Finite Group Databasebeta

← #7633248

Subgroups — #7633248

Order 512.

4908 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C214yesmaximal, complemented
2#36286C214yesmaximal, complemented
3#36280C218yesmaximal
4#35936C218yesmaximal, complemented
5#35743C218yesmaximal
6#35662C212yesmaximal
7#35583C218yesmaximal, complemented
8#35555C212yesmaximal
9#35281C211yesmaximal, characteristic
10#35208C214yesmaximal, complemented
11#35179C211yesmaximal, characteristic
12#34950C214yesmaximal, complemented
13C2×#2567C211yesmaximal, characteristic
14#34938C211yesmaximal, characteristic
15#34254C212yesmaximal
16#34216C211yesmaximal, characteristic
17#34214C212yesmaximal
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#2862C2212yes—
29#2859C2218yes—
30#2855C2214yes—
31#2666C22116yes—
32#2660C2214yes—
33#2627C2214yes—
34#2612C22116yes—
35#2606C22116yes—
36#2605C22116yes—
37#2588C22116yes—
38#2586C22116yes—
39#2585C22116yes—
40#2582C2218yes—
41#2582C2214yes—
42#2581C2218yes—
43#2581C2214yes—
44#2570C2218yes—
45#2568C2212yes—
46#2567C2214yes—
47#2567C2212yes—
48#2567C2212yes—
49#2567C2212yes—
50#2567C2214yes—
51#2567C2214yes—
52#2567C2214yes—
53#2565C2214yes—
54#2552C2218yes—
55#2552C2212yes—
56#2551C2212yes—
57#2550C2214yes—
58#2550C22116yes—
59#2549C2214yes—
60#2546C2214yes—
61#2511C2218yes—
62#2507C2218yes—
63#2507C2218yes—
64#2507C2218yes—
65#2506C2218yes—
66#2502C2214yes—
67#2473C22116yes—
68#2462C22116yes—
69#2462C22116yes—
70#2453C2218yes—
71#2451C2218yes—
72#2451C2218yes—
73#2451C22116yes—
74#2440C2214yes—
75#2438C22116yes—
76#2434C2214yes—
77#2399C2214yes—
78#2399C22116yes—
79C2×#395C2214yes—
80#2370C2214yes—
81#2369C2212yes—
82#2369C2214yes—
83#2368C2214yes—
84#2365C2214yes—
85#2365C2212yes—
86#2360C2214yes—
87#2360C22116yes—
88#2356C2218yes—
89#2355C2218yes—
90#2355C2218yes—
91Q8×C4⋊C4C2218yes—
92#2345C2214yes—
93#2343C22116yes—
94Q8×C22⋊C4C2214yes—
95#2331C2214yes—
96#2330C2218yes—
97#2320C2218yes—
98#2317C2212yes—
99#2317C2212yes—
100#2317C2212yes—
101#2316C2212yes—
102#2310C2212yes—
103C4×C4⋊Q8C2214yes—
104C4×C4⋊Q8C2214yes—
105C4×C4⋊Q8C2218yes—
106C4×C4⋊Q8C2218yes—
107#2296C2212yes—
108#2296C2212yes—
109#2295C2212yes—
110#2295C2214yes—
111#2295C2218yes—
112#2295C2218yes—
113C2×#391C2212yes—
114C2×#389C2214yes—
115#2285C2211yescharacteristic
116C2×#384C2212yes—
117#2276C2214yes—
118#2276C2214yes—
119#2274C2214yes—
120#2274C2211yescharacteristic
121#2274C2212yes—
122#2274C2211yescharacteristic
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⋊Q8C2314yes—
137C2×C4⋊Q8C2312yes—
138C2×C4⋊Q8C2314yes—
139C2×C4⋊Q8C2312yes—
140C2×C4⋊Q8C2312yes—
141C2×C4⋊Q8C2311yescharacteristic
142C2×C4⋊Q8C2312yes—
143C2×C4⋊Q8C2311yescharacteristic
144C2×C4⋊Q8C2318yes—
145C2×#125C2312yes—
146C2×#125C2314yes—
147C2×#125C23116yes—
