Finite Group Databasebeta

← #776744

Subgroups — #776744

Order 512.

3600 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic, complemented
2C2×#2489C212yesmaximal
3C2×#2429C211yesmaximal, characteristic
4#16506C212yesmaximal
5#15881C214yesmaximal, complemented
6#15880C214yesmaximal
7#15230C212yesmaximal, complemented
8#15201C211yesmaximal, characteristic, complemented
9#15191C211yesmaximal, characteristic, complemented
10#15125C211yesmaximal, characteristic
11#14884C212yesmaximal, complemented
12#14188C212yesmaximal
13#14054C212yesmaximal, complemented
14#14024C212yesmaximal, complemented
15#13969C212yesmaximal
16C4×#393C211yesmaximal, characteristic
17#13817C211yesmaximal, characteristic, complemented
18C2×#557—216no—
19C2×#557—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2214yes—
24C22×#125C2212yes—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2212yes—
29C22×C4×Q8C2212yes—
30C22×C4×Q8C2211yescharacteristic
31#2489—216no—
32#2429—28no—
33C2×#400C2214yes—
34C2×#398C2212yes—
35C2×#397C2214yes—
36C2×#396C2212yes—
37C2×#393C2212yes—
38C2×#393C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#391C2212yes—
43C2×#389C2212yes—
44C2×#389C2211yescharacteristic
45C2×#388C2214yes—
46C2×#388C2212yes—
47C2×#382C2214yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52#1446C2212yes—
53#1446C2211yescharacteristic
54#1446C2212yes—
55#1446C2212yes—
56#1446C2212yes—
57#1445C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2212yes—
62#1442C2211yescharacteristic
63#1442C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2211yescharacteristic
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C24×C4C2311yescharacteristic
92#557—2128no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—216no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—28no—
99C2×#125—216no—
100C2×#125—216no—
101C2×#125—216no—
102C2×#125—28no—
103C2×#125—28no—
104C2×#125D4116yes—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—28no—
109C2×C4×Q8—28no—
110C2×C4×Q8—28no—
111C2×C4×Q8—216no—
112C2×C4×Q8—28no—
113C2×C4×Q8—28no—
114C2×C4×Q8—24no—
115C2×C4×Q8—28no—
116C2×C4×Q8—28no—
117C2×C4×Q8D418yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2311yescharacteristic
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2311yescharacteristic
141C22×C42C2314yes—
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144C22×C42C2314yes—
145C22×C42C2312yes—
146C22×C42C2312yes—
147C22×C42C2311yescharacteristic
148#400—216no—
149#398—28no—
150#397—216no—
151#396—28no—
152#393—28no—
153#393—24no—
154#391—28no—
155#391—28no—
156#391—24no—
157#391—28no—
158#389—24no—
159#389—28no—
160#388—216no—
161#388—28no—
162#382—216no—
163#380—28no—
164C4×C4⋊C4—28no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—28no—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2311yescharacteristic
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2314yes—
197C25C2411yescharacteristic
198C22×Q8—24no—
199C22×Q8—28no—
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—44no—
215C22×Q8—42no—
216C22×Q8C2×D414yes—
217C22×Q8#4118yes—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C4⋊Q8—264no—
240C4⋊Q8—232no—
241C4⋊Q8—416no—
242#125—2128no—
243#125—264no—
244C22⋊Q8—416no—
245C22⋊Q8—416no—
246C4×Q8—264no—
247C4×Q8—264no—
248C4×Q8—232no—
249C4×Q8—232no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4C2×D4116yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D4116yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D414yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4#4114yes—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42C2×D418yes—
324C2×C42C2×D414yes—
325C2×C42C2×D418yes—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—28no—
343#95—22no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—42no—
350#95—44no—
351#95—44no—
352#95—42no—
353#95—44no—
354#95—44no—
355#95—44no—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—21no—
366C24C2×#4112yes—
367C24C2511yescharacteristic
368C2×Q8—216no—
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—216no—
373C2×Q8—216no—
374C2×Q8—416no—
375C2×Q8—48no—
376C2×Q8—416no—
377C2×Q8—48no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8#124116yes—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4C22×D414yes—
467C22×C4C22×D418yes—
468C22×C4C22×D412yes—
469C22×C4C22×D414yes—
470C22×C4C22×D418yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4118yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4114yes—
480C22×C42−1+414yes—
481C22×C42−1+412yes—
482C22×C42−1+412yes—
483C22×C42−1+412yes—
484C22×C42−1+412yes—
485C4⋊C4—28no—
486C4⋊C4—216no—
487C4⋊C4—232no—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—232no—
491C4⋊C4—216no—
492C4⋊C4—232no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4#120116yes—
501C22⋊C4—28no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C42—232no—
509C42—216no—
510C42—216no—
511C42—28no—
512C42—416no—
513C42—44no—
514C42—48no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—42no—
538C23C23×D411yescharacteristic
539C23C23×D412yes—
540C23C22×#4112yes—
541C23C22×#4111yescharacteristic
542C23C2×2−1+411yescharacteristic
543C23C2×2−1+411yescharacteristic
544C23C2×2−1+411yescharacteristic
545C23C2×2−1+412yes—
546C23C2×2−1+411yescharacteristic
547C23#58512yes—
548C23#58511yescharacteristic
549Q8—232no—
550Q8—432no—
551Q8—416no—
552Q8—88no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—88no—
638C2×C4C2×#12018yes—
639C2×C4C2×#12014yes—
640C2×C4C2×#12418yes—
641C2×C4C2×#12414yes—
642C2×C4#53614yes—
643C2×C4#54814yes—
644C2×C4#54818yes—
645C2×C4#54818yes—
646C22—44no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—81no—
652C22—82no—
653C22—81no—
654C22—81no—
655C22—82no—
656C22—81no—
657C22C22×#12011yescharacteristic
658C22C22×#12411yescharacteristic
659C22C2×#53611yescharacteristic
660C22C2×#54812yes—
661C22C2×#54812yes—
662C22C2×#54811yescharacteristic
663C22D4×#4112yes—
664C22#348212yes—
665C22#348711yescharacteristic
666C22#348911yescharacteristic
667C22#348912yes—
668C22#348912yes—
669C22#349912yes—
670C22#349911yescharacteristic
671C22#350711yescharacteristic
672C22#350911yescharacteristic
673C22#351612yes—
674C22#352312yes—
675C22#352412yes—
676C22#355512yes—
677C22#355511yescharacteristic
678C22#355611yescharacteristic
679C22#356312yes—
680C4—24no—
681C4—28no—
682C4—28no—
683C4—42no—
684C4—48no—
685C4—44no—
686C4—42no—
687C4—44no—
688C4—44no—
689C4—42no—
690C4—82no—
691C4—82no—
692C4—81no—
693C4—82no—
694C4—84no—
695C4#243618yes—
696C2—82no—
697C2C2×#243611yescharacteristic
698C2#3585411yescharacteristic
699C2#3608511yescharacteristic
700C2#3642811yescharacteristic
701C2#3644611yescharacteristic
702C2#3972411yescharacteristic
703C2#4287911yescharacteristic
704C2#4530012yes—
705C2#4534312yes—
706C2#5908212yes—
707C2#5965612yes—
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. C22×#557C2semidirect
5. #15881C2semidirect
7. #15230C2semidirect
8. #15201C2semidirect
9. #15191C2semidirect
11. #14884C2semidirect
13. #14054C2semidirect
14. #14024C2semidirect
17. #13817C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#24892
C2×#24292
#165062
#158812
#158802
#152302
#152012
#151912
#151252
#148842
#141882
#140542
#140242
#139692
C4×#3932
#138172