Finite Group Databasebeta

← #7633296

Subgroups — #7633296

Order 512.

4768 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C214yesmaximal, complemented
2#36286C214yesmaximal, complemented
3#36279C218yesmaximal
4#36035C218yesmaximal
5#35958C218yesmaximal, complemented
6#35662C212yesmaximal
7#35643C218yesmaximal, complemented
8#35627C212yesmaximal
9#35281C211yesmaximal, characteristic
10#35199C214yesmaximal, complemented
11#35179C211yesmaximal, characteristic
12#34951C214yesmaximal, complemented
13C2×#2568C211yesmaximal, characteristic
14#34938C211yesmaximal, characteristic
15#34254C212yesmaximal
16#34216C211yesmaximal, characteristic
17#34215C212yesmaximal
18#34145C211yesmaximal, characteristic
19C4×#519C211yesmaximal, characteristic
20C2×#533C2212yes—
21C2×#533C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26#2868C2212yes—
27#2863C2218yes—
28#2862C2218yes—
29#2862C2212yes—
30#2855C2214yes—
31#2666C22116yes—
32#2660C2214yes—
33#2627C2214yes—
34#2612C22116yes—
35#2605C22116yes—
36#2605C22116yes—
37#2588C22116yes—
38#2586C22116yes—
39#2585C22116yes—
40#2582C2218yes—
41#2582C2214yes—
42#2581C2214yes—
43#2581C2218yes—
44#2571C2218yes—
45#2570C2218yes—
46#2570C22116yes—
47#2568C2214yes—
48#2568C2212yes—
49#2568C2214yes—
50#2568C2212yes—
51#2568C2212yes—
52#2568C2214yes—
53#2567C2214yes—
54#2567C2212yes—
55#2565C2214yes—
56#2552C2212yes—
57#2551C2212yes—
58#2550C2214yes—
59#2549C2214yes—
60#2546C2214yes—
61#2520C22116yes—
62#2520C22116yes—
63#2511C22116yes—
64#2511C2218yes—
65#2509C2218yes—
66#2508C2218yes—
67#2505C22116yes—
68#2502C2214yes—
69#2453C2218yes—
70#2451C2218yes—
71#2451C2218yes—
72#2446C2218yes—
73#2446C2218yes—
74#2440C2214yes—
75#2438C22116yes—
76#2434C2214yes—
77#2402C22116yes—
78#2399C2214yes—
79C2×#400C2214yes—
80#2370C2214yes—
81#2369C2212yes—
82#2369C2214yes—
83#2368C2214yes—
84#2368C22116yes—
85#2365C2212yes—
86#2365C2214yes—
87#2360C2214yes—
88#2356C2218yes—
89#2355C2218yes—
90#2355C2218yes—
91#2355C2218yes—
92#2345C2214yes—
93#2343C22116yes—
94Q8×C22⋊C4C2214yes—
95#2331C2214yes—
96#2331C2218yes—
97#2320C2218yes—
98#2317C2212yes—
99#2317C2212yes—
100#2317C2212yes—
101#2316C2212yes—
102#2310C2212yes—
103C4×C4⋊Q8C2214yes—
104C4×C4⋊Q8C2214yes—
105C4×C4⋊Q8C2218yes—
106C4×#125C2218yes—
107#2296C2212yes—
108#2296C2212yes—
109#2295C2214yes—
110#2295C2218yes—
111#2295C2212yes—
112#2295C2218yes—
113C2×#391C2212yes—
114C2×#391C2214yes—
115#2285C2211yescharacteristic
116C2×#382C2212yes—
117#2276C2214yes—
118#2276C2214yes—
119#2274C2211yescharacteristic
120#2274C2211yescharacteristic
121#2274C2212yes—
122#2274C2214yes—
123C42×Q8C2212yes—
124C42×Q8C2214yes—
125C42×Q8C2214yes—
126C4×#117C2212yes—
127C4×#117C2211yescharacteristic
128C4×#117C2211yescharacteristic
129C4×#117C2214yes—
130C2×C4×C4⋊C4C2211yescharacteristic
131C23×Q8C2311yescharacteristic
