Finite Group Databasebeta

← #554197

Subgroups — #554197

Order 512.

3102 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C2×#2325C211yesmaximal, characteristic
3C22×#387C211yesmaximal, characteristic, complemented
4#14696C212yesmaximal, complemented
5#14582C214yesmaximal, complemented
6#14573C214yesmaximal, complemented
7#14433C214yesmaximal, complemented
8#14413C214yesmaximal, complemented
9#14259C212yesmaximal, complemented
10#14137C211yesmaximal, characteristic
11#14016C212yesmaximal, complemented
12#14016C212yesmaximal, complemented
13#13923C211yesmaximal, characteristic, complemented
14#13880C211yesmaximal, characteristic, complemented
15#13796C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2591C4116yescomplemented
22C2×#400C2212yes—
23C2×#394C2212yes—
24#2325—28no—
25C2×#390C2212yes—
26C2×#388C2212yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#387—28no—
30C2×#387C2212yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#384C2212yes—
43C2×#380C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1444C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2212yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C2×#121—24no—
89C2×#121—24no—
90C2×#121C2×C418yes—
91C2×C4×D4—28no—
92C2×C4×D4—44no—
93C2×C4×D4—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4C2×C418yescomplemented
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42—24no—
121C22×C42—44no—
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126#400C2×C4116yescomplemented
127#394C2×C4116yescomplemented
128#390—44no—
129#388—28no—
130#388—48no—
131#388C2×C4132yescomplemented
132#387—28no—
133#387—216no—
134#387—216no—
135#386—28no—
136#386—28no—
137#386—44no—
138#386—44no—
139#386C2×C4116yescomplemented
140#386C2×C4116yescomplemented
141#385—28no—
142#385—44no—
143#385—44no—
144#384—44no—
145#380—24no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4C2×C418yescomplemented
148C2×#95—24no—
149C2×#95—24no—
150C2×#95—24no—
151C2×#95D418yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C25C2412yes—
177C25C2411yescharacteristic
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4C22×C418yes—
189C22×D4C22×C414yes—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C22×C414yes—
220C23×C4C22×C414yes—
221C23×C4C22×C414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4#4114yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239#121—232no—
240C4×D4—432no—
241C2×C4⋊C4—232no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—416no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C22×C418yes—
269C2×C4⋊C4C22×C418yes—
270C2×C4⋊C4C22×C414yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
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—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4C22×C418yes—
299C2×C22⋊C4C22×C418yes—
300C2×C22⋊C4C22×C4116yes—
301C2×C22⋊C4C22×C4116yes—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—44no—
310C2×C42—42no—
311C2×C42—416no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42C22×C418yes—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95C22×C418yes—
335#95C22×C4116yes—
336#95C22×C4116yes—
337#95C2×D418yes—
338#95C2×D418yes—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24C23×C414yes—
354C24C23×C412yes—
355C24C22×D412yes—
356C24C2×#4112yes—
357C24H(5,𝔽2)14yes—
358C242−1+414yes—
359C24C2511yescharacteristic
360C2×D4—216no—
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—28no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—84no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
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—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
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—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4C4×D4116yes—
467C22×C4C23×C414yes—
468C22×C4C23×C418yes—
469C22×C4C23×C414yes—
470C22×C4C23×C414yes—
471C22×C4C23×C412yes—
472C22×C4C23×C412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)14yes—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C42—48no—
502C42—416no—
503C42—48no—
504C42—84no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—28no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23C2×C4×D414yes—
541C23#52018yes—
542C23#53514yes—
543C23#53614yes—
544C23#54218yes—
545C23C24×C411yescharacteristic
546C23C23×D411yescharacteristic
547C23C22×#4111yescharacteristic
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)12yes—
550C23C2×2−1+412yes—
551C23#58512yes—
552C23#58512yes—
553C23#58512yes—
554D4—432no—
555D4—88no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—416no—
584C2×C4—416no—
585C2×C4—416no—
586C2×C4—416no—
587C2×C4—416no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—88no—
632C2×C4—816no—
633C2×C4—88no—
634C2×C4C2×C4×D414yes—
635C2×C4C2×C4×D414yes—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—216no—
642C22—216no—
643C22—24no—
644C22—44no—
645C22—44no—
646C22—48no—
647C22—48no—
648C22—44no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—84no—
654C22—84no—
655C22#232618yes—
656C22C22×C4×D411yescharacteristic
657C22C2×#52012yes—
658C22#342812yes—
659C22C4×H(5,𝔽2)12yes—
660C22C2×#53511yescharacteristic
661C22C2×#53611yescharacteristic
662C22C2×#54212yes—
663C22#348411yescharacteristic
664C22#348511yescharacteristic
665C22#348511yescharacteristic
666C22#348711yescharacteristic
667C22#349312yes—
668C22#350512yes—
669C22#353414yes—
670C22#354114yes—
671C22#355314yes—
672C22#356014yes—
673C4—24no—
674C4—24no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—48no—
681C4—44no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—84no—
686C2—28no—
687C2—84no—
688C2C2×#232611yescharacteristic
689C2#3420111yescharacteristic
690C2#3426411yescharacteristic
691C2#3449714yes—
692C2#3670511yescharacteristic
693C2#3671011yescharacteristic
694C2#3678114yes—
695C2#3696911yescharacteristic
696C2#3712011yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#387C2semidirect
4. #14696C2semidirect
5. #14582C2semidirect
6. #14573C2semidirect
7. #14433C2semidirect
8. #14413C2semidirect
9. #14259C2semidirect
11. #14016C2semidirect
12. #14016C2semidirect
13. #13923C2semidirect
14. #13880C2semidirect
15. #13796C2semidirect
21. #2591C4semidirect
98. C22×C4⋊C4C2×C4semidirect
126. #400C2×C4semidirect
127. #394C2×C4semidirect
131. #388C2×C4semidirect
139. #386C2×C4semidirect
140. #386C2×C4semidirect
147. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C2×#23252
C22×#3872
#146962
#145822
#145732
#144332
#144132
#142592
#141372
#140162
#139232
#138802
#137962