Finite Group Databasebeta

← #6559847

Subgroups — #6559847

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36556C211yesmaximal, characteristic
2#36313C211yesmaximal, characteristic
3#30387C212yesmaximal
4#30281C211yesmaximal, characteristic
5#29583C212yesmaximal
6#29522C212yesmaximal
7#29349C211yesmaximal, characteristic
8#29222C211yesmaximal, characteristic
9#28971C212yesmaximal
10#28969C212yesmaximal
11#28684C211yesmaximal, characteristic
12#28683C211yesmaximal, characteristic
13#28647C211yesmaximal, characteristic
14#28646C211yesmaximal, characteristic
15#28053C212yesmaximal
16#28047C212yesmaximal
17#23836C211yesmaximal, characteristic
18#23785C212yesmaximal
19#23222C211yesmaximal, characteristic
20#23221C211yesmaximal, characteristic
21#23163C211yesmaximal, characteristic
22#23161C211yesmaximal, characteristic
23#20570C211yesmaximal, characteristic
24C2×#500C2211yescharacteristic
25C2×#498C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#487C2211yescharacteristic
28C2×#446C2211yescharacteristic
29C2×#446C2211yescharacteristic
30#2864C2212yes—
31#2863C2212yes—
32#2862C2211yescharacteristic
33#2862C2211yescharacteristic
34#2588—22no—
35#2568C2211yescharacteristic
36#2567C2211yescharacteristic
37#2546C2211yescharacteristic
38#2546C2211yescharacteristic
39#2528—22no—
40#2524—22no—
41#2520—22no—
42#2514—21no—
43#2514C2212yes—
44#2511—21no—
45#2509C2212yes—
46#2508C2212yes—
47#2453C2212yes—
48#2446—21no—
49#2446—21no—
50#2369C2211yescharacteristic
51#2365—21no—
52#2365—21no—
53#2365C2211yescharacteristic
54#2365C2211yescharacteristic
55#2365C2211yescharacteristic
56#2356—21no—
57#2355—21no—
58C4×#125C2211yescharacteristic
59C4×#125C2211yescharacteristic
60C4×#125C2212yes—
61#2295C2212yes—
62C42×Q8C2211yescharacteristic
63#2101C2212yes—
64#2101C2212yes—
65#2097C2212yes—
66#2097C2212yes—
67#2097C2212yes—
68#2097C2212yes—
69#2088C2212yes—
70#2088C2212yes—
71#2087C2212yes—
72#2087C2212yes—
73#2085C2211yescharacteristic
74#2085C2211yescharacteristic
75#2085C2211yescharacteristic
76#2085C2211yescharacteristic
77#2074C2212yes—
78#2068C2212yes—
79#2068C2212yes—
80#2066C2212yes—
81#2066C2212yes—
82#2059C2211yescharacteristic
83#2059C2211yescharacteristic
84#2059C2211yescharacteristic
85#2059C2211yescharacteristic
86#2026C2212yes—
87#1991C2212yes—
88#1988C2211yescharacteristic
89#1988C2211yescharacteristic
90#1988C2211yescharacteristic
91#1988C2211yescharacteristic
92#1987C2211yescharacteristic
93#1987C2211yescharacteristic
94#1981C2211yescharacteristic
95#1981C2211yescharacteristic
96#1944C2212yes—
97#1944C2212yes—
98#1944C2212yes—
99#1943C2212yes—
100#1943C2212yes—
101#1943C2212yes—
102#1935C2212yes—
103#1935C2212yes—
104#1935C2212yes—
105#1935C2212yes—
106#1935C2212yes—
107#1935C2212yes—
108#1933C2212yes—
109#1933C2212yes—
110#1928C2212yes—
111#1927C2212yes—
112#1921C2212yes—
113#1921C2212yes—
114#1920C2212yes—
115#1919C2212yes—
116#1865C2212yes—
117#1863C2212yes—
118#1835C2212yes—
119#1776C2212yes—
120#1776C2212yes—
121C4×#107C2211yescharacteristic
122C4×#107C2211yescharacteristic
123C4×#106C2211yescharacteristic
124C4×#106C2211yescharacteristic
125#1771C2212yes—
126#1771C2212yes—
127#1770C2211yescharacteristic
128#1770C2211yescharacteristic
129#1767C2212yes—
130C4×C4⋊C8C2211yescharacteristic
131#1763C2212yes—
132#1763C2212yes—
133#1745C2212yes—
134#1744C2211yescharacteristic
135C2×C4⋊Q8—21no—
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8C2311yescharacteristic
138C2×C4⋊Q8C2311yescharacteristic
139C2×C4⋊Q8C2311yescharacteristic
140C2×C4⋊Q8C2311yescharacteristic
141C2×#125C2311yescharacteristic
142C2×#125C2311yescharacteristic
143C2×C4×Q8—21no—
144C2×C4×Q8—21no—
145C2×C4×Q8D411yescharacteristic
146C2×C4×Q8D411yescharacteristic
147C2×C4×Q8C2312yes—
148C2×C4×Q8C2311yescharacteristic
149#500—24no—
150#498—24no—
151#487—24no—
152#487—24no—
153#446—24no—
154#446—24no—
155C2×#107C2311yescharacteristic
156C2×#107C2311yescharacteristic
