Finite Group Databasebeta

← #780310

Subgroups — #780310

Order 512.

3728 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic, complemented
2C2×#2421C211yesmaximal, characteristic
3C2×#2429C212yesmaximal
4#16605C212yesmaximal, complemented
5#15880C214yesmaximal
6#15847C214yesmaximal, complemented
7#15188C211yesmaximal, characteristic, complemented
8#15178C211yesmaximal, characteristic
9#15158C211yesmaximal, characteristic, complemented
10#14933C212yesmaximal
11#14877C212yesmaximal
12#14186C212yesmaximal, complemented
13#14024C212yesmaximal, complemented
14#13978C212yesmaximal, complemented
15#13969C212yesmaximal
16C4×#393C211yesmaximal, characteristic
17Q8×#95C211yesmaximal, characteristic, complemented
18C2×#557—216no—
19C2×#557—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C22×#125C2214yes—
23C22×#125C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×C4×Q8C2212yes—
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2212yes—
29C22×C4×Q8C2212yes—
30C22×C4×Q8C2211yescharacteristic
31#2429—216no—
32#2421—28no—
33C2×#400C2212yes—
34C2×#398C2214yes—
35C2×#397C2212yes—
36C2×#396C2214yes—
37C2×#395C2212yes—
38C2×#393C2211yescharacteristic
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#391C2211yescharacteristic
42C2×#391C2212yes—
43C2×#389C2211yescharacteristic
44C2×#389C2212yes—
45C2×#389C2214yes—
46C2×#388C2214yes—
47C2×#388C2212yes—
48C2×#387C2212yes—
49C2×#382C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53#1446C2211yescharacteristic
54#1444C2212yes—
55#1444C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2211yescharacteristic
59#1442C2212yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2211yescharacteristic
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2312yes—
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C24×C4C2311yescharacteristic
91#557—2128no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8D4116yes—
97C2×#125—216no—
98C2×#125—216no—
99C2×#125—216no—
100C2×#125—216no—
101C2×#125—28no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8—44no—
107C2×C4×Q8—28no—
108C2×C4×Q8—28no—
109C2×C4×Q8—28no—
110C2×C4×Q8—216no—
111C2×C4×Q8—28no—
112C2×C4×Q8—28no—
113C2×C4×Q8—28no—
114C2×C4×Q8—28no—
115C2×C4×Q8—28no—
116C2×C4×Q8—24no—
117C2×C4×Q8—44no—
118C2×C4×Q8D418yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2311yescharacteristic
142C22×C22⋊C4C2312yes—
143C22×C42C2314yes—
144C22×C42C2312yes—
145C22×C42C2311yescharacteristic
146C22×C42C2314yes—
147C22×C42C2312yes—
148C22×C42C2312yes—
149C22×C42C2311yescharacteristic
150#400—28no—
151#398—216no—
152#397—28no—
153#396—216no—
154#395—28no—
155#393—24no—
156#391—24no—
157#391—28no—
158#391—28no—
159#391—44no—
160#389—216no—
161#389—24no—
162#389—28no—
163#388—216no—
164#388—28no—
165#387—28no—
166#382—24no—
167#380—24no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4—28no—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2311yescharacteristic
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C25C2411yescharacteristic
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—22no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—24no—
214C22×Q8C2×D418yes—
215C22×Q8C2×D418yes—
216C22×Q8C2×D414yes—
217C22×Q8#4114yes—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C4⋊Q8—232no—
240C4⋊Q8—264no—
241#125—2128no—
242#125—264no—
243C22⋊Q8—432no—
244C22⋊Q8—416no—
245C4×Q8—264no—
246C4×Q8—264no—
247C4×Q8—232no—
248C4×Q8—232no—
249C4×Q8—432no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D4116yes—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×D4116yes—
293C2×C4⋊C4#4118yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
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—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—41no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C42—28no—