148C2×C22⋊Q8C2312yes—
149C2×C22⋊Q8C2318yes—
150C2×C22⋊Q8C2312yes—
151#519—28no—
152#519Q8116yescomplemented
153C2×C4×Q8—28no—
154C2×C4×Q8C2318yes—
155C2×C4×Q8C2312yes—
156C2×C4×Q8C2312yes—
157C2×C4×Q8C2312yes—
158C2×C4×Q8C2312yes—
159C2×C4×Q8C2312yes—
160C2×C4×Q8C2314yes—
161C2×C4×Q8C2312yes—
162C2×C4×Q8C2314yes—
163C2×C4×Q8C2314yes—
164C2×C4×Q8C23116yes—
165C2×C4×Q8C2318yes—
166C2×C4×Q8C2314yes—
167C2×C4×Q8C2314yes—
168C2×#117C2312yes—
169C2×#117C2312yes—
170C2×#117C2311yescharacteristic
171C2×#117C2314yes—
172C2×#117C2311yescharacteristic
173C2×#117C2314yes—
174C2×#117C2311yescharacteristic
175C2×#117C2311yescharacteristic
176C22×C4⋊C4C2312yes—
177C22×C4⋊C4C2311yescharacteristic
178C22×C4⋊C4C2312yes—
179C22×C4⋊C4C2312yes—
180C22×C4⋊C4C2312yes—
181C22×C4⋊C4C2318yes—
182C22×C42C2312yes—
183C22×C42C2312yes—
184C22×C42C2311yescharacteristic
185#400C2318yes—
186#400C23116yes—
187#400C23116yes—
188#398C23116yes—
189#398C23116yes—
190#398C23116yes—
191#398C23116yes—
192#398C2318yes—
193#398C23116yes—
194#398C23116yes—
195#398C23116yes—
196#398C2318yes—
197#395C2318yes—
198#395C23116yes—
199#395C2318yes—
200#395C2318yes—
201#395C23116yes—
202#395C23116yes—
203#395C2318yes—
204#395C2318yes—
205#391C23116yes—
206#391C2318yes—
207#391C23116yes—
208#391C2318yes—
209#391C2314yes—
210#391C2314yes—
211#391C2318yes—
212#391C2314yes—
213#391C2318yes—
214#391C2314yes—
215#391C2318yes—
216#391C2318yes—
217#391C23116yes—
218#391C2318yes—
219#391C2314yes—
220#391C2318yes—
221#389C2318yes—
222#389C2318yes—
223#389C2318yes—
224#389C2318yes—
225#389C2318yes—
226#389C23116yes—
227#389C23116yes—
228#389C2318yes—
229#389C2318yes—
230#389C2314yes—
231#389C2314yes—
232#389C23116yes—
233#389C2318yes—
234#389C2318yes—
235#389C2314yes—
236#389C2318yes—
237#389C2314yes—
238#389C2314yes—
239#389C23116yes—
240#389C23116yes—
241#389C23116yes—
242#389C2314yes—
243#389C2318yes—
244#389C2314yes—
245#387C23116yes—
246#387C23116yes—
247#387C23116yes—
248#387C23116yes—
249#387C23116yes—
250#387C23116yes—
251#387C23116yes—
252#387C2318yes—
253#387C23116yes—
254#387C23116yes—
255#387C23116yes—
256#387C23116yes—
257#387C23116yes—
258#387C2318yes—
259#387C2318yes—
260#385C2318yes—
261#385C2318yes—
262#385C2318yes—
263#385C2318yes—
264#385C2318yes—
265#384C2314yes—
266#384C2312yes—
267#384C2318yes—
268#384C2314yes—
269#384C2314yes—
270#384C2318yes—
271#384C2314yes—
272#384C2314yes—
273#384C2314yes—
274#383C23116yes—
275#382C2314yes—
276#382C2314yes—
277#382C2314yes—
278#382C23116yes—
279#382C2312yes—
280#381C2318yes—
281#380C2318yes—
282#380C2312yes—
283#380C2312yes—
284C4×C4⋊C4C2312yes—
285C4×C4⋊C4C2312yes—
286C4×C4⋊C4C2318yes—
287C4×C4⋊C4C2314yes—
288C4×C4⋊C4C2318yes—
289C4×C4⋊C4C2314yes—
290C4×C4⋊C4C2318yes—
291C4×C4⋊C4C23116yes—
292C4×C4⋊C4C23116yes—
