Finite Group Databasebeta

← #7672583

Subgroups — #7672583

Order 512.

4996 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59333C218yesmaximal, complemented
2#59332C218yesmaximal, complemented
3#39473C218yesmaximal, complemented
4#37884C218yesmaximal, complemented
5#36625C211yesmaximal, characteristic, complemented
6#36206C218yesmaximal, complemented
7#36171C218yesmaximal, complemented
8#35629C212yesmaximal, complemented
9#35627C212yesmaximal, complemented
10#35339C211yesmaximal, characteristic, complemented
11#35334C211yesmaximal, characteristic, complemented
12#35227C212yesmaximal
13#34215C211yesmaximal, characteristic, complemented
14#34189C211yesmaximal, characteristic, complemented
15#34178C212yesmaximal, complemented
16C4×#525C211yesmaximal, characteristic
17C22×#382C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#525C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C42C2211yescharacteristic
23#2864C2214yescomplemented
24#2857C2214yes—
25#2855C2214yes—
26#2851C2214yes—
27#2851C2214yescomplemented
28#2849C2218yes—
29#2848C2214yes—
30#2773C22116yescomplemented
31#2767C22116yescomplemented
32#2762C22116yescomplemented
33#2754C22116yescomplemented
34#2647C2214yescomplemented
35#2628C2214yescomplemented
36#2623C2218yes—
37#2581C2214yescomplemented
38#2580C2214yescomplemented
39#2575C22116yescomplemented
40#2571C2218yescomplemented
41#2570C2218yescomplemented
42#2570C2218yescomplemented
43#2569C2218yes—
44#2564C2218yes—
45#2528C22116yescomplemented
46#2524C22116yescomplemented
47#2517C22116yescomplemented
48#2514C2218yescomplemented
49#2513C2218yescomplemented
50#2510C22116yes—
51#2509C2218yescomplemented
52#2508C2218yescomplemented
53#2505C22116yes—
54#2504C2214yes—
55#2503C2214yes—
56#2502C2214yes—
57#2501C2214yes—
58#2498C22116yes—
59#2495C2218yescomplemented
60#2494C2218yescomplemented
61#2493C22116yes—
62#2493C22116yes—
63#2488C22116yes—
64#2487C2218yes—
65#2486C2218yes—
66#2438C22116yes—
67#2419C22116yes—
68#2368C2218yes—
69#2359C22116yes—
70#2357C2214yescomplemented
71#2356C2214yescomplemented
72#2355C2218yescomplemented
73#2353C2218yes—
74#2345C2218yes—
75#2345C2218yes—
76#2344C2218yes—
77#2344C2218yes—
78#2343C22116yes—
79#2331C2214yescomplemented
80#2317C2212yes—
81#2317C2212yes—
82#2316C2212yes—
83#2316C2212yes—
84#2314C2214yescomplemented
85#2313C2218yes—
86#2311C2212yes—
87#2311C2212yes—
88C4×#125C2214yescomplemented
89C4×C22⋊Q8C2218yes—
90C4×C22⋊Q8C2214yes—
91C4×#123C2214yes—
92C4×#121C2214yes—
93C4×#120C2214yes—
94#2297C2211yescharacteristic
95#2296C2212yes—
96#2296C2211yescharacteristic
97#2290C2214yes—
98#2285C2211yescharacteristic
99#2283C2214yescomplemented
100C2×#382C2211yescharacteristic
101C2×#382C2212yes—
102C2×#382C2214yes—
103C2×#382C2214yes—
104C2×#382C2211yescharacteristic
105C2×#382C2218yes—
106C2×#382C2214yes—
107#2280C2212yes—
108#2280C2218yes—
109#2280C2211yescharacteristic
110#2274C2212yes—
111#2274C2212yes—
112C42×D4C2214yes—
113C4×#117C2211yescharacteristic
114C22×#95C2214yes—
115C22×#41C2311yescharacteristic
116C24×C4C2311yescharacteristic
117C24×C4C2312yes—
118C24×C4C2314yes—
119#533—28no—
120C2×C4⋊Q8C2312yes—
121C2×#125C2312yes—
122#525—28no—
123C2×#123C2312yes—
124C2×#123C23116yes—
125C2×C22⋊Q8C2314yes—
126C2×C22⋊Q8C2312yes—
127C2×C22⋊Q8C2318yes—
128C2×#121C2312yes—
129C2×#120C2312yes—
130C4×#41—28no—
131C2×C4×Q8C2314yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4C2314yes—
134C2×C4×D4C2312yes—
135C2×C4×D4C2314yes—
136C2×C4×D4C2318yes—
137C2×#117C2312yes—
138C2×#117C2312yes—
139C2×#117C2312yes—
