Finite Group Databasebeta

← #6560752

Subgroups — #6560752

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36556C211yesmaximal, characteristic
2#36285C211yesmaximal, characteristic
3#30380C212yesmaximal
4#30229C211yesmaximal, characteristic
5#29583C212yesmaximal
6#29522C212yesmaximal
7#29321C211yesmaximal, characteristic
8#29030C212yesmaximal
9#29026C212yesmaximal
10#28522C211yesmaximal, characteristic
11#27998C212yesmaximal
12#27997C212yesmaximal
13#25480C211yesmaximal, characteristic
14#25480C211yesmaximal, characteristic
15#25479C211yesmaximal, characteristic
16#25479C211yesmaximal, characteristic
17#23820C211yesmaximal, characteristic
18#23779C212yesmaximal
19#23063C211yesmaximal, characteristic
20#23062C211yesmaximal, characteristic
21#22965C211yesmaximal, characteristic
22#22964C211yesmaximal, characteristic
23#20573C211yesmaximal, characteristic
24C2×#477C2211yescharacteristic
25C2×#477C2211yescharacteristic
26C2×#475C2211yescharacteristic
27C2×#475C2211yescharacteristic
28C2×#446C2211yescharacteristic
29C2×#446C2211yescharacteristic
30#2864C2212yes—
31#2863C2212yes—
32#2862C2211yescharacteristic
33#2859C2211yescharacteristic
34#2606—22no—
35#2590—22no—
36#2588—22no—
37#2582C2211yescharacteristic
38#2582C2211yescharacteristic
39#2570—21no—
40#2567C2211yescharacteristic
41#2565—21no—
42#2546C2211yescharacteristic
43#2520—22no—
44#2511—21no—
45#2509C2212yes—
46#2508C2212yes—
47#2507C2212yes—
48#2446—21no—
49#2446C2212yes—
50#2370—21no—
51#2369C2211yescharacteristic
52#2365—21no—
53#2365C2211yescharacteristic
54#2356—21no—
55#2355—21no—
56#2355C2211yescharacteristic
57Q8×C4⋊C4C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×#125C2211yescharacteristic
60C4×#125C2212yes—
61#2295C2212yes—
62C42×Q8C2211yescharacteristic
63#2101C2212yes—
64#2101C2212yes—
65#2101C2212yes—
66#2101C2212yes—
67#2099C2211yescharacteristic
68#2099C2211yescharacteristic
69#2097C2212yes—
70#2097C2212yes—
71#2088C2212yes—
72#2088C2212yes—
73#2086C2212yes—
74#2086C2212yes—
75#2085C2212yes—
76#2074C2211yescharacteristic
77#2074C2211yescharacteristic
78#2068C2212yes—
79#2068C2212yes—
80#2066C2212yes—
81#2066C2212yes—
82#2059C2212yes—
83#2043C2211yescharacteristic
84#2043C2211yescharacteristic
85#2026C2211yescharacteristic
86#2026C2211yescharacteristic
87#1990C2211yescharacteristic
88#1990C2211yescharacteristic
89#1990C2211yescharacteristic
90#1990C2211yescharacteristic
91#1960C2212yes—
92#1948C2211yescharacteristic
93#1948C2211yescharacteristic
94#1947C2211yescharacteristic
95#1947C2211yescharacteristic
96#1944C2212yes—
97#1944C2212yes—
98#1944C2212yes—
99#1943C2212yes—
100#1943C2212yes—
101#1943C2212yes—
102#1942C2212yes—
103#1941C2212yes—
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#1847C2212yes—
119C4×#107C2212yes—
120C4×#106C2212yes—
121#1771C2212yes—
122#1771C2212yes—
123C8×C4⋊C4C2211yescharacteristic
124C8×C4⋊C4C2211yescharacteristic
125#1767C2211yescharacteristic
126C4×C4⋊C8C2212yes—
127C4×#103C2211yescharacteristic
128C4×#103C2211yescharacteristic
129C4×#103C2211yescharacteristic
130C4×#103C2211yescharacteristic
131C4×#103C2212yes—
132C4×#103C2212yes—
133#1745C2212yes—
134#1744C2211yescharacteristic
135C2×C4⋊Q8—21no—
136C2×C4⋊Q8C2311yescharacteristic
137C2×C4⋊Q8C2311yescharacteristic
138C2×C4⋊Q8C2311yescharacteristic
139C2×#125—21no—
140C2×#125C2311yescharacteristic
141C2×C4×Q8—21no—
142C2×C4×Q8D411yescharacteristic
143C2×C4×Q8D411yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2312yes—
147C2×C4×Q8C2311yescharacteristic
148#477—24no—
149#477—24no—
150#475—24no—
151#475—24no—
152#446—24no—
153#446—24no—
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×#106C2312yes—
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2311yescharacteristic
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2311yescharacteristic
171C2×#103C2311yescharacteristic
172C2×#103C2311yescharacteristic
173C2×#103C2311yescharacteristic
174C2×#103C2311yescharacteristic
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#97C2312yes—
179C2×#97C2312yes—
180C2×#97C2312yes—
