Finite Group Databasebeta

← #6693353

Subgroups — #6693353

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38986C211yesmaximal, characteristic
2#36447C211yesmaximal, characteristic
3#33202C211yesmaximal, characteristic
4#32497C212yesmaximal
5#32493C212yesmaximal
6#32339C212yesmaximal
7#32327C212yesmaximal
8#30229C211yesmaximal, characteristic
9#29997C211yesmaximal, characteristic
10#27738C211yesmaximal, characteristic
11#27737C211yesmaximal, characteristic
12#27575C212yesmaximal
13#27570C212yesmaximal
14#26513C211yesmaximal, characteristic
15#26512C211yesmaximal, characteristic
16#26123C212yesmaximal
17#26120C212yesmaximal
18#24818C211yesmaximal, characteristic
19#24817C211yesmaximal, characteristic
20#22971C211yesmaximal, characteristic
21#22970C211yesmaximal, characteristic
22#22303C211yesmaximal, characteristic
23#21119C211yesmaximal, characteristic
24C2×#477C2211yescharacteristic
25C2×#475C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27#2862C2211yescharacteristic
28#2859—21no—
29#2859C2211yescharacteristic
30#2789—22no—
31#2758—22no—
32#2645—21no—
33#2571—21no—
34#2571C2212yes—
35#2571C2211yescharacteristic
36#2570C2212yes—
37#2568C2211yescharacteristic
38#2567C2211yescharacteristic
39#2552—21no—
40#2552C2211yescharacteristic
41#2514—21no—
42#2514C2212yes—
43#2514C2212yes—
44#2508—22no—
45#2508C2212yes—
46#2506—22no—
47#2506C2212yes—
48#2453—21no—
49#2369C2211yescharacteristic
50#2365C2211yescharacteristic
51#2351C2212yes—
52Q8×C4⋊C4C2212yes—
53Q8×C4⋊C4C2211yescharacteristic
54Q8×C4⋊C4C2211yescharacteristic
55#2320—21no—
56C4×#125—21no—
57C4×#125C2211yescharacteristic
58#2295C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2212yes—
61#2101C2212yes—
62#2101C2212yes—
63#2101C2212yes—
64#2101C2212yes—
65#2101C2212yes—
66#2100C2212yes—
67#2099C2212yes—
68#2088C2212yes—
69#2087C2212yes—
70#2087C2212yes—
71#2086C2212yes—
72#2085C2212yes—
73#2083C2212yes—
74#2074C2212yes—
75#2072C2212yes—
76#2068C2212yes—
77#2068C2212yes—
78#2066C2212yes—
79#2066C2212yes—
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1990C2211yescharacteristic
83#1990C2211yescharacteristic
84#1988C2211yescharacteristic
85#1987C2211yescharacteristic
86#1949C2212yes—
87#1949C2212yes—
88#1949C2212yes—
89#1948C2211yescharacteristic
90#1947C2212yes—
91#1947C2211yescharacteristic
92#1944C2212yes—
93#1943C2212yes—
94#1940C2212yes—
95#1940C2212yes—
96#1939C2211yescharacteristic
97#1939C2211yescharacteristic
98#1935C2212yes—
99#1935C2212yes—
100#1928C2211yescharacteristic
101#1928C2211yescharacteristic
102#1921C2211yescharacteristic
103#1921C2212yes—
104#1921C2211yescharacteristic
105#1919C2212yes—
106#1919C2212yes—
107#1918C2212yes—
108#1917C2212yes—
109#1917C2212yes—
110#1878C2211yescharacteristic
111#1878C2211yescharacteristic
112#1878C2212yes—
113#1878C2212yes—
114#1865C2212yes—
115#1863C2211yescharacteristic
116#1863C2212yes—
117#1863C2211yescharacteristic
118#1842C2211yescharacteristic
119#1836C2212yes—
120#1836C2212yes—
121#1805C2212yes—
122#1805C2212yes—
123#1805C2212yes—
124#1803C2212yes—
125#1803C2211yescharacteristic
126#1803C2211yescharacteristic
127C4×#107C2211yescharacteristic
128#1770C2211yescharacteristic
129#1770C2211yescharacteristic
130#1763C2212yes—
131#1763C2212yes—
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134#1744C2211yescharacteristic
135C22×Q16C2311yescharacteristic
136C2×C4⋊Q8C2311yescharacteristic
137C2×C4⋊Q8C2311yescharacteristic
138C2×#125—21no—
139C2×#125—22no—
140C2×#125—21no—
141C2×#125D411yescharacteristic
142C2×C4×Q8—21no—
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2312yes—
