Finite Group Databasebeta

← #6586381

Subgroups — #6586381

Order 512.

1346 conjugacy classes of subgroups.

The subgroup lattice has 771 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36565C211yesmaximal, characteristic
2#36447C211yesmaximal, characteristic
3#33263C211yesmaximal, characteristic
4#33073C211yesmaximal, characteristic
5#33067C211yesmaximal, characteristic
6#28970C212yesmaximal
7#28956C212yesmaximal
8#28799C211yesmaximal, characteristic
9#28791C211yesmaximal, characteristic
10#26877C212yesmaximal
11#26876C212yesmaximal
12#26393C212yesmaximal
13#26389C212yesmaximal
14#26093C211yesmaximal, characteristic
15#26088C211yesmaximal, characteristic
16#26046C211yesmaximal, characteristic
17#26042C211yesmaximal, characteristic
18#22981C212yesmaximal
19#22971C212yesmaximal
20#22771C211yesmaximal, characteristic
21#22763C211yesmaximal, characteristic
22#21675C211yesmaximal, characteristic
23C4×#441C211yesmaximal, characteristic
24C2×#477C2212yes—
25C2×#475C2212yes—
26C2×#441C2211yescharacteristic
27C2×#441C2211yescharacteristic
28C2×C4×Q16C2211yescharacteristic
29#2862—21no—
30#2859—21no—
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2586—22no—
34#2586—22no—
35#2571—21no—
36#2571C2211yescharacteristic
37#2552—21no—
38#2552C2211yescharacteristic
39#2551C2211yescharacteristic
40#2551C2211yescharacteristic
41#2550—21no—
42#2550—21no—
43#2508—22no—
44#2508C2212yes—
45#2506—22no—
46#2506C2212yes—
47#2451C2212yes—
48#2451C2212yes—
49#2369C2212yes—
50#2369C2212yes—
51#2355C2211yescharacteristic
52#2355C2211yescharacteristic
53#2351C2212yes—
54Q8×C4⋊C4C2212yes—
55Q8×C4⋊C4C2211yescharacteristic
56Q8×C4⋊C4C2211yescharacteristic
57C4×#125—21no—
58C4×#125—21no—
59C4×#125C2211yescharacteristic
60C42×Q8C2211yescharacteristic
61C42×Q8C2211yescharacteristic
62#2097C2212yes—
63#2096C2212yes—
64#2088C2212yes—
65#2087C2212yes—
66#2085C2212yes—
67#2084C2212yes—
68#2068C2212yes—
69#2066C2212yes—
70#2058C2212yes—
71#2058C2212yes—
72#2058C2212yes—
73#2057C2212yes—
74#1993C2212yes—
75#1990C2212yes—
76#1949C2212yes—
77#1948C2211yescharacteristic
78#1948C2211yescharacteristic
79#1948C2211yescharacteristic
80#1948C2212yes—
81#1947C2211yescharacteristic
82#1944C2212yes—
83#1944C2212yes—
84#1943C2212yes—
85#1943C2212yes—
86#1943C2212yes—
87#1941C2212yes—
88#1939C2211yescharacteristic
89#1939C2211yescharacteristic
90#1935C2212yes—
91#1933C2212yes—
92#1921C2212yes—
93#1920C2212yes—
94#1919C2212yes—
95#1919C2212yes—
96#1919C2212yes—
97#1918C2212yes—
98#1918C2212yes—
99#1918C2212yes—
100#1917C2212yes—
101#1917C2212yes—
102#1878C2212yes—
103#1878C2212yes—
104#1865C2212yes—
105#1865C2212yes—
106#1865C2212yes—
107#1865C2212yes—
108#1863C2212yes—
109#1863C2212yes—
110#1845C2211yescharacteristic
111#1844C2211yescharacteristic
112#1834C2211yescharacteristic
113#1834C2211yescharacteristic
114#1831C2211yescharacteristic
115#1803C2211yescharacteristic
116#1803C2211yescharacteristic
117#1803C2211yescharacteristic
118#1803C2211yescharacteristic
119#1776C2211yescharacteristic
120#1776C2211yescharacteristic
121C4×#106C2211yescharacteristic
122#1771C2212yes—
123#1770C2212yes—
124#1770C2211yescharacteristic
125C8×C4⋊C4C2211yescharacteristic
126#1763C2211yescharacteristic
127#1763C2211yescharacteristic
128#1763C2211yescharacteristic
129#1763C2211yescharacteristic
130#1763C2212yes—
