Finite Group Databasebeta

← #6591324

Subgroups — #6591324

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 774 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2#36285C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic
4#33200C211yesmaximal, characteristic
5#33196C211yesmaximal, characteristic
6#28196C211yesmaximal, characteristic
7#28187C211yesmaximal, characteristic
8#27746C212yesmaximal
9#27741C212yesmaximal
10#26522C212yesmaximal
11#26520C212yesmaximal
12#24997C211yesmaximal, characteristic
13#24996C211yesmaximal, characteristic
14#24994C211yesmaximal, characteristic
15#24994C211yesmaximal, characteristic
16#24698C212yesmaximal
17#24691C212yesmaximal
18#23078C212yesmaximal
19#23077C212yesmaximal
20#22742C211yesmaximal, characteristic
21#22738C211yesmaximal, characteristic
22#21714C211yesmaximal, characteristic
23C4×#441C211yesmaximal, characteristic
24C2×#479C2212yes—
25C2×#475C2212yes—
26C2×#446C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28C2×#441C2211yescharacteristic
29C2×#441C2211yescharacteristic
30C2×C4×Q16C2211yescharacteristic
31#2868C2211yescharacteristic
32#2863—21no—
33#2859—21no—
34#2859C2211yescharacteristic
35#2606—22no—
36#2590—22no—
37#2582C2211yescharacteristic
38#2582C2211yescharacteristic
39#2582C2211yescharacteristic
40#2582C2211yescharacteristic
41#2570C2212yes—
42#2567C2212yes—
43#2565C2212yes—
44#2552C2212yes—
45#2550—22no—
46#2507—22no—
47#2507—21no—
48#2451—21no—
49#2451—21no—
50#2446—21no—
51#2370C2212yes—
52#2369C2212yes—
53#2356C2212yes—
54#2355C2212yes—
55#2355C2211yescharacteristic
56Q8×C4⋊C4C2211yescharacteristic
57Q8×C4⋊C4C2211yescharacteristic
58Q8×C4⋊C4C2211yescharacteristic
59C4×C4⋊Q8—21no—
60C4×C4⋊Q8C2211yescharacteristic
61#2295—21no—
62C42×Q8C2211yescharacteristic
63C42×Q8C2211yescharacteristic
64#2101C2212yes—
65#2100C2212yes—
66#2099C2212yes—
67#2099C2212yes—
68#2087C2212yes—
69#2086C2212yes—
70#2074C2212yes—
71#2072C2212yes—
72#2072C2212yes—
73#2072C2212yes—
74#2056C2212yes—
75#2056C2212yes—
76#1993C2212yes—
77#1993C2212yes—
78#1992C2212yes—
79#1992C2212yes—
80#1989C2212yes—
81#1988C2212yes—
82#1984C2211yescharacteristic
83#1984C2211yescharacteristic
84#1949C2212yes—
85#1949C2212yes—
86#1948C2212yes—
87#1948C2212yes—
88#1948C2212yes—
89#1948C2212yes—
90#1948C2211yescharacteristic
91#1948C2211yescharacteristic
92#1948C2211yescharacteristic
93#1947C2211yescharacteristic
94#1940C2212yes—
95#1940C2212yes—
96#1935C2212yes—
97#1935C2212yes—
98#1933C2212yes—
99#1933C2212yes—
100#1928C2212yes—
101#1927C2212yes—
102#1927C2212yes—
103#1927C2212yes—
104#1921C2212yes—
105#1920C2212yes—
106#1920C2212yes—
107#1919C2212yes—
108#1842C2211yescharacteristic
109#1836C2212yes—
110#1834C2211yescharacteristic
111#1834C2211yescharacteristic
112#1834C2211yescharacteristic
113#1834C2211yescharacteristic
114#1834C2212yes—
115#1805C2212yes—
116#1803C2211yescharacteristic
117#1803C2211yescharacteristic
118#1803C2211yescharacteristic
119#1803C2211yescharacteristic
120#1803C2212yes—
121#1776C2211yescharacteristic
122#1776C2211yescharacteristic
123#1776C2212yes—
124C4×#106C2211yescharacteristic
125C4×#106C2212yes—
126#1771C2212yes—
127#1771C2212yes—
128#1763C2211yescharacteristic
129#1763C2211yescharacteristic
130#1763C2211yescharacteristic
131#1763C2211yescharacteristic
