Finite Group Databasebeta

← #8069510

Subgroups — #8069510

Order 512.

4001 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59681C214yesmaximal
2#59459C214yesmaximal
3#59445C214yesmaximal
4#51082C218yesmaximal
5#50910C218yesmaximal
6#50908C218yesmaximal
7#46243C212yesmaximal
8#46009C214yesmaximal
9#46007C211yesmaximal, characteristic
10#46006C212yesmaximal
11#39655C214yesmaximal
12#39655C214yesmaximal
13#39653C214yesmaximal
14#36556C212yesmaximal
15Q8×#125C211yesmaximal, characteristic
16#34526C212yesmaximal
17C4×#564C211yesmaximal, characteristic
18C2×#564C2211yescharacteristic
19#2864C2214yes—
20#2863C2214yes—
21#2862C2212yes—
22#2859C2211yescharacteristic
23#2859C2212yes—
24#2845C2212yes—
25#2844C2214yes—
26#2806C2214yes—
27#2805C2218yes—
28#2805C2218yes—
29#2805C2214yes—
30#2802C2218yes—
31#2802C2218yes—
32#2802C2218yes—
33#2802C2218yes—
34#2802C2218yes—
35#2792C2218yes—
36#2792C2218yes—
37#2792C2218yes—
38#2792C2218yes—
39#2791C2218yes—
40#2791C2218yes—
41#2774C2218yes—
42#2774C2218yes—
43#2774C2218yes—
44#2755C2218yes—
45#2755C2218yes—
46#2755C2218yes—
47#2734C2214yes—
48#2713C2218yes—
49#2713C2218yes—
50#2666C2218yes—
51#2666C2218yes—
52#2666C2218yes—
53#2590C2218yes—
54#2590C2218yes—
55#2588C2218yes—
56#2588C2218yes—
57#2588C2218yes—
58#2586C2218yes—
59#2586C2218yes—
60#2585C2218yes—
61#2585C2218yes—
62#2584C2218yes—
63#2570C2218yes—
64#2570C2218yes—
65#2567C2214yes—
66#2567C2214yes—
67#2567C2212yes—
68#2567C2218yes—
69#2552C2212yes—
70#2552C2214yes—
71#2552C2214yes—
72#2551C2214yes—
73#2550C2218yes—
74#2546C2212yes—
75#2546C2214yes—
76#2546C2214yes—
77#2520C2218yes—
78#2511C2214yes—
79#2511C2214yes—
80#2511C2214yes—
81#2511C2214yes—
82#2509C2218yes—
83#2509C2214yes—
84#2508C2218yes—
85#2508C2214yes—
86#2507C2218yes—
87#2507C2218yes—
88#2507C2214yes—
89#2507C2214yes—
90#2506C2218yes—
91#2462C2218yes—
92#2462C2218yes—
93#2462C2218yes—
94#2451C2214yes—
95#2451C2214yes—
96#2451C2214yes—
97#2451C2214yes—
98#2446C2214yes—
99#2446C2214yes—
100#2446C2214yes—
101#2446C2218yes—
102#2446C2214yes—
103#2446C2214yes—
104#2382C2214yes—
105#2382C2211yescharacteristic
106#2382C2212yes—
107#2382C2214yes—
108#2382C2214yes—
109#2382C2212yes—
110#2382C2214yes—
111#2370C2218yes—
112#2370C2218yes—
113#2370C2218yes—
114#2369C2212yes—
115#2365C2214yes—
116#2365C2214yes—
117#2365C2212yes—
118#2355C2214yes—
119#2355C2214yes—
120Q8×C4⋊C4C2214yes—
121Q8×C4⋊C4C2214yes—
122Q8×C4⋊C4C2212yes—
123#2336C2218yes—
124#2336C2218yes—
125#2336C2218yes—
126C4×C4⋊Q8C2214yes—
