Finite Group Databasebeta

← #556951

Subgroups — #556951

Order 512.

2598 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2817C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×C4×C22⋊Q8C211yesmaximal, characteristic
4#14674C211yesmaximal, characteristic
5#14660C211yesmaximal, characteristic
6#14660C211yesmaximal, characteristic
7#14327C212yesmaximal, complemented
8#14326C212yesmaximal, complemented
9#14307C212yesmaximal, complemented
10#14304C214yesmaximal
11#14302C212yesmaximal, complemented
12#14242C212yesmaximal, complemented
13#14241C212yesmaximal, complemented
14#14239C212yesmaximal, complemented
15#14202C214yesmaximal
16#14174C212yesmaximal, complemented
17C4×#381C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2817C4116yescomplemented
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#397C2214yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C4×C22⋊Q8—28no—
30C4×C22⋊Q8—28no—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#387C2214yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#381C2211yescharacteristic
41C2×#381C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51#1446C2212yes—
52#1446C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1443C2214yes—
58#1442C2212yes—
59#1442C2212yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79#1434C2212yes—
80#1434C2214yes—
81#1434C2214yes—
82#1434C2212yes—
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8C2×C418yes—
96C2×C22⋊Q8C2×C418yes—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—44no—
100C2×C4×Q8—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42—44no—
122C22×C42D418yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#400C2×C4116yescomplemented
133#400C2×C4116yescomplemented
134#397C2×C4132yes—
135#395C2×C4116yescomplemented
136#395C2×C4116yescomplemented
137#391—24no—
138#391—24no—
139#389—24no—
140#389—24no—
141#387—28no—
142#387—28no—
143#387—48no—
144#385—28no—
145#385—28no—
146#381—24no—
147#381C2×C418yes—
148#380—24no—
149#380—24no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—44no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C25C2411yescharacteristic
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8C22×C414yes—
198C22×Q8C22×C414yes—
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×C4C22×C414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4#4114yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C22⋊Q8—232no—
229C22⋊Q8—232no—
230C4×Q8—416no—
231C4×Q8—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C22×C418yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C4116yes—
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—24no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4C22×C418yes—
282C2×C22⋊C4C22×C418yes—
283C2×C22⋊C4C22×C418yes—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—416no—
293C2×C42—44no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—44no—
321#95—44no—
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#95C22×C418yes—
333#95C22×C418yes—
334#95C22×C4116yes—
335#95C22×C4116yes—
336#95C22×C4116yes—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24C23×C412yes—
344C24C22×D412yes—
345C24C2×#4112yes—
346C24C2511yescharacteristic
347C2×Q8—28no—
348C2×Q8—28no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—48no—
356C2×Q8—48no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
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—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4C4×D4116yes—
436C22×C4C23×C418yes—
437C22×C4C23×C414yes—
438C22×C4C23×C414yes—
439C22×C4C23×C414yes—
440C22×C4C23×C414yes—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C42−1+414yes—
453C22×C42−1+414yes—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—416no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—88no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C42—28no—
475C42—28no—
476C42—44no—
477C42—44no—
478C42—48no—
479C42—48no—
480C42—48no—
481C42—48no—
482C42—88no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23C2×C4×D414yes—
500C23C24×C411yescharacteristic
501C23C23×D411yescharacteristic
502C23C22×#4111yescharacteristic
503C23C2×H(5,𝔽2)12yes—
504C23C2×2−1+412yes—
505C23C2×2−1+412yes—
506C23#58512yes—
507C23#58512yes—
508C23#58512yes—
509Q8—416no—
510Q8—416no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—416no—
545C2×C4—416no—
546C2×C4—416no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
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×C4C2×C4×D414yes—
594C2×C4C2×C4×D414yes—
595C2×C4#53414yes—
596C2×C4#53614yes—
597C2×C4#53614yes—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—48no—
605C22—42no—
606C22—48no—
607C22C22×C4×D411yescharacteristic
608C22C4×H(5,𝔽2)12yes—
609C22C4×2−1+412yes—
610C22C4×2−1+412yes—
611C22C2×#53411yescharacteristic
612C22C2×#53611yescharacteristic
613C22C2×#53611yescharacteristic
614C22#344911yescharacteristic
615C22#344911yescharacteristic
616C22#344911yescharacteristic
617C22#345912yes—
618C22#346112yes—
619C22#346112yes—
620C22#353514yes—
621C22#354114yes—
622C22#355514yes—
623C22#356614yes—
624C4—24no—
625C4—24no—
626C4—24no—
627C4—24no—
628C4—24no—
629C4—44no—
630C4—44no—
631C4—44no—
632C4—44no—
633C4—42no—
634C4—48no—
635C4—82no—
636C4—82no—
637C4—84no—
638C4—82no—
639C4—82no—
640C2—44no—
641C2C4×#53411yescharacteristic
642C2C4×#53611yescharacteristic
643C2C4×#53611yescharacteristic
644C2#3451614yes—
645C2#3743811yescharacteristic
646C2#3744711yescharacteristic
647C2#3753214yes—
648C2#3755211yescharacteristic
649C2#3763511yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #14327C2semidirect
8. #14326C2semidirect
9. #14307C2semidirect
11. #14302C2semidirect
12. #14242C2semidirect
13. #14241C2semidirect
14. #14239C2semidirect
16. #14174C2semidirect
23. #2817C4semidirect
132. #400C2×C4semidirect
133. #400C2×C4semidirect
135. #395C2×C4semidirect
136. #395C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28172
C2×C4×C22⋊Q82
#146742
#146602
#143272
#143262
#143072
#143042
#143022
#142422
#142412
#142392
#142022
#141742
C4×#3812