157C2×#107C2311yescharacteristic
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#107C2311yescharacteristic
161C2×#106C2311yescharacteristic
162C2×#106C2311yescharacteristic
163C2×#106C2311yescharacteristic
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2311yescharacteristic
167C2×C4⋊C8C2312yes—
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2311yescharacteristic
170C2×C4⋊C8C2311yescharacteristic
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#97C2312yes—
180C2×#97C2312yes—
181C2×#97C2312yes—
182C2×#97C2312yes—
183C2×C4×C8C2312yes—
184C2×C4×C8C2311yescharacteristic
185C2×C4×C8C2311yescharacteristic
186#401—22no—
187#400—22no—
188#400—22no—
189#400—22no—
190#400—22no—
191#400—22no—
192#400—22no—
193#398—22no—
194#398—22no—
195#398—22no—
196#398—22no—
197#395—22no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—22no—
211#391C2312yes—
212#391C2312yes—
213#391C2312yes—
214#391C2312yes—
215#391C2312yes—
216#389—22no—
217#389—22no—
218#389—21no—
219#389—21no—
220#389C2312yes—
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—21no—
236#387—22no—
237#387—22no—
238#387—21no—
239#387—21no—
240#387—21no—
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#387—22no—
253#387—22no—
254#387—22no—
255#384—21no—
256#383—22no—
257#383—22no—
258#382—21no—
259#382—21no—
260#382—21no—
261#382—22no—
262#382C2311yescharacteristic
263#382C2311yescharacteristic
264#382C2312yes—
265#382C2312yes—
266C4×C4⋊C4—21no—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—22no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—22no—
274C4×C4⋊C4—22no—
275C4×C4⋊C4D412yes—
276C4×C4⋊C4D411yescharacteristic
277C4×C4⋊C4D411yescharacteristic
278C4×C4⋊C4D412yes—
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2312yes—
282C4×C4⋊C4C2311yescharacteristic
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2311yescharacteristic
285C4×C4⋊C4C2312yes—
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2312yes—
288#376—22no—
289C43C2311yescharacteristic
290C43C2312yes—
291#340C2311yescharacteristic
292#340C2311yescharacteristic
293#340C2311yescharacteristic
294#340C2311yescharacteristic
295#340C2312yes—
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2311yescharacteristic
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#340C2312yes—
306#340C2312yes—
307#340C2312yes—
308#340C2312yes—
309#340C2312yes—
310#340C2312yes—
311#336C2311yescharacteristic
312#336C2311yescharacteristic
313#336C2311yescharacteristic
314#336C2311yescharacteristic
315#336C2312yes—
316#336C2312yes—
317#336C2312yes—
318C22×Q8—21no—
319C22×Q8—21no—
320C22×Q8—21no—
321C22×Q8C2×D411yescharacteristic
322C22×Q8C2412yes—
323C22×C8C2412yes—
324C22×C8C2412yes—
325C22×C8C2411yescharacteristic
326C22×C8C2411yescharacteristic
327C22×C8C2411yescharacteristic
328C22×C8C2411yescharacteristic
329C4⋊Q8—22no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8—22no—
335C4⋊Q8—22no—
336C4⋊Q8—22no—
337C4⋊Q8—44no—
338C4⋊Q8—44no—
339#125—22no—
340#125—22no—
341#125—22no—
342#125—41no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—24no—
346C4×Q8—42no—
347C4×Q8—42no—
348C4×Q8—42no—
349C4×Q8—44no—
350C4×Q8—44no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
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—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4C2×D411yescharacteristic
375C2×C4⋊C4C2×D411yescharacteristic
376C2×C4⋊C4C2×D411yescharacteristic
377C2×C4⋊C4C2×D412yes—
378C2×C4⋊C4C2×D412yes—
379C2×C4⋊C4C2×D412yes—
380C2×C4⋊C4C2411yescharacteristic
381C2×C4⋊C4C2412yes—
382C2×C4⋊C4C2412yes—
383C2×C4⋊C4C2412yes—
384C2×C4⋊C4C2411yescharacteristic
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2411yescharacteristic
387C2×C4⋊C4C2411yescharacteristic
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2411yescharacteristic
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—22no—
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D412yes—
402C2×C42C2×D412yes—