127C4×C4⋊Q8C2214yes—
128C4×C4⋊Q8C2214yes—
129C4×#125C2214yes—
130C4×#125C2214yes—
131C4×#125C2214yes—
132C4×#125C2212yes—
133C4×#125C2211yescharacteristic
134C4×#125C2212yes—
135C4×#125C2214yes—
136C42×Q8C2211yescharacteristic
137#564Q8116yescomplemented
138C2×C4⋊Q8C2314yes—
139C2×C4⋊Q8C2314yes—
140C2×C4⋊Q8C2314yes—
141C2×C4⋊Q8C2312yes—
142C2×C4⋊Q8C2312yes—
143C2×C4⋊Q8C2318yes—
144C2×#125C2318yes—
145C2×#125C2312yes—
146C2×#125C2314yes—
147C2×#125C2312yes—
148C2×#125C2311yescharacteristic
149C2×#125C2314yes—
150C2×#125C2314yes—
151C2×#125C2314yes—
152C2×C4×Q8C2311yescharacteristic
153C2×C4×Q8C2314yes—
154C2×C4×Q8C2314yes—
155C2×C4×Q8C2314yes—
156C2×C4×Q8C2314yes—
157C2×C4×Q8C2312yes—
158C2×C4×Q8C2318yes—
159#400C2318yes—
160#400C2318yes—
161#400C2318yes—
162#400C2318yes—
163#400C2318yes—
164#400C2318yes—
165#400C2318yes—
166#400C2318yes—
167#400C2318yes—
168#400C2318yes—
169#400C2318yes—
170#400C2318yes—
171#400C2318yes—
172#398C2318yes—
173#398C2318yes—
174#398C2318yes—
175#398C2318yes—
176#398C2318yes—
177#398C2318yes—
178#398C2318yes—
179#398C2318yes—
180#398C2318yes—
181#398C2318yes—
182#398C2318yes—
183#398C2318yes—
184#398C2318yes—
185#398C2318yes—
186#398C2318yes—
187#398C2318yes—
188#398C2318yes—
189#398C2318yes—
190#398C2318yes—
191#398C2318yes—
192#398C2318yes—
193#395C2318yes—
194#395C2318yes—
195#395C2318yes—
196#395C2318yes—
197#395C2318yes—
198#395C2318yes—
199#395C2318yes—
200#395C2318yes—
201#395C2318yes—
202#395C2318yes—
203#395C2318yes—
204#391C2314yes—
205#391C2314yes—
206#391C2318yes—
207#391C2314yes—
208#391C2314yes—
209#391C2318yes—
210#391C2318yes—
211#391C2314yes—
212#391C2314yes—
213#391C2314yes—
214#391C2318yes—
215#391C2314yes—
216#391C2318yes—
217#391C2318yes—
218#391C2314yes—
219#391C2314yes—
220#391C2314yes—
221#389C2314yes—
222#389C2314yes—
223#389C2314yes—
224#389C2314yes—
225#389C2314yes—
226#389C2314yes—
227#389C2312yes—
228#389C2314yes—
229#389C2314yes—
230#389C2314yes—
231#389C2312yes—
232#389C2312yes—
233#389C2318yes—
234#389C2318yes—
235#389C2314yes—
236#389C2314yes—
237#389C2318yes—
238#389C2318yes—
239#389C2318yes—
240#389C2318yes—
241#389C2318yes—
242#389C2314yes—
243#389C2318yes—
244#389C2318yes—
245#389C2318yes—
246#389C2318yes—
247#389C2318yes—
248#389C2318yes—
249#389C2318yes—
250#389C2318yes—
251#389C2318yes—
252#389C2318yes—
253#389C2318yes—
254#389C2318yes—
255#389C2318yes—
256#389C2318yes—
257#389C2318yes—
258#389C2314yes—
259#389C2318yes—
260#389C2318yes—
261#389C2314yes—
262#389C2314yes—
263#389C2314yes—
