Finite Group Databasebeta

← #776123

Subgroups — #776123

Order 512.

3670 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic, complemented
2C2×#2442C212yesmaximal
3C2×#2360C211yesmaximal, characteristic
4#16560C212yesmaximal, complemented
5#15996C214yesmaximal
6#15847C214yesmaximal, complemented
7#15230C211yesmaximal, characteristic, complemented
8#15202C211yesmaximal, characteristic, complemented
9#15125C212yesmaximal
10#15125C211yesmaximal, characteristic
11#14878C212yesmaximal, complemented
12#14172C212yesmaximal
13#14054C212yesmaximal, complemented
14#13978C212yesmaximal, complemented
15#13969C212yesmaximal
16#13935C211yesmaximal, characteristic, complemented
17C4×#393C211yesmaximal, characteristic
18C2×#557—216no—
19C2×#557—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C22×#125C2214yes—
23C22×#125C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2212yes—
28C22×C4×Q8C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30#2442—216no—
31C2×#400C2214yes—
32C2×#398C2214yes—
33C2×#393C2211yescharacteristic
34C2×#393C2212yes—
35#2360—28no—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#389C2211yescharacteristic
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#387C2212yes—
43C2×#385C2212yes—
44C2×#385C2214yes—
45C2×#382C2212yes—
46C2×#380C2214yes—
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52#1446C2212yes—
53#1446C2212yes—
54#1446C2212yes—
55#1446C2211yescharacteristic
56#1446C2212yes—
57#1445C2214yes—
58#1444C2212yes—
59#1444C2214yes—
60#1442C2214yes—
61#1442C2212yes—
62#1442C2211yescharacteristic
63#1442C2212yes—
64#1442C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2211yescharacteristic
72#1439C2212yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84#1434C2212yes—
85C4×#95C2211yescharacteristic
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C23×Q8C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92#557—2128no—
93C2×C4⋊Q8—216no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—24no—
97C2×#125—216no—
98C2×#125—216no—
99C2×#125—216no—
100C2×#125—28no—
101C2×#125—28no—
102C2×#125D4116yes—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8Q818yescomplemented
106C2×C4×Q8—28no—
107C2×C4×Q8—28no—
108C2×C4×Q8—216no—
109C2×C4×Q8—28no—
110C2×C4×Q8—28no—
111C2×C4×Q8—28no—
112C2×C4×Q8—24no—
113C2×C4×Q8—28no—
114C2×C4×Q8D418yes—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—28no—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2311yescharacteristic
140C22×C42C2314yes—
141C22×C42C2311yescharacteristic
142C22×C42C2312yes—
143C22×C42C2312yes—
144C22×C42C2311yescharacteristic
145C22×C42C2312yes—
146C22×C42C2312yes—
147#400—216no—
148#398—216no—
149#393—24no—
150#393—28no—
151#391—24no—
152#391—28no—
153#389—28no—
154#389—28no—
155#389—24no—
156#389—28no—
157#387—28no—
158#385—28no—
159#385—216no—
160#382—28no—
161#380—216no—
162#380—28no—
163C4×C4⋊C4—24no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—28no—
166C4×C22⋊C4—28no—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2311yescharacteristic
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C25C2411yescharacteristic
198C22×Q8—24no—
199C22×Q8—28no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—24no—
208C22×Q8—42no—
209C22×Q8—44no—
210C22×Q8C2×D414yes—
211C22×Q8C2×Q814yes—
212C22×Q8#4114yes—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4C2×Q814yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C4⋊Q8—264no—
240C4⋊Q8—232no—
241#125—2128no—
242#125—264no—
243C22⋊Q8—232no—
244C4×Q8—264no—
245C4×Q8—264no—
246C4×Q8—232no—
247C4×Q8—232no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—416no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4C2×D4116yes—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D4116yes—
289C2×C4⋊C4C2×D418yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×Q814yes—
292C2×C4⋊C4C2×Q818yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4C2×Q818yes—
310C2×C22⋊C4#4114yes—
311C2×C42—28no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—28no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—44no—
325C2×C42C2×D418yes—
326C2×C42C2×D418yes—
327C2×C42C2×D414yes—
