Finite Group Databasebeta

← #6555969

Subgroups — #6555969

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36584C211yesmaximal, characteristic
2#34328C211yesmaximal, characteristic
3#29988C211yesmaximal, characteristic
4#29988C212yesmaximal
5#29279C212yesmaximal
6#29274C211yesmaximal, characteristic
7#29214C212yesmaximal
8#29207C211yesmaximal, characteristic
9#26525C212yesmaximal
10#26523C212yesmaximal
11#26513C211yesmaximal, characteristic
12#26509C211yesmaximal, characteristic
13#26105C212yesmaximal
14#26105C212yesmaximal
15#26093C211yesmaximal, characteristic
16#26089C211yesmaximal, characteristic
17#23785C212yesmaximal
18#23779C211yesmaximal, characteristic
19#22736C211yesmaximal, characteristic
20#22729C211yesmaximal, characteristic
21#22728C211yesmaximal, characteristic
22#22524C211yesmaximal, characteristic
23#22524C211yesmaximal, characteristic
24C2×#557C2211yescharacteristic
25C2×#446C2211yescharacteristic
26C2×#446C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2212yes—
29#2571C2211yescharacteristic
30#2571C2211yescharacteristic
31#2571C2212yes—
32#2571C2212yes—
33#2509—22no—
34#2509—21no—
35#2509—22no—
36#2509—21no—
37#2371—22no—
38#2370C2212yes—
39#2365—21no—
40#2365C2211yescharacteristic
41#2356—21no—
42#2356C2212yes—
43#2355C2211yescharacteristic
44#2355C2211yescharacteristic
45#2351—21no—
46#2351—21no—
47Q8×C4⋊C4C2211yescharacteristic
48Q8×C4⋊C4C2211yescharacteristic
49#2336—22no—
50#2327—21no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×#125C2212yes—
53C4×#125C2211yescharacteristic
54C4×#125C2212yes—
55#2295C2211yescharacteristic
56#2295C2212yes—
57#2276—21no—
58C42×Q8C2211yescharacteristic
59#2101C2212yes—
60#2101C2212yes—
61#2101C2212yes—
62#2101C2212yes—
63#2101C2211yescharacteristic
64#2097C2212yes—
65#2097C2212yes—
66#2097C2212yes—
67#2097C2211yescharacteristic
68#2088C2212yes—
69#2088C2211yescharacteristic
70#2087C2212yes—
71#2086C2211yescharacteristic
72#2068C2211yescharacteristic
73#2068C2212yes—
74#2066C2211yescharacteristic
75#2066C2212yes—
76#2059C2212yes—
77#2059C2212yes—
78#2059C2211yescharacteristic
79#2059C2212yes—
80#2059C2212yes—
81#2026C2211yescharacteristic
82#2026C2212yes—
83#1992C2212yes—
84#1992C2212yes—
85#1990C2211yescharacteristic
86#1990C2211yescharacteristic
87#1989C2212yes—
88#1989C2212yes—
89#1988C2211yescharacteristic
90#1988C2211yescharacteristic
91#1949C2212yes—
92#1949C2212yes—
93#1949C2212yes—
94#1949C2212yes—
95#1949C2212yes—
96#1949C2212yes—
97#1949C2212yes—
98#1949C2212yes—
99#1948C2211yescharacteristic
100#1948C2212yes—
101#1948C2211yescharacteristic
102#1947C2211yescharacteristic
103#1947C2212yes—
104#1947C2211yescharacteristic
105#1940C2212yes—
106#1940C2212yes—
107#1939C2211yescharacteristic
108#1939C2211yescharacteristic
109#1921C2212yes—
110#1921C2212yes—
111#1920C2212yes—
112#1919C2212yes—
113#1919C2212yes—
114#1918C2212yes—
115#1847C2212yes—
116#1847C2212yes—
117#1845C2211yescharacteristic
118#1844C2211yescharacteristic
119#1831C2211yescharacteristic
120#1831C2212yes—
121#1803C2211yescharacteristic
122#1803C2211yescharacteristic
123#1803C2212yes—
124#1803C2212yes—
125#1776C2212yes—
