Finite Group Databasebeta

← #7672433

Subgroups — #7672433

Order 512.

4872 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59331C218yesmaximal, complemented
2#59330C218yesmaximal, complemented
3#39472C218yesmaximal, complemented
4#37887C218yesmaximal, complemented
5#36625C211yesmaximal, characteristic, complemented
6#36205C218yesmaximal, complemented
7#36201C218yesmaximal, complemented
8#35557C212yesmaximal, complemented
9#35555C212yesmaximal, complemented
10#35339C211yesmaximal, characteristic, complemented
11#35334C211yesmaximal, characteristic, complemented
12#35157C212yesmaximal
13#34215C211yesmaximal, characteristic, complemented
14#34189C211yesmaximal, characteristic, complemented
15#34179C212yesmaximal, complemented
16C4×#525C211yesmaximal, characteristic
17C2×#2285C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#525C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C22×#117C2212yes—
22C23×C42C2211yescharacteristic
23#2864C2214yescomplemented
24#2857C2214yes—
25#2855C2214yes—
26#2853C2218yes—
27#2851C2214yes—
28#2851C2214yescomplemented
29#2848C2214yes—
30#2747C22116yes—
31#2741C22116yes—
32#2702C22116yes—
33#2690C22116yes—
34#2647C2214yescomplemented
35#2628C2214yescomplemented
36#2622C2218yes—
37#2581C2214yescomplemented
38#2580C2214yescomplemented
39#2577C22116yes—
40#2570C2218yescomplemented
41#2570C2218yescomplemented
42#2569C2218yes—
43#2564C2218yes—
44#2552C2218yescomplemented
45#2533C22116yescomplemented
46#2532C22116yescomplemented
47#2530C22116yes—
48#2524C22116yescomplemented
49#2520C22116yescomplemented
50#2519C22116yescomplemented
51#2515C22116yescomplemented
52#2512C2218yescomplemented
53#2511C2218yescomplemented
54#2507C2218yescomplemented
55#2506C2218yescomplemented
56#2500C2214yes—
57#2500C2214yes—
58#2499C2214yes—
59#2499C2214yes—
60#2497C2218yescomplemented
61#2496C2218yescomplemented
62#2485C2218yes—
63#2485C2218yes—
64#2481C22116yes—
65#2473C22116yes—
66#2468C22116yes—
67#2460C22116yes—
68#2373C22116yescomplemented
69#2371C22116yescomplemented
70#2368C2218yes—
71#2357C2214yescomplemented
72#2356C2214yescomplemented
73#2355C2218yescomplemented
74#2353C2218yes—
75#2347C2218yes—
76#2347C2218yes—
77#2346C2218yes—
78#2346C2218yes—
79#2331C2214yescomplemented
80#2317C2212yes—
81#2317C2212yes—
82#2316C2212yes—
83#2316C2212yes—
84#2314C2214yescomplemented
85#2312C2218yes—
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#2285C2214yes—
100#2285C2218yes—
101#2285C2214yes—
102C2×#384C2212yes—
103#2283C2218yes—
104#2283C2214yescomplemented
105C2×#382C2211yescharacteristic
106C2×#382C2211yescharacteristic
107#2280C2212yes—
108#2280C2211yescharacteristic
109C2×#380C2214yes—
110#2274C2212yes—
111#2274C2212yes—
112C42×D4C2214yes—
113C4×#117C2211yescharacteristic
114C2×C4×C4⋊C4C2214yes—
115C22×#41C2311yescharacteristic
116C24×C4C2311yescharacteristic
117C24×C4C2312yes—
118#533—28no—
119C2×C4⋊Q8C2312yes—
120C2×#126C23116yes—
121C2×#125C2312yes—
122C2×#125C23116yes—
123#525—28no—
124C2×#123C2312yes—
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×#117—216no—
138C2×#117C2312yes—
139C2×#117C2312yes—
140C2×#117C2312yes—
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C2×#117C2314yes—
145C2×#117C2314yes—
146C2×#117C2314yes—
147C2×#117C2314yes—
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2312yes—