328#95—22no—
329#95—24no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—28no—
345#95—22no—
346#95—24no—
347#95—22no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—42no—
353#95—44no—
354#95—42no—
355#95—42no—
356#95—42no—
357#95—42no—
358C24—21no—
359C24—22no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—22no—
364C24—21no—
365C24—22no—
366C24C22×Q812yes—
367C24C2×#4112yes—
368C242−1+412yes—
369C24C2511yescharacteristic
370C2×Q8—216no—
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—216no—
375C2×Q8—216no—
376C2×Q8—416no—
377C2×Q8—48no—
378C2×Q8—416no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—44no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8C22⋊Q8116yes—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—28no—
427C22×C4—216no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—48no—
460C22×C4—416no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—84no—
466C22×C4—88no—
467C22×C4C22×D418yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×D418yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D414yes—
477C22×C4C22×Q814yes—
478C22×C4C22×Q812yes—
479C22×C4C22×Q814yes—
480C22×C4C22×Q812yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C42−1+412yes—
486C22×C42−1+414yes—
487C4⋊C4—28no—
488C4⋊C4—216no—
489C4⋊C4—232no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—232no—
493C4⋊C4—216no—
494C4⋊C4—232no—
495C4⋊C4—28no—
496C4⋊C4—216no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—88no—
501C4⋊C4—84no—
502C4⋊C4—88no—
503C4⋊C4#120116yes—
504C22⋊C4—216no—
505C22⋊C4—28no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C42—232no—
511C42—216no—
512C42—216no—
513C42—28no—
514C42—44no—
515C42—48no—
516C42—84no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23#55714yescomplemented
540C23C23×D411yescharacteristic
541C23C23×D412yes—
542C23C23×Q811yescharacteristic
543C23C22×#4111yescharacteristic
544C23C22×#4111yescharacteristic
545C23C2×2−1+412yes—
546C23C2×2−1+411yescharacteristic
547C23C2×2−1+411yescharacteristic
548C23#58512yes—
549C23#58511yescharacteristic
550C23#58512yes—
551Q8—232no—
552Q8—432no—
553Q8—416no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
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—44no—
606C2×C4—42no—
607C2×C4—416no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—816no—
640C2×C4C2×#12018yes—
641C2×C4C2×#12014yes—
642C2×C4C2×C22⋊Q818yes—
643C2×C4C2×C22⋊Q814yes—
644C2×C4#53614yes—
645C2×C4#54814yes—
646C2×C4#54818yes—
647C2×C4D4×Q818yes—
648C22—28no—
649C22—24no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—44no—
659C22C22×#12011yescharacteristic
660C22C22×C22⋊Q811yescharacteristic
661C22C2×#53611yescharacteristic
662C22C2×#54811yescharacteristic
663C22C2×#54812yes—
664C22C2×D4×Q812yes—
665C22D4×#4112yes—
666C22C2×#55711yescharacteristic
667C22#348011yescharacteristic
668C22#348711yescharacteristic
669C22#348912yes—
670C22#348912yes—
671C22#348911yescharacteristic
672C22#350011yescharacteristic
673C22#350012yes—
674C22#350211yescharacteristic
675C22#350412yes—
676C22#351512yes—
677C22#352311yescharacteristic
678C22#352412yes—
679C22#355512yes—
680C22#356312yes—
681C22#356512yes—
682C4—24no—
683C4—28no—
684C4—28no—
685C4—42no—
686C4—48no—
687C4—44no—
688C4—42no—
689C4—44no—
690C4—44no—
691C4—42no—
692C4—82no—
693C4—84no—
694C4—81no—
695C4—82no—
696C4—82no—
697C4#240918yes—
698C2—44no—
699C2C2×#240911yescharacteristic
700C2#3604011yescharacteristic
701C2#3607811yescharacteristic
702C2#3642811yescharacteristic
703C2#3657711yescharacteristic
704C2#3917311yescharacteristic
705C2#4288811yescharacteristic
706C2#4527412yes—
707C2#4530012yes—
708C2#5915412yes—
709C2#5965812yes—
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
4. #16560C2semidirect
6. #15847C2semidirect
7. #15230C2semidirect
8. #15202C2semidirect
11. #14878C2semidirect
13. #14054C2semidirect
14. #13978C2semidirect
16. #13935C2semidirect
105. C2×C22⋊Q8Q8semidirect
539. C23#557semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#24422
C2×#23602
#165602
#159962
#158472
#152302
#152022
#151252
#148782
#141722
#140542
#139782
#139692
#139352
C4×#3932