264#389C2314yes—
265#389C2314yes—
266#389C2314yes—
267#389C2314yes—
268#387C2314yes—
269#387C2314yes—
270#387C2314yes—
271#387C2314yes—
272#387C2314yes—
273#387C2314yes—
274#387C2314yes—
275#387C2318yes—
276#387C2314yes—
277#387C2314yes—
278#387C2318yes—
279#387C2318yes—
280#387C2314yes—
281#387C2314yes—
282#387C2314yes—
283#387C2314yes—
284#387C2314yes—
285#387C2314yes—
286#387C2318yes—
287#387C2318yes—
288#387C2318yes—
289#387C2318yes—
290#387C2318yes—
291#387C2318yes—
292#387C2318yes—
293#387C2318yes—
294#387C2318yes—
295#387C2318yes—
296#387C2318yes—
297#387C2318yes—
298#387C2318yes—
299#387C2318yes—
300#387C2318yes—
301#387C2318yes—
302#387C2318yes—
303#387C2318yes—
304#387C2318yes—
305#387C2318yes—
306#387C2318yes—
307#387C2318yes—
308#387C2318yes—
309#387C2318yes—
310#387C2318yes—
311#387C2318yes—
312#387C2318yes—
313#387C2318yes—
314#387C2318yes—
315#387C2318yes—
316#387C2318yes—
317#387C2318yes—
318#387C2318yes—
319#384C2314yes—
320#384C2314yes—
321#384C2314yes—
322#384C2312yes—
323#384C2318yes—
324#384C2318yes—
325#384C2314yes—
326#383C2318yes—
327#382C2314yes—
328#382C2314yes—
329#382C2312yes—
330#382C2312yes—
331#382C2314yes—
332#382C2311yescharacteristic
333#382C2314yes—
334#382C2318yes—
335#382C2318yes—
336#382C2314yes—
337C4×C4⋊C4C2314yes—
338C4×C4⋊C4C2314yes—
339C4×C4⋊C4C2312yes—
340C4×C4⋊C4C2312yes—
341C4×C4⋊C4C2314yes—
342C4×C4⋊C4C2314yes—
343C4×C4⋊C4C2314yes—
344C4×C4⋊C4C2314yes—
345C4×C4⋊C4C2312yes—
346C4×C4⋊C4C2318yes—
347C4×C4⋊C4C2314yes—
348C4×C4⋊C4C2318yes—
349C4×C4⋊C4C2314yes—
350C4×C4⋊C4C2318yes—
351C4×C4⋊C4C2314yes—
352C4×C4⋊C4C2318yes—
353C4×C4⋊C4C2318yes—
354C4×C4⋊C4C2318yes—
355C4×C4⋊C4C2318yes—
356C4×C4⋊C4C2318yes—
357C4×C4⋊C4C2314yes—
358C4×C4⋊C4C2314yes—
359C4×C4⋊C4C2318yes—
360C4×C4⋊C4C2318yes—
361C4×C4⋊C4C2312yes—
362C4×C4⋊C4C2318yes—
363C4×C4⋊C4C2311yescharacteristic
364C4×C4⋊C4C2312yes—
365C4×C4⋊C4C2314yes—
366C43C2314yes—
367C43C2311yescharacteristic
368C43C2312yes—
369C22×Q8C2414yes—
370C22×Q8C2411yescharacteristic
371C22×Q8C2418yes—
372C22×Q8C2414yes—
373C4⋊Q8—216no—
374C4⋊Q8—48no—
375C4⋊Q8—44no—
376C4⋊Q8—48no—
377C4⋊Q8—44no—
378C4⋊Q8—88no—
379#125—88no—
380#125—84no—
381#125—82no—
382#125C2×Q8116yes—
383#125C2×Q818yes—
384#125C2×Q8132yes—
385#125C2×Q8132yes—
386#125C2×Q8132yes—
387C4×Q8—24no—
388C4×Q8—48no—
389C4×Q8—48no—
390C4×Q8—48no—
391C4×Q8—48no—
392C4×Q8—44no—
393C4×Q8—88no—
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2414yes—
396C2×C4⋊C4C2414yes—
397C2×C4⋊C4C2411yescharacteristic
