Finite Group Databasebeta

← #763972

Subgroups — #763972

Order 512.

3620 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic, complemented
3C2×#2322C211yesmaximal, characteristic
4C2×#2316C211yesmaximal, characteristic, complemented
5#15860C214yesmaximal, complemented
6#15633C214yesmaximal, complemented
7#15623C214yesmaximal, complemented
8#15614C214yesmaximal, complemented
9#14729C212yesmaximal
10#14728C212yesmaximal, complemented
11#14149C211yesmaximal, characteristic, complemented
12#14135C212yesmaximal
13#13962C212yesmaximal, complemented
14#13960C211yesmaximal, characteristic
15#13922C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23#2409—28no—
24C2×#398C2214yes—
25C2×#396C2214yes—
26C2×#395—28no—
27C2×#395C2212yes—
28C2×#395C2214yes—
29C2×#395C2214yes—
30C2×#393C2212yes—
31#2322—28no—
32#2316—28no—
33C2×#391C2212yes—
34C2×#391C2214yes—
35C2×#389C2214yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#388C2212yes—
39C2×#387C2212yes—
40C2×#386C2211yescharacteristic
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#385C2211yescharacteristic
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#381C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2212yes—
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C22×#95C2212yes—
53C22×#95C2211yescharacteristic
54#1446C2214yes—
55#1444C2214yes—
56#1442C2214yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2214yes—
72#1434C2214yes—
73#1434C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—48no—
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#123—24no—
93C2×#123—24no—
94C2×#123Q818yescomplemented
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—48no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×#117—28no—
101C2×#117—28no—
102C2×#117—44no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—48no—
107C22×C4⋊C4D418yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2314yes—
133C22×C42C2312yes—
134#398—48no—
135#396—48no—
136#395—216no—
137#395—216no—
138#395—28no—
139#393—28no—
140#391—28no—
141#391—48no—
142#389—216no—
143#389—28no—
144#389—216no—
145#388—28no—
146#387—28no—
147#386—24no—
148#386—28no—
149#386—44no—
150#385—24no—
151#385—28no—
152#385—28no—
153#385—44no—
154#381—24no—
155#380—24no—
156#380—28no—
157#380D418yes—
158C4×C22⋊C4—24no—
159C2×#95—28no—
160C2×#95—24no—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C25C2411yescharacteristic
186C25C2411yescharacteristic
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—28no—
194C22×Q8—28no—
195C22×Q8—44no—
196C22×Q8—44no—
197C22×Q8—432no—
198C22×Q8—432no—
199C22×Q8—432no—
200C22×Q8—416no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4C2×Q814yes—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—28no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×Q814yes—
235C23×C4#4114yes—
236C23×C4#4114yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2414yes—
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250#123—232no—
251C22⋊Q8—432no—
252C4×D4—416no—
253#117—432no—
254C2×C4⋊C4—232no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—432no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2×D418yes—
289C2×C4⋊C4C2×D418yes—
290C2×C4⋊C4C2×D418yes—
291C2×C4⋊C4C2×Q818yescomplemented
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4C2×D418yes—
319C2×C22⋊C4C2×Q814yes—
320C2×C22⋊C4C2×Q818yescomplemented
321C2×C42—28no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—28no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—44no—
328C2×C42—44no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—28no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95C2×D418yes—
353C24—24no—
354C24—24no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—24no—
360C24—21no—
361C24—24no—
362C24—21no—
363C24C22×D412yes—
364C24C22×D412yes—
365C24C22×Q812yes—
366C24C22×Q812yes—
367C24C2×#4112yes—
368C24C2×#4112yes—
369C24H(5,𝔽2)12yes—
370C242−1+412yes—
371C24C2511yescharacteristic
372C2×Q8—216no—
373C2×Q8—216no—
374C2×Q8—416no—
375C2×Q8—416no—
376C2×Q8—416no—
377C2×Q8—416no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—4128no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—28no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—44no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4#121116yes—
476C22×C4C22⋊Q8116yescomplemented
477C22×C4C22×D418yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C22×Q814yes—
485C22×C4C22×Q812yes—
486C22×C4C22×Q814yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4114yes—
490C22×C4H(5,𝔽2)18yes—
491C22×C42−1+418yes—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—416no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—44no—
501C4⋊C4—84no—
502C4⋊C4—88no—
503C4⋊C4—84no—
504C4⋊C4—88no—
505C22⋊C4—28no—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—416no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—84no—
514C22⋊C4—88no—
515C42—416no—
516C42—44no—
517C42—84no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—28no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23C2×#12114yes—
546C23C2×C22⋊Q814yes—
547C23#53714yes—
548C23#54614yes—
549C23D4×Q814yes—
550C23#55214yes—
551C23#55414yes—
552C23C23×D411yescharacteristic
553C23C23×D411yescharacteristic
554C23C23×Q811yescharacteristic
555C23C22×#4111yescharacteristic
556C23C22×#4111yescharacteristic
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×H(5,𝔽2)14yes—
559C23C2×2−1+411yescharacteristic
560C23C2×2−1+414yes—
561Q8—432no—
562Q8—432no—
563D4—416no—
564C2×C4—24no—
565C2×C4—232no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—416no—
595C2×C4—432no—
596C2×C4—432no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—88no—
639C2×C4—88no—
640C2×C4C2×#12114yes—
641C2×C4C2×#12114yes—
642C2×C4C2×C22⋊Q818yes—
643C22—24no—
644C22—24no—
645C22—28no—
646C22—28no—
647C22—28no—
648C22—28no—
649C22—28no—
650C22—28no—
651C22—44no—
652C22—42no—
653C22—44no—
654C22—44no—
655C22—81no—
656C22—81no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22#241718yes—
661C22C22×#12111yescharacteristic
662C22C22×C22⋊Q811yescharacteristic
663C22C2×#53711yescharacteristic
664C22C2×#54611yescharacteristic
665C22C2×D4×Q811yescharacteristic
666C22C2×#55211yescharacteristic
667C22C2×#55411yescharacteristic
668C22#348412yes—
669C22#348412yes—
670C22#348612yes—
671C22#349012yes—
672C22#349912yes—
673C22#350314yes—
674C22#350912yes—
675C22#351114yes—
676C22#353614yes—
677C22#353814yes—
678C4—24no—
679C4—24no—
680C4—28no—
681C4—48no—
682C4—44no—
683C4—48no—
684C4—48no—
685C4—42no—
686C4—44no—
687C4—81no—
688C4—81no—
689C4—82no—
690C4—84no—
691C2—28no—
692C2—82no—
693C2C2×#241711yescharacteristic
694C2#3534811yescharacteristic
695C2#3557412yes—
696C2#3566711yescharacteristic
697C2#3671912yes—
698C2#4388812yes—
699C2#4389112yes—
700C2#4395812yes—
701C2#4396112yes—
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
2. C22×#395C2semidirect
4. C2×#2316C2semidirect
5. #15860C2semidirect
6. #15633C2semidirect
7. #15623C2semidirect
8. #15614C2semidirect
10. #14728C2semidirect
11. #14149C2semidirect
13. #13962C2semidirect
15. #13922C2semidirect
94. C2×#123Q8semidirect
291. C2×C4⋊C4C2×Q8semidirect
320. C2×C22⋊C4C2×Q8semidirect
476. C22×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C22×#3952
C2×#23222
C2×#23162
#158602
#156332
#156232
#156142
#147292
#147282
#141492
#141352
#139622
#139602
#139222