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134C4×#97C2211yescharacteristic
135C22×Q16C2311yescharacteristic
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8D411yescharacteristic
139C2×C4⋊Q8C2312yes—
140C2×#125—22no—
141C2×#125C2312yes—
142C2×C4×Q8—21no—
143C2×C4×Q8—21no—
144C2×C4×Q8—21no—
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2311yescharacteristic
150C2×C4×Q8C2311yescharacteristic
151C2×C4×Q8C2311yescharacteristic
152C2×C4×Q8C2311yescharacteristic
153#479—28no—
154#475—28no—
155#446—24no—
156C8×Q8—24no—
157#441—24no—
158#441—24no—
159C4×Q16—24no—
160C2×#107C2312yes—
161C2×#107C2311yescharacteristic
162C2×#106C2312yes—
163C2×#106C2312yes—
164C2×#106C2312yes—
165C2×#106C2311yescharacteristic
166C2×#106C2311yescharacteristic
167C2×C4⋊C8C2312yes—
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2311yescharacteristic
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2311yescharacteristic
178C2×#103C2311yescharacteristic
179C2×#103C2311yescharacteristic
180C2×#103C2311yescharacteristic
181C2×#103C2311yescharacteristic
182C2×#103C2311yescharacteristic
183C2×#97C2312yes—
184C2×#97C2311yescharacteristic
185C2×#97C2311yescharacteristic
186C2×#97C2311yescharacteristic
187C2×#97C2311yescharacteristic
188C2×C4×C8C2311yescharacteristic
189C2×C4×C8C2312yes—
190C2×C4×C8C2311yescharacteristic
191#400—22no—
192#398—22no—
193#398—22no—
194#398—22no—
195#398—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#398—22no—
200#398—22no—
201#395—22no—
202#395—22no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—22no—
209#391—22no—
210#391D411yescharacteristic
211#391D411yescharacteristic
212#391C2312yes—
213#391C2312yes—
214#391C2312yes—
215#391C2312yes—
216#389—22no—
217#389—22no—
218#389—22no—
219#389—21no—
220#389—21no—
221#389—21no—
222#389—21no—
223#389—22no—
224#389—21no—
225#389—21no—
226#389—22no—
227#389—21no—
228#389—22no—
229#389—22no—
230#389—22no—
231#389—21no—
232#389—21no—
233#389—21no—
234#389—21no—
235#389D412yes—
236#389D412yes—
237#389C2312yes—
238#389C2312yes—
239#389C2312yes—
240#389C2312yes—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#387—22no—
247#387—22no—
248#387—22no—
249#387—22no—
250#387—22no—
251#387—22no—
252#384—21no—
253#384—21no—
254#384D411yescharacteristic
255#384C2312yes—
256#384C2312yes—
257#384C2312yes—
258#384C2311yescharacteristic
259#384C2311yescharacteristic
260#384C2311yescharacteristic
261#384C2311yescharacteristic
262#384C2312yes—
263#383—22no—
264#382—22no—
265#382C2312yes—
266#382C2312yes—
267C4×C4⋊C4—22no—
268C4×C4⋊C4—22no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4—22no—
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2311yescharacteristic
285C4×C4⋊C4C2311yescharacteristic
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2312yes—
288C4×C4⋊C4C2312yes—
289#376—22no—
290C43—21no—
291C43—21no—
292C43C2311yescharacteristic
293#340C2312yes—
294#340C2312yes—
295#340C2312yes—
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2312yes—
302#340C2312yes—
303#340C2312yes—
304#340C2312yes—
305#340C2312yes—
306#340C2312yes—
307#340C2311yescharacteristic
308#340C2312yes—
309#340C2312yes—
310#340C2311yescharacteristic
311#340C2311yescharacteristic
312#340C2311yescharacteristic
313#340C2311yescharacteristic
314#340C2311yescharacteristic
315#336C2312yes—
316#336C2312yes—
317#336C2312yes—
318#336C2312yes—