126#1776C2212yes—
127C4×#107C2211yescharacteristic
128C4×#106C2211yescharacteristic
129#1767C2212yes—
130C4×C4⋊C8C2211yescharacteristic
131#1763C2212yes—
132#1763C2212yes—
133C4×#103C2211yescharacteristic
134C4×#103C2211yescharacteristic
135#557—24no—
136#557Q818yes—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8C2311yescharacteristic
139C2×#125—22no—
140C2×#125C2312yes—
141C2×#125C2311yescharacteristic
142C2×#125C2312yes—
143C2×C4×Q8—21no—
144C2×C4×Q8—21no—
145C2×C4×Q8—21no—
146C2×C4×Q8—21no—
147C2×C4×Q8—21no—
148C2×C4×Q8D411yescharacteristic
149C2×C4×Q8D411yescharacteristic
150C2×C4×Q8C2311yescharacteristic
151C2×C4×Q8C2312yes—
152C2×C4×Q8C2311yescharacteristic
153#446—24no—
154#446—24no—
155C2×#107C2311yescharacteristic
156C2×#107C2312yes—
157C2×#107C2311yescharacteristic
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#106C2311yescharacteristic
162C2×#106C2312yes—
163C2×#106C2311yescharacteristic
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2312yes—
167C2×C4⋊C8C2311yescharacteristic
168C2×C4⋊C8C2311yescharacteristic
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2312yes—
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#103C2312yes—
179C2×#103C2311yescharacteristic
180C2×#103C2312yes—
181C2×#97C2312yes—
182C2×#97C2312yes—
183C2×C4×C8C2311yescharacteristic
184C2×C4×C8C2311yescharacteristic
185#400—22no—
186#400—22no—
187#400—22no—
188#400—22no—
189#400—22no—
190#400—22no—
191#391—21no—
192#391—21no—
193#391—21no—
194#391C2312yes—
195#391C2311yescharacteristic
196#391C2311yescharacteristic
197#389—21no—
198#389—21no—
199#389—22no—
200#389—22no—
201#389—22no—
202#389—21no—
203#389—21no—
204#389—22no—
205#389—22no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—22no—
210#389—22no—
211#389—22no—
212#389—22no—
213#389—21no—
214#389—21no—
215#389C2311yescharacteristic
216#389C2312yes—
217#389C2311yescharacteristic
218#389C2312yes—
219#389C2312yes—
220#389C2311yescharacteristic
221#389C2311yescharacteristic
222#389C2312yes—
223#387—22no—
224#387—22no—
225#387—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#387—22no—
230#387—22no—
231#387—21no—
232#387—21no—
233#387—22no—
234#387—21no—
235#387—22no—
236#387—21no—
237#387—22no—
238#387—22no—
239#387—22no—
240#384C2311yescharacteristic
241#384C2311yescharacteristic
242#384C2311yescharacteristic
243#384C2312yes—
244#384C2312yes—
245#383—22no—
246#382—21no—
247#382—21no—
248#382—22no—
249#382—22no—
250#382—21no—
251#382C2311yescharacteristic
252#382C2312yes—
253#382C2312yes—
254#382C2312yes—
255#382C2312yes—
256C4×C4⋊C4—21no—
257C4×C4⋊C4—21no—
258C4×C4⋊C4—22no—
259C4×C4⋊C4—21no—
260C4×C4⋊C4—21no—
261C4×C4⋊C4—22no—
262C4×C4⋊C4—22no—
263C4×C4⋊C4—22no—
264C4×C4⋊C4—21no—
265C4×C4⋊C4—22no—
266C4×C4⋊C4—21no—
267C4×C4⋊C4—22no—
268C4×C4⋊C4D412yes—
269C4×C4⋊C4D411yescharacteristic
270C4×C4⋊C4D412yes—
271C4×C4⋊C4D411yescharacteristic
272C4×C4⋊C4C2311yescharacteristic
273C4×C4⋊C4C2311yescharacteristic
274C4×C4⋊C4C2312yes—
275C4×C4⋊C4C2312yes—
276C4×C4⋊C4C2312yes—
277C4×C4⋊C4C2312yes—
278C4×C4⋊C4C2312yes—
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2312yes—
283#376—21no—
284#376—22no—
285#376—22no—
