Finite Group Databasebeta

← #760379

Subgroups — #760379

Order 512.

3930 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3C2×#2546C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊Q8C211yesmaximal, characteristic
5#16683C214yesmaximal, complemented
6#15236C212yesmaximal
7#15179C214yesmaximal
8#15178C212yesmaximal
9#14906C214yesmaximal
10#14155C211yesmaximal, characteristic, complemented
11#14155C211yesmaximal, characteristic, complemented
12#14024C212yesmaximal, complemented
13#14017C214yesmaximal, complemented
14#13978C212yesmaximal, complemented
15C4×#380C211yesmaximal, characteristic
16C2×#533—28no—
17C2×#533—28no—
18C2×#533—28no—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125C2212yes—
23C22×C22⋊Q8C2211yescharacteristic
24C22×C22⋊Q8C2212yes—
25C22×C22⋊Q8C2212yes—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2212yes—
28C22×C4×Q8C2212yes—
29C22×#117C2212yes—
30C23×C42C2211yescharacteristic
31#2568—28no—
32#2546—28no—
33C2×#400C2214yes—
34C2×#398C2214yes—
35C4×C22⋊Q8—28no—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#391C2214yes—
39C2×#391C2214yes—
40C2×#391C2212yes—
41C2×#391C2211yescharacteristic
42C2×#391C2212yes—
43C2×#389C2214yes—
44C2×#389C2211yescharacteristic
45C2×#387C2212yes—
46C2×#385C2212yes—
47C2×#382C2212yes—
48C2×#380C2212yes—
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×C4×C22⋊C4C2212yes—
56#1445C2214yes—
57#1445C2214yes—
58#1444C2218yes—
59#1442C2218yes—
60#1442C2218yes—
61#1441C2218yes—
62#1440C2218yes—
63#1439C2218yes—
64#1436C2218yes—
65#1436C2218yes—
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2212yes—
71C22⋊#95C2212yes—
72C4×#95C2212yes—
73C23×Q8C2312yes—
74C23×Q8C2312yes—
75C23×Q8C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79#533—2128no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—44no—
87C2×C4⋊Q8D418yes—
88C2×C4⋊Q8D418yes—
89C2×#125—28no—
90C2×#125—28no—
91C2×#125—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—24no—
101C2×C4×Q8—28no—
102C2×C4×Q8—28no—
103C2×C4×Q8—28no—
104C2×C4×Q8—28no—
105C2×C4×Q8—28no—
106C2×C4×Q8—28no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×#117—28no—
111C2×#117—28no—
112C2×#117—28no—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2318yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C42—24no—
133C22×C42—24no—
134C22×C42D418yes—
135C22×C42C2314yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#400—216no—
145#398—216no—
146#391—216no—
147#391—28no—
148#391—28no—
149#391—28no—
150#391—216no—
151#391—24no—
152#391—44no—
153#389—216no—
154#389—24no—
155#387—28no—
156#385—28no—
157#382—28no—
158#380—24no—
159#380—28no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—24no—
164C4×C22⋊C4—28no—
165C4×C22⋊C4—44no—
166C2×#95C2312yes—
167C2×#95C2318yes—
168C2×#95C2312yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C2×#95C2318yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2318yes—
183C25C2411yescharacteristic
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—28no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—28no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8C2×D418yes—
201C22×Q8C2×D418yes—
202C22×Q8#4114yes—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—232no—
235C4⋊Q8—232no—
236C4⋊Q8—432no—
237#125—264no—
238C22⋊Q8—232no—
239C22⋊Q8—264no—
240C22⋊Q8—264no—
241C4×Q8—264no—
242C4×Q8—264no—
243C4×Q8—416no—
244#117—264no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—216no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—216no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4#4114yes—