398C2×C4⋊C4C2412yes—
399C2×C4⋊C4C2418yes—
400C2×C4⋊C4C2418yes—
401C2×C4⋊C4C2418yes—
402C2×C4⋊C4C2418yes—
403C2×C4⋊C4C2418yes—
404C2×C4⋊C4C2414yes—
405C2×C4⋊C4C2418yes—
406C2×C4⋊C4C2418yes—
407C2×C4⋊C4C2418yes—
408C2×C4⋊C4C2418yes—
409C2×C4⋊C4C2414yes—
410C2×C4⋊C4C2412yes—
411C2×C4⋊C4C2414yes—
412C2×C4⋊C4C2418yes—
413C2×C4⋊C4C2418yes—
414C2×C4⋊C4C2418yes—
415C2×C4⋊C4C2418yes—
416C2×C4⋊C4C2414yes—
417C2×C4⋊C4C2414yes—
418C2×C4⋊C4C2414yes—
419C2×C4⋊C4C2418yes—
420C2×C4⋊C4C2418yes—
421C2×C4⋊C4C2418yes—
422C2×C4⋊C4C2414yes—
423C2×C4⋊C4C2418yes—
424C2×C4⋊C4C2418yes—
425C2×C4⋊C4C2414yes—
426C2×C4⋊C4C2414yes—
427C2×C4⋊C4C2414yes—
428C2×C4⋊C4C2414yes—
429C2×C4⋊C4C2412yes—
430C2×C4⋊C4C2412yes—
431C2×C4⋊C4C2418yes—
432C2×C4⋊C4C2418yes—
433C2×C4⋊C4C2414yes—
434C2×C4⋊C4C2418yes—
435C2×C4⋊C4C2414yes—
436C2×C4⋊C4C2412yes—
437C2×C4⋊C4C2412yes—
438C2×C4⋊C4C2414yes—
439C2×C4⋊C4C2414yes—
440C2×C4⋊C4C2412yes—
441C2×C4⋊C4C2414yes—
442C2×C4⋊C4C2414yes—
443C2×C4⋊C4C2414yes—
444C2×C42C2414yes—
445C2×C42C2414yes—
446C2×C42C2412yes—
447C2×C42C2411yescharacteristic
448C2×C42C2412yes—
449C2×C42C2414yes—
450C2×C42C2414yes—
451C2×C42C2418yes—
452C2×C42C2418yes—
453C2×C42C2414yes—
454C2×C42C2418yes—
455C2×C42C2418yes—
456C2×C42C2418yes—
457C2×C42C2418yes—
458C2×C42C2412yes—
459C2×C42C2411yescharacteristic
460C2×C42C2414yes—
461C2×C42C2412yes—
462C2×C42C2414yes—
463C2×C42C2414yes—
464C2×C42C2411yescharacteristic
465C2×C42C2414yes—
466#95C2412yes—
467#95C2418yes—
468#95C2418yes—
469#95C2418yes—
470#95C2414yes—
471#95C2418yes—
472#95C2418yes—
473#95C2414yes—
474#95C2414yes—
475#95C2414yes—
476#95C2414yes—
477#95C2414yes—
478#95C2414yes—
479#95C2414yes—
480#95C2414yes—
481#95C2414yes—
482#95C2414yes—
483#95C2418yes—
484#95C2418yes—
485#95C2414yes—
486#95C2414yes—
487#95C2418yes—
488#95C2414yes—
489#95C2414yes—
490#95C2414yes—
491#95C2418yes—
492#95C2418yes—
493#95C2418yes—
494#95C2418yes—
495#95C2418yes—
496#95C2418yes—
497#95C2418yes—
498#95C2418yes—
499#95C2418yes—
500#95C2418yes—
501#95C2418yes—
502#95C2418yes—
503#95C2418yes—
504#95C2418yes—
505#95C2418yes—
506#95C2418yes—
507#95C2418yes—
508#95C2418yes—
509C2×Q8—28no—
510C2×Q8—24no—
511C2×Q8—28no—
512C2×Q8—28no—
513C2×Q8—48no—
514C2×Q8—48no—
515C2×Q8—48no—
516C2×Q8—44no—
517C2×Q8—44no—
518C2×Q8—44no—
519C2×Q82−1+414yes—
520C22×C4C2511yescharacteristic
521C22×C4C2512yes—
522C22×C4C2514yes—
523C22×C4C2518yes—
524C22×C4C2512yes—
525C22×C4C2514yes—
526C22×C4C2518yes—
527C22×C4C2518yes—
528C22×C4C2514yes—
529C22×C4C2514yes—
