Finite Group Databasebeta

← #3233668

Subgroups — #3233668

Order 512.

1588 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17263C211yesmaximal, characteristic, complemented
2#17221C211yesmaximal, characteristic, complemented
3#17051C212yesmaximal, complemented
4#16738C212yesmaximal
5#16492C212yesmaximal
6#16201C212yesmaximal, complemented
7#16088C211yesmaximal, characteristic, complemented
8#15989C212yesmaximal
9#15976C211yesmaximal, characteristic, complemented
10#15767C212yesmaximal, complemented
11#15653C212yesmaximal
12#15490C212yesmaximal
13#15444C212yesmaximal
14#15016C211yesmaximal, characteristic
15#14791C212yesmaximal, complemented
16#14491C212yesmaximal, complemented
17#14480C212yesmaximal, complemented
18#14295C211yesmaximal, characteristic
19#14225C211yesmaximal, characteristic
20C2×#400C2212yes—
21C2×#399C2211yescharacteristic
22C2×#398C2212yes—
23C2×#397C2211yescharacteristic
24C2×#397C2211yescharacteristic
25C2×#389C2212yes—
26C2×#388C2212yes—
27C2×#387C2212yes—
28C2×#387C2212yes—
29C2×#387C2212yes—
30C2×#387C2212yes—
31C2×#382C2211yescharacteristic
32C2×#381C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#1445C2211yescharacteristic
37#1445C2212yes—
38#1445C2212yes—
39#1445C2211yescharacteristic
40#1445C2211yescharacteristic
41#1445C2212yes—
42#1445C2212yes—
43#1445C2212yes—
44#1444C2212yes—
45#1444C2212yes—
46#1444C2211yescharacteristic
47#1444C2212yes—
48#1444C2211yescharacteristic
49#1444C2212yes—
50#1444C2211yescharacteristic
51#1443C2212yes—
52#1443C2212yes—
53#1443C2212yes—
54#1443C2212yes—
55#1443C2212yes—
56#1443C2212yes—
57#1443C2212yes—
58#1442C2212yes—
59#1442C2212yes—
60#1442C2212yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2212yes—
65#1442C2211yescharacteristic
66#1442C2212yes—
67#1442C2211yescharacteristic
68#1441C2212yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2212yes—
73#1441C2212yes—
74#1441C2212yes—
75#1440C2212yes—
76#1439C2212yes—
77#1439C2212yes—
78#1439C2212yes—
79#1439C2212yes—
80#1439C2212yes—
81#1439C2212yes—
82#1439C2212yes—
83#1439C2211yescharacteristic
84#1439C2212yes—
85#1437C2212yes—
86#1437C2212yes—
87#1436C2212yes—
88#1436C2212yes—
89#1436C2212yes—
90#1436C2212yes—
91#1436C2212yes—
92#1436C2212yes—
93#1436C2212yes—
94#1436C2212yes—
95C4⋊#95C2212yes—
96C4⋊#95C2212yes—
97#1434C2211yescharacteristic
98#1434C2212yes—
99#1434C2212yes—
100#1434C2212yes—
101#1434C2212yes—
102#1434C2212yes—
103#1434C2212yes—
104#1434C2211yescharacteristic
105C22⋊#95C2212yes—
106C22⋊#95C2212yes—
107C24×C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#400—44no—
138#399—42no—
139#398—44no—
140#397—24no—
141#397—42no—
142#389—44no—
143#388—44no—
144#387—28no—
145#387—28no—
146#387—44no—
147#387—44no—
148#382—42no—
149#381—24no—
150#380—42no—
151C4×C4⋊C4—42no—
152C4×C22⋊C4—42no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2311yescharacteristic
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C2×#95C2312yes—
207C2×#95C2311yescharacteristic
208C2×#95C2311yescharacteristic
209C2×#95C2311yescharacteristic
210C25C2411yescharacteristic
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—42no—
316#95—24no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—42no—
336#95—42no—
337#95—42no—
338#95—42no—
339#95—42no—
340#95—42no—
341#95—42no—
342#95—41no—
343#95—42no—
344#95—42no—
345#95—42no—
346#95—42no—
347#95—42no—
348#95—42no—
349#95—42no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—42no—
354#95—41no—
355#95—41no—
356#95—42no—
357#95—42no—
358#95—42no—
359#95—42no—
360#95—42no—
361#95—42no—
362#95—42no—
363#95—42no—
364#95—41no—
365#95—42no—
366#95—42no—
367#95—42no—
368#95—41no—
369#95—42no—
370#95—42no—
371#95#4114yes—
372#95#4114yes—
373C24—21no—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—21no—
378C24—22no—
379C24—21no—
380C24—22no—
381C24—22no—
382C24—22no—
383C242−1+412yes—
384C24C2511yescharacteristic
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—84no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—84no—
496C22×C4C2×#4112yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4114yes—
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C42−1+414yes—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—44no—
529C42—44no—
530C42—84no—
531C42—84no—
532C42—84no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23C22×#4111yescharacteristic
557C23C22×#4111yescharacteristic
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×2−1+411yescharacteristic
562C23C2×2−1+412yes—
563C23#58511yescharacteristic
564C23#58512yes—
565C23#58512yes—
566C23#58512yes—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—88no—
682C22—42no—
683C22—44no—
684C22—42no—
685C22—42no—
686C22—44no—
687C22—81no—
688C22—82no—
689C22—81no—
690C22—82no—
691C22—81no—
692C22—81no—
693C22#346011yescharacteristic
694C22#348311yescharacteristic
695C22#349211yescharacteristic
696C22#349512yes—
697C22#349512yes—
698C22#351411yescharacteristic
699C22#352011yescharacteristic
700C22#352112yes—
701C22#352112yes—
702C22#353311yescharacteristic
703C22#353311yescharacteristic
704C22#353312yes—
705C22#353512yes—
706C22#353711yescharacteristic
707C22#353912yes—
708C22#354011yescharacteristic
709C22#355011yescharacteristic
710C22#355412yes—
711C22#355712yes—
712C22#356111yescharacteristic
713C22#356312yes—
714C22#356412yes—
715C22#356712yes—
716C4—42no—
717C4—42no—
718C4—42no—
719C4—44no—
720C4—44no—
721C4—42no—
722C4—44no—
723C4—42no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—81no—
732C4—82no—
733C4—82no—
734C2—82no—
735C2#3710011yescharacteristic
736C2#3882311yescharacteristic
737C2#4221111yescharacteristic
738C2#4387812yes—
739C2#4715111yescharacteristic
740C2#4778911yescharacteristic
741C2#5025712yes—
742C2#5126512yes—
743C2#5381511yescharacteristic
744C2#5579312yes—
745C2#5666111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17263C2semidirect
2. #17221C2semidirect
3. #17051C2semidirect
6. #16201C2semidirect
7. #16088C2semidirect
9. #15976C2semidirect
10. #15767C2semidirect
15. #14791C2semidirect
16. #14491C2semidirect
17. #14480C2semidirect

Maximal subgroups

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

subgroupindex
#172632
#172212
#170512
#167382
#164922
#162012
#160882
#159892
#159762
#157672
#156532
#154902
#154442
#150162
#147912
#144912
#144802
#142952
#142252