yes—
122#1917C2212yes—
123#1847C2211yescharacteristic
124#1836C2212yes—
125#1836C2212yes—
126#1834C2212yes—
127C4×#107C2211yescharacteristic
128C4×#107C2211yescharacteristic
129C4×#106C2211yescharacteristic
130C4×#106C2211yescharacteristic
131#1768C2212yes—
132C4×C4⋊C8C2211yescharacteristic
133#1746C2212yes—
134C4×#97C2211yescharacteristic
135C2×C4⋊Q8—21no—
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8C2311yescharacteristic
140C2×C4⋊Q8C2311yescharacteristic
141C2×C4⋊Q8C2311yescharacteristic
142C2×#125—21no—
143C2×#125C2312yes—
144C2×#125C2311yescharacteristic
145C2×C4×Q8—21no—
146C2×C4×Q8—21no—
147C2×C4×Q8—21no—
148C2×C4×Q8—21no—
149C2×C4×Q8C2311yescharacteristic
150#501—24no—
151#499—28no—
152C2×#107C2311yescharacteristic
153C2×#107C2312yes—
154C2×#107C2312yes—
155C2×#107C2311yescharacteristic
156C2×#107C2311yescharacteristic
157C2×#107C2311yescharacteristic
158C2×#106C2311yescharacteristic
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×#106C2311yescharacteristic
162C2×#106C2311yescharacteristic
163C2×#106C2311yescharacteristic
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2311yescharacteristic
168C2×#97C2311yescharacteristic
169C2×#97C2311yescharacteristic
170C2×#97C2311yescharacteristic
171C2×#97C2311yescharacteristic
172C2×C4×C8C2312yes—
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175#400—22no—
176#400—22no—
177#400—22no—
178#400—22no—
179#400—22no—
180#400—22no—
181#398—22no—
182#398—22no—
183#398—22no—
184#398—22no—
185#398—22no—
186#398—22no—
187#398—22no—
188#398—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#398—22no—
193#395—22no—
194#395—22no—
195#391—21no—
196#391—21no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391D411yescharacteristic
202#391D411yescharacteristic
203#389—21no—
204#389—22no—
205#389—21no—
206#389—21no—
207#389—22no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—22no—
214#389—22no—
215#389—22no—
216#389—22no—
217#389—22no—
218#389—22no—
219#389D411yescharacteristic
220#389D411yescharacteristic
221#389C2311yescharacteristic
222#389C2311yescharacteristic
223#389C2311yescharacteristic
224#389C2311yescharacteristic
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#389C2312yes—
229#389C2312yes—
230#389C2312yes—
231#389C2312yes—
232#389C2312yes—
233#387—22no—
234#387—22no—
235#387—22no—
236#387—21no—
237#387—21no—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#387D412yes—
247#387D412yes—
248#384—21no—
249#384—21no—
250#384C2311yescharacteristic
251#384C2312yes—
252#384C2312yes—
253#384C2311yescharacteristic
254#384C2312yes—
255#384C2311yescharacteristic
256#384C2312yes—
257#382—21no—
258#382—22no—
259#382C2311yescharacteristic
260#382C2312yes—
261#382C2312yes—
262#382C2312yes—
263C4×C4⋊C4—21no—
264C4×C4⋊C4—21no—
265C4×C4⋊C4—21no—
266C4×C4⋊C4—21no—
267C4×C4⋊C4C2311yescharacteristic
268C4×C4⋊C4C2311yescharacteristic
269C4×C4⋊C4C2311yescharacteristic
270C4×C4⋊C4C2311yescharacteristic
271C4×C4⋊C4C2312yes—
272C4×C4⋊C4C2312yes—
273C4×C4⋊C4C2312yes—
274C4×C4⋊C4C2312yes—
275C4×C4⋊C4C2312yes—
276C4×C4⋊C4C2311yescharacteristic
277#376C2312yes—
278C43C2311yescharacteristic
279#340C2312yes—
280#340C2312yes—
281#340C2312yes—
282#340C2312yes—
283#340C2312yes—
284#340C2312yes—
285#340C2312yes—
286#340C2312yes—
287#340C2312yes—
288#340C2312yes—
289#340C2312yes—
290#340C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2312yes—
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2311yescharacteristic
300#340C2311yescharacteristic
301#340C2311yescharacteristic
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#336C2312yes—
308#336C2312yes—
309#336C2312yes—
310#336C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314C22×Q8—21no—
315C22×Q8—21no—
316C22×Q8—21no—
317C22×Q8C2×D411yescharacteristic
318C22×C8C2411yescharacteristic
319C22×C8C2411yescharacteristic
320C22×C8C2412yes—
321C22×C8C2412yes—
322C22×C8C2411yescharacteristic
323C22×C8C2411yescharacteristic
324C4⋊Q8—22no—
325C4⋊Q8—22no—
326C4⋊Q8—22no—
327C4⋊Q8—22no—
328C4⋊Q8—22no—
329C4⋊Q8—22no—
330C4⋊Q8—82no—
331C4⋊Q8—82no—
332C4⋊Q8—82no—
333C4⋊Q8—82no—
334#125—24no—
335#125—22no—
336#125—24no—
337#125—82no—