530C22×C4C2511yescharacteristic
531C22×C4C2514yes—
532C22×C4C2514yes—
533C22×C4C2514yes—
534C22×C4C2512yes—
535C22×C4C2512yes—
536C22×C4C2511yescharacteristic
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—216no—
540C4⋊C4—216no—
541C4⋊C4—24no—
542C4⋊C4—28no—
543C4⋊C4—28no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—48no—
551C4⋊C4—48no—
552C4⋊C4—48no—
553C4⋊C4—48no—
554C4⋊C4—44no—
555C4⋊C4—48no—
556C4⋊C4—48no—
557C4⋊C4—44no—
558C4⋊C4—48no—
559C4⋊C4—48no—
560C4⋊C4—48no—
561C4⋊C4—48no—
562C4⋊C4—48no—
563C4⋊C4—48no—
564C4⋊C4—44no—
565C4⋊C4—42no—
566C4⋊C4—44no—
567C4⋊C4—48no—
568C4⋊C4—48no—
569C4⋊C4—48no—
570C4⋊C4—48no—
571C4⋊C4—48no—
572C4⋊C4—48no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—48no—
576C4⋊C4—44no—
577C4⋊C4—44no—
578C4⋊C4C22×Q8116yes—
579C4⋊C4C22×Q818yes—
580C4⋊C4C22×Q8116yes—
581C4⋊C4C22×Q818yes—
582C4⋊C4C22×Q8116yes—
583C4⋊C4C22×Q8116yes—
584C4⋊C4C22×Q818yes—
585C4⋊C4C22×Q8116yes—
586C4⋊C4C22×Q8116yes—
587C42—24no—
588C42—22no—
589C42—28no—
590C42—28no—
591C42—28no—
592C42—24no—
593C42—44no—
594C42—42no—
595C42—44no—
596C42—44no—
597C42—48no—
598C42—48no—
599C42—48no—
600C42—48no—
601C42—44no—
602C42—44no—
603C42—48no—
604C42—48no—
605C42—42no—
606C42—41no—
607C42C22×Q814yes—
608C42C22×Q8116yes—
609C23C2611yescharacteristic
610Q8—28no—
611Q8—416no—
612Q8—816no—
613Q8—164no—
614C2×C4—21no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—24no—
628C2×C4—28no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—24no—
638C2×C4—28no—
639C2×C4—24no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—28no—
652C2×C4—28no—
653C2×C4C23×Q812yes—
654C2×C4C23×Q818yes—
655C2×C4C23×Q818yes—
656C2×C4C23×Q818yes—
657C2×C4C23×Q814yes—
658C2×C4C2×2−1+414yes—
659C2×C4C2×2−1+412yes—
660C2×C4#58514yes—
661C22C24×Q811yescharacteristic
662C22C22×2−1+411yescharacteristic
663C22C2×#58512yes—
664C22H(7,𝔽2)11yescharacteristic
665C222−1+612yes—
666C4—22no—
667C4—28no—
668C4—28no—
669C4—28no—
670C4—24no—
671C4—24no—
672C4—48no—
673C4—44no—
674C4—44no—
675C4—48no—
676C4—44no—
677C4—48no—
678C4—42no—
679C4—42no—
680C4—41no—
681C4—44no—
682C4#355314yes—
683C2C2×#355311yescharacteristic
684C2Q8×2−1+411yescharacteristic
685C2#6300412yes—
686C2#6305612yes—
687C2#6305711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
137. #564Q8semidirect

Maximal subgroups

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

subgroupindex
#596812
#594592
#594452
#510822
#509102
#509082
#462432
#460092
#460072
#460062
#396552
#396532
#365562
Q8×#1252
#345262
C4×#5642