286C43C2312yes—
287C43C2311yescharacteristic
288#340C2312yes—
289#340C2312yes—
290#340C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2311yescharacteristic
295#340C2312yes—
296#340C2312yes—
297#340C2311yescharacteristic
298#340C2311yescharacteristic
299#340C2312yes—
300#340C2312yes—
301#340C2311yescharacteristic
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2312yes—
305#340C2312yes—
306#340C2312yes—
307#340C2311yescharacteristic
308#340C2312yes—
309#340C2311yescharacteristic
310#336C2311yescharacteristic
311#336C2311yescharacteristic
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2312yes—
316C22×Q8—21no—
317C22×Q8C2×D411yescharacteristic
318C22×Q8C2411yescharacteristic
319C22×C8C2411yescharacteristic
320C22×C8C2411yescharacteristic
321C22×C8C2412yes—
322C22×C8C2412yes—
323C22×C8C2411yescharacteristic
324C22×C8C2411yescharacteristic
325C4⋊Q8—28no—
326C4⋊Q8—22no—
327C4⋊Q8C2×Q814yes—
328#125—216no—
329#125—24no—
330#125—41no—
331#125—42no—
332#125—42no—
333#125—41no—
334#125C2×Q818yes—
335C4×Q8—28no—
336C4×Q8—24no—
337C4×Q8—22no—
338C4×Q8—42no—
339C4×Q8—42no—
340C4×Q8—44no—
341C4×Q8—44no—
342C4×Q8—44no—
343C4×Q8—44no—
344C4×Q8C2×Q818yes—
345C4×Q8C2×Q814yes—
346C4×Q8C2×Q814yes—
347C4×Q8#4114yes—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
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—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4C2×D411yescharacteristic
376C2×C4⋊C4C2×D411yescharacteristic
377C2×C4⋊C4C2×D412yes—
378C2×C4⋊C4C2×D412yes—
379C2×C4⋊C4C2×D411yescharacteristic
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2412yes—
382C2×C4⋊C4C2411yescharacteristic
383C2×C4⋊C4C2411yescharacteristic
384C2×C4⋊C4C2412yes—
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2411yescharacteristic
387C2×C4⋊C4C2411yescharacteristic
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2412yes—
392C2×C42—22no—
393C2×C42—21no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D412yes—
406C2×C42C2×D412yes—
407C2×C42C2×D412yes—
408C2×C42C2×D412yes—
409C2×C42C2×D411yescharacteristic
410C2×C42C2×D411yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2412yes—
413C2×C42C2411yescharacteristic
414C2×C42C2412yes—
415C2×C42C2411yescharacteristic
416#107—22no—
417#107—41no—
418#107—42no—
419#107—42no—
420#107—42no—
421#107—42no—
422#106—22no—
423#106—42no—
424#106—41no—
425#106—42no—
426#106—42no—
427#106—42no—
428C4⋊C8—22no—
429C4⋊C8—24no—
430C4⋊C8—24no—
431C4⋊C8—22no—
432C4⋊C8—22no—
433C4⋊C8—22no—
434C4⋊C8—24no—
435C4⋊C8—24no—
436#103—22no—
437#103—22no—
438#103—41no—
439#103—42no—
440#103—42no—
441#103—41no—
442#97—24no—
443#97—24no—
444C4×C8—22no—
445C4×C8—22no—
446#95—21no—
447#95—21no—
448#95—22no—
449#95—21no—
450#95—21no—
451#95—22no—
452#95—21no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95—21no—
457#95—21no—
458#95—22no—
459#95—21no—
460#95—21no—
461#95—22no—
462#95—22no—
463#95—21no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—21no—
468#95—22no—
469#95—22no—
470#95—22no—
471#95—21no—
472#95—21no—
473#95—22no—
474#95—22no—
475#95—22no—
476#95—22no—
477#95—22no—
478#95—22no—
479#95C2×D412yes—
480#95C2×D412yes—
481#95C2×D412yes—