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C23116yes—
144C22×C4⋊C4—216no—
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2314yes—
149C22×C4⋊C4C2314yes—
150C22×C4⋊C4C2314yes—
151C22×C4⋊C4C2314yes—
152C22×C42—28no—
153C22×C42C2312yes—
154C22×C42C2312yes—
155C22×C42C2312yes—
156C22×C42C2312yes—
157C22×C42C2311yescharacteristic
158C22×C42C2314yes—
159C22×C42C2311yescharacteristic
160#401C23116yes—
161#400C2318yes—
162#400C23116yes—
163#400C23116yes—
164#400C23116yes—
165#400C23116yes—
166#400C23116yes—
167#400C23116yes—
168#399C2318yes—
169#399C23116yes—
170#398C23116yes—
171#398C23116yes—
172#398C23116yes—
173#397C23116yes—
174#397C23116yes—
175#397C23116yes—
176#395C2318yes—
177#393C2318yes—
178#391C2314yes—
179#391C2314yes—
180#391C23116yes—
181#391C2318yes—
182#391C2312yes—
183#390C2314yes—
184#390C2314yes—
185#390C2312yes—
186#389C2318yes—
187#389C2314yes—
188#389C2318yes—
189#389C2318yes—
190#389C23116yes—
191#389C2318yes—
192#389C2318yes—
193#389C2318yes—
194#389C2314yes—
195#389C2314yes—
196#389C2318yes—
197#388C23116yes—
198#388C23116yes—
199#388C23116yes—
200#388C2318yes—
201#387C23116yes—
202#387C23116yes—
203#387C23116yes—
204#387C23116yes—
205#387C23116yes—
206#387C23116yes—
207#387C23116yes—
208#387C23116yes—
209#387C23116yes—
210#387C23116yes—
211#387C23116yes—
212#387C2318yes—
213#387C23116yes—
214#387C2318yes—
215#387C23116yes—
216#387C2318yes—
217#386C2318yes—
218#386C2318yes—
219#386C23116yes—
220#386C2318yes—
221#386C2314yes—
222#385C23116yes—
223#385C2318yes—
224#385C2318yes—
225#385C2318yes—
226#385C2318yes—
227#385C2318yes—
228#385C23116yes—
229#385C2318yes—
230#385C2314yes—
231#384C2312yes—
232#384C2314yes—
233#384C2314yes—
234#383C23116yes—
235#383C2318yes—
236#382C2312yes—
237#382C2312yes—
238#382C2314yes—
239#382C2314yes—
240#382C2318yes—
241#382C2314yes—
242#382C23116yes—
243#382C23116yes—
244#382C2312yes—
245#382C2312yes—
246#382C2314yes—
247#382C2312yes—
248#382C2314yes—
249#382C2314yes—
250#382C23116yes—
251#381C2312yes—
252#381C2314yes—
253#381C2318yes—
254#381C23116yes—
255#381C23116yes—
256#380C2318yes—
257#380C2318yes—
258#380C2318yes—
259#380C2318yes—
260#380C2314yes—
261#380C2318yes—
262#380C2314yes—
263#380C2314yes—
264#380C2314yes—
265#380C23116yes—
266#379C2314yes—
267#379C2312yes—
268C4×C4⋊C4C23116yes—
269C4×C4⋊C4C2318yes—
270C4×C4⋊C4C2318yes—
271C4×C4⋊C4C2314yes—
272C4×C4⋊C4C2314yes—
273C4×C4⋊C4C2318yes—
274C4×C4⋊C4C2318yes—
275C4×C4⋊C4C2318yes—
276C4×C4⋊C4C2314yes—
277C4×C4⋊C4C2318yes—
278C4×C22⋊C4C2318yes—
279C4×C22⋊C4C2318yes—
280C4×C22⋊C4C2314yes—
281C4×C22⋊C4C2318yes—
282C4×C22⋊C4C2318yes—
283C4×C22⋊C4C23116yes—
284C4×C22⋊C4C2314yes—
285C4×C22⋊C4C2314yes—
286C4×C22⋊C4C2314yes—
287C4×C22⋊C4C2318yes—
288C4×C22⋊C4C2314yes—
289#376C2312yes—
290#376C2318yes—
291C2×#95C2314yes—
292C2×#95C23116yes—
293C2×#95C2314yes—
294C2×#95C23116yes—
295C2×#95C2318yes—
296C43C2312yes—
297C25C2411yescharacteristic
298C2×#41—24no—
299C2×#41—24no—
300C2×#41—24no—
301C22×Q8C2411yescharacteristic
302C22×Q8C2414yes—
303C22×D4C2414yes—
304C22×D4C2412yes—
305C22×D4C2411yescharacteristic
306C23×C4—24no—
307C23×C4—24no—
308C23×C4—216no—
309C23×C4—28no—
310C23×C4—48no—
311C23×C4—416no—
312C23×C4#4118yes—
313C23×C4C2418yes—
314C23×C4C2412yes—
315C23×C4C2414yes—
316C23×C4C2414yes—
317C23×C4C2412yes—
318C23×C4C2412yes—
319C23×C4C2411yescharacteristic
320C23×C4C2414yes—
321C23×C4C2418yes—
322C23×C4C2414yes—
323C23×C4C2414yes—
324C23×C4C2412yes—
325C23×C4C2412yes—
326C23×C4C2411yescharacteristic
327C4⋊Q8—28no—
328#125—28no—