293C4×C4⋊C4C2312yes—
294C4×C4⋊C4C2312yes—
295C4×C4⋊C4C2311yescharacteristic
296C4×C4⋊C4C2312yes—
297C4×C4⋊C4C2311yescharacteristic
298C4×C4⋊C4C2312yes—
299C4×C4⋊C4C2314yes—
300C4×C4⋊C4C2318yes—
301C4×C4⋊C4C2318yes—
302C4×C4⋊C4C2318yes—
303C4×C4⋊C4C2318yes—
304C4×C4⋊C4C2318yes—
305C4×C4⋊C4C2314yes—
306C4×C4⋊C4C2314yes—
307C4×C4⋊C4C2318yes—
308C4×C4⋊C4C2314yes—
309C4×C4⋊C4C2312yes—
310C4×C4⋊C4C2312yes—
311C4×C4⋊C4C2318yes—
312C4×C4⋊C4C2314yes—
313C4×C4⋊C4C2318yes—
314C4×C4⋊C4C2314yes—
315C4×C4⋊C4C2314yes—
316C4×C4⋊C4C2312yes—
317C4×C22⋊C4C2312yes—
318C4×C22⋊C4C2318yes—
319C4×C22⋊C4C2318yes—
320C4×C22⋊C4C2312yes—
321C4×C22⋊C4C2314yes—
322C4×C22⋊C4C2312yes—
323C4×C22⋊C4C2312yes—
324C4×C22⋊C4C2314yes—
325#376C2314yes—
326#376C2312yes—
327#376C23116yes—
328#376C2318yes—
329#376C2312yes—
330C2×#95C2318yes—
331C2×#95C2312yes—
332C43C2312yes—
333C43C2312yes—
334C43C2314yes—
335C43C2318yes—
336C22×Q8—24no—
337C22×Q8—24no—
338C22×Q8—28no—
339C22×Q8—84no—
340C22×Q8C2×Q818yes—
341C22×Q8C2412yes—
342C22×Q8C2418yes—
343C22×Q8C2412yes—
344C22×Q8C2412yes—
345C22×Q8C2414yes—
346C22×Q8C2412yes—
347C22×Q8C2414yes—
348C23×C4C2414yes—
349C23×C4C2412yes—
350C23×C4C2411yescharacteristic
351C23×C4C2412yes—
352C23×C4C2414yes—
353C23×C4C2411yescharacteristic
354C23×C4C2411yescharacteristic
355C4⋊Q8—216no—
356C4⋊Q8—216no—
357C4⋊Q8—28no—
358C4⋊Q8—28no—
359C4⋊Q8—28no—
360C4⋊Q8—28no—
361C4⋊Q8—24no—
362C4⋊Q8—24no—
363C4⋊Q8—88no—
364#125—216no—
365#125—28no—
366#125—816no—
367C22⋊Q8—28no—
368C22⋊Q8—28no—
369C22⋊Q8—232no—
370C4×Q8—216no—
371C4×Q8—216no—
372C4×Q8—28no—
373C4×Q8—28no—
374C4×Q8—28no—
375C4×Q8—28no—
376C4×Q8—216no—
377C4×Q8—232no—
378C4×Q8—28no—
379C4×Q8—28no—
380C4×Q8—816no—
381C4×Q8—88no—
382C4×Q8C2×Q8132yescomplemented
383C4×Q8C2×Q8132yescomplemented
384#117—24no—
385#117—28no—
386#117—28no—
387#117—216no—
388#117C2×Q8132yes—
389#117C2×Q818yes—
390#117C2×Q818yes—
391#117#4118yes—
392C2×C4⋊C4—28no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—28no—
395C2×C4⋊C4—28no—
396C2×C4⋊C4—28no—
397C2×C4⋊C4—88no—
398C2×C4⋊C4C2412yes—
399C2×C4⋊C4C2412yes—
400C2×C4⋊C4C2414yes—
401C2×C4⋊C4C2414yes—
402C2×C4⋊C4C2414yes—
403C2×C4⋊C4C2414yes—
404C2×C4⋊C4C2414yes—
405C2×C4⋊C4C2414yes—
406C2×C4⋊C4C2412yes—
407C2×C4⋊C4C2414yes—
408C2×C4⋊C4C2412yes—
409C2×C4⋊C4C2418yes—
410C2×C4⋊C4C2412yes—
411C2×C4⋊C4C2414yes—
412C2×C4⋊C4C2412yes—
413C2×C4⋊C4C2414yes—
414C2×C4⋊C4C2418yes—
415C2×C4⋊C4C2414yes—
416C2×C4⋊C4C2418yes—
417C2×C4⋊C4C2412yes—
418C2×C4⋊C4C2418yes—
419C2×C4⋊C4C2412yes—
420C2×C4⋊C4C2418yes—
421C2×C4⋊C4C2414yes—
422C2×C4⋊C4C2418yes—
423C2×C4⋊C4C24116yes—
424C2×C4⋊C4C24116yes—
425C2×C4⋊C4C2414yes—
426C2×C4⋊C4C24116yes—
427C2×C4⋊C4C24116yes—
428C2×C4⋊C4C24116yes—
429C2×C4⋊C4C2418yes—
430C2×C4⋊C4C2418yes—
431C2×C4⋊C4C24116yes—
432C2×C4⋊C4C2412yes—
