Finite Group Databasebeta

← #518717

Subgroups — #518717

Order 512.

3926 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#2364C211yesmaximal, characteristic
3C2×#2359C214yesmaximal, complemented
4C2×#2344C211yesmaximal, characteristic, complemented
5#14322C212yesmaximal, complemented
6#14271C211yesmaximal, characteristic
7#14231C212yesmaximal, complemented
8#14230C212yesmaximal, complemented
9#14084C214yesmaximal, complemented
10#14049C214yesmaximal, complemented
11#13996C214yesmaximal, complemented
12#13965C211yesmaximal, characteristic, complemented
13C4×#399C212yesmaximal, complemented
14C2×#1445C211yesmaximal, characteristic, complemented
15#13749C211yesmaximal, characteristic, complemented
16C22×#123C2214yes—
17C22×#121C2211yescharacteristic
18C22×C4×D4C2212yes—
19C22×#117C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21#2501C4116yescomplemented
22C2×#399C2212yes—
23C2×#399C2212yes—
24#2364—28no—
25#2359—232no—
26#2344—28no—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#389C2214yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#388C2212yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#387C2214yes—
36C2×#387C2212yes—
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#386C2212yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#384C2212yes—
43C2×#381C2212yes—
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2214yes—
47C2×C4×C22⋊C4C2214yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C22×#95C2214yes—
54C22×#95C2211yescharacteristic
55C22×#95C2211yescharacteristic
56#1445C2214yes—
57#1445C2211yescharacteristic
58#1445C2211yescharacteristic
59#1445C2211yescharacteristic
60#1445C2211yescharacteristic
61#1445C2214yes—
62#1445C2214yes—
63#1445C2212yes—
64#1444C2212yes—
65#1444C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1438C2212yes—
71#1437C2214yes—
72#1437C2214yes—
73C4⋊#95C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2214yes—
76C4×#95C2214yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C2×#123—216no—
87C2×#123—216no—
88C2×#123C2×C4132yescomplemented
89C2×#121—24no—
90C2×#121—24no—
91C2×#121C2×C418yes—
92C2×C4×D4—216no—
93C2×C4×D4—28no—
94C2×#117—28no—
95C2×#117—216no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4C2×C418yescomplemented
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128#399C2×C4116yescomplemented
129#399C2×C4116yescomplemented
130#390C2×C4116yescomplemented
131#389—28no—
132#389—216no—
133#388—216no—
134#388—28no—
135#388—216no—
136#387—216no—
137#387—216no—
138#387—216no—
139#387—28no—
140#386—216no—
141#386—28no—
142#386—28no—
143#385—216no—
144#385—216no—
145#384C2×C4116yescomplemented
146#381—28no—
147#380—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—216no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—216no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4C2×C418yescomplemented
154C4×C22⋊C4D418yes—
155C4×C22⋊C4D4116yescomplemented
156C2×#95—216no—
157C2×#95—24no—
158C2×#95—24no—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—44no—
191C22×D4C22×C418yes—
192C22×D4C22×C414yes—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4C22×C418yes—
213C23×C4C22×C414yes—
214C23×C4C2×D418yes—
215C23×C4C2×D414yes—
216C23×C4#4114yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230#123—2128no—
231#121—232no—
232C4×D4—432no—
233#117—432no—
234C2×C4⋊C4—232no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
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—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C22×C418yes—
261C2×C4⋊C4C22×C414yes—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C22×C4116yes—
265C2×C4⋊C4C22×C4116yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C22×C418yes—
289C2×C22⋊C4C22×C4116yes—
290C2×C22⋊C4C22×C4116yes—
291C2×C22⋊C4C22×C418yes—
292C2×C22⋊C4C2×D414yes—
293C2×C22⋊C4C2×D418yes—
294C2×C22⋊C4C2×D414yes—
295C2×C22⋊C4C2×D418yes—
296C2×C22⋊C4#4118yes—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—28no—
302C2×C42—28no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42C22×C418yes—