132C23×Q8C2312yes—
133#533—216no—
134#533—28no—
135C2×C4⋊Q8—28no—
136C2×C4⋊Q8C2312yes—
137C2×C4⋊Q8C2312yes—
138C2×C4⋊Q8C2311yescharacteristic
139C2×C4⋊Q8C2312yes—
140C2×C4⋊Q8C2311yescharacteristic
141C2×C4⋊Q8C2312yes—
142C2×C4⋊Q8C2314yes—
143C2×C4⋊Q8C2314yes—
144C2×C4⋊Q8C2318yes—
145C2×#125C2314yes—
146C2×#125C2312yes—
147C2×C22⋊Q8C2312yes—
148C2×C22⋊Q8C2312yes—
149C2×C22⋊Q8C2318yes—
150#519—28no—
151#519Q8116yescomplemented
152C2×C4×Q8—28no—
153C2×C4×Q8C2318yes—
154C2×C4×Q8C2312yes—
155C2×C4×Q8C2312yes—
156C2×C4×Q8C2312yes—
157C2×C4×Q8C2312yes—
158C2×C4×Q8C2312yes—
159C2×C4×Q8C2314yes—
160C2×C4×Q8C2312yes—
161C2×C4×Q8C2314yes—
162C2×C4×Q8C2318yes—
163C2×C4×Q8C2314yes—
164C2×C4×Q8C2314yes—
165C2×C4×Q8C2314yes—
166C2×#117C2312yes—
167C2×#117C2312yes—
168C2×#117C2311yescharacteristic
169C2×#117C2314yes—
170C2×#117C2311yescharacteristic
171C2×#117C2314yes—
172C2×#117C2311yescharacteristic
173C2×#117C2311yescharacteristic
174C22×C4⋊C4C2312yes—
175C22×C4⋊C4C2311yescharacteristic
176C22×C4⋊C4C2312yes—
177C22×C4⋊C4C2312yes—
178C22×C4⋊C4C2312yes—
179C22×C42C2312yes—
180C22×C42C2312yes—
181C22×C42C2311yescharacteristic
182#400C23116yes—
183#400C2318yes—
184#400C2318yes—
185#400C2318yes—
186#400C23116yes—
187#400C23116yes—
188#400C23116yes—
189#400C23116yes—
190#398C23116yes—
191#398C23116yes—
192#398C23116yes—
193#398C23116yes—
194#398C2318yes—
195#398C23116yes—
196#398C2318yes—
197#395C2318yes—
198#395C23116yes—
199#395C23116yes—
200#395C2318yes—
201#395C2318yes—
202#391C2318yes—
203#391C2318yes—
204#391C2318yes—
205#391C2318yes—
206#391C2314yes—
207#391C2314yes—
208#391C2318yes—
209#391C2318yes—
210#391C2314yes—
211#391C2318yes—
212#391C23116yes—
213#391C2318yes—
214#391C2314yes—
215#391C23116yes—
216#391C23116yes—
217#391C23116yes—
218#391C23116yes—
219#391C2318yes—
220#391C2314yes—
221#391C2318yes—
222#389C2318yes—
223#389C2318yes—
224#389C2318yes—
225#389C23116yes—
226#389C2318yes—
227#389C23116yes—
228#389C2318yes—
229#389C2314yes—
230#389C2314yes—
231#389C2318yes—
232#389C2318yes—
233#389C2314yes—
234#389C2318yes—
235#389C2314yes—
236#389C2314yes—
237#389C2318yes—
238#389C2314yes—
239#389C2314yes—
240#387C23116yes—
241#387C23116yes—
242#387C23116yes—
243#387C2318yes—
244#387C23116yes—
245#387C23116yes—
246#387C23116yes—
247#387C23116yes—
248#387C23116yes—
249#387C23116yes—
250#387C23116yes—
251#387C23116yes—
252#387C23116yes—
253#387C23116yes—
254#387C23116yes—
255#387C2318yes—
256#387C2318yes—
257#385C2318yes—
258#385C2318yes—
259#385C2318yes—
260#385C2318yes—
261#385C2318yes—
262#384C2314yes—
263#384C2312yes—
264#384C2314yes—
265#384C2318yes—
266#384C2314yes—
267#384C2314yes—
268#383C23116yes—
269#382C2314yes—
270#382C2314yes—
271#382C2318yes—
272#382C2314yes—
273#382C2314yes—
274#382C23116yes—
275#382C23116yes—
276#382C2314yes—
277#382C2312yes—
278#381C2318yes—
