Finite Group Databasebeta

← #647641

Subgroups — #647641

Order 512.

2570 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C22×#387C211yesmaximal, characteristic
3#17044C214yesmaximal, complemented
4#16565C212yesmaximal, complemented
5#16517C212yesmaximal, complemented
6#15088C212yesmaximal, complemented
7#15019C211yesmaximal, characteristic
8#14935C212yesmaximal
9#14934C211yesmaximal, characteristic
10#14879C212yesmaximal, complemented
11#14821C211yesmaximal, characteristic
12#14728C212yesmaximal
13#14720C211yesmaximal, characteristic
14#14682C211yesmaximal, characteristic
15#14316C214yesmaximal, complemented
16#14040C212yesmaximal
17#13992C212yesmaximal
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C2×#398C2214yes—
24C2×#398C2212yes—
25C2×#395C2212yes—
26C4×C22⋊Q8—28no—
27C2×#391C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#387—28no—
30C2×#387C2212yes—
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#382C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1446C2212yes—
51#1445C2212yes—
52#1445C2214yes—
53#1444C2214yes—
54#1444C2214yes—
55#1443C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2212yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74#1434C2212yes—
75#1434C2212yes—
76#1434C2214yes—
77#1434C2214yes—
78C22⋊#95C2211yescharacteristic
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2211yescharacteristic
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—28no—
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42—44no—
119C22×C42D418yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#398—216no—
127#398—44no—
128#395—44no—
129#391—24no—
130#389—24no—
131#387—28no—
132#387—28no—
133#387—28no—
134#387—28no—
135#387—28no—
136#385—28no—
137#385—28no—
138#385—44no—
139#382—44no—
140#381—28no—
141#381—44no—
142#380—24no—
143#380—44no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—24no—
146C4×C22⋊C4—28no—
147C2×#95—24no—
148C2×#95—24no—
149C2×#95—24no—
150C2×#95—24no—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—42no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C22⋊Q8—232no—
238C4×Q8—416no—
239C2×C4⋊C4—232no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—416no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4#4118yes—
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—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4#4118yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—416no—
293C2×C42—44no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—44no—
323#95—42no—
324#95—42no—
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—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24C22×D412yes—
339C24C2×#4114yes—
340C24H(5,𝔽2)12yes—
341C242−1+414yes—
342C242−1+412yes—
343C24C2511yescharacteristic
344C2×Q8—28no—
345C2×Q8—416no—
346C2×Q8—44no—
347C2×Q8—44no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—416no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—88no—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C42−1+414yes—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C22⋊C4—216no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C42—28no—
479C42—28no—
480C42—44no—
481C42—44no—
482C42—48no—
483C42—48no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23#52514yes—
497C23#53614yes—
498C23#53714yes—
499C23#54118yes—
500C23#54218yes—
501C23C23×D411yescharacteristic
502C23C22×#4112yes—
503C23C22×#4112yes—
504C23C2×H(5,𝔽2)12yes—
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×2−1+412yes—
507C23C2×2−1+412yes—
508C23C2×2−1+411yescharacteristic
509C23#58512yes—
510Q8—416no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—416no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—88no—
596C2×C4—88no—
597C2×C4#52514yes—
598C2×C4#53414yes—
599C2×C4#53614yes—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—216no—
605C22—216no—
606C22—24no—
607C22—24no—
608C22#263718yes—
609C22C2×#52511yescharacteristic
610C22C2×#52511yescharacteristic
611C22C2×#53411yescharacteristic
612C22C2×#53611yescharacteristic
613C22C2×#53611yescharacteristic
614C22C2×#53711yescharacteristic
615C22#344911yescharacteristic
616C22C2×#54112yes—
617C22C2×#54212yes—
618C22#348214yes—
619C22#349514yes—
620C22#351314yes—
621C22#351912yes—
622C22#352012yes—
623C22#353814yes—
624C22#355912yes—
625C22#356312yes—
626C4—24no—
627C4—24no—
628C4—24no—
629C4—42no—
630C4—44no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—44no—
635C4—44no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—84no—
640C4—84no—
641C4—82no—
642C2—28no—
643C2C2×#263711yescharacteristic
644C2#3497111yescharacteristic
645C2#3506211yescharacteristic
646C2#3507711yescharacteristic
647C2#3532511yescharacteristic
648C2#3745311yescharacteristic
649C2#3754911yescharacteristic
650C2#3861014yes—
651C2#3861114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17044C2semidirect
4. #16565C2semidirect
5. #16517C2semidirect
6. #15088C2semidirect
10. #14879C2semidirect
15. #14316C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C22×#3872
#170442
#165652
#165172
#150882
#150192
#149352
#149342
#148792
#148212
#147282
#147202
#146822
#143162
#140402
#139922