Finite Group Databasebeta

← #533181

Subgroups — #533181

Order 512.

3536 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2362C212yesmaximal, complemented
3C2×C4×C22⋊Q8C212yesmaximal, complemented
4#14543C212yesmaximal, complemented
5#14424C212yesmaximal, complemented
6#14194C212yesmaximal, complemented
7#14178C214yesmaximal, complemented
8#14036C214yesmaximal
9#14024C212yesmaximal, complemented
10#13997C212yesmaximal, complemented
11#13997C212yesmaximal, complemented
12#13871C211yesmaximal, characteristic
13#13821C212yesmaximal, complemented
14C4⋊C42C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2542C4116yescomplemented
24C2×#399C2212yes—
25C2×#399C2212yes—
26#2362—216no—
27C4×C22⋊Q8—216no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×#387C2214yes—
38C2×#386C2212yes—
39C2×#385C2214yes—
40C2×#384C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2214yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2214yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2214yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2214yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52#1445C2214yes—
53#1445C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1437C2212yes—
61#1437C2214yes—
62#1437C2211yescharacteristic
63#1437C2214yes—
64#1437C2211yescharacteristic
65#1436C2212yes—
66#1436C2211yescharacteristic
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2211yescharacteristic
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2214yes—
74C4⋊#95C2214yes—
75#1434C2214yes—
76C22⋊#95C2212yes—
77C4×#95C2212yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C4⋊Q8—42no—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8C2×C418yescomplemented
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8C2×C4116yes—
91C2×C22⋊Q8C2×C4116yescomplemented
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—44no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C2×C4×D4C2×C418yes—
98C2×#117—28no—
99C2×#117—42no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C42—28no—
122C22×C42—42no—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2314yes—
129C22×C42C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2312yes—
133#399C2×C4116yes—
134#399C2×C4116yescomplemented
135#391—28no—
136#391—28no—
137#391—44no—
138#390C2×C4116yes—
139#389—28no—
140#389D4116yescomplemented
141#388—216no—
142#388—216no—
143#387—216no—
144#387—216no—
145#386—28no—
146#385—216no—
147#384C2×C4116yescomplemented
148#380—28no—
149#380—42no—
150C4×C4⋊C4—216no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—216no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—216no—
155C4×C4⋊C4C2×C418yescomplemented
156C4×C22⋊C4—28no—
157C4×C22⋊C4—216no—
158C4×C22⋊C4—42no—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8C22×C418yes—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4C22×C414yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4C22×C414yes—
212C23×C4C22×C414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C4⋊Q8—416no—
227C22⋊Q8—264no—
228C22⋊Q8—264no—
229C4×Q8—432no—
230C4×D4—416no—
231#117—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4C22×C4116yes—
256C2×C4⋊C4C22×C4116yes—
257C2×C4⋊C4C22×C414yes—
258C2×C4⋊C4C22×C418yes—
259C2×C4⋊C4C22×C418yes—
260C2×C4⋊C4C22×C418yes—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×D4116yes—
263C2×C4⋊C4C2×D418yes—
264C2×C4⋊C4#4118yes—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4C22×C4116yes—
290C2×C22⋊C4C22×C4116yes—
291C2×C22⋊C4C22×C414yes—
292C2×C22⋊C4C22×C414yes—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—416no—
311C2×C42—42no—
312C2×C42—44no—
313C2×C42—41no—
314C2×C42—41no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42C22×C414yes—
318C2×C42C22×C418yes—
319C2×C42C2×D418yes—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—28no—
332#95—24no—
333#95—44no—
334#95—42no—
335#95—41no—
336#95—41no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95C22×C418yes—
341#95C2×D4116yes—
342#95#4118yes—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—21no—
347C24—21no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24C23×C412yes—
359C24C23×C412yes—
360C242−1+414yes—
361C24C2511yescharacteristic
362C2×Q8—216no—
363C2×Q8—216no—
364C2×Q8—216no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—216no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4C23×C412yes—
453C22×C4C23×C418yes—
454C22×C4C23×C412yes—
455C22×C4C23×C412yes—
456C22×C4C23×C418yes—
457C22×C4C23×C412yes—
458C22×C4C23×C412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D418yes—
461C22×C4C22×D418yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4118yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C42−1+414yes—
472C4⋊C4—232no—
473C4⋊C4—232no—
474C4⋊C4—216no—
475C4⋊C4—232no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—416no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4C4×D4132yes—
483C22⋊C4—232no—
484C22⋊C4—232no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—84no—
493C42—216no—
494C42—28no—
495C42—216no—
496C42—416no—
497C42—416no—
498C42—44no—
499C42—44no—
500C42—48no—
501C42—416no—
502C42—48no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23#51914yes—
545C23C24×C411yescharacteristic
546C23C23×D412yes—
547C23C22×#4112yes—
548C23C22×#4112yes—
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×2−1+412yes—
552C23C2×2−1+412yes—
553C23#58512yes—
554Q8—432no—
555D4—416no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—432no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—416no—
617C2×C4—416no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4C2×C4×D4116yes—
633C2×C4C2×C4×D418yes—
634C2×C4C4×#4114yes—
635C2×C4#51814yes—
636C2×C4D4214yes—
637C2×C4#54718yes—
638C2×C4#55018yes—
639C2×C4#55914yes—
640C22—24no—
641C22—28no—
642C22—42no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—44no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22C22×C4×D412yes—
659C22C2×C4×#4111yescharacteristic
660C22C2×#51811yescharacteristic
661C22C2×#51911yescharacteristic
662C22C4×2−1+412yes—
663C22C2×D4211yescharacteristic
664C22C2×#54712yes—
665C22C2×#55012yes—
666C22D4×#4112yes—
667C22#347512yes—
668C22C2×#55911yescharacteristic
669C22#348212yes—
670C22#348612yes—
671C22#348912yes—
672C22#351612yes—
673C22#352012yes—
674C22#352012yes—
675C22#352312yes—
676C22#354112yes—
677C22#355011yescharacteristic
678C22#355111yescharacteristic
679C4—216no—
680C4—28no—
681C4—24no—
682C4—42no—
683C4—42no—
684C4—48no—
685C4—42no—
686C4—48no—
687C4—44no—
688C4—44no—
689C4—82no—
690C4—84no—
691C4#236618yes—
692C2—44no—
693C2—82no—
694C2C2×#236611yescharacteristic
695C2C4×#54712yes—
696C2#3424511yescharacteristic
697C2#3429112yes—
698C2#3441311yescharacteristic
699C2#5900212yes—
700C2#5905512yes—
701C2#5905912yes—
702C2#5959312yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2362C2semidirect
3. C2×C4×C22⋊Q8C2semidirect
4. #14543C2semidirect
5. #14424C2semidirect
6. #14194C2semidirect
7. #14178C2semidirect
9. #14024C2semidirect
10. #13997C2semidirect
11. #13997C2semidirect
13. #13821C2semidirect
14. C4⋊C42C2semidirect
23. #2542C4semidirect
85. C2×C4⋊Q8C2×C4semidirect
91. C2×C22⋊Q8C2×C4semidirect
134. #399C2×C4semidirect
140. #389D4semidirect
147. #384C2×C4semidirect
155. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#23622
C2×C4×C22⋊Q82
#145432
#144242
#141942
#141782
#140362
#140242
#139972
#138712
#138212
C4⋊C422
C4×#3862