329#123—28no—
330#123—432no—
331C22⋊Q8—28no—
332C22⋊Q8—216no—
333C22⋊Q8—416no—
334#121—28no—
335#120—28no—
336C4×Q8—28no—
337C4×Q8—216no—
338C4×D4—216no—
339C4×D4—216no—
340C4×D4—28no—
341C4×D4—416no—
342#117—24no—
343#117—24no—
344#117—28no—
345#117—28no—
346#117—24no—
347#117—28no—
348#117—432no—
349C2×C4⋊C4—216no—
350C2×C4⋊C4—216no—
351C2×C4⋊C4—216no—
352C2×C4⋊C4—216no—
353C2×C4⋊C4—216no—
354C2×C4⋊C4—28no—
355C2×C4⋊C4—28no—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2414yes—
359C2×C4⋊C4C2414yes—
360C2×C4⋊C4C2418yes—
361C2×C4⋊C4C2418yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2414yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2414yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2414yes—
368C2×C4⋊C4C2414yes—
369C2×C4⋊C4C2418yes—
370C2×C4⋊C4C2414yes—
371C2×C4⋊C4C24116yes—
372C2×C4⋊C4C24116yes—
373C2×C4⋊C4C2418yes—
374C2×C4⋊C4C2418yes—
375C2×C4⋊C4C2418yes—
376C2×C4⋊C4C2418yes—
377C2×C4⋊C4C24116yes—
378C2×C4⋊C4C24116yes—
379C2×C22⋊C4C2418yes—
380C2×C22⋊C4C2418yes—
381C2×C22⋊C4C2412yes—
382C2×C22⋊C4C2414yes—
383C2×C22⋊C4C2412yes—
384C2×C22⋊C4C2414yes—
385C2×C22⋊C4C2412yes—
386C2×C22⋊C4C2414yes—
387C2×C22⋊C4C2414yes—
388C2×C22⋊C4C2414yes—
389C2×C22⋊C4C2418yes—
390C2×C22⋊C4C2418yes—
391C2×C22⋊C4C24116yes—
392C2×C22⋊C4C24116yes—
393C2×C42—28no—
394C2×C42—24no—
395C2×C42—28no—
396C2×C42—28no—
397C2×C42—28no—
398C2×C42—216no—
399C2×C42#4118yes—
400C2×C42C2412yes—
401C2×C42C2412yes—
402C2×C42C2414yes—
403C2×C42C2412yes—
404C2×C42C2414yes—
405C2×C42C2414yes—
406C2×C42C2414yes—
407C2×C42C2412yes—
408C2×C42C2412yes—
409C2×C42C2414yes—
410C2×C42C24116yes—
411C2×C42C24116yes—
412C2×C42C2418yes—
413C2×C42C2411yescharacteristic
414C2×C42C2412yes—
415C2×C42C2411yescharacteristic
416#95C2418yes—
417#95C24116yes—
418#95C2418yes—
419#95C2418yes—
420#95C24116yes—
421#95C2418yes—
422#95C2414yes—
423#95C2418yes—
424#95C24116yes—
425#95C2418yes—
426#95C2414yes—
427#95C2418yes—
428#95C2418yes—
429#95C24116yes—
430#95C24116yes—
431#95C24116yes—
432#95C24116yes—
433#95C2418yes—
434#95C2418yes—
435#95C2418yes—
436#95C2418yes—
437#95C24116yes—
438#95C24116yes—
439#95C24116yes—
440C24—22no—
441C24—22no—
442C24—28no—
443C24C2×#4114yes—
444C24C2512yes—
445C24C2511yescharacteristic
446C24C2512yes—
447#41—416no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—28no—
451C2×Q8—22no—
452C2×Q8—48no—
453C2×D4—24no—
454C2×D4—28no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—216no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—232no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—216no—
470C22×C4—232no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—216no—
476C22×C4—28no—
477C22×C4—28no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—28no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—416no—
489C22×C4—48no—
490C22×C4—432no—
491C22×C4—832no—
492C22×C4C2×#4118yes—
493C22×C4C2×#4118yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4118yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4114yes—
498C22×C42−1+414yes—
499C22×C4C2512yes—
500C22×C4C2514yes—
501C22×C4C2518yes—
502C22×C4C2518yes—
503C22×C4C2514yes—
504C22×C4C2518yes—
505C22×C4C2514yes—
506C22×C4C2518yes—
507C22×C4C2514yes—
508C22×C4C2512yes—
509C22×C4C2512yes—
510C22×C4C2512yes—
511C22×C4C2511yescharacteristic
512C22×C4C2511yescharacteristic
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—28no—
516C4⋊C4—28no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—24no—
520C4⋊C4—28no—
521C4⋊C4—24no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—216no—