310C2×C42C2×D418yes—
311C2×C42C2×D4116yes—
312#95—22no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—28no—
317#95—22no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—28no—
332#95C22×C418yes—
333#95C2×D418yes—
334#95C2×D4116yes—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24C23×C414yes—
349C24C23×C412yes—
350C24C22×D414yes—
351C24C22×D412yes—
352C24C2×#4114yes—
353C24C2×#4112yes—
354C24C2511yescharacteristic
355C2×D4—216no—
356C2×D4—232no—
357C2×D4—28no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—416no—
362C2×D4—84no—
363C2×D4—84no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—216no—
374C22×C4—232no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—416no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—88no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—88no—
436C22×C4C4×D4116yes—
437C22×C4C23×C414yes—
438C22×C4C23×C414yes—
439C22×C4C23×C414yes—
440C22×C4C23×C412yes—
441C22×C4C23×C418yes—
442C22×C4C23×C412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D418yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D418yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4118yes—
454C22×C42−1+414yes—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—216no—
458C4⋊C4—28no—
459C4⋊C4—232no—
460C4⋊C4—232no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—88no—
466C22⋊C4—28no—
467C22⋊C4—216no—
468C22⋊C4—232no—
469C22⋊C4—232no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—84no—
475C22⋊C4—88no—
476C22⋊C4C4×D4132yes—
477C22⋊C4#127116yes—
478C42—48no—
479C42—48no—
480C42—416no—
481C42—416no—
482C42—48no—
483C42—88no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—42no—
501C23—44no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23C2×C4×D418yes—
519C23C2×C4×D414yes—
520C23C2×#12714yes—
521C23#53214yes—
522C23#53218yes—
523C23C24×C411yescharacteristic
524C23C23×D411yescharacteristic
525C23C23×D412yes—
526C23C22×#4111yescharacteristic
527C23C22×#4112yes—
528C23C2×2−1+412yes—
529C23#58514yes—
530C23#58512yes—
531D4—432no—
532D4—88no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—416no—
565C2×C4—432no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—816no—
599C2×C4—816no—
600C2×C4—816no—
601C2×C4C2×C4×D4116yes—
602C2×C4C2×C4×D414yes—
603C2×C4C2×C4×D414yes—
604C2×C4C2×#12718yes—
605C22—24no—
606C22—28no—
607C22—28no—
608C22—28no—
609C22—24no—
610C22—28no—
611C22—216no—
612C22—44no—
613C22—44no—
614C22—48no—
615C22—48no—
616C22—44no—
617C22—84no—
618C22—84no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C4×#12718yes—
624C22C22×C4×D411yescharacteristic
625C22C22×C4×D412yes—
626C22#342812yes—
627C22C4×2−1+412yes—
628C22C22×#12711yescharacteristic
629C22C2×#53212yes—
630C22C2×#53211yescharacteristic
631C22#348512yes—
632C22#348514yes—
633C22#348914yes—
634C22#348912yes—
635C22#349112yes—
636C22#350911yescharacteristic
637C22#351011yescharacteristic
638C22#351514yes—
639C22#352114yes—
640C4—24no—
641C4—24no—
642C4—28no—
643C4—216no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—84no—
652C4—82no—
653C2—28no—
654C2—84no—
655C2C2×C4×#12711yescharacteristic
656C2#3419911yescharacteristic
657C2#3426811yescharacteristic
658C2#3428914yes—
659C2#3601212yes—
660C2#3602412yes—
661C2#3604512yes—
662C2#3606012yes—
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. C2×#2359C2semidirect
4. C2×#2344C2semidirect
5. #14322C2semidirect
7. #14231C2semidirect
8. #14230C2semidirect
9. #14084C2semidirect
10. #14049C2semidirect
11. #13996C2semidirect
12. #13965C2semidirect
13. C4×#399C2semidirect
14. C2×#1445C2semidirect
15. #13749C2semidirect
21. #2501C4semidirect
88. C2×#123C2×C4semidirect
98. C22×C4⋊C4C2×C4semidirect
128. #399C2×C4semidirect
129. #399C2×C4semidirect
130. #390C2×C4semidirect
145. #384C2×C4semidirect
153. C4×C22⋊C4C2×C4semidirect
155. C4×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#23642
C2×#23592
C2×#23442
#143222
#142712
#142312
#142302
#140842
#140492
#139962
#139652
C4×#3992
C2×#14452
#137492