Finite Group Databasebeta

← #615227

Subgroups — #615227

Order 512.

2994 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic
3#15206C212yesmaximal, complemented
4#15164C212yesmaximal, complemented
5#15075C214yesmaximal, complemented
6#15015C211yesmaximal, characteristic
7#14914C211yesmaximal, characteristic
8#14822C211yesmaximal, characteristic
9#14720C211yesmaximal, characteristic
10#14695C212yesmaximal
11#14695C212yesmaximal
12#14682C211yesmaximal, characteristic
13#14239C212yesmaximal, complemented
14#14174C212yesmaximal, complemented
15#14149C212yesmaximal
16#14069C214yesmaximal, complemented
17#13923C212yesmaximal
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C23×C42C2211yescharacteristic
24C23×C42C2211yescharacteristic
25C2×#400C2212yes—
26C2×#395C2212yes—
27C4×C22⋊Q8—28no—
28C2×#391C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#387C2214yes—
31C2×#387C2212yes—
32C2×#387C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#382C2212yes—
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4—28no—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1446C2212yes—
50#1445C2212yes—
51#1444C2212yes—
52#1444C2214yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73#1434C2212yes—
74#1434C2212yes—
75#1434C2214yes—
76C22⋊#95C2211yescharacteristic
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
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—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4Q818yescomplemented
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42—24no—
120C22×C42—28no—
121C22×C42—24no—
122C22×C42—44no—
123C22×C42—44no—
124C22×C42D418yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#400—28no—
136#395—28no—
137#391—24no—
138#389—24no—
139#387—28no—
140#387—216no—
141#387—48no—
142#385—28no—
143#385—28no—
144#385—28no—
145#382—44no—
146#380—24no—
147#380—28no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C22⋊C4—28no—
155C2×#95—24no—
156C2×#95—24no—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25C2411yescharacteristic
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—42no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×Q814yes—
220C23×C4C2×Q814yes—
221C23×C4C2×Q814yes—
222C23×C4#4114yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C22⋊Q8—232no—
240C4×Q8—416no—
241C2×C4⋊C4—232no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×Q818yes—
275C2×C4⋊C4C2×Q818yes—
276C2×C4⋊C4#4118yes—
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—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4#4118yes—
288C2×C42—232no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—416no—
302C2×C42—44no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—416no—
308C2×C42C2×D414yes—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42#4118yes—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—42no—
339#95—42no—
340#95—44no—
341#95—44no—
342#95—44no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24C22×D412yes—
348C24C22×Q812yes—
349C24C2×#4112yes—
350C24C2×#4114yes—
351C242−1+414yes—
352C24C2511yescharacteristic
353C2×Q8—28no—
354C2×Q8—416no—
355C2×Q8—44no—
356C2×Q8—44no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—216no—
360C22×C4—232no—
361C22×C4—28no—
362C22×C4—28no—
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—22no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—216no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
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—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
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—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4C22⋊Q8116yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×Q814yes—
454C22×C4C22×Q812yes—
455C22×C4C22×Q812yes—
456C22×C4C22×Q814yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4114yes—
467C22×C4H(5,𝔽2)14yes—
468C4⋊C4—216no—
469C4⋊C4—28no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—416no—
475C4⋊C4—416no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—88no—
481C22⋊C4—216no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C42—216no—
486C42—216no—
487C42—28no—
488C42—28no—
489C42—44no—
490C42—44no—
491C42—48no—
492C42—48no—
493C42—48no—
494C42—48no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23C2×C22⋊Q814yes—
508C23#52514yes—
509C23#53614yes—
510C23#55618yes—
511C23#55718yes—
512C23C23×D411yescharacteristic
513C23C23×Q811yescharacteristic
514C23C22×#4111yescharacteristic
515C23C22×#4112yes—
516C23C22×#4112yes—
517C23C22×#4112yes—
518C23C2×H(5,𝔽2)12yes—
519C23C2×2−1+412yes—
520C23#58512yes—
521Q8—416no—
522C2×C4—232no—
523C2×C4—232no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—416no—
564C2×C4—416no—
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—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4C2×C22⋊Q814yes—
604C2×C4C2×C22⋊Q814yes—
605C2×C4#52514yes—
606C2×C4#52514yes—
607C2×C4#53318yes—
608C2×C4#53414yes—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—216no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—216no—
617C22#258118yes—
618C22C22×C22⋊Q811yescharacteristic
619C22C2×#52511yescharacteristic
620C22C2×#52511yescharacteristic
621C22C2×#52511yescharacteristic
622C22C2×#53312yes—
623C22#344412yes—
624C22C2×#53411yescharacteristic
625C22C2×#53611yescharacteristic
626C22#344911yescharacteristic
627C22C2×#55612yes—
628C22#347514yes—
629C22C2×#55712yes—
630C22#348114yes—
631C22#349214yes—
632C22#350112yes—
633C22#350512yes—
634C22#351614yes—
635C4—28no—
636C4—24no—
637C4—24no—
638C4—24no—
639C4—24no—
640C4—24no—
641C4—44no—
642C4—44no—
643C4—42no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—82no—
648C4—82no—
649C4—84no—
650C4—84no—
651C2—28no—
652C2#3487511yescharacteristic
653C2C2×#258111yescharacteristic
654C2#3495611yescharacteristic
655C2#3501011yescharacteristic
656C2#3505711yescharacteristic
657C2#3524611yescharacteristic
658C2#3633714yes—
659C2#3651914yes—
660C2#3662511yescharacteristic
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. #15206C2semidirect
4. #15164C2semidirect
5. #15075C2semidirect
13. #14239C2semidirect
14. #14174C2semidirect
16. #14069C2semidirect
101. C22×C4⋊C4Q8semidirect

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×C4×C4⋊C42
#152062
#151642
#150752
#150152
#149142
#148222
#147202
#146952
#146822
#142392
#141742
#141492
#140692
#139232