319C22×Q8—22no—
320C22×Q8C2×D411yescharacteristic
321C22×Q8C2×D411yescharacteristic
322C22×Q8C2411yescharacteristic
323C22×Q8C2411yescharacteristic
324C2×Q16—22no—
325C2×Q16—22no—
326C2×Q16—22no—
327C22×C8C2411yescharacteristic
328C22×C8C2411yescharacteristic
329C22×C8C2412yes—
330C22×C8C2412yes—
331C22×C8C2411yescharacteristic
332C22×C8C2411yescharacteristic
333C4⋊Q8—24no—
334C4⋊Q8—24no—
335C4⋊Q8—24no—
336C4⋊Q8—82no—
337C4⋊Q8—82no—
338#125—24no—
339#125—24no—
340#125—84no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×Q8—44no—
356C4×Q8—82no—
357C4×Q8—82no—
358C4×Q8#4114yes—
359C4×Q8#4114yes—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4C2×D411yescharacteristic
390C2×C4⋊C4C2×D411yescharacteristic
391C2×C4⋊C4C2×D412yes—
392C2×C4⋊C4C2×D412yes—
393C2×C4⋊C4C2×D412yes—
394C2×C4⋊C4C2×D412yes—
395C2×C4⋊C4C2×D412yes—
396C2×C4⋊C4C2×D412yes—
397C2×C4⋊C4C2411yescharacteristic
398C2×C4⋊C4C2411yescharacteristic
399C2×C4⋊C4C2411yescharacteristic
400C2×C4⋊C4C2411yescharacteristic
401C2×C4⋊C4C2412yes—
402C2×C4⋊C4C2412yes—
403C2×C4⋊C4C2412yes—
404C2×C4⋊C4C2412yes—
405C2×C4⋊C4C2411yescharacteristic
406C2×C4⋊C4C2411yescharacteristic
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—22no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—22no—
419C2×C42—22no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—22no—
423C2×C42C2×D411yescharacteristic
424C2×C42C2×D411yescharacteristic
425C2×C42C2×D411yescharacteristic
426C2×C42C2×D411yescharacteristic
427C2×C42C2412yes—
428C2×C42C2412yes—
429C2×C42C2411yescharacteristic
430C2×C42C2411yescharacteristic
431C2×C42C2411yescharacteristic
432#107—22no—
433#107—24no—
434#106—22no—
435#106—22no—
436#106—24no—
437#106—24no—
438#106—24no—
439C4⋊C8—24no—
440C4⋊C8—24no—
441C4⋊C8—22no—
442C4⋊C8—22no—
443C4⋊C8—24no—
444C4⋊C8—24no—
445#103—22no—
446#103—22no—
447#103—24no—
448#103—24no—
449#103—24no—
450#103—24no—
451#103—22no—
452#103—22no—
453#103—22no—
454#103—22no—
455#97—24no—
456#97—22no—
457#97—22no—
458#97—22no—
459#97—22no—
460C4×C8—24no—
461C4×C8—22no—
462C4×C8#4114yes—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—22no—
473#95—22no—
474#95—22no—
475#95—22no—
476#95—22no—
477#95—22no—
478#95—22no—
479#95—22no—
480#95—22no—
481#95—22no—
482#95—22no—
483#95—22no—
484#95—22no—
485#95—22no—
486#95—22no—
487#95—22no—
488#95—22no—
489#95C2×D412yes—
490#95C2×D412yes—
491#95C2×D412yes—
492#95C2×D412yes—
493C2×Q8—22no—
494C2×Q8—22no—
495C2×Q8—22no—
496C2×Q8—22no—
497C2×Q8—22no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—21no—
508C2×Q8—44no—
509C2×Q8—82no—
510C2×Q8—82no—
511C2×Q8#13714yes—
512C2×Q8C2×#4112yes—
513C2×Q8C2×#4112yes—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—22no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—21no—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D411yescharacteristic
533C22×C4C22×D411yescharacteristic
534C22×C4C22×D412yes—
535C22×C4C22×D411yescharacteristic
536C22×C4C2511yescharacteristic
537Q16—28no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—22no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—21no—
554C2×C8C2×#4112yes—
555C2×C8C2×#4112yes—
556C4⋊C4—24no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—24no—
560C4⋊C4—24no—
561C4⋊C4—24no—
562C4⋊C4—24no—
563C4⋊C4—24no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—22no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—22no—