131C4×#103C2211yescharacteristic
132C4×#103C2211yescharacteristic
133C4×#103C2212yes—
134C4×#97C2211yescharacteristic
135C22×Q16C2311yescharacteristic
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8C2312yes—
139C2×C4⋊Q8C2312yes—
140C2×#125—21no—
141C2×#125—22no—
142C2×#125—21no—
143C2×#125—21no—
144C2×#125D411yescharacteristic
145C2×C4×Q8C2312yes—
146C2×C4×Q8C2312yes—
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2311yescharacteristic
150C2×C4×Q8C2311yescharacteristic
151C2×C4×Q8C2311yescharacteristic
152C2×C4×Q8C2311yescharacteristic
153#477—28no—
154#475—28no—
155#441—24no—
156#441—24no—
157C4×Q16Q818yescomplemented
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#107C2311yescharacteristic
162C2×#106C2312yes—
163C2×#106C2312yes—
164C2×#106C2311yescharacteristic
165C2×#106C2311yescharacteristic
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×#106C2312yes—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2311yescharacteristic
172C2×C4⋊C8C2311yescharacteristic
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2311yescharacteristic
180C2×#103C2311yescharacteristic
181C2×#103C2311yescharacteristic
182C2×#103C2311yescharacteristic
183C2×#97C2311yescharacteristic
184C2×#97C2311yescharacteristic
185C2×#97C2312yes—
186C2×#97C2311yescharacteristic
187C2×#97C2311yescharacteristic
188C2×C4×C8C2311yescharacteristic
189C2×C4×C8C2312yes—
190C2×C4×C8C2311yescharacteristic
191#400—22no—
192#400—22no—
193#400—22no—
194#398—22no—
195#398—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#395—22no—
200#395—22no—
201#395—22no—
202#395—22no—
203#391—22no—
204#391—22no—
205#391—22no—
206#391—22no—
207#391—22no—
208#391—22no—
209#391C2312yes—
210#391C2312yes—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—22no—
216#389—22no—
217#389—22no—
218#389—22no—
219#389—21no—
220#389—21no—
221#389—21no—
222#389—21no—
223#389—22no—
224#389—22no—
225#389—22no—
226#389—22no—
227#389D411yescharacteristic
228#389D411yescharacteristic
229#389C2312yes—
230#389C2312yes—
231#389C2312yes—
232#389C2312yes—
233#389C2312yes—
234#389C2312yes—
235#389C2312yes—
236#389C2312yes—
237#389C2312yes—
238#389C2312yes—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387D412yes—
246#387D412yes—
247#384—21no—
248#384—21no—
249#384—22no—
250#384C2311yescharacteristic
251#384C2311yescharacteristic
252#384C2311yescharacteristic
253#384C2311yescharacteristic
254#384C2312yes—
255#382—21no—
256#382—21no—
257#382—21no—
258#382—21no—
259#382—21no—
260#382—22no—
261#382—22no—
262#382D411yescharacteristic
263#382C2312yes—
264C4×C4⋊C4—21no—
265C4×C4⋊C4—22no—
266C4×C4⋊C4—22no—
267C4×C4⋊C4—22no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4—21no—
277C4×C4⋊C4—22no—
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2311yescharacteristic
282C4×C4⋊C4C2311yescharacteristic
283C4×C4⋊C4C2311yescharacteristic
284C4×C4⋊C4C2311yescharacteristic
285C4×C4⋊C4C2311yescharacteristic
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2311yescharacteristic
288C4×C4⋊C4C2312yes—
289C4×C4⋊C4C2312yes—
290C43—21no—
291C43—21no—
292C43C2311yescharacteristic
293#340C2311yescharacteristic
294#340C2312yes—
295#340C2311yescharacteristic
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2312yes—
302#340C2312yes—
303#340C2312yes—
304#340C2312yes—
305#340C2312yes—
306#340C2312yes—
307#340C2311yescharacteristic
