Finite Group Databasebeta

← #6693385

Subgroups — #6693385

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39769C211yesmaximal, characteristic
2#36447C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#32498C212yesmaximal
5#32476C212yesmaximal
6#32336C212yesmaximal
7#32322C212yesmaximal
8#30230C211yesmaximal, characteristic
9#29997C211yesmaximal, characteristic
10#27737C211yesmaximal, characteristic
11#27730C211yesmaximal, characteristic
12#27580C212yesmaximal
13#27556C212yesmaximal
14#26510C211yesmaximal, characteristic
15#26509C211yesmaximal, characteristic
16#26121C212yesmaximal
17#26116C212yesmaximal
18#24818C211yesmaximal, characteristic
19#24818C211yesmaximal, characteristic
20#22970C211yesmaximal, characteristic
21#22964C211yesmaximal, characteristic
22#22301C211yesmaximal, characteristic
23#21116C211yesmaximal, characteristic
24C2×#475C2211yescharacteristic
25C2×#475C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27#2868C2211yescharacteristic
28#2859—21no—
29#2859C2211yescharacteristic
30#2794—22no—
31#2794—22no—
32#2642—21no—
33#2571—21no—
34#2571C2211yescharacteristic
35#2570C2212yes—
36#2570C2212yes—
37#2568C2211yescharacteristic
38#2568C2211yescharacteristic
39#2552—21no—
40#2552C2211yescharacteristic
41#2511C2212yes—
42#2511C2212yes—
43#2508—22no—
44#2508C2212yes—
45#2506—22no—
46#2506C2212yes—
47#2446—21no—
48#2446—21no—
49#2365C2211yescharacteristic
50#2365C2211yescharacteristic
51#2351C2212yes—
52Q8×C4⋊C4C2212yes—
53Q8×C4⋊C4C2211yescharacteristic
54Q8×C4⋊C4C2211yescharacteristic
55#2327—21no—
56C4×#125—21no—
57C4×#125C2211yescharacteristic
58#2295C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2212yes—
61#2101C2212yes—
62#2101C2212yes—
63#2100C2212yes—
64#2100C2212yes—
65#2100C2212yes—
66#2100C2212yes—
67#2100C2212yes—
68#2088C2212yes—
69#2088C2212yes—
70#2088C2212yes—
71#2086C2212yes—
72#2085C2212yes—
73#2085C2212yes—
74#2074C2212yes—
75#2072C2212yes—
76#2068C2212yes—
77#2066C2212yes—
78#2066C2212yes—
79#2066C2212yes—
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1990C2211yescharacteristic
83#1990C2211yescharacteristic
84#1988C2211yescharacteristic
85#1988C2211yescharacteristic
86#1949C2212yes—
87#1948C2212yes—
88#1948C2211yescharacteristic
89#1948C2212yes—
90#1948C2212yes—
91#1947C2211yescharacteristic
92#1944C2212yes—
93#1941C2212yes—
94#1940C2212yes—
95#1939C2212yes—
96#1939C2211yescharacteristic
97#1939C2211yescharacteristic
98#1935C2212yes—
99#1933C2212yes—
100#1927C2211yescharacteristic
101#1927C2211yescharacteristic
102#1921C2212yes—
103#1920C2212yes—
104#1920C2212yes—
105#1919C2212yes—
106#1918C2211yescharacteristic
107#1918C2211yescharacteristic
108#1917C2212yes—
109#1916C2212yes—
110#1878C2211yescharacteristic
111#1878C2211yescharacteristic
112#1878C2212yes—
113#1878C2212yes—
114#1865C2211yescharacteristic
115#1865C2211yescharacteristic
116#1865C2212yes—
117#1863C2212yes—
118#1842C2211yescharacteristic
119#1836C2212yes—
120#1835C2212yes—
121#1805C2212yes—
122#1805C2212yes—
123#1803C2212yes—
124#1803C2212yes—
125#1803C2211yescharacteristic
126#1803C2211yescharacteristic
127C4×#106C2211yescharacteristic
128#1770C2211yescharacteristic
129C8×C4⋊C4C2211yescharacteristic
130#1763C2212yes—
131C4×#103C2212yes—
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134#1744C2211yescharacteristic
135C22×Q16C2311yescharacteristic
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8C2311yescharacteristic
139C2×C4⋊Q8C2311yescharacteristic
140C2×#125—21no—
141C2×#125—22no—
142C2×#125—21no—
143C2×#125D411yescharacteristic
144C2×C4×Q8—21no—
145C2×C4×Q8—21no—
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2312yes—
150C2×C4×Q8C2312yes—
151C2×C4×Q8C2311yescharacteristic
152#475—24no—
153#475—24no—
154C4×Q16—24no—
155C2×#107C2311yescharacteristic
156C2×#107C2312yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×#106C2311yescharacteristic