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—22no—
579C4⋊C4—22no—
580C4⋊C4—22no—
581C4⋊C4—22no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—44no—
585C4⋊C4—42no—
586C4⋊C4—42no—
587C4⋊C4—42no—
588C4⋊C4—42no—
589C4⋊C4—44no—
590C4⋊C4—44no—
591C4⋊C4—44no—
592C4⋊C4—44no—
593C4⋊C4—42no—
594C4⋊C4—42no—
595C4⋊C4—42no—
596C4⋊C4—42no—
597C4⋊C4—81no—
598C4⋊C4—81no—
599C4⋊C4—82no—
600C4⋊C4—82no—
601C4⋊C4—82no—
602C4⋊C4—82no—
603C4⋊C4—82no—
604C4⋊C4—82no—
605C4⋊C4—82no—
606C4⋊C4—82no—
607C4⋊C4—82no—
608C4⋊C4—81no—
609C4⋊C4—81no—
610C4⋊C4—82no—
611C4⋊C4C2×#4112yes—
612C4⋊C4C2×#4112yes—
613C42—24no—
614C42—24no—
615C42—22no—
616C42—22no—
617C42—22no—
618C42—22no—
619C42—22no—
620C42—22no—
621C42—41no—
622C42—41no—
623C42—42no—
624C42—42no—
625C42—42no—
626C42—42no—
627C42—42no—
628C42—42no—
629C42—44no—
630C42—42no—
631C42—42no—
632C42—81no—
633C42—81no—
634C42—82no—
635C42—82no—
636C42—82no—
637C42C2×#4112yes—
638C42C2×#4112yes—
639C42C2×#4112yes—
640C422−1+412yes—
641C422−1+412yes—
642C422−1+412yes—
643C23C23×D411yescharacteristic
644Q8—28no—
645Q8—22no—
646Q8—22no—
647Q8—22no—
648Q8—22no—
649Q8—44no—
650Q8—41no—
651Q8—41no—
652Q8—41no—
653Q8—41no—
654Q8—164no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—22no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—22no—
674C2×C4—22no—
675C2×C4—22no—
676C2×C4—22no—
677C2×C4—22no—
678C2×C4—22no—
679C2×C4—22no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—41no—
696C2×C4—41no—
697C2×C4—42no—
698C2×C4—41no—
699C2×C4—41no—
700C2×C4—41no—
701C2×C4—41no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—42no—
706C2×C4—41no—
707C2×C4—42no—
708C2×C4—41no—
709C2×C4—42no—
710C2×C4—42no—
711C2×C4—42no—
712C2×C4#54712yes—
713C2×C4#54712yes—
714C2×C4#54812yes—
715C2×C4#54812yes—
716C2×C4#54812yes—
717C2×C4#54812yes—
718C2×C4C2×#13714yes—
719C2×C4C2×#13712yes—
720C2×C4#57712yes—
721C2×C4C22×#4111yescharacteristic
722C2×C4C22×#4111yescharacteristic
723C2×C4C22×#4111yescharacteristic
724C2×C4C2×2−1+411yescharacteristic
725C2×C4C2×2−1+411yescharacteristic
726C2×C4C2×2−1+411yescharacteristic
727C8—22no—
728C8—22no—
729C8—42no—
730C8—42no—
731C8—41no—
732C8—41no—
733C22C2×#54711yescharacteristic
734C22C2×#54811yescharacteristic
735C22C2×#54811yescharacteristic
736C22C22×#13711yescharacteristic
737C22C2×#57711yescharacteristic
738C22#358511yescharacteristic
739C22#358611yescharacteristic
740C4—24no—
741C4—22no—
742C4—22no—
743C4—22no—
744C4—22no—
745C4—22no—
746C4—42no—
747C4—42no—
748C4—42no—
749C4—42no—
750C4—42no—
751C4—42no—
752C4—42no—
753C4—81no—
754C4—81no—
755C4—81no—
756C4—82no—
757C4—81no—
758C4—82no—
759C4—82no—
760C4—82no—
761C4#334414yes—
762C4#344411yescharacteristic
763C4#350911yescharacteristic
764C4#350911yescharacteristic
765C4#350911yescharacteristic
766C2#3615111yescharacteristic
767C2C2×#334411yescharacteristic
768C2#6129411yescharacteristic
769C2#6131911yescharacteristic
770C2#6132211yescharacteristic
771C2#6134111yescharacteristic
772C2#6134611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ 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
#365632
#362852
#332972
#332002
#331962
#281962
#281872
#277462
#277412
#265222
#265202
#249972
#249962
#249942
#246982
#246912
#230782
#230772
#227422
#227382
#217142
C4×#4412