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2311yescharacteristic
183C2×C4×C8C2312yes—
184C2×C4×C8C2312yes—
185#400—22no—
186#400—22no—
187#400—22no—
188#400—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#398—22no—
193#398—22no—
194#398—22no—
195#398—22no—
196#395—22no—
197#391—21no—
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—22no—
207#391C2311yescharacteristic
208#391C2311yescharacteristic
209#391C2312yes—
210#391C2312yes—
211#391C2312yes—
212#389—21no—
213#389—21no—
214#389—22no—
215#389—21no—
216#389—21no—
217#389—21no—
218#389—21no—
219#389—21no—
220#389—22no—
221#389—22no—
222#389C2312yes—
223#389C2312yes—
224#389C2311yescharacteristic
225#389C2311yescharacteristic
226#389C2312yes—
227#389C2312yes—
228#389C2312yes—
229#389C2312yes—
230#389C2312yes—
231#389C2312yes—
232#389C2312yes—
233#389C2312yes—
234#389C2312yes—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—21no—
239#387—22no—
240#387—22no—
241#387—21no—
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#384—21no—
257#384C2311yescharacteristic
258#384C2311yescharacteristic
259#384C2311yescharacteristic
260#383—22no—
261#382—21no—
262#382—21no—
263#382—22no—
264#382C2311yescharacteristic
265#382C2312yes—
266#382C2312yes—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—22no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—22no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4—22no—
277C4×C4⋊C4D411yescharacteristic
278C4×C4⋊C4D412yes—
279C4×C4⋊C4D412yes—
280C4×C4⋊C4D411yescharacteristic
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2311yescharacteristic
285C4×C4⋊C4C2312yes—
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2312yes—
288#376—22no—
289C43C2311yescharacteristic
290C43C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2312yes—
296#340C2312yes—
297#340C2311yescharacteristic
298#340C2311yescharacteristic
299#340C2311yescharacteristic
300#340C2311yescharacteristic
301#340C2312yes—
302#340C2312yes—
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#340C2312yes—
308#340C2312yes—
309#340C2312yes—
310#340C2312yes—
311#336C2312yes—
312#336C2311yescharacteristic
313#336C2311yescharacteristic
314#336C2311yescharacteristic
315#336C2311yescharacteristic
316#336C2312yes—
317#336C2312yes—
318C22×Q8—21no—
319C22×Q8—21no—
320C22×Q8C2×D411yescharacteristic
321C22×Q8C2411yescharacteristic
322C22×Q8C2411yescharacteristic
323C22×C8C2411yescharacteristic
324C22×C8C2411yescharacteristic
325C22×C8C2411yescharacteristic
326C22×C8C2411yescharacteristic
327C22×C8C2412yes—
328C22×C8C2412yes—
329C4⋊Q8—22no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8—41no—
335C4⋊Q8—44no—
336#125—22no—
337#125—22no—
338#125—44no—
339C4×Q8—24no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—24no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—42no—
348C4×Q8—42no—
349C4×Q8—44no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
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—22no—
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⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4C2×D411yescharacteristic
379C2×C4⋊C4C2×D412yes—
380C2×C4⋊C4C2×D411yescharacteristic
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D412yes—
383C2×C4⋊C4C2×D411yescharacteristic
384C2×C4⋊C4C2412yes—
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2411yescharacteristic
387C2×C4⋊C4C2412yes—
388C2×C4⋊C4C2412yes—
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2411yescharacteristic
394C2×C42—22no—
395C2×C42—21no—
396C2×C42—22no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D412yes—
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D412yes—
406C2×C42C2×D412yes—
407C2×C42C2×D412yes—
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D411yescharacteristic
410C2×C42C2412yes—
411C2×C42C2412yes—
412C2×C42C2411yescharacteristic
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415#107—24no—
416#107—24no—
417#107—24no—
418#107—42no—
419#106—24no—
420#106—24no—
421#106—24no—
422#106—42no—
423C4⋊C8—24no—
424C4⋊C8—24no—
425C4⋊C8#4114yes—
426C4⋊C8#4114yes—
427#103—22no—
428#103—22no—
429#103—24no—
