Finite Group Databasebeta

← #611395

Subgroups — #611395

Order 512.

2754 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2399C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×C4×C22⋊Q8C211yesmaximal, characteristic
4C2×#2283C211yesmaximal, characteristic
5#16677C212yesmaximal, complemented
6#16663C212yesmaximal, complemented
7#16565C212yesmaximal, complemented
8#16517C212yesmaximal, complemented
9#16399C212yesmaximal
10#16393C212yesmaximal
11#16264C212yesmaximal
12#16262C212yesmaximal
13C4×#398C214yesmaximal, complemented
14C4×#397C214yesmaximal
15#13854C211yesmaximal, characteristic, complemented
16#13850C211yesmaximal, characteristic, complemented
17#13850C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2399—28no—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#397C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C4×C22⋊Q8—28no—
31C4×C22⋊Q8—28no—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#387C2212yes—
37C2×#387C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40#2283—28no—
41C2×#383C2212yes—
42C2×#382C2212yes—
43C2×#381C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2214yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×C4×C22⋊C4C2212yes—
56C2×#376C2211yescharacteristic
57C2×#376C2211yescharacteristic
58#1446C2212yes—
59#1446C2212yes—
60#1445C2214yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2214yes—
64#1444C2214yes—
65#1443C2214yes—
66#1443C2214yes—
67#1442C2212yes—
68#1442C2214yes—
69#1442C2212yes—
70#1442C2214yes—
71#1442C2214yes—
72#1442C2212yes—
73#1442C2212yes—
74#1441C2212yes—
75#1441C2212yes—
76#1441C2214yes—
77#1441C2214yes—
78#1441C2212yes—
79#1441C2212yes—
80#1441C2214yes—
81#1440C2214yes—
82#1440C2214yes—
83#1439C2214yes—
84C4×#95C2214yes—
85C4×#95C2214yes—
86C4×#95C2214yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42—24no—
123C22×C42—24no—
124C22×C42D418yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#398—216no—
136#398—28no—
137#398—28no—
138#397—216no—
139#395—28no—
140#395—28no—
141#391—24no—
142#391—24no—
143#389—24no—
144#389—24no—
145#387—28no—
146#387—28no—
147#385—28no—
148#385—28no—
149#383—28no—
150#382—28no—
151#381—28no—
152#380—24no—
153#380—24no—
154#380—28no—
155C4×C4⋊C4—216no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—24no—
161C4×C22⋊C4—28no—
162C4×C22⋊C4—28no—
163C4×C22⋊C4—28no—
164#376—24no—
165#376D418yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—42no—
206C22×Q8—42no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C22⋊Q8—232no—
239C22⋊Q8—232no—
240C4×Q8—416no—
241C4×Q8—416no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4#4118yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4#4118yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—232no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42C2×D418yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42#4114yes—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95C2×D418yes—
352#95C2×D418yes—
353C24—21no—
354C24—22no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24C22×D412yes—
361C24C2×#4114yes—
362C24C2511yescharacteristic
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
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—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4C22×D412yes—
457C22×C4C22×D418yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4112yes—
474C22×C42−1+414yes—
475C22×C42−1+414yes—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—416no—
481C4⋊C4—416no—
482C4⋊C4—416no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C42—28no—
497C42—28no—
498C42—28no—
499C42—216no—
500C42—48no—
501C42—48no—
502C42—48no—
503C42—48no—
504C42—48no—
505C42#121116yes—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—28no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23#52514yes—
523C23C23×D411yescharacteristic
524C23C23×D411yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4112yes—
527C23C22×#4112yes—
528C23C2×2−1+412yes—
529C23C2×2−1+412yes—
530C23#58512yes—
531C23#58512yes—
532Q8—416no—
533Q8—416no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—416no—
566C2×C4—416no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—416no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
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×C4C2×#12114yes—
618C2×C4C2×#12114yes—
619C2×C4C2×#12114yes—
620C2×C4#52514yes—
621C2×C4#53618yes—
622C2×C4#53614yes—
623C2×C4#53614yes—
624C2×C4#54118yes—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—48no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—48no—
634C22C22×#12111yescharacteristic
635C22C2×#52511yescharacteristic
636C22C2×#52511yescharacteristic
637C22C2×#53611yescharacteristic
638C22C2×#53611yescharacteristic
639C22C2×#53612yes—
640C22#344911yescharacteristic
641C22#344911yescharacteristic
642C22C2×#54112yes—
643C22#346114yes—
644C22#346114yes—
645C22D4×#4112yes—
646C22D4×#4112yes—
647C22#347512yes—
648C22#347512yes—
649C22#348214yes—
650C22#353114yes—
651C4—24no—
652C4—24no—
653C4—24no—
654C4—42no—
655C4—44no—
656C4—44no—
657C4—42no—
658C4—44no—
659C4—44no—
660C4—48no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4#262118yes—
667C2—44no—
668C2#3460211yescharacteristic
669C2C2×#262111yescharacteristic
670C2#3462711yescharacteristic
671C2#3473711yescharacteristic
672C2#3473711yescharacteristic
673C2#3497111yescharacteristic
674C2#3669911yescharacteristic
675C2#5216614yes—
676C2#5225114yes—
subgroupG/H
centerC23×C4C24
commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2399C2semidirect
5. #16677C2semidirect
6. #16663C2semidirect
7. #16565C2semidirect
8. #16517C2semidirect
13. C4×#398C2semidirect
15. #13854C2semidirect
16. #13850C2semidirect
17. #13850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23992
C2×C4×C22⋊Q82
C2×#22832
#166772
#166632
#165652
#165172
#163992
#163932
#162642
#162622
C4×#3982
C4×#3972
#138542
#138502