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2312yes—
148C2×C4×Q8C2311yescharacteristic
149#477—24no—
150#475—24no—
151C4×Q16—24no—
152C2×#107C2312yes—
153C2×#107C2311yescharacteristic
154C2×#107C2312yes—
155C2×#107C2311yescharacteristic
156C2×#107C2311yescharacteristic
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2311yescharacteristic
168C2×#103C2311yescharacteristic
169C2×#103C2311yescharacteristic
170C2×#103C2311yescharacteristic
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2311yescharacteristic
174C2×#103C2311yescharacteristic
175C2×#97C2312yes—
176C2×#97C2312yes—
177C2×C4×C8C2311yescharacteristic
178C2×C4×C8C2311yescharacteristic
179#401—22no—
180#401—22no—
181#400—22no—
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#398—22no—
191#398—22no—
192#398—22no—
193#398—22no—
194#395—22no—
195#391—21no—
196#391—21no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—22no—
203#391C2312yes—
204#391C2312yes—
205#391C2311yescharacteristic
206#391C2311yescharacteristic
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—22no—
212#389—22no—
213#389—22no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389—22no—
218#389—22no—
219#389—22no—
220#389—22no—
221#389D411yescharacteristic
222#389D411yescharacteristic
223#389C2311yescharacteristic
224#389C2312yes—
225#389C2311yescharacteristic
226#389C2312yes—
227#389C2312yes—
228#389C2312yes—
229#389C2312yes—
230#389C2312yes—
231#389C2312yes—
232#389C2312yes—
233#387—22no—
234#387—22no—
235#387—22no—
236#387—21no—
237#387—21no—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387D412yes—
243#387D412yes—
244#384—21no—
245#384—21no—
246#384—22no—
247#384C2312yes—
248#384C2311yescharacteristic
249#384C2311yescharacteristic
250#384C2312yes—
251#383—22no—
252#383—22no—
253#382—21no—
254#382—22no—
255#382—21no—
256#382—22no—
257#382—22no—
258#382D411yescharacteristic
259#382C2312yes—
260#382C2312yes—
261#382C2312yes—
262#382C2312yes—
263C4×C4⋊C4—21no—
264C4×C4⋊C4—22no—
265C4×C4⋊C4—21no—
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#340C2311yescharacteristic
301#340C2312yes—
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—
332#125—42no—
333#125—44no—
334#125—82no—
335#125—84no—
336C4×Q8—22no—
337C4×Q8—24no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—24no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—42no—
344C4×Q8—42no—
345C4×Q8—41no—
346C4×Q8—41no—
347C4×Q8—44no—
348C4×Q8#4114yes—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4C2×D411yescharacteristic
376C2×C4⋊C4C2×D411yescharacteristic
377C2×C4⋊C4C2×D411yescharacteristic
378C2×C4⋊C4C2×D411yescharacteristic
379C2×C4⋊C4C2×D412yes—
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D412yes—
383C2×C4⋊C4C2411yescharacteristic
384C2×C4⋊C4C2411yescharacteristic
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2412yes—
387C2×C4⋊C4C2412yes—
388C2×C4⋊C4C2412yes—
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2411yescharacteristic
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—22no—
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D411yescharacteristic
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2412yes—
412C2×C42C2412yes—
413C2×C42C2411yescharacteristic
414#107—22no—
415#107—24no—
416#107—41no—
417#107—42no—
418#107—41no—
419#106—24no—
420#106—42no—
421C4⋊C8—22no—
422C4⋊C8—22no—
423C4⋊C8—42no—
424C4⋊C8—42no—
425C4⋊C8—42no—
426C4⋊C8—42no—
427#103—22no—
428#103—24no—
429#103—24no—
430#103—22no—
431#103—22no—
432#103—22no—
433#103—42no—
434#103—41no—
435#103—41no—
436#103—42no—
437#97—42no—