433C2×C4⋊C4C2418yes—
434C2×C4⋊C4C2412yes—
435C2×C4⋊C4C2412yes—
436C2×C4⋊C4C2412yes—
437C2×C22⋊C4C2412yes—
438C2×C22⋊C4C2414yes—
439C2×C22⋊C4C2414yes—
440C2×C22⋊C4C2412yes—
441C2×C22⋊C4C2412yes—
442C2×C22⋊C4C2412yes—
443C2×C42—28no—
444C2×C42—28no—
445C2×C42#4118yes—
446C2×C42C2412yes—
447C2×C42C2418yes—
448C2×C42C2414yes—
449C2×C42C2414yes—
450C2×C42C2414yes—
451C2×C42C2414yes—
452C2×C42C2414yes—
453C2×C42C2414yes—
454C2×C42C2418yes—
455C2×C42C2412yes—
456C2×C42C2412yes—
457C2×C42C2414yes—
458C2×C42C2418yes—
459C2×C42C2414yes—
460C2×C42C2418yes—
461C2×C42C24116yes—
462C2×C42C2418yes—
463C2×C42C2414yes—
464C2×C42C2412yes—
465C2×C42C2418yes—
466C2×C42C2412yes—
467C2×C42C2412yes—
468C2×C42C2411yescharacteristic
469C2×C42C2411yescharacteristic
470C2×C42C2411yescharacteristic
471C2×C42C2411yescharacteristic
472#95C24116yes—
473#95C2412yes—
474#95C2412yes—
475#95C2414yes—
476#95C24116yes—
477#95C2414yes—
478#95C2414yes—
479#95C2418yes—
480#95C2418yes—
481#95C2414yes—
482#95C2414yes—
483#95C2418yes—
484#95C2418yes—
485#95C2418yes—
486#95C24116yes—
487#95C24116yes—
488#95C24116yes—
489#95C2418yes—
490#95C2414yes—
491#95C2414yes—
492#95C24116yes—
493#95C24116yes—
494#95C24116yes—
495#95C24116yes—
496#95C2418yes—
497#95C2418yes—
498C24C2511yescharacteristic
499C2×Q8—232no—
500C2×Q8—24no—
501C2×Q8—24no—
502C2×Q8—24no—
503C2×Q8—24no—
504C2×Q8—216no—
505C2×Q8—24no—
506C2×Q8—28no—
507C2×Q8—24no—
508C2×Q8—28no—
509C2×Q8—28no—
510C2×Q8—216no—
511C2×Q8—48no—
512C2×Q8—48no—
513C2×Q8—44no—
514C2×Q8—48no—
515C2×Q8—816no—
516C2×Q8C22×Q818yes—
517C2×Q8C22×Q818yes—
518C22×C4—24no—
519C22×C4—24no—
520C22×C4—24no—
521C22×C4—28no—
522C22×C4—24no—
523C22×C4—28no—
524C22×C4—28no—
525C22×C4—44no—
526C22×C4—42no—
527C22×C4—44no—
528C22×C4—48no—
529C22×C4C22×Q818yes—
530C22×C4C22×Q814yes—
531C22×C4C2×#4114yes—
532C22×C4C2×#4114yes—
533C22×C4C2×#4114yes—
534C22×C4C2×#4114yes—
535C22×C4C2514yes—
536C22×C4C2518yes—
537C22×C4C2514yes—
538C22×C4C2514yes—
539C22×C4C2518yes—
540C22×C4C2514yes—
541C22×C4C2514yes—
542C22×C4C2514yes—
543C22×C4C2514yes—
544C22×C4C2514yes—
545C22×C4C2511yescharacteristic
546C22×C4C2511yescharacteristic
547C22×C4C2514yes—
548C22×C4C2511yescharacteristic
549C22×C4C2514yes—
550C22×C4C2511yescharacteristic
551C22×C4C2511yescharacteristic
552C22×C4C2511yescharacteristic
553C4⋊C4—28no—
554C4⋊C4—28no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—28no—
558C4⋊C4—28no—
559C4⋊C4—28no—
560C4⋊C4—28no—
561C4⋊C4—216no—
562C4⋊C4—24no—
563C4⋊C4—28no—
564C4⋊C4—24no—
565C4⋊C4—24no—
566C4⋊C4—216no—
567C4⋊C4—216no—
568C4⋊C4—28no—
569C4⋊C4—28no—
570C4⋊C4—216no—
571C4⋊C4—24no—
572C4⋊C4—24no—
573C4⋊C4—28no—
574C4⋊C4—24no—
575C4⋊C4—216no—
576C4⋊C4—216no—
577C4⋊C4—28no—
578C4⋊C4—416no—
579C4⋊C4—416no—
580C4⋊C4—44no—
581C4⋊C4—416no—