315C2×C42—28no—
316C2×C42—28no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—28no—
323C2×C42—22no—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—44no—
329C2×C42C2×D418yes—
330C2×C42C2×D418yes—
331C2×C42C2×D414yes—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—42no—
357#95—44no—
358#95—44no—
359#95—42no—
360#95—44no—
361#95—44no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—21no—
372C242−1+412yes—
373C24C2511yescharacteristic
374C2×Q8—28no—
375C2×Q8—216no—
376C2×Q8—28no—
377C2×Q8—216no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—216no—
381C2×Q8—216no—
382C2×Q8—216no—
383C2×Q8—216no—
384C2×Q8—216no—
385C2×Q8—28no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8#120116yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D418yes—
475C22×C4C22×D414yes—
476C22×C4C22×D418yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D414yes—
481C22×C4C2×#4118yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C42−1+414yes—
487C22×C42−1+414yes—
488C22×C42−1+412yes—
489C22×C42−1+412yes—
490C4⋊C4—216no—
491C4⋊C4—28no—
492C4⋊C4—232no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—216no—
497C4⋊C4—232no—
498C4⋊C4—232no—
499C4⋊C4—216no—
500C4⋊C4—416no—
501C4⋊C4—416no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—416no—
506C4⋊C4—48no—
507C22⋊C4—216no—
508C22⋊C4—416no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C42—232no—
515C42—216no—
516C42—216no—
517C42—28no—
518C42—416no—
519C42—44no—
520C42—48no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23C23×D411yescharacteristic
545C23C23×D412yes—
546C23C22×#4112yes—
547C23C22×#4111yescharacteristic
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×2−1+411yescharacteristic
550C23C2×2−1+412yes—
551C23C2×2−1+411yescharacteristic
552C23C2×2−1+412yes—
553C23C2×2−1+411yescharacteristic
554C23#58511yescharacteristic
555Q8—232no—
556Q8—232no—
557Q8—432no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—88no—
645C2×C4C2×#12018yes—
646C2×C4C2×#12014yes—
647C2×C4#53214yes—
648C2×C4#53714yes—
649C2×C4#54618yes—
650C2×C4#54718yes—
651C2×C4#54818yes—
652C2×C4#54814yes—
653C22—44no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—82no—
659C22—82no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22C22×#12011yescharacteristic
665C22C2×#53211yescharacteristic
666C22C2×#53711yescharacteristic
667C22C2×#54612yes—
668C22C2×#54712yes—
669C22C2×#54812yes—
670C22C2×#54811yescharacteristic
671C22#348212yes—
672C22#348612yes—
673C22#348612yes—
674C22#348611yescharacteristic
675C22#348911yescharacteristic
676C22#349712yes—
677C22#349911yescharacteristic
678C22#351512yes—
679C22#351711yescharacteristic
680C22#352312yes—
681C22#352312yes—
682C22#352311yescharacteristic
683C22#355211yescharacteristic
684C22#355212yes—
685C22#355511yescharacteristic
686C22#355612yes—
687C4—28no—
688C4—28no—
689C4—24no—
690C4—44no—
691C4—48no—
692C4—44no—
693C4—44no—
694C4—42no—
695C4—48no—
696C4—82no—
697C4—81no—
698C4—81no—
699C4—81no—
700C4—82no—
701C4—82no—
702C4#243018yes—
703C2—82no—
704C2C2×#243011yescharacteristic
705C2#3568711yescharacteristic
706C2#3594911yescharacteristic
707C2#3650811yescharacteristic
708C2#3657511yescharacteristic
709C2#3912711yescharacteristic
710C2#4279111yescharacteristic
711C2#4532412yes—
712C2#4533612yes—
713C2#5907112yes—
714C2#5941812yes—
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. #16605C2semidirect
6. #15847C2semidirect
7. #15188C2semidirect
9. #15158C2semidirect
12. #14186C2semidirect
13. #14024C2semidirect
14. #13978C2semidirect
17. Q8×#95C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#24212
C2×#24292
#166052
#158802
#158472
#151882
#151782
#151582
#149332
#148772
#141862
#140242
#139782
#139692
C4×#3932
Q8×#952