438#97—42no—
439C4×C8—22no—
440C4×C8—22no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—22no—
449#95—22no—
450#95—22no—
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—22no—
460#95—22no—
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#95C2×D412yes—
474#95C2×D412yes—
475#95C2×D412yes—
476#95C2×D412yes—
477#95C2×D412yes—
478#95C2×D412yes—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—42no—
489C2×Q8—42no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8—81no—
493C2×Q8—81no—
494C2×Q8C2×#4112yes—
495C2×Q82−1+412yes—
496C2×Q82−1+412yes—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—22no—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D412yes—
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C2511yescharacteristic
520Q16—44no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—22no—
524C2×C8—22no—
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—21no—
538C2×C8—21no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—42no—
568C4⋊C4—42no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—44no—
572C4⋊C4—44no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—42no—
576C4⋊C4—42no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—81no—
581C4⋊C4—82no—
582C4⋊C4—81no—
583C4⋊C4—82no—
584C4⋊C4—82no—
585C4⋊C4—81no—
586C4⋊C4—81no—
587C4⋊C4—82no—
588C4⋊C4—82no—
589C4⋊C4—82no—
590C4⋊C4—82no—
591C4⋊C4—82no—
592C4⋊C4C2×#4112yes—
593C42—24no—
594C42—22no—
595C42—22no—
596C42—21no—
597C42—21no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—21no—
603C42—21no—
604C42—41no—
605C42—41no—
606C42—42no—
607C42—42no—
608C42—44no—
609C42—44no—
610C42—42no—
611C42—44no—
612C42—42no—
613C42—42no—
614C42—42no—
615C42—42no—
616C42—82no—
617C42C2×#4112yes—
618C422−1+412yes—
619C23C23×D411yescharacteristic
620Q8—22no—
621Q8—22no—
622Q8—22no—
623Q8—41no—
624Q8—41no—
625Q8—41no—
626Q8—41no—
627Q8—84no—
628Q8—162no—
629Q8#54114yes—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—22no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—41no—
679C2×C4—42no—
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—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4#54812yes—
693C2×C4#54812yes—
694C2×C4#57512yes—
695C2×C4#57812yes—
696C2×C4C22×#4111yescharacteristic
697C2×C4C2×2−1+411yescharacteristic
698C2×C4C2×2−1+411yescharacteristic
699C2×C4C2×2−1+411yescharacteristic
700C2×C4#58511yescharacteristic
701C2×C4#58511yescharacteristic
702C8—41no—
703C8—41no—
704C8—42no—
705C8—42no—
706C8—41no—
707C8—41no—
708C22C2×#54811yescharacteristic
709C22#348911yescharacteristic
710C22#348911yescharacteristic
711C22C2×#57511yescharacteristic
712C22C2×#57811yescharacteristic
713C22#358511yescharacteristic
714C22#358611yescharacteristic
715C4—24no—
716C4—22no—
717C4—42no—
718C4—41no—
719C4—42no—
720C4—41no—
721C4—42no—
722C4—41no—
723C4—42no—
724C4—41no—
725C4—42no—
726C4—81no—
727C4—82no—
728C4—81no—
729C4—82no—
730C4—81no—
731C4—81no—
732C4—82no—
733C4—81no—
734C4—81no—
735C4—82no—
736C4#337714yes—
737C4C2×#54111yescharacteristic
738C4#351211yescharacteristic
739C4#356311yescharacteristic
740C4#356311yescharacteristic
741C2#3770511yescharacteristic
742C2C2×#337711yescharacteristic
743C2#6133611yescharacteristic
744C2#6212311yescharacteristic
745C2#6212411yescharacteristic
746C2#6219211yescharacteristic
747C2#6219311yescharacteristic
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
#389862
#364472
#332022
#324972
#324932
#323392
#323272
#302292
#299972
#277382
#277372
#275752
#275702
#265132
#265122
#261232
#261202
#248182
#248172
#229712
#229702
#223032
#211192