582C4⋊C4—416no—
583C4⋊C4—416no—
584C4⋊C4—48no—
585C4⋊C4—48no—
586C4⋊C4—416no—
587C4⋊C4C22×Q8132yes—
588C4⋊C4C22×Q818yes—
589C4⋊C4C22×Q818yes—
590C4⋊C4C2×#4118yes—
591C4⋊C42−1+418yes—
592C22⋊C4—24no—
593C22⋊C4—28no—
594C22⋊C4—28no—
595C22⋊C4C22×Q818yes—
596C22⋊C4C22×Q818yes—
597C22⋊C4C2×#4118yes—
598C42—28no—
599C42—28no—
600C42—28no—
601C42—24no—
602C42—24no—
603C42—216no—
604C42—24no—
605C42—28no—
606C42—28no—
607C42—28no—
608C42—216no—
609C42—28no—
610C42—48no—
611C42—48no—
612C42—44no—
613C42—416no—
614C42—48no—
615C42—44no—
616C42C22×Q8132yes—
617C42C22×Q818yes—
618C42C22×Q818yes—
619C42C2×#4114yes—
620C42C2×#4114yes—
621C42C2×#4114yes—
622C42C2×#4114yes—
623C42C2×#4118yes—
624C23—22no—
625C23—22no—
626C23C23×Q812yes—
627C23C22×#4112yes—
628C23C22×#4112yes—
629C23C2611yescharacteristic
630Q8—216no—
631Q8—216no—
632Q8—432no—
633Q8—416no—
634Q8—48no—
635Q8—84no—
636Q8—88no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—22no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—216no—
651C2×C4—28no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—28no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—28no—
659C2×C4—24no—
660C2×C4—28no—
661C2×C4—24no—
662C2×C4—28no—
663C2×C4—24no—
664C2×C4—28no—
665C2×C4—28no—
666C2×C4—28no—
667C2×C4—216no—
668C2×C4—24no—
669C2×C4—44no—
670C2×C4—416no—
671C2×C4—84no—
672C2×C4—88no—
673C2×C4#533116yes—
674C2×C4C23×Q818yes—
675C2×C4C23×Q818yes—
676C2×C4C23×Q818yes—
677C2×C4C23×Q812yes—
678C2×C4C23×Q812yes—
679C2×C4C22×#4112yes—
680C2×C4C22×#4118yes—
681C2×C4C22×#4112yes—
682C2×C4C22×#4112yes—
683C2×C4C22×#4112yes—
684C2×C4C22×#4112yes—
685C2×C4C22×#4112yes—
686C2×C4C22×#4112yes—
687C2×C4C22×#4112yes—
688C2×C4C2×2−1+418yes—
689C2×C4C2×2−1+414yes—
690C22—24no—
691C22—24no—
692C22—22no—
693C22—22no—
694C22C2×#53314yes—
695C22C24×Q811yescharacteristic
696C22C23×#4111yescharacteristic
697C22C23×#4111yescharacteristic
698C22C22×2−1+412yes—
699C22H(7,𝔽2)12yes—
700C4—28no—
701C4—28no—
702C4—28no—
703C4—28no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—28no—
708C4—44no—
709C4—44no—
710C4—48no—
711C4—44no—
712C4—44no—
713C4—48no—
714C4—44no—
715C4C2×#53314yes—
716C4C2×#53314yes—
717C4#350914yes—
718C2—24no—
719C2C22×#53311yescharacteristic
720C2C2×#350911yescharacteristic
721C2#6285511yescharacteristic
722C2Q8×2−1+412yes—
723C2#6300012yes—
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. #36563C2semidirect
2. #36286C2semidirect
4. #35936C2semidirect
7. #35583C2semidirect
10. #35208C2semidirect
12. #34950C2semidirect
152. #519Q8semidirect
382. C4×Q8C2×Q8semidirect
383. C4×Q8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#362862
#362802
#359362
#357432
#356622
#355832
#355552
#352812
#352082
#351792
#349502
C2×#25672
#349382
#342542
#342162
#342142
#341452
C4×#5192