Finite Group Databasebeta

← #4306620

Subgroups — #4306620

Order 512.

2626 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#15244C211yesmaximal, characteristic
4#15231C212yesmaximal, complemented
5#15222C212yesmaximal
6#15157C211yesmaximal, characteristic
7#15152C211yesmaximal, characteristic, complemented
8#15100C212yesmaximal, complemented
9#14960C214yesmaximal, complemented
10#14887C212yesmaximal, complemented
11#14866C212yesmaximal, complemented
12#14774C214yesmaximal, complemented
13C4×#395C211yesmaximal, characteristic, complemented
14#13852C212yesmaximal, complemented
15#13821C211yesmaximal, characteristic, complemented
16#13820C212yesmaximal
17#13804C212yesmaximal, complemented
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C2×#400C2212yes—
22C2×#395C2211yescharacteristic
23C2×#393C2212yes—
24#2355—28no—
25#2354—28no—
26C2×#391C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#389C2212yes—
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#387C2214yes—
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#382C2212yes—
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2214yes—
49#1446C2211yescharacteristic
50#1445C2212yes—
51#1444C2212yes—
52#1444C2211yescharacteristic
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2211yescharacteristic
58#1441C2212yes—
59#1441C2214yes—
60#1441C2212yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2214yes—
80#1434C2212yes—
81C22⋊#95C2214yes—
82C4×#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4×Q8—24no—
89C2×C4×Q8—28no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4Q818yescomplemented
93C2×#117—28no—
94C2×#117—28no—
95C2×#117—44no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2314yes—
130#400—44no—
131#395—24no—
132#393—44no—
133#391—28no—
134#390—24no—
135#390—24no—
136#390—28no—
137#390—28no—
138#389—28no—
139#388—28no—
140#388—28no—
141#387—216no—
142#384—24no—
143#384—24no—
144#382—28no—
145#382—28no—
146#380—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—44no—
154C4×C4⋊C4D418yes—
155C4×C22⋊C4—48no—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C25C2412yes—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4C2×Q814yes—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4C2×Q818yes—
202C23×C4C2411yescharacteristic
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C4×Q8—232no—
219C4×D4—232no—
220#117—432no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×Q814yes—
262C2×C4⋊C4#4114yes—
263C2×C4⋊C4#4114yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C2×Q818yes—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—42no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42C2×D418yes—
310C2×C42C2×D414yes—
311C2×C42C2×Q814yes—
312C2×C42#4118yes—
313C2×C42#4118yes—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—22no—
318#95—22no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—22no—
331#95—24no—
332#95—42no—
333#95—44no—
334#95—42no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—42no—
340#95—44no—
341#95C2×D418yes—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24C22×Q814yes—
353C24C2511yescharacteristic
354C2×Q8—28no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×D4—28no—
360C2×D4—216no—
361C2×D4—44no—
362C2×D4—48no—
363C2×D4—44no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
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—24no—
413C22×C4—21no—
414C22×C4—216no—
415C22×C4—216no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q814yes—
452C22×C4C22×Q812yes—
453C22×C4C22×Q812yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4114yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)12yes—
467C4⋊C4—28no—
468C4⋊C4—216no—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—416no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4C22⋊Q8116yes—
483C22⋊C4—216no—
484C22⋊C4—48no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C22⋊C4—88no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4—88no—
492C42—28no—
493C42—28no—
494C42—216no—
495C42—216no—
496C42—216no—
497C42—48no—
498C42—44no—
499C42—88no—
500C42—88no—
501C23—24no—
502C23—28no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—44no—
524C23C23×D411yescharacteristic
525C23C23×Q811yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4112yes—
528C23C22×#4112yes—
529C23C22×#4111yescharacteristic
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×H(5,𝔽2)11yescharacteristic
532C23#58512yes—
533C23#58511yescharacteristic
534C23#58512yes—
535Q8—416no—
536D4—416no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
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—24no—
575C2×C4—432no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4C2×C22⋊Q818yes—
627C2×C4C2×C22⋊Q814yes—
628C2×C4#52514yes—
629C2×C4#52918yes—
630C2×C4#53214yes—
631C2×C4#53318yes—
632C2×C4#54614yes—
633C2×C4#55414yes—
634C22—44no—
635C22—48no—
636C22—48no—
637C22—44no—
638C22—44no—
639C22—82no—
640C22—84no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—84no—
645C22C22×C22⋊Q811yescharacteristic
646C22C2×#52511yescharacteristic
647C22C2×#52912yes—
648C22C2×#53211yescharacteristic
649C22C2×#53312yes—
650C22C2×#54611yescharacteristic
651C22C2×#55411yescharacteristic
652C22#347511yescharacteristic
653C22#347512yes—
654C22#348012yes—
655C22#348711yescharacteristic
656C22#348811yescharacteristic
657C22#349312yes—
658C22#349412yes—
659C22#350011yescharacteristic
660C22#350111yescharacteristic
661C22#350412yes—
662C22#350612yes—
663C22#350612yes—
664C22#350612yes—
665C22#351012yes—
666C22#351911yescharacteristic
667C22#352412yes—
668C4—24no—
669C4—28no—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—42no—
674C4—44no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—44no—
679C4—82no—
680C4—84no—
681C4—81no—
682C4—84no—
683C4#239618yes—
684C2—84no—
685C2C2×#239611yescharacteristic
686C2#3575611yescharacteristic
687C2#3578512yes—
688C2#3595212yes—
689C2#3595611yescharacteristic
690C2#3604011yescharacteristic
691C2#3615411yescharacteristic
692C2#3641612yes—
693C2#3641911yescharacteristic
694C2#3642012yes—
695C2#3642611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2355C2semidirect
4. #15231C2semidirect
7. #15152C2semidirect
8. #15100C2semidirect
9. #14960C2semidirect
10. #14887C2semidirect
11. #14866C2semidirect
12. #14774C2semidirect
13. C4×#395C2semidirect
14. #13852C2semidirect
15. #13821C2semidirect
17. #13804C2semidirect
92. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#23542
#152442
#152312
#152222
#151572
#151522
#151002
#149602
#148872
#148662
#147742
C4×#3952
#138522
#138212
#138202
#138042