338#125#4114yes—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—82no—
342C4×Q8—82no—
343C4×Q8—82no—
344C4×Q8—82no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4C2×D411yescharacteristic
379C2×C4⋊C4C2×D411yescharacteristic
380C2×C4⋊C4C2×D411yescharacteristic
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D412yes—
383C2×C4⋊C4C2×D412yes—
384C2×C4⋊C4C2412yes—
385C2×C4⋊C4C2412yes—
386C2×C4⋊C4C2412yes—
387C2×C4⋊C4C2412yes—
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2411yescharacteristic
395C2×C4⋊C4C2411yescharacteristic
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2412yes—
409C2×C42C2411yescharacteristic
410C2×C42C2412yes—
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413#107—22no—
414#107—22no—
415#107—24no—
416#107—24no—
417#107—22no—
418#107—22no—
419#106—22no—
420#106—22no—
421#106—24no—
422#106—24no—
423#106—22no—
424#106—22no—
425C4⋊C8—24no—
426C4⋊C8—24no—
427C4⋊C8—22no—
428C4⋊C8—22no—
429#97—22no—
430#97—22no—
431#97—22no—
432#97—22no—
433C4×C8—24no—
434C4×C8—22no—
435C4×C8—22no—
436#95—21no—
437#95—22no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—22no—
446#95—22no—
447#95—21no—
448#95—22no—
449#95—22no—
450#95—21no—
451#95—22no—
452#95—22no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—22no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95C2×D412yes—
465#95C2×D412yes—
466#95C2×D412yes—
467#95C2×D412yes—
468#95C2×D412yes—
469#95C2×D412yes—
470#95C2×D412yes—
471C2×Q8—22no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—42no—
477C2×Q8—42no—
478C2×Q8—42no—
479C2×Q8—42no—
480C2×Q8—42no—
481C2×Q8—81no—
482C2×Q8—81no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C2511yescharacteristic
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C82−1+412yes—
523C2×C82−1+412yes—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—81no—
564C4⋊C4—81no—
565C4⋊C4—81no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4—81no—
570C4⋊C4—81no—
571C4⋊C4—81no—
572C4⋊C4—82no—
573C4⋊C4—82no—
574C4⋊C4—82no—
575C4⋊C4—82no—
576C4⋊C4—82no—
577C4⋊C4—82no—
578C4⋊C4—82no—
579C4⋊C4—82no—
580C4⋊C4—82no—
581C4⋊C4—82no—
582C4⋊C4—82no—
583C4⋊C4—82no—
584C4⋊C4—82no—
585C4⋊C4C2×#4112yes—
586C4⋊C4C2×#4112yes—
587C42—22no—
588C42—22no—
589C42—21no—
590C42—21no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—41no—
596C42—41no—
597C42—42no—
598C42—42no—
599C42—42no—
600C42—42no—
601C42—82no—
602C42—82no—
603C42—82no—
604C42—82no—
605C42C2×#4112yes—
606C422−1+412yes—
607C422−1+412yes—
608C422−1+412yes—
609C23C23×D411yescharacteristic
610Q8—44no—
611Q8—84no—
612Q8—84no—
613Q8—84no—
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—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4#53612yes—
675C2×C4#53612yes—
676C2×C4#54812yes—
677C2×C4#54812yes—
678C2×C4#57612yes—
679C2×C4#57712yes—
680C2×C4C22×#4111yescharacteristic
681C2×C4C2×2−1+411yescharacteristic
682C2×C4C2×2−1+411yescharacteristic
683C2×C4C2×2−1+411yescharacteristic
684C2×C4C2×2−1+411yescharacteristic
685C2×C4C2×2−1+411yescharacteristic
686C8—22no—
687C8—22no—
688C8—42no—
689C8—42no—
690C8—41no—
691C8—41no—
692C22C2×#53611yescharacteristic
693C22C2×#54811yescharacteristic
694C22#349911yescharacteristic
695C22C2×#57611yescharacteristic
696C22C2×#57711yescharacteristic
697C22#358611yescharacteristic
698C22#358611yescharacteristic
699C4—22no—
700C4—22no—
701C4—22no—
702C4—41no—
703C4—42no—
704C4—41no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—81no—
711C4—81no—
712C4—81no—
713C4—81no—
714C4—82no—
715C4—81no—
716C4—81no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4#324114yes—
721C4#352311yescharacteristic
722C4#352311yescharacteristic
723C4#355611yescharacteristic
724C4#355611yescharacteristic
725C2#3917011yescharacteristic
726C2C2×#324111yescharacteristic
727C2#6088411yescharacteristic
728C2#6134511yescharacteristic
729C2#6134611yescharacteristic
730C2#6252011yescharacteristic
731C2#6252211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2295 ◁ 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
#397542
#396412
#332942
#323412
#323382
#323372
#323342
#302322
#292192
#291882
#286982
#286912
#265132
#265102
#265092
#265062
#262832
#262812
#262782
#262762
#222522
C4×#5012
#213432