308#340C2311yescharacteristic
309#340C2311yescharacteristic
310#340C2311yescharacteristic
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2312yes—
316#336C2312yes—
317C22×Q8—22no—
318C22×Q8—22no—
319C22×Q8C2411yescharacteristic
320C22×Q8C2411yescharacteristic
321C2×Q16—22no—
322C2×Q16—22no—
323C2×Q16C2×Q814yes—
324C22×C8C2412yes—
325C22×C8C2412yes—
326C22×C8C2411yescharacteristic
327C22×C8C2411yescharacteristic
328C22×C8C2411yescharacteristic
329C22×C8C2411yescharacteristic
330C4⋊Q8—24no—
331C4⋊Q8—24no—
332C4⋊Q8—42no—
333C4⋊Q8—42no—
334C4⋊Q8—82no—
335C4⋊Q8—82no—
336#125—24no—
337#125—44no—
338#125—82no—
339#125—82no—
340#125—84no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—24no—
346C4×Q8—24no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—42no—
350C4×Q8—41no—
351C4×Q8—41no—
352C4×Q8—41no—
353C4×Q8—41no—
354C4×Q8—42no—
355C4×Q8C2×Q814yes—
356C4×Q8C2×Q814yes—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2×D411yescharacteristic
389C2×C4⋊C4C2×D412yes—
390C2×C4⋊C4C2×D411yescharacteristic
391C2×C4⋊C4C2×D411yescharacteristic
392C2×C4⋊C4C2×D412yes—
393C2×C4⋊C4C2×D412yes—
394C2×C4⋊C4C2×D411yescharacteristic
395C2×C4⋊C4C2411yescharacteristic
396C2×C4⋊C4C2411yescharacteristic
397C2×C4⋊C4C2412yes—
398C2×C4⋊C4C2412yes—
399C2×C4⋊C4C2412yes—
400C2×C4⋊C4C2412yes—
401C2×C4⋊C4C2411yescharacteristic
402C2×C4⋊C4C2411yescharacteristic
403C2×C4⋊C4C2411yescharacteristic
404C2×C4⋊C4C2411yescharacteristic
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—22no—
412C2×C42—22no—
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—22no—
420C2×C42—22no—
421C2×C42C2×D411yescharacteristic
422C2×C42C2×D411yescharacteristic
423C2×C42C2×D411yescharacteristic
424C2×C42C2×D411yescharacteristic
425C2×C42C2411yescharacteristic
426C2×C42C2411yescharacteristic
427C2×C42C2412yes—
428C2×C42C2412yes—
429C2×C42C2411yescharacteristic
430#107—24no—
431#107—24no—
432#107—42no—
433#107C2×Q814yes—
434#106—24no—
435#106—22no—
436#106—22no—
437#106—24no—
438#106—42no—
439#106—42no—
440#106—42no—
441C4⋊C8—24no—
442C4⋊C8—24no—
443C4⋊C8—22no—
444C4⋊C8—22no—
445#103—24no—
446#103—24no—
447#103—22no—
448#103—22no—
449#103—42no—
450#103—42no—
451#103C2×Q814yes—
452#103C2×Q814yes—
453#103#4114yes—
454#103#4114yes—
455#97—22no—
456#97—22no—
457#97—41no—
458#97—41no—
459#97—42no—
460C4×C8—22no—
461C4×C8—42no—
462C4×C8C2×Q814yes—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
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#95—22no—
482#95—22no—
483#95—22no—
484#95—22no—
485#95—22no—
486#95—22no—
487#95C2×D412yes—
488#95C2×D412yes—
489#95C2×D412yes—
490#95C2×D412yes—
491#95C2×D412yes—
492#95C2×D412yes—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—44no—
502C2×Q8—44no—
503C2×Q8—82no—
504C2×Q8—82no—
505C2×Q8—82no—
506C2×Q8—82no—
507C2×Q8C22×Q812yes—
508C2×Q8C22×Q812yes—
509C2×Q8C2×#4112yes—
510C2×Q8C2×#4112yes—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—21no—
522C22×C4—22no—
523C22×C4C22×D412yes—
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D412yes—
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D412yes—
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D412yes—
532C22×C4C22×D411yescharacteristic
533C22×C4C2511yescharacteristic
534Q16—28no—
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×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8C22×Q812yes—