161C2×#106C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2312yes—
167C2×C4⋊C8C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2311yescharacteristic
171C2×#103C2311yescharacteristic
172C2×#103C2311yescharacteristic
173C2×#103C2311yescharacteristic
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#97C2312yes—
179C2×C4×C8C2311yescharacteristic
180C2×C4×C8C2311yescharacteristic
181C2×C4×C8C2312yes—
182#400—22no—
183#400—22no—
184#400—22no—
185#400—22no—
186#400—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#398—22no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391C2312yes—
205#391C2312yes—
206#391C2311yescharacteristic
207#391C2311yescharacteristic
208#389—21no—
209#389—21no—
210#389—22no—
211#389—22no—
212#389—21no—
213#389—21no—
214#389—22no—
215#389—22no—
216#389—22no—
217#389—22no—
218#389D411yescharacteristic
219#389D411yescharacteristic
220#389C2311yescharacteristic
221#389C2311yescharacteristic
222#389C2312yes—
223#389C2312yes—
224#389C2312yes—
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#389C2312yes—
229#389C2312yes—
230#387—22no—
231#387—22no—
232#387—22no—
233#387—21no—
234#387—21no—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387D412yes—
245#387D412yes—
246#384—21no—
247#384—21no—
248#384—22no—
249#384—21no—
250#384—21no—
251#384C2312yes—
252#384C2312yes—
253#384C2311yescharacteristic
254#384C2311yescharacteristic
255#384C2312yes—
256#382—21no—
257#382—21no—
258#382—22no—
259#382—22no—
260#382D411yescharacteristic
261#382C2312yes—
262#382C2312yes—
263#382C2312yes—
264C4×C4⋊C4—21no—
265C4×C4⋊C4—22no—
266C4×C4⋊C4—21no—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—22no—
272C4×C4⋊C4C2312yes—
273C4×C4⋊C4C2312yes—
274C4×C4⋊C4C2311yescharacteristic
275C4×C4⋊C4C2311yescharacteristic
276C4×C4⋊C4C2311yescharacteristic
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2312yes—
282#376—21no—
283C43—21no—
284C43C2311yescharacteristic
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#340C2312yes—
301#340C2311yescharacteristic
302#340C2312yes—
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#336C2312yes—
308#336C2312yes—
309#336C2312yes—
310#336C2312yes—
311#336C2311yescharacteristic
312#336C2311yescharacteristic
313#336C2311yescharacteristic
314#336C2311yescharacteristic
315C22×Q8—21no—
316C22×Q8—21no—
317C22×Q8C2411yescharacteristic
318C22×Q8C2411yescharacteristic
319C2×Q16—22no—
320C2×Q16—22no—
321C2×Q16—22no—
322C22×C8C2411yescharacteristic
323C22×C8C2411yescharacteristic
324C22×C8C2411yescharacteristic
325C22×C8C2411yescharacteristic
326C22×C8C2412yes—
327C22×C8C2412yes—
328C4⋊Q8—22no—
329C4⋊Q8—22no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—44no—
333C4⋊Q8—82no—
334#125—42no—
335#125—44no—
336#125—82no—
337#125—84no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—24no—
342C4×Q8—24no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—42no—
346C4×Q8—42no—
347C4×Q8—41no—
348C4×Q8—41no—
349C4×Q8—82no—
350C4×Q8—82no—
351C4×Q8#4114yes—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
381C2×C4⋊C4C2×D411yescharacteristic
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D412yes—
384C2×C4⋊C4C2×D411yescharacteristic
385C2×C4⋊C4C2×D412yes—
386C2×C4⋊C4C2×D412yes—
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2412yes—
392C2×C4⋊C4C2412yes—
393C2×C4⋊C4C2412yes—
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2411yescharacteristic
396C2×C4⋊C4C2411yescharacteristic
397C2×C4⋊C4C2411yescharacteristic
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42C2×D411yescharacteristic
411C2×C42C2×D411yescharacteristic
412C2×C42C2×D411yescharacteristic
413C2×C42C2×D411yescharacteristic
414C2×C42C2411yescharacteristic
415C2×C42C2411yescharacteristic
416C2×C42C2412yes—
417C2×C42C2412yes—
418C2×C42C2411yescharacteristic