482C2×Q8—24no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8C22⋊Q814yes—
493C2×Q8C22×Q812yes—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D412yes—
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C2511yescharacteristic
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—21no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—21no—
533C2×C82−1+412yes—
534C2×C82−1+412yes—
535C4⋊C4—28no—
536C4⋊C4—24no—
537C4⋊C4—28no—
538C4⋊C4—24no—
539C4⋊C4—28no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—22no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—21no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—44no—
566C4⋊C4—44no—
567C4⋊C4—44no—
568C4⋊C4—44no—
569C4⋊C4—44no—
570C4⋊C4—42no—
571C4⋊C4—44no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—42no—
575C4⋊C4—42no—
576C4⋊C4—82no—
577C4⋊C4—81no—
578C4⋊C4—81no—
579C4⋊C4—82no—
580C4⋊C4—81no—
581C4⋊C4—82no—
582C4⋊C4—81no—
583C4⋊C4—81no—
584C4⋊C4—82no—
585C4⋊C4—81no—
586C4⋊C4C22⋊Q814yes—
587C4⋊C4C22⋊Q818yes—
588C4⋊C4C22×Q812yes—
589C4⋊C4C22×Q814yes—
590C42—28no—
591C42—24no—
592C42—22no—
593C42—22no—
594C42—21no—
595C42—21no—
596C42—22no—
597C42—22no—
598C42—21no—
599C42—21no—
600C42—42no—
601C42—41no—
602C42—42no—
603C42—41no—
604C42—42no—
605C42—42no—
606C42—44no—
607C42—44no—
608C42—44no—
609C42—44no—
610C42—42no—
611C42—42no—
612C42—42no—
613C42—82no—
614C42—82no—
615C42C22×Q814yes—
616C42C22×Q812yes—
617C42C2×#4112yes—
618C42C2×#4114yes—
619C23C23×D411yescharacteristic
620Q8—28no—
621Q8—22no—
622Q8—22no—
623Q8—48no—
624Q8—41no—
625Q8—41no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—41no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—41no—
686C2×C4C2×C22⋊Q814yes—
687C2×C4C2×C22⋊Q812yes—
688C2×C4C2×C22⋊Q814yes—
689C2×C4C2×C22⋊Q812yes—
690C2×C4#57512yes—
691C2×C4#57512yes—
692C2×C4C23×Q811yescharacteristic
693C2×C4C22×#4111yescharacteristic
694C2×C4C2×2−1+411yescharacteristic
695C2×C4C2×2−1+411yescharacteristic
696C2×C4#58511yescharacteristic
697C2×C4#58511yescharacteristic
698C8—22no—
699C8—22no—
700C8—41no—
701C8—41no—
702C8—42no—
703C8—42no—
704C22C22×C22⋊Q811yescharacteristic
705C22#348711yescharacteristic
706C22#349911yescharacteristic
707C22C2×#57511yescharacteristic
708C22C2×#57511yescharacteristic
709C22#358411yescharacteristic
710C22#358611yescharacteristic
711C4—24no—
712C4—24no—
713C4—22no—
714C4—42no—
715C4—42no—
716C4—41no—
717C4—41no—
718C4—42no—
719C4—44no—
720C4—42no—
721C4—41no—
722C4—41no—
723C4—81no—
724C4—81no—
725C4—81no—
726C4—81no—
727C4—82no—
728C4—81no—
729C4—82no—
730C4—81no—
731C4—82no—
732C4#308014yes—
733C4#350211yescharacteristic
734C4#350211yescharacteristic
735C4#351611yescharacteristic
736C4#351611yescharacteristic
737C2#3584511yescharacteristic
738C2C2×#308011yescharacteristic
739C2#6049911yescharacteristic
740C2#6189711yescharacteristic
741C2#6190811yescharacteristic
742C2#6248511yescharacteristic
743C2#6250711yescharacteristic
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
#365842
#343282
#299882
#292792
#292742
#292142
#292072
#265252
#265232
#265132
#265092
#261052
#260932
#260892
#237852
#237792
#227362
#227292
#227282
#225242