550C2×C8C22×Q812yes—
551C2×C8C2×#4112yes—
552C2×C8C2×#4112yes—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—22no—
571C4⋊C4—22no—
572C4⋊C4—21no—
573C4⋊C4—21no—
574C4⋊C4—22no—
575C4⋊C4—22no—
576C4⋊C4—42no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—44no—
581C4⋊C4—44no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—44no—
585C4⋊C4—44no—
586C4⋊C4—44no—
587C4⋊C4—44no—
588C4⋊C4—44no—
589C4⋊C4—42no—
590C4⋊C4—42no—
591C4⋊C4—42no—
592C4⋊C4—42no—
593C4⋊C4—81no—
594C4⋊C4—81no—
595C4⋊C4—81no—
596C4⋊C4—81no—
597C4⋊C4—82no—
598C4⋊C4—82no—
599C4⋊C4—82no—
600C4⋊C4—82no—
601C4⋊C4—82no—
602C4⋊C4—82no—
603C4⋊C4—82no—
604C4⋊C4—82no—
605C4⋊C4—82no—
606C4⋊C4#13714yes—
607C4⋊C4C22×Q812yes—
608C4⋊C4C22×Q812yes—
609C4⋊C4C2×#4112yes—
610C4⋊C4C2×#4112yes—
611C42—24no—
612C42—24no—
613C42—22no—
614C42—22no—
615C42—21no—
616C42—21no—
617C42—22no—
618C42—22no—
619C42—22no—
620C42—21no—
621C42—21no—
622C42—22no—
623C42—41no—
624C42—41no—
625C42—44no—
626C42—42no—
627C42—42no—
628C42—44no—
629C42—42no—
630C42—42no—
631C42—42no—
632C42—42no—
633C42—42no—
634C42—42no—
635C42—82no—
636C42—81no—
637C42—82no—
638C42—81no—
639C42C22×Q812yes—
640C422−1+412yes—
641C23C23×D411yescharacteristic
642Q8—22no—
643Q8—22no—
644Q8—22no—
645Q8—22no—
646Q8—22no—
647Q8—41no—
648Q8—41no—
649Q8—164no—
650Q8—164no—
651Q8#53314yes—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—22no—
674C2×C4—22no—
675C2×C4—22no—
676C2×C4—22no—
677C2×C4—22no—
678C2×C4—21no—
679C2×C4—21no—
680C2×C4—22no—
681C2×C4—21no—
682C2×C4—21no—
683C2×C4—22no—
684C2×C4—22no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—41no—
700C2×C4—42no—
701C2×C4—41no—
702C2×C4—41no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—42no—
706C2×C4—42no—
707C2×C4—41no—
708C2×C4—41no—
709C2×C4—42no—
710C2×C4—42no—
711C2×C4—42no—
712C2×C4—42no—
713C2×C4D4×Q812yes—
714C2×C4D4×Q812yes—
715C2×C4C2×#13714yes—
716C2×C4C2×#13712yes—
717C2×C4#57712yes—
718C2×C4C23×Q811yescharacteristic
719C2×C4C22×#4111yescharacteristic
720C2×C4C22×#4111yescharacteristic
721C2×C4C2×2−1+411yescharacteristic
722C2×C4#58511yescharacteristic
723C2×C4#58511yescharacteristic
724C8—22no—
725C8—22no—
726C8—22no—
727C8—22no—
728C8—42no—
729C8—42no—
730C22C2×D4×Q811yescharacteristic
731C22D4×#4111yescharacteristic
732C22D4×#4111yescharacteristic
733C22C22×#13711yescharacteristic
734C22C2×#57711yescharacteristic
735C22#358511yescharacteristic
736C22#358611yescharacteristic
737C4—24no—
738C4—24no—
739C4—22no—
740C4—41no—
741C4—41no—
742C4—42no—
743C4—42no—
744C4—41no—
745C4—42no—
746C4—42no—
747C4—41no—
748C4—42no—
749C4—42no—
750C4—42no—
751C4—42no—
752C4—81no—
753C4—81no—
754C4—82no—
755C4—82no—
756C4—82no—
757C4—82no—
758C4#338714yes—
759C4C2×#53311yescharacteristic
760C4#350411yescharacteristic
761C4#351511yescharacteristic
762C4#351511yescharacteristic
763C2#3613111yescharacteristic
764C2C2×#338711yescharacteristic
765C2#6138011yescharacteristic
766C2#6145711yescharacteristic
767C2#6146911yescharacteristic
768C2#6154011yescharacteristic
769C2#6155411yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
157. C4×Q16Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#365652
#364472
#332632
#330732
#330672
#289702
#289562
#287992
#287912
#268772
#268762
#263932
#263892
#260932
#260882
#260462
#260422
#229812
#229712
#227712
#227632
#216752
C4×#4412