279#380C2318yes—
280#380C2312yes—
281#380C2312yes—
282C4×C4⋊C4C2314yes—
283C4×C4⋊C4C2318yes—
284C4×C4⋊C4C23116yes—
285C4×C4⋊C4C23116yes—
286C4×C4⋊C4C2312yes—
287C4×C4⋊C4C2312yes—
288C4×C4⋊C4C2318yes—
289C4×C4⋊C4C23116yes—
290C4×C4⋊C4C2318yes—
291C4×C4⋊C4C2312yes—
292C4×C4⋊C4C2312yes—
293C4×C4⋊C4C2312yes—
294C4×C4⋊C4C2311yescharacteristic
295C4×C4⋊C4C2312yes—
296C4×C4⋊C4C2311yescharacteristic
297C4×C4⋊C4C2314yes—
298C4×C4⋊C4C2318yes—
299C4×C4⋊C4C2318yes—
300C4×C4⋊C4C2318yes—
301C4×C4⋊C4C2318yes—
302C4×C4⋊C4C2318yes—
303C4×C4⋊C4C2314yes—
304C4×C4⋊C4C2314yes—
305C4×C4⋊C4C2318yes—
306C4×C4⋊C4C2314yes—
307C4×C4⋊C4C2314yes—
308C4×C4⋊C4C2312yes—
309C4×C4⋊C4C2312yes—
310C4×C4⋊C4C2318yes—
311C4×C4⋊C4C2318yes—
312C4×C4⋊C4C2314yes—
313C4×C4⋊C4C2314yes—
314C4×C4⋊C4C2314yes—
315C4×C4⋊C4C2312yes—
316C4×C22⋊C4C2312yes—
317C4×C22⋊C4C2318yes—
318C4×C22⋊C4C2318yes—
319C4×C22⋊C4C2312yes—
320C4×C22⋊C4C2314yes—
321C4×C22⋊C4C2312yes—
322C4×C22⋊C4C2312yes—
323C4×C22⋊C4C2314yes—
324#376C2314yes—
325#376C2312yes—
326#376C23116yes—
327#376C2318yes—
328#376C2312yes—
329C2×#95C2318yes—
330C2×#95C2312yes—
331C2×#95C2318yes—
332C43C2314yes—
333C43C2312yes—
334C43C2312yes—
335C43C2318yes—
336C22×Q8—24no—
337C22×Q8—24no—
338C22×Q8—28no—
339C22×Q8—48no—
340C22×Q8C2×Q818yes—
341C22×Q8C2412yes—
342C22×Q8C2414yes—
343C22×Q8C2412yes—
344C22×Q8C2412yes—
345C22×Q8C2418yes—
346C22×Q8C2412yes—
347C22×Q8C2414yes—
348C23×C4C2412yes—
349C23×C4C2411yescharacteristic
350C23×C4C2414yes—
351C23×C4C2412yes—
352C23×C4C2414yes—
353C23×C4C2411yescharacteristic
354C23×C4C2411yescharacteristic
355C4⋊Q8—216no—
356C4⋊Q8—28no—
357C4⋊Q8—28no—
358C4⋊Q8—28no—
359C4⋊Q8—28no—
360C4⋊Q8—24no—
361C4⋊Q8—24no—
362C4⋊Q8—48no—
363C4⋊Q8—416no—
364#125—216no—
365#125—28no—
366C22⋊Q8—28no—
367C22⋊Q8—28no—
368C22⋊Q8—232no—
369C4×Q8—216no—
370C4×Q8—28no—
371C4×Q8—28no—
372C4×Q8—28no—
373C4×Q8—28no—
374C4×Q8—216no—
375C4×Q8—232no—
376C4×Q8—28no—
377C4×Q8—28no—
378C4×Q8—48no—
379C4×Q8—416no—
380C4×Q8C2×Q8132yescomplemented
381C4×Q8C2×Q8132yescomplemented
382#117—24no—
383#117—28no—
384#117—28no—
385#117—216no—
386#117C2×Q8132yes—
387#117C2×Q818yes—
388#117C2×Q818yes—
389#117#4118yes—
390C2×C4⋊C4—28no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—28no—
393C2×C4⋊C4—28no—
394C2×C4⋊C4—44no—
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2412yes—
398C2×C4⋊C4C2414yes—
399C2×C4⋊C4C2414yes—
400C2×C4⋊C4C2414yes—
401C2×C4⋊C4C2414yes—
402C2×C4⋊C4C2412yes—
403C2×C4⋊C4C2414yes—
404C2×C4⋊C4C2414yes—
405C2×C4⋊C4C2412yes—
406C2×C4⋊C4C2418yes—
407C2×C4⋊C4C2412yes—
408C2×C4⋊C4C2412yes—
409C2×C4⋊C4C2414yes—
410C2×C4⋊C4C2414yes—
411C2×C4⋊C4C2418yes—
412C2×C4⋊C4C2418yes—
413C2×C4⋊C4C2414yes—
414C2×C4⋊C4C2414yes—
415C2×C4⋊C4C2412yes—
416C2×C4⋊C4C2412yes—
417C2×C4⋊C4C2418yes—
418C2×C4⋊C4C2414yes—
419C2×C4⋊C4C2418yes—
420C2×C4⋊C4C24116yes—
421C2×C4⋊C4C2418yes—
