Finite Group Databasebeta

← #525268

Subgroups — #525268

Order 512.

3158 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic, complemented
3C2×#2323C212yesmaximal, complemented
4C2×#2293C211yesmaximal, characteristic
5#14707C211yesmaximal, characteristic, complemented
6#14563C214yesmaximal
7#14548C214yesmaximal, complemented
8#14264C212yesmaximal, complemented
9#14146C214yesmaximal, complemented
10#14010C214yesmaximal, complemented
11#13973C212yesmaximal, complemented
12#13969C212yesmaximal, complemented
13#13905C211yesmaximal, characteristic
14#13804C211yesmaximal, characteristic, complemented
15C4×#391C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2212yes—
17C22×C4×Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21#2536C4116yescomplemented
22C2×#398C2212yes—
23C2×#394C2212yes—
24#2351—28no—
25#2323—216no—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28#2293—28no—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#388C2214yes—
40C2×#387C2214yes—
41C2×#382C2211yescharacteristic
42C2×#382C2212yes—
43C2×#382C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×C43C2211yescharacteristic
53#1445C2212yes—
54#1444C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2211yescharacteristic
75#1434C2212yes—
76#1434C2214yes—
77C4×#95C2214yes—
78C4×#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8C2×C4116yes—
85C2×C4×Q8—28no—
86C2×C4×Q8—24no—
87C2×C4×Q8—28no—
88C2×C4×Q8—24no—
89C2×C4×Q8—44no—
90C2×C4×Q8C2×C418yes—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4C2×C418yes—
94C2×#117—28no—
95C2×#117—28no—
96C2×#117—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#398C2×C4116yes—
131#394C2×C4116yes—
132#391—24no—
133#391D418yescomplemented
134#390—28no—
135#389—216no—
136#389—28no—
137#389—28no—
138#389—28no—
139#389—216no—
140#388—216no—
141#388—216no—
142#388—28no—
143#388—28no—
144#388C2×C4132yes—
145#387—216no—
146#382—24no—
147#382—28no—
148#382C2×C418yes—
149#380—28no—
150#380—44no—
151#380C2×C4116yes—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4D418yes—
156C4×C22⋊C4—28no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2314yes—
179C2×#95C2311yescharacteristic
180C43—24no—
181C25C2412yes—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8C22×C414yes—
187C22×Q8C2×D414yes—
188C22×Q8#4114yes—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4C22×C414yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4C22×C418yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C22⋊Q8—264no—
216C4×Q8—232no—
217C4×Q8—432no—
218C4×D4—232no—
219#117—432no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C22×C414yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4C22×C418yes—
258C2×C4⋊C4C22×C418yes—
259C2×C4⋊C4C22×C4116yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D418yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C22×C418yes—
287C2×C22⋊C4C22×C4116yes—
288C2×C22⋊C4C22×C4116yes—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—44no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42C22×C418yes—
313C2×C42C22×C414yes—
314C2×C42C2×D414yes—
315C2×C42C2×D418yes—
316C2×C42C2×D414yes—
317C2×C42#4114yes—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95C22×C4116yes—
336#95C22×C418yes—
337#95C22×C418yes—
338#95C2×D414yes—
339#95C2×D414yes—
340#95C2×D4116yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24C23×C414yes—
356C24C2511yescharacteristic
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8C4×D4116yes—
364C2×D4—28no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—44no—
368C2×D4—44no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—216no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4C23×C418yes—
441C22×C4C23×C414yes—
442C22×C4C23×C414yes—
443C22×C4C23×C412yes—
444C22×C4C23×C412yes—
445C22×C4C23×C414yes—
446C22×C4C23×C412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+414yes—
463C22×C42−1+412yes—
464C4⋊C4—28no—
465C4⋊C4—216no—
466C4⋊C4—28no—
467C4⋊C4—216no—
468C4⋊C4—232no—
469C4⋊C4—28no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—416no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—88no—
478C22⋊C4—216no—
479C22⋊C4—232no—
480C22⋊C4—416no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C42—216no—
488C42—28no—
489C42—216no—
490C42—48no—
491C42—416no—
492C42—48no—
493C42—44no—
494C42—84no—
495C42—88no—
496C42C4×D4116yes—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C24×C411yescharacteristic
533C23C23×D411yescharacteristic
534C23C23×D411yescharacteristic
535C23C22×#4111yescharacteristic
536C23C22×#4111yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×2−1+411yescharacteristic
539C23C2×2−1+412yes—
540C23#58512yes—
541C23#58512yes—
542C23#58512yes—
543Q8—232no—
544D4—416no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
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—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
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—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—432no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4C2×C4×D414yes—
625C2×C4C2×C4×D414yes—
626C2×C4C2×C4×D418yes—
627C2×C4C2×C4×D414yes—
628C2×C4C2×C4×D414yes—
629C2×C4#52014yes—
630C2×C4D4214yes—
631C2×C4#54714yes—
632C2×C4#54814yes—
633C2×C4#55314yes—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22C22×C4×D411yescharacteristic
650C22C22×C4×D411yescharacteristic
651C22C2×#52011yescharacteristic
652C22#342812yes—
653C22C4×2−1+412yes—
654C22#344912yes—
655C22#345511yescharacteristic
656C22#345511yescharacteristic
657C22C2×D4211yescharacteristic
658C22C2×#54711yescharacteristic
659C22C2×#54811yescharacteristic
660C22C2×#55311yescharacteristic
661C22#348912yes—
662C22#349712yes—
663C22#350012yes—
664C22#350911yescharacteristic
665C22#351011yescharacteristic
666C22#351612yes—
667C22#352412yes—
668C22#354712yes—
669C22#355112yes—
670C22#356312yes—
671C22#356512yes—
672C4—24no—
673C4—28no—
674C4—24no—
675C4—48no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—82no—
684C4—84no—
685C4#236118yes—
686C2—84no—
687C2#3406511yescharacteristic
688C2C2×#236111yescharacteristic
689C2#3426811yescharacteristic
690C2#3431212yes—
691C2#3448212yes—
692C2#3911912yes—
693C2#3918611yescharacteristic
694C2#3918811yescharacteristic
695C2#3933512yes—
696C2#3937311yescharacteristic
697C2#3937411yescharacteristic
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×#2351C2semidirect
3. C2×#2323C2semidirect
5. #14707C2semidirect
7. #14548C2semidirect
8. #14264C2semidirect
9. #14146C2semidirect
10. #14010C2semidirect
11. #13973C2semidirect
12. #13969C2semidirect
14. #13804C2semidirect
15. C4×#391C2semidirect
21. #2536C4semidirect
133. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
C2×#23512
C2×#23232
C2×#22932
#147072
#145632
#145482
#142642
#141462
#140102
#139732
#139692
#139052
#138042
C4×#3912