150C22×C4⋊C4C2318yes—
151C22×C22⋊C4C2314yes—
152C22×C42—28no—
153C22×C42C2312yes—
154C22×C42C2312yes—
155C22×C42C2314yes—
156C22×C42C2312yes—
157C22×C42C2312yes—
158C22×C42C2311yescharacteristic
159C22×C42C2314yes—
160C22×C42C2311yescharacteristic
161#400C2318yes—
162#400C23116yes—
163#400C23116yes—
164#400C23116yes—
165#400C23116yes—
166#399C2318yes—
167#398C23116yes—
168#398C23116yes—
169#398C23116yes—
170#398C23116yes—
171#397C23116yes—
172#397C23116yes—
173#397C23116yes—
174#396C23116yes—
175#395C2318yes—
176#394C23116yes—
177#394C23116yes—
178#393C2318yes—
179#391C2314yes—
180#391C2314yes—
181#391C23116yes—
182#391C2318yes—
183#391C2312yes—
184#390C2314yes—
185#390C2314yes—
186#390C2312yes—
187#389C2318yes—
188#389C2318yes—
189#389C2314yes—
190#389C2318yes—
191#389C2318yes—
192#389C2318yes—
193#389C2318yes—
194#389C2314yes—
195#389C2314yes—
196#389C2318yes—
197#389C23116yes—
198#388C23116yes—
199#388C23116yes—
200#388C23116yes—
201#388C23116yes—
202#388C23116yes—
203#388C23116yes—
204#388C23116yes—
205#388C23116yes—
206#388C2318yes—
207#387C23116yes—
208#387C23116yes—
209#387C23116yes—
210#387C23116yes—
211#387C23116yes—
212#387C23116yes—
213#387C23116yes—
214#387C23116yes—
215#387C23116yes—
216#387C23116yes—
217#387C2318yes—
218#387C2318yes—
219#387C23116yes—
220#387C23116yes—
221#387C2318yes—
222#386C2318yes—
223#386C2318yes—
224#386C2318yes—
225#386C2314yes—
226#385C23116yes—
227#385C2318yes—
228#385C2318yes—
229#385C2318yes—
230#385C2318yes—
231#385C2318yes—
232#385C2318yes—
233#385C2314yes—
234#384C2312yes—
235#384C2314yes—
236#384C2318yes—
237#384C2314yes—
238#384C2314yes—
239#384C2314yes—
240#383C2318yes—
241#383C23116yes—
242#382C2312yes—
243#382C2312yes—
244#382C2314yes—
245#382C2314yes—
246#382C2314yes—
247#382C2312yes—
248#382C2312yes—
249#382C23116yes—
250#382C2314yes—
251#382C2312yes—
252#381C2314yes—
253#381C2318yes—
254#381C2312yes—
255#380C2318yes—
256#380C2318yes—
257#380C2318yes—
258#380C2318yes—
259#380C2314yes—
260#380C23116yes—
261#380C2318yes—
262#380C2314yes—
263#380C2314yes—
264#380C2314yes—
265#380C23116yes—
266#379C2312yes—
267#379C2314yes—
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⋊C4C23116yes—
277C4×C4⋊C4C2318yes—
278C4×C4⋊C4C23116yes—
279C4×C4⋊C4C2314yes—
280C4×C4⋊C4C23116yes—
281C4×C22⋊C4C2318yes—
282C4×C22⋊C4C2318yes—
283C4×C22⋊C4C2314yes—
284C4×C22⋊C4C2318yes—
285C4×C22⋊C4C2314yes—
286C4×C22⋊C4C2314yes—
287C4×C22⋊C4C2314yes—
288C4×C22⋊C4C2318yes—
289C4×C22⋊C4C2318yes—
290C4×C22⋊C4C2314yes—
291#376C2312yes—
292#376C23116yes—
293#376C2318yes—
294C2×#95C2314yes—
295C2×#95C2314yes—
296C43C2312yes—
297C25C2411yescharacteristic
298C2×#41—24no—
299C2×#41—24no—
300C2×#41—24no—
301C22×Q8C2411yescharacteristic
302C22×Q8C2414yes—
303C22×D4C2412yes—
304C22×D4C2411yescharacteristic
305C22×D4C2414yes—
306C23×C4—24no—
307C23×C4—24no—
308C23×C4—28no—
309C23×C4—48no—
310C23×C4#4118yes—
311C23×C4C2412yes—
312C23×C4C2412yes—
313C23×C4C2414yes—
314C23×C4C2414yes—
315C23×C4C2412yes—
316C23×C4C2412yes—
317C23×C4C2411yescharacteristic
318C23×C4C2414yes—
319C23×C4C2414yes—
320C23×C4C2414yes—
321C23×C4C2412yes—
322C23×C4C2411yescharacteristic
323C4⋊Q8—28no—
324#126—432no—
325#125—28no—
326#125—432no—
327#123—28no—
328C22⋊Q8—28no—
329C22⋊Q8—216no—
330C22⋊Q8—416no—
331#121—28no—
332#120—28no—
333C4×Q8—28no—
334C4×Q8—216no—