419#107—22no—
420#107—42no—
421#106—24no—
422#106—24no—
423#106—42no—
424#106—41no—
425#106—41no—
426C4⋊C8—22no—
427C4⋊C8—22no—
428C4⋊C8—42no—
429C4⋊C8—42no—
430C4⋊C8—42no—
431C4⋊C8—42no—
432#103—22no—
433#103—24no—
434#103—24no—
435#103—22no—
436#103—22no—
437#103—22no—
438#103—41no—
439#103—41no—
440#103—42no—
441#103—42no—
442#97—42no—
443C4×C8—22no—
444C4×C8—22no—
445C4×C8—42no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95—21no—
457#95—21no—
458#95—22no—
459#95—22no—
460#95—21no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95—22no—
470#95—22no—
471#95—22no—
472#95—22no—
473#95—22no—
474#95—22no—
475#95—22no—
476#95C2×D412yes—
477#95C2×D412yes—
478#95C2×D412yes—
479#95C2×D412yes—
480#95C2×D412yes—
481#95C2×D412yes—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8C2×#4112yes—
498C2×Q82−1+412yes—
499C2×Q82−1+412yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C2511yescharacteristic
523Q16—44no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—22no—
536C2×C8—21no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—21no—
541C2×C8—21no—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—44no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—44no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—42no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—44no—
585C4⋊C4—42no—
586C4⋊C4—82no—
587C4⋊C4—82no—
588C4⋊C4—81no—
589C4⋊C4—81no—
590C4⋊C4—82no—
591C4⋊C4—82no—
592C4⋊C4—82no—
593C4⋊C4—82no—
594C4⋊C4—82no—
595C4⋊C4—82no—
596C4⋊C4—82no—
597C4⋊C4C2×#4112yes—
598C42—24no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—21no—
604C42—21no—
605C42—22no—
606C42—22no—
607C42—22no—
608C42—22no—
609C42—21no—
610C42—21no—
611C42—42no—
612C42—42no—
613C42—44no—
614C42—44no—
615C42—42no—
616C42—42no—
617C42—42no—
618C42—42no—
619C42—42no—
620C42—82no—
621C42—82no—
622C42C2×#4112yes—
623C422−1+412yes—
624C23C23×D411yescharacteristic
625Q8—22no—
626Q8—22no—
627Q8—22no—
628Q8—41no—
629Q8—41no—
630Q8—41no—
631Q8—41no—
632Q8—84no—
633Q8—84no—
634Q8#54114yes—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—22no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—21no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—21no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—41no—
690C2×C4—42no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4#54812yes—
696C2×C4#54812yes—
697C2×C4#57512yes—
698C2×C4#57712yes—
699C2×C4#57712yes—
700C2×C4#57812yes—
701C2×C4C22×#4111yescharacteristic
702C2×C4C2×2−1+411yescharacteristic
703C2×C4C2×2−1+411yescharacteristic
704C2×C4C2×2−1+411yescharacteristic
705C2×C4#58511yescharacteristic
706C2×C4#58511yescharacteristic
707C8—41no—
708C8—41no—
709C8—42no—
710C8—42no—
711C8—41no—
712C8—41no—
713C22C2×#54811yescharacteristic
714C22#348911yescharacteristic
715C22#348911yescharacteristic
716C22C2×#57511yescharacteristic
717C22C2×#57711yescharacteristic
718C22C2×#57711yescharacteristic
719C22C2×#57811yescharacteristic
720C4—24no—
721C4—22no—
722C4—22no—
723C4—22no—
724C4—42no—
725C4—41no—
726C4—42no—
727C4—42no—
728C4—41no—
729C4—42no—
730C4—42no—
731C4—81no—
732C4—82no—
733C4—82no—
734C4—81no—
735C4—81no—
736C4—82no—
737C4—81no—
738C4—81no—
739C4—82no—
740C4—81no—
741C4#337114yes—
742C4C2×#54111yescharacteristic
743C4#351211yescharacteristic
744C4#356311yescharacteristic
745C4#356311yescharacteristic
746C2#3770511yescharacteristic
747C2C2×#337111yescharacteristic
748C2#6133011yescharacteristic
749C2#6213511yescharacteristic
750C2#6214611yescharacteristic
751C2#6220311yescharacteristic
752C2#6220511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#125 ◁ 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
#397692
#364472
#331962
#324982
#324762
#323362
#323222
#302302
#299972
#277372
#277302
#275802
#275562
#265102
#265092
#261212
#261162
#248182
#229702
#229642
#223012
#211162