525C4⋊C4—28no—
526C4⋊C4—24no—
527C4⋊C4—28no—
528C4⋊C4—416no—
529C4⋊C4—416no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—416no—
535C4⋊C4—416no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—24no—
541C22⋊C4—28no—
542C22⋊C4—216no—
543C22⋊C4—216no—
544C22⋊C4—216no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—48no—
548C22⋊C4—416no—
549C22⋊C4—416no—
550C42—28no—
551C42—28no—
552C42—24no—
553C42—24no—
554C42—24no—
555C42—28no—
556C42—24no—
557C42—28no—
558C42—28no—
559C42—24no—
560C42—416no—
561C42—416no—
562C42—48no—
563C42C2×#4114yes—
564C42C2×#4114yes—
565C42C2×#4118yes—
566C23—216no—
567C23—216no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—28no—
573C23—216no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—28no—
578C23C22×#4112yes—
579C23C22×#4112yes—
580C23C22×#4114yes—
581C23C2×2−1+412yes—
582C23#58518yes—
583C23C2611yescharacteristic
584Q8—44no—
585Q8—88no—
586D4—48no—
587D4—416no—
588D4—44no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—216no—
595C2×C4—216no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—216no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—216no—
617C2×C4—24no—
618C2×C4—48no—
619C2×C4—416no—
620C2×C4—416no—
621C2×C4—48no—
622C2×C4—432no—
623C2×C4—48no—
624C2×C4—88no—
625C2×C4—88no—
626C2×C4—816no—
627C2×C4—88no—
628C2×C4#552116yes—
629C2×C4C22×#4114yes—
630C2×C4C22×#4114yes—
631C2×C4C22×#4114yes—
632C2×C4C22×#4112yes—
633C2×C4C22×#4112yes—
634C2×C4C22×#4112yes—
635C2×C4C22×#4112yes—
636C2×C4C2×2−1+412yes—
637C2×C4C2×2−1+412yes—
638C22—232no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—24no—
644C22—28no—
645C22—24no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22C2×#55214yes—
651C22#351018yes—
652C22C23×#4111yescharacteristic
653C22C23×#4111yescharacteristic
654C22C22×2−1+411yescharacteristic
655C22C2×#58514yes—
656C4—24no—
657C4—24no—
658C4—24no—
659C4—48no—
660C4—48no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—48no—
665C4—44no—
666C4—48no—
667C4—42no—
668C4C2×#55214yes—
669C4C2×#55214yes—
670C2—24no—
671C2—28no—
672C2—82no—
673C2C22×#55211yescharacteristic
674C2C2×#351012yes—
675C2#6281412yes—
676C2#6282512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC2×#525C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59333C2semidirect
2. #59332C2semidirect
3. #39473C2semidirect
4. #37884C2semidirect
5. #36625C2semidirect
6. #36206C2semidirect
7. #36171C2semidirect
8. #35629C2semidirect
9. #35627C2semidirect
10. #35339C2semidirect
11. #35334C2semidirect
13. #34215C2semidirect
14. #34189C2semidirect
15. #34178C2semidirect
17. C22×#382C2semidirect
23. #2864C22semidirect
27. #2851C22semidirect
30. #2773C22semidirect
31. #2767C22semidirect
32. #2762C22semidirect
33. #2754C22semidirect
34. #2647C22semidirect
35. #2628C22semidirect
37. #2581C22semidirect
38. #2580C22semidirect
39. #2575C22semidirect
40. #2571C22semidirect
41. #2570C22semidirect
42. #2570C22semidirect
45. #2528C22semidirect
46. #2524C22semidirect
47. #2517C22semidirect
48. #2514C22semidirect
49. #2513C22semidirect
51. #2509C22semidirect
52. #2508C22semidirect
59. #2495C22semidirect
60. #2494C22semidirect
70. #2357C22semidirect
71. #2356C22semidirect
72. #2355C22semidirect
79. #2331C22semidirect
84. #2314C22semidirect
88. C4×#125C22semidirect
99. #2283C22semidirect

Maximal subgroups

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

subgroupindex
#593332
#593322
#394732
#378842
#366252
#362062
#361712
#356292
#356272
#353392
#353342
#352272
#342152
#341892
#341782
C4×#5252
C22×#3822