422C2×C4⋊C4C24116yes—
423C2×C4⋊C4C24116yes—
424C2×C4⋊C4C2414yes—
425C2×C4⋊C4C2418yes—
426C2×C4⋊C4C24116yes—
427C2×C4⋊C4C2418yes—
428C2×C4⋊C4C2412yes—
429C2×C4⋊C4C2418yes—
430C2×C4⋊C4C2412yes—
431C2×C4⋊C4C2412yes—
432C2×C22⋊C4C2412yes—
433C2×C22⋊C4C2414yes—
434C2×C22⋊C4C2414yes—
435C2×C22⋊C4C2412yes—
436C2×C22⋊C4C2412yes—
437C2×C22⋊C4C2412yes—
438C2×C42—28no—
439C2×C42—28no—
440C2×C42#4118yes—
441C2×C42C2412yes—
442C2×C42C2418yes—
443C2×C42C2414yes—
444C2×C42C2414yes—
445C2×C42C2414yes—
446C2×C42C2414yes—
447C2×C42C2414yes—
448C2×C42C2414yes—
449C2×C42C2412yes—
450C2×C42C2414yes—
451C2×C42C2412yes—
452C2×C42C2418yes—
453C2×C42C24116yes—
454C2×C42C2418yes—
455C2×C42C2414yes—
456C2×C42C2418yes—
457C2×C42C2418yes—
458C2×C42C2414yes—
459C2×C42C2412yes—
460C2×C42C2418yes—
461C2×C42C2412yes—
462C2×C42C2412yes—
463C2×C42C2411yescharacteristic
464C2×C42C2411yescharacteristic
465C2×C42C2411yescharacteristic
466C2×C42C2411yescharacteristic
467#95C24116yes—
468#95C2412yes—
469#95C24116yes—
470#95C2412yes—
471#95C2414yes—
472#95C24116yes—
473#95C24116yes—
474#95C2414yes—
475#95C2414yes—
476#95C2418yes—
477#95C2418yes—
478#95C2414yes—
479#95C2414yes—
480#95C2418yes—
481#95C2418yes—
482#95C2418yes—
483#95C24116yes—
484#95C24116yes—
485#95C24116yes—
486#95C2418yes—
487#95C2414yes—
488#95C2414yes—
489#95C24116yes—
490#95C24116yes—
491#95C24116yes—
492#95C2418yes—
493#95C24116yes—
494#95C2418yes—
495C24C2511yescharacteristic
496C2×Q8—232no—
497C2×Q8—24no—
498C2×Q8—24no—
499C2×Q8—24no—
500C2×Q8—24no—
501C2×Q8—24no—
502C2×Q8—28no—
503C2×Q8—216no—
504C2×Q8—24no—
505C2×Q8—28no—
506C2×Q8—28no—
507C2×Q8—216no—
508C2×Q8—48no—
509C2×Q8—48no—
510C2×Q8—48no—
511C2×Q8—44no—
512C2×Q8—816no—
513C2×Q8C22×Q818yes—
514C2×Q8C22×Q818yes—
515C22×C4—24no—
516C22×C4—24no—
517C22×C4—24no—
518C22×C4—24no—
519C22×C4—28no—
520C22×C4—28no—
521C22×C4—42no—
522C22×C4—44no—
523C22×C4—44no—
524C22×C4—44no—
525C22×C4—48no—
526C22×C4C22×Q818yes—
527C22×C4C22×Q814yes—
528C22×C4C2×#4114yes—
529C22×C4C2×#4114yes—
530C22×C4C2×#4114yes—
531C22×C4C2×#4114yes—
532C22×C4C2514yes—
533C22×C4C2514yes—
534C22×C4C2514yes—
535C22×C4C2518yes—
536C22×C4C2514yes—
537C22×C4C2518yes—
538C22×C4C2514yes—
539C22×C4C2514yes—
540C22×C4C2514yes—
541C22×C4C2514yes—
542C22×C4C2514yes—
543C22×C4C2511yescharacteristic
544C22×C4C2511yescharacteristic
545C22×C4C2514yes—
546C22×C4C2511yescharacteristic
547C22×C4C2511yescharacteristic
548C22×C4C2511yescharacteristic
549C22×C4C2511yescharacteristic
550C4⋊C4—28no—
551C4⋊C4—28no—
552C4⋊C4—24no—
553C4⋊C4—24no—
554C4⋊C4—28no—
555C4⋊C4—28no—
556C4⋊C4—28no—
557C4⋊C4—28no—
558C4⋊C4—216no—
559C4⋊C4—24no—
560C4⋊C4—28no—
561C4⋊C4—24no—
562C4⋊C4—24no—
563C4⋊C4—216no—
564C4⋊C4—216no—
565C4⋊C4—28no—
566C4⋊C4—28no—
567C4⋊C4—216no—
568C4⋊C4—24no—
569C4⋊C4—24no—
570C4⋊C4—24no—
571C4⋊C4—24no—
572C4⋊C4—28no—
