Finite Group Databasebeta

← #645672

Subgroups — #645672

Order 512.

2398 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#17058C211yesmaximal, characteristic
4#16983C212yesmaximal, complemented
5#16804C212yesmaximal, complemented
6#16366C211yesmaximal, characteristic
7#16365C211yesmaximal, characteristic
8#16254C211yesmaximal, characteristic
9#15205C214yesmaximal
10#15108C212yesmaximal, complemented
11#15097C212yesmaximal, complemented
12#14965C214yesmaximal
13#14684C211yesmaximal, characteristic
14#14308C212yesmaximal, complemented
15#14302C212yesmaximal, complemented
16#14239C212yesmaximal, complemented
17#14174C212yesmaximal, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C2×#400C2212yes—
24C2×#398C2212yes—
25C2×#395C2212yes—
26C2×#395C2212yes—
27C4×C22⋊Q8—28no—
28C4×C22⋊Q8—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#387C2214yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#381C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47#1446C2212yes—
48#1446C2212yes—
49#1445C2214yes—
50#1445C2212yes—
51#1445C2212yes—
52#1444C2212yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1434C2212yes—
78#1434C2212yes—
79#1434C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82C22⋊#95C2212yes—
83C22⋊#95C2211yescharacteristic
84C22⋊#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C2×C4×Q8—44no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
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×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#400—28no—
129#398—28no—
130#395—28no—
131#395—28no—
132#391—24no—
133#391—24no—
134#389—24no—
135#389—24no—
136#387—28no—
137#387—28no—
138#387—48no—
139#385—28no—
140#385—28no—
141#381—44no—
142#380—24no—
143#380—24no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—28no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
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—
182C25C2411yescharacteristic
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8—42no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—42no—
197C23×C4—41no—
198C23×C4—41no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C22⋊Q8—232no—
221C22⋊Q8—232no—
222C4×Q8—416no—
223C4×Q8—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4#4118yes—
257C2×C4⋊C4#4118yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—416no—
282C2×C42—44no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42C2×D414yes—
290C2×C42C2×D414yes—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—21no—
331C24—21no—
332C24C22×D412yes—
333C242−1+414yes—
334C24C2511yescharacteristic
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—416no—
338C2×Q8—416no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
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—22no—
378C22×C4—21no—
379C22×C4—21no—
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—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
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—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—416no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—416no—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4114yes—
438C22×C4H(5,𝔽2)14yes—
439C22×C42−1+414yes—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—416no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—88no—
454C22⋊C4—216no—
455C22⋊C4—216no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C42—28no—
461C42—28no—
462C42—44no—
463C42—44no—
464C42—48no—
465C42—48no—
466C42—48no—
467C42—48no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—24no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23#53614yes—
485C23C23×D411yescharacteristic
486C23C22×#4112yes—
487C23C22×#4112yes—
488C23C2×H(5,𝔽2)12yes—
489C23C2×2−1+412yes—
490C23C2×2−1+412yes—
491C23#58511yescharacteristic
492C23#58511yescharacteristic
493C23#58512yes—
494Q8—416no—
495Q8—416no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
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—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—416no—
526C2×C4—416no—
527C2×C4—416no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
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—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4#52514yes—
580C2×C4#52514yes—
581C2×C4#53414yes—
582C2×C4#53614yes—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—48no—
591C22—48no—
592C22C2×#52511yescharacteristic
593C22C2×#52511yescharacteristic
594C22C2×#53411yescharacteristic
595C22C2×#53611yescharacteristic
596C22C2×#53611yescharacteristic
597C22#344911yescharacteristic
598C22#344911yescharacteristic
599C22#346112yes—
600C22#346112yes—
601C22#348114yes—
602C22#351012yes—
603C22#351212yes—
604C22#351614yes—
605C22#352314yes—
606C22#353512yes—
607C22#353512yes—
608C22#354114yes—
609C4—24no—
610C4—24no—
611C4—24no—
612C4—24no—
613C4—44no—
614C4—42no—
615C4—44no—
616C4—42no—
617C4—44no—
618C4—44no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—84no—
624C4—84no—
625C2—44no—
626C2#3462711yescharacteristic
627C2#3495611yescharacteristic
628C2#3524611yescharacteristic
629C2#3524811yescharacteristic
630C2#3534211yescharacteristic
631C2#3743811yescharacteristic
632C2#3744711yescharacteristic
633C2#4037514yes—
634C2#4040714yes—
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
4. #16983C2semidirect
5. #16804C2semidirect
10. #15108C2semidirect
11. #15097C2semidirect
14. #14308C2semidirect
15. #14302C2semidirect
16. #14239C2semidirect
17. #14174C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#170582
#169832
#168042
#163662
#163652
#162542
#152052
#151082
#150972
#149652
#146842
#143082
#143022
#142392
#141742