288C2×C4⋊C4#4118yes—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—44no—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—28no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—232no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—216no—
335#95—216no—
336#95—216no—
337#95—28no—
338#95—216no—
339#95—216no—
340#95—216no—
341#95—28no—
342#95—44no—
343#95—48no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—21no—
353C24C22×D412yes—
354C242−1+414yes—
355C24C2511yescharacteristic
356C2×Q8—216no—
357C2×Q8—216no—
358C2×Q8—28no—
359C2×Q8—28no—
360C2×Q8—216no—
361C2×Q8—216no—
362C2×Q8—232no—
363C2×Q8—28no—
364C2×Q8—44no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—416no—
373C2×Q8—48no—
374C2×Q8—44no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4C22×D418yes—
451C22×C4C22×D412yes—
452C22×C4C22×D418yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C2×#4118yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4114yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C42−1+414yes—
472C22×C42−1+414yes—
473C22×C42−1+412yes—
474C4⋊C4—232no—
475C4⋊C4—232no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—28no—
482C4⋊C4—216no—
483C4⋊C4—432no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C22⋊C4—232no—
490C22⋊C4—216no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C42—232no—
494C42—28no—
495C42—28no—
496C42—28no—
497C42—48no—
498C42—48no—
499C42—48no—
500C42—48no—
501C42#120116yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23#53614yes—
525C23C23×D411yescharacteristic
526C23C23×D411yescharacteristic
527C23C23×D411yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4112yes—
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×2−1+412yes—
533C23C2×2−1+412yes—
534C23C2×2−1+412yes—
535C23C2×2−1+411yescharacteristic
536Q8—432no—
537Q8—416no—
538Q8—432no—
539C2×C4—232no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—216no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4C2×#12014yes—
611C2×C4C2×#12014yes—
612C2×C4C2×#12014yes—
613C2×C4#52514yes—
614C2×C4#53214yes—
615C2×C4#53614yes—
616C2×C4#53614yes—
617C2×C4#53714yes—
618C2×C4#54614yes—
619C2×C4#54714yes—
620C2×C4#54718yes—
621C2×C4#54814yes—
622C2×C4#54818yes—
623C2×C4#55014yes—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—42no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22C22×#12011yescharacteristic
636C22C2×#52511yescharacteristic
637C22C2×#53211yescharacteristic
638C22C2×#53611yescharacteristic
639C22C2×#53611yescharacteristic
640C22C2×#53611yescharacteristic
641C22C2×#53711yescharacteristic
642C22C2×#54611yescharacteristic
643C22C2×#54711yescharacteristic
644C22C2×#54712yes—
645C22C2×#54811yescharacteristic
646C22C2×#54812yes—
647C22C2×#55011yescharacteristic
648C22#348212yes—
649C22#348212yes—
650C22#348612yes—
651C22#348612yes—
652C22#352012yes—
653C22#352312yes—
654C22#352312yes—
655C22#352312yes—
656C22#355012yes—
657C22#355612yes—
658C4—24no—
659C4—24no—
660C4—42no—
661C4—48no—
662C4—44no—
663C4—42no—
664C4—48no—
665C4—48no—
666C4—44no—
667C4—82no—
668C4—82no—
669C4—84no—
670C4#240718yes—
671C4#243218yes—
672C2—44no—
673C2C2×#240711yescharacteristic
674C2C2×#243211yescharacteristic
675C2#3497911yescharacteristic
676C2#3524111yescharacteristic
677C2#3524211yescharacteristic
678C2#3524811yescharacteristic
679C2#3764111yescharacteristic
680C2#5904212yes—
681C2#5905512yes—
682C2#5907112yes—
683C2#5907112yes—
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
2. C2×#2568C2semidirect
3. C2×#2546C2semidirect
5. #16683C2semidirect
10. #14155C2semidirect
11. #14155C2semidirect
12. #14024C2semidirect
13. #14017C2semidirect
14. #13978C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#25682
C2×#25462
C2×C4×C22⋊Q82
#166832
#152362
#151792
#151782
#149062
#141552
#140242
#140172
#139782
C4×#3802