335C4×D4—216no—
336C4×D4—216no—
337C4×D4—28no—
338C4×D4—416no—
339#117—216no—
340#117—216no—
341#117—216no—
342#117—216no—
343#117—24no—
344#117—24no—
345#117—28no—
346#117—28no—
347#117—24no—
348#117—28no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—28no—
351C2×C4⋊C4—416no—
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2414yes—
354C2×C4⋊C4C2414yes—
355C2×C4⋊C4C2414yes—
356C2×C4⋊C4C24116yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2414yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2414yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2414yes—
363C2×C4⋊C4C2414yes—
364C2×C4⋊C4C2418yes—
365C2×C4⋊C4C2418yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2418yes—
368C2×C4⋊C4C2418yes—
369C2×C4⋊C4C24116yes—
370C2×C4⋊C4C2418yes—
371C2×C4⋊C4C24116yes—
372C2×C4⋊C4C24116yes—
373C2×C4⋊C4C24116yes—
374C2×C4⋊C4C24116yes—
375C2×C22⋊C4—216no—
376C2×C22⋊C4C2418yes—
377C2×C22⋊C4C2418yes—
378C2×C22⋊C4C2412yes—
379C2×C22⋊C4C2414yes—
380C2×C22⋊C4C2412yes—
381C2×C22⋊C4C2414yes—
382C2×C22⋊C4C2412yes—
383C2×C22⋊C4C2414yes—
384C2×C22⋊C4C2414yes—
385C2×C22⋊C4C2414yes—
386C2×C22⋊C4C2418yes—
387C2×C22⋊C4C2418yes—
388C2×C22⋊C4C2418yes—
389C2×C22⋊C4C2418yes—
390C2×C22⋊C4C24116yes—
391C2×C22⋊C4C24116yes—
392C2×C42—216no—
393C2×C42—28no—
394C2×C42—24no—
395C2×C42—28no—
396C2×C42—28no—
397C2×C42—28no—
398C2×C42—216no—
399C2×C42#4118yes—
400C2×C42C2418yes—
401C2×C42C2412yes—
402C2×C42C2418yes—
403C2×C42C2412yes—
404C2×C42C2414yes—
405C2×C42C2412yes—
406C2×C42C2414yes—
407C2×C42C2414yes—
408C2×C42C2414yes—
409C2×C42C2412yes—
410C2×C42C2412yes—
411C2×C42C2414yes—
412C2×C42C2418yes—
413C2×C42C24116yes—
414C2×C42C24116yes—
415C2×C42C2411yescharacteristic
416C2×C42C2412yes—
417C2×C42C2411yescharacteristic
418#95C2418yes—
419#95C2418yes—
420#95C2418yes—
421#95C2418yes—
422#95C2418yes—
423#95C2418yes—
424#95C2414yes—
425#95C2418yes—
426#95C24116yes—
427#95C24116yes—
428#95C24116yes—
429#95C2414yes—
430#95C2418yes—
431#95C2418yes—
432#95C2418yes—
433#95C24116yes—
434#95C24116yes—
435#95C2418yes—
436#95C2418yes—
437#95C24116yes—
438#95C24116yes—
439#95C24116yes—
440C24—22no—
441C24—22no—
442C24—44no—
443C24C2×#4114yes—
444C24C2512yes—
445C24C2512yes—
446C24C2511yescharacteristic
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—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—48no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—232no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—28no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—432no—
487C22×C4C2×#4114yes—
488C22×C4C2×#4118yes—
489C22×C4C2×#4118yes—
490C22×C4C2×#4118yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C42−1+414yes—
494C22×C4C2512yes—
495C22×C4C2514yes—
496C22×C4C2518yes—
497C22×C4C2518yes—
498C22×C4C2514yes—
499C22×C4C2518yes—
500C22×C4C2514yes—
501C22×C4C2518yes—
502C22×C4C2512yes—
503C22×C4C2512yes—
504C22×C4C2512yes—
505C22×C4C2514yes—
506C22×C4C2511yescharacteristic
507C22×C4C2511yescharacteristic
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—24no—
513C4⋊C4—28no—
514C4⋊C4—24no—
515C4⋊C4—28no—
516C4⋊C4—28no—
517C4⋊C4—216no—
518C4⋊C4—28no—
519C4⋊C4—24no—
520C4⋊C4—28no—
521C4⋊C4—232no—
522C4⋊C4—216no—
523C4⋊C4—216no—
524C4⋊C4—216no—
525C4⋊C4—416no—
526C4⋊C4—48no—
527C4⋊C4—416no—
528C4⋊C4—416no—
529C4⋊C4—416no—
530C4⋊C4—416no—
531C22⋊C4—216no—