403C2×C42C2×D412yes—
404C2×C42C2×D412yes—
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2412yes—
408C2×C42C2412yes—
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412#107—24no—
413#107—24no—
414#107—22no—
415#107—22no—
416#107—22no—
417#107—41no—
418#106—24no—
419#106—24no—
420#106—22no—
421#106—22no—
422#106—22no—
423#106—41no—
424C4⋊C8—24no—
425C4⋊C8—24no—
426C4⋊C8—22no—
427C4⋊C8—22no—
428#103—24no—
429#103—24no—
430#103—24no—
431#103—24no—
432#103—24no—
433#103—24no—
434#103—42no—
435#103—42no—
436#97—24no—
437#97—24no—
438#97—42no—
439#97—42no—
440C4×C8—42no—
441C4×C8#4114yes—
442C4×C8#4114yes—
443#95—21no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95—22no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—22no—
453#95—22no—
454#95—21no—
455#95—21no—
456#95—22no—
457#95—21no—
458#95—21no—
459#95—22no—
460#95—22no—
461#95—22no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—21no—
469#95—22no—
470#95—22no—
471#95—22no—
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#95C2×D412yes—
482#95C2×D412yes—
483#95C2×D412yes—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—22no—
491C2×Q8—22no—
492C2×Q8—22no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8—42no—
498C2×Q8—42no—
499C2×Q8—42no—
500C2×Q8—81no—
501C2×Q8—81no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C2511yescharacteristic
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8C2×#4112yes—
540C2×C8C2×#4112yes—
541C2×C8C2×#4112yes—
542C2×C8C2×#4112yes—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—44no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—42no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—44no—
578C4⋊C4—44no—
579C4⋊C4—44no—
580C4⋊C4—44no—
581C4⋊C4—42no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—81no—
587C4⋊C4—81no—
588C4⋊C4—82no—
589C4⋊C4—82no—
590C4⋊C4—82no—
591C4⋊C42−1+412yes—
592C4⋊C42−1+412yes—
593C4⋊C42−1+412yes—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—21no—
600C42—21no—
601C42—21no—
602C42—21no—
603C42—22no—
604C42—41no—
605C42—41no—
606C42—42no—
607C42—42no—
608C42—42no—
609C42—42no—
610C42—44no—
611C42—44no—
612C42—44no—
613C42—42no—
614C42—42no—
615C42—42no—
616C42—42no—
617C42—82no—
618C42C2×#4112yes—
619C42C2×#4112yes—
620C23C23×D411yescharacteristic
621Q8—44no—
622Q8—42no—
623Q8—42no—
624Q8—42no—
625Q8—42no—
626Q8—84no—
627Q8—84no—
628Q8—162no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—22no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4#53212yes—
692C2×C4#53212yes—
693C2×C4#57512yes—
694C2×C4#57512yes—
695C2×C4C22×#4111yescharacteristic
696C2×C4C22×#4111yescharacteristic
697C2×C4C2×2−1+411yescharacteristic
698C2×C4C2×2−1+411yescharacteristic
699C2×C4C2×2−1+411yescharacteristic
700C2×C4#58511yescharacteristic
701C8—22no—
702C8—22no—
703C8—22no—
704C8—22no—
705C8—42no—
706C8—42no—
707C22C2×#53211yescharacteristic
708C22#348611yescharacteristic
709C22#348911yescharacteristic
710C22C2×#57511yescharacteristic
711C22C2×#57511yescharacteristic
712C22#358611yescharacteristic
713C22#358611yescharacteristic
714C4—22no—
715C4—42no—
716C4—42no—
717C4—41no—
718C4—41no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—41no—
723C4—42no—
724C4—42no—
725C4—41no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—81no—
731C4—81no—
732C4—81no—
733C4—81no—
734C4—81no—
735C4#318314yes—
736C4#350711yescharacteristic
737C4#350711yescharacteristic
738C4#350911yescharacteristic
739C4#350911yescharacteristic
740C2#3593811yescharacteristic
741C2C2×#318311yescharacteristic
742C2#6072011yescharacteristic
743C2#6183411yescharacteristic
744C2#6183411yescharacteristic
745C2#6212111yescharacteristic
746C2#6212411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×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
#365562
#363132
#303872
#302812
#295832
#295222
#293492
#292222
#289712
#289692
#286842
#286832
#286472
#286462
#280532
#280472
#238362
#237852
#232222
#232212
#231632
#231612
#205702