430#103—24no—
431#103—24no—
432#103—24no—
433#103—22no—
434#103—22no—
435#103—22no—
436#103—22no—
437#103—41no—
438#103—41no—
439#97—24no—
440#97—24no—
441#97—42no—
442C4×C8—22no—
443C4×C8—22no—
444C4×C8—42no—
445C4×C8—42no—
446#95—21no—
447#95—22no—
448#95—22no—
449#95—22no—
450#95—21no—
451#95—22no—
452#95—21no—
453#95—21no—
454#95—22no—
455#95—22no—
456#95—21no—
457#95—22no—
458#95—22no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—21no—
467#95—22no—
468#95—22no—
469#95—21no—
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#95C2×D412yes—
481#95C2×D412yes—
482#95C2×D412yes—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—42no—
498C2×Q8—42no—
499C2×Q8—42no—
500C2×Q8—42no—
501C2×Q8—81no—
502C2×Q8—81no—
503C2×Q82−1+412yes—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—22no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D412yes—
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D412yes—
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D412yes—
525C22×C4C22×D411yescharacteristic
526C22×C4C2511yescharacteristic
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8C2×#4114yes—
544C2×C8C2×#4114yes—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—24no—
549C4⋊C4—24no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—22no—
569C4⋊C4—42no—
570C4⋊C4—44no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—44no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—81no—
585C4⋊C4—81no—
586C4⋊C4—81no—
587C4⋊C4—81no—
588C4⋊C4—82no—
589C4⋊C4—81no—
590C4⋊C4—82no—
591C4⋊C4—82no—
592C4⋊C4—81no—
593C4⋊C4—82no—
594C4⋊C4—82no—
595C4⋊C4—82no—
596C4⋊C4#13714yes—
597C4⋊C42−1+412yes—
598C4⋊C42−1+412yes—
599C42—22no—
600C42—24no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—21no—
606C42—21no—
607C42—21no—
608C42—22no—
609C42—21no—
610C42—42no—
611C42—41no—
612C42—41no—
613C42—42no—
614C42—42no—
615C42—44no—
616C42—42no—
617C42—42no—
618C42—42no—
619C42—44no—
620C42—42no—
621C42—82no—
622C42—82no—
623C42C2×#4112yes—
624C42C2×#4112yes—
625C23C23×D411yescharacteristic
626Q8—22no—
627Q8—22no—
628Q8—44no—
629Q8—41no—
630Q8—41no—
631Q8—41no—
632Q8—41no—
633Q8—41no—
634Q8—41no—
635Q8—84no—
636Q8—162no—
637C2×C4—22no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—21no—
646C2×C4—22no—
647C2×C4—21no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—21no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—22no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—22no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—41no—
688C2×C4—42no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—42no—
694C2×C4—41no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4#53212yes—
699C2×C4#53212yes—
700C2×C4C2×#13714yes—
701C2×C4C2×#13712yes—
702C2×C4#57512yes—
703C2×C4C22×#4111yescharacteristic
704C2×C4C22×#4111yescharacteristic
705C2×C4C2×2−1+411yescharacteristic
706C2×C4C2×2−1+411yescharacteristic
707C2×C4C2×2−1+411yescharacteristic
708C2×C4#58511yescharacteristic
709C8—24no—
710C8—24no—
711C8—41no—
712C8—41no—
713C8—41no—
714C8—41no—
715C22C2×#53211yescharacteristic
716C22#348611yescharacteristic
717C22#348911yescharacteristic
718C22C22×#13711yescharacteristic
719C22C2×#57511yescharacteristic
720C22#358411yescharacteristic
721C22#358411yescharacteristic
722C4—22no—
723C4—24no—
724C4—42no—
725C4—42no—
726C4—42no—
727C4—41no—
728C4—42no—
729C4—41no—
730C4—42no—
731C4—41no—
732C4—41no—
733C4—42no—
734C4—82no—
735C4—81no—
736C4—81no—
737C4—82no—
738C4—81no—
739C4—82no—
740C4—81no—
741C4—82no—
742C4—81no—
743C4#317214yes—
744C4#351511yescharacteristic
745C4#351511yescharacteristic
746C4#352311yescharacteristic
747C4#352311yescharacteristic
748C2#3594911yescharacteristic
749C2C2×#317211yescharacteristic
750C2#6072411yescharacteristic
751C2#6184511yescharacteristic
752C2#6184611yescharacteristic
753C2#6213911yescharacteristic
754C2#6214211yescharacteristic
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
#362852
#303802
#302292
#295832
#295222
#293212
#290302
#290262
#285222
#279982
#279972
#254802
#254792
#238202
#237792
#230632
#230622
#229652
#229642
#205732