532C22⋊C4—216no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—28no—
536C22⋊C4—28no—
537C22⋊C4—24no—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—28no—
541C22⋊C4—216no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—416no—
546C22⋊C4—416no—
547C42—28no—
548C42—28no—
549C42—24no—
550C42—24no—
551C42—216no—
552C42—216no—
553C42—24no—
554C42—28no—
555C42—24no—
556C42—28no—
557C42—28no—
558C42—24no—
559C42—48no—
560C42—416no—
561C42—416no—
562C42C2×#4114yes—
563C42C2×#4114yes—
564C42C2×#4118yes—
565C23—24no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—24no—
570C23—28no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—28no—
575C23—48no—
576C23—48no—
577C23—48no—
578C23C22×#4114yes—
579C23C22×#4112yes—
580C23C22×#4112yes—
581C23C2×2−1+412yes—
582C23C2611yescharacteristic
583Q8—44no—
584Q8—88no—
585D4—416no—
586D4—48no—
587D4—44no—
588C2×C4—216no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—216no—
597C2×C4—216no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—216no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—216no—
616C2×C4—28no—
617C2×C4—48no—
618C2×C4—416no—
619C2×C4—416no—
620C2×C4—48no—
621C2×C4—88no—
622C2×C4—88no—
623C2×C4—88no—
624C2×C4#552116yes—
625C2×C4C22×#4114yes—
626C2×C4C22×#4114yes—
627C2×C4C22×#4114yes—
628C2×C4C22×#4112yes—
629C2×C4C22×#4112yes—
630C2×C4C22×#4112yes—
631C2×C4C22×#4112yes—
632C2×C4C2×2−1+412yes—
633C2×C4C2×2−1+412yes—
634C2×C4#58518yes—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—416no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—42no—
648C22C2×#55214yes—
649C22C23×#4111yescharacteristic
650C22C23×#4111yescharacteristic
651C22C22×2−1+411yescharacteristic
652C22C2×#58514yes—
653C4—28no—
654C4—24no—
655C4—24no—
656C4—24no—
657C4—44no—
658C4—48no—
659C4—48no—
660C4—48no—
661C4—44no—
662C4—44no—
663C4—48no—
664C4—42no—
665C4C2×#55214yes—
666C4C2×#55214yes—
667C2—24no—
668C2—44no—
669C2—82no—
670C2C22×#55211yescharacteristic
671C2#6281412yes—
672C2#6281512yes—
673C2#6283812yes—
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. #59331C2semidirect
2. #59330C2semidirect
3. #39472C2semidirect
4. #37887C2semidirect
5. #36625C2semidirect
6. #36205C2semidirect
7. #36201C2semidirect
8. #35557C2semidirect
9. #35555C2semidirect
10. #35339C2semidirect
11. #35334C2semidirect
13. #34215C2semidirect
14. #34189C2semidirect
15. #34179C2semidirect
17. C2×#2285C2semidirect
23. #2864C22semidirect
28. #2851C22semidirect
34. #2647C22semidirect
35. #2628C22semidirect
37. #2581C22semidirect
38. #2580C22semidirect
40. #2570C22semidirect
41. #2570C22semidirect
44. #2552C22semidirect
45. #2533C22semidirect
46. #2532C22semidirect
48. #2524C22semidirect
49. #2520C22semidirect
50. #2519C22semidirect
51. #2515C22semidirect
52. #2512C22semidirect
53. #2511C22semidirect
54. #2507C22semidirect
55. #2506C22semidirect
60. #2497C22semidirect
61. #2496C22semidirect
68. #2373C22semidirect
69. #2371C22semidirect
71. #2357C22semidirect
72. #2356C22semidirect
73. #2355C22semidirect
79. #2331C22semidirect
84. #2314C22semidirect
88. C4×#125C22semidirect
104. #2283C22semidirect

Maximal subgroups

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

subgroupindex
#593312
#593302
#394722
#378872
#366252
#362052
#362012
#355572
#355552
#353392
#353342
#351572
#342152
#341892
#341792
C4×#5252
C2×#22852