573C4⋊C4—44no—
574C4⋊C4—48no—
575C4⋊C4—44no—
576C4⋊C4—48no—
577C4⋊C4—416no—
578C4⋊C4—48no—
579C4⋊C4—416no—
580C4⋊C4—48no—
581C4⋊C4—416no—
582C4⋊C4—416no—
583C4⋊C4C22×Q8132yes—
584C4⋊C4C22×Q818yes—
585C4⋊C4C22×Q818yes—
586C4⋊C4C2×#4118yes—
587C22⋊C4—24no—
588C22⋊C4—28no—
589C22⋊C4—28no—
590C22⋊C4C22×Q818yes—
591C22⋊C4C22×Q818yes—
592C22⋊C4C2×#4118yes—
593C42—28no—
594C42—28no—
595C42—28no—
596C42—24no—
597C42—24no—
598C42—216no—
599C42—24no—
600C42—28no—
601C42—28no—
602C42—28no—
603C42—28no—
604C42—48no—
605C42—44no—
606C42—48no—
607C42—44no—
608C42—48no—
609C42—48no—
610C42—416no—
611C42C22×Q8132yes—
612C42C22×Q818yes—
613C42C22×Q818yes—
614C42C2×#4114yes—
615C42C2×#4114yes—
616C42C2×#4114yes—
617C42C2×#4114yes—
618C42C2×#4118yes—
619C23—22no—
620C23—22no—
621C23C23×Q812yes—
622C23C22×#4112yes—
623C23C22×#4112yes—
624C23C2611yescharacteristic
625Q8—216no—
626Q8—216no—
627Q8—48no—
628Q8—416no—
629Q8—816no—
630Q8—84no—
631Q8—88no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—216no—
644C2×C4—28no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—28no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—28no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—28no—
656C2×C4—28no—
657C2×C4—216no—
658C2×C4—24no—
659C2×C4—24no—
660C2×C4—24no—
661C2×C4—28no—
662C2×C4—28no—
663C2×C4—28no—
664C2×C4—24no—
665C2×C4—44no—
666C2×C4—88no—
667C2×C4—84no—
668C2×C4—88no—
669C2×C4#533116yes—
670C2×C4C23×Q818yes—
671C2×C4C23×Q818yes—
672C2×C4C23×Q818yes—
673C2×C4C23×Q812yes—
674C2×C4C23×Q812yes—
675C2×C4C22×#4112yes—
676C2×C4C22×#4118yes—
677C2×C4C22×#4112yes—
678C2×C4C22×#4112yes—
679C2×C4C22×#4112yes—
680C2×C4C22×#4112yes—
681C2×C4C22×#4112yes—
682C2×C4C22×#4112yes—
683C2×C4C22×#4112yes—
684C2×C4#58514yes—
685C22—24no—
686C22—24no—
687C22—22no—
688C22—22no—
689C22C2×#53314yes—
690C22C24×Q811yescharacteristic
691C22C23×#4111yescharacteristic
692C22C23×#4111yescharacteristic
693C22C2×#58512yes—
694C222−1+612yes—
695C4—28no—
696C4—28no—
697C4—28no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—28no—
702C4—44no—
703C4—44no—
704C4—44no—
705C4—48no—
706C4—48no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4C2×#53314yes—
711C4C2×#53314yes—
712C4#351014yes—
713C2—24no—
714C2C22×#53311yescharacteristic
715C2C2×#351011yescharacteristic
716C2#6286011yescharacteristic
717C2#6300412yes—
718C2#6300912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC24C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
2. #36286C2semidirect
5. #35958C2semidirect
7. #35643C2semidirect
10. #35199C2semidirect
12. #34951C2semidirect
151. #519Q8semidirect
380. C4×Q8C2×Q8semidirect
381. C4×Q8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
#362862
#362792
#360352
#359582
#356622
#356432
#356272
#352812
#351992
#351792
#349512
C2×#25682
#349382
#342542
#342162
#342152
#341452
C4×#5192