Finite Group Databasebeta

← #645291

Subgroups — #645291

Order 512.

2414 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#17057C211yesmaximal, characteristic
4#16981C212yesmaximal, complemented
5#16806C212yesmaximal, complemented
6#16365C211yesmaximal, characteristic
7#16365C211yesmaximal, characteristic
8#16254C211yesmaximal, characteristic
9#15204C214yesmaximal
10#15101C212yesmaximal, complemented
11#14879C212yesmaximal, complemented
12#14872C212yesmaximal, complemented
13#14818C212yesmaximal, complemented
14#14297C214yesmaximal
15#14241C212yesmaximal, complemented
16#14226C212yesmaximal, complemented
17C4×#380C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2855—28no—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#395C2212yes—
26C2×#395C2212yes—
27C4×C22⋊Q8—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2214yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#387C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1446C2212yes—
50#1446C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1444C2212yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2212yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2211yescharacteristic
84C4×#95C2212yes—
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—44no—
94C2×C4×Q8—24no—
95C2×C4×Q8—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42—44no—
120C22×C42D418yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#398—28no—
129#398—44no—
130#395—28no—
131#395—44no—
132#391—24no—
133#391—24no—
134#391—24no—
135#389—24no—
136#389—24no—
137#389—24no—
138#389—48no—
139#387—28no—
140#387—28no—
141#387—28no—
142#385—28no—
143#385—28no—
144#381—28no—
145#380—24no—
146#380—24no—
147#380Q8116yescomplemented
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—44no—
151C4×C22⋊C4—28no—
152C4×C22⋊C4—44no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
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—
183C25C2411yescharacteristic
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—42no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×Q818yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C22⋊Q8—232no—
222C22⋊Q8—416no—
223C4×Q8—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×Q8116yes—
267C2×C4⋊C4#4118yes—
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—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4C2×Q818yes—
283C2×C22⋊C4C2×Q818yes—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—28no—
289C2×C42—416no—
290C2×C42—44no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
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—24no—
314#95—28no—
315#95—24no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—44no—
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#95C2×Q8116yes—
331C24—22no—
332C24—21no—
333C24—22no—
334C24—21no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24C22×D412yes—
339C24C22×Q814yes—
340C24C2511yescharacteristic
341C2×Q8—28no—
342C2×Q8—416no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—88no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
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—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4C4⋊Q8116yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×Q818yes—
437C22×C4C22×Q818yes—
438C22×C4C22×Q814yes—
439C22×C4C22×Q814yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4114yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C42−1+412yes—
446C4⋊C4—28no—
447C4⋊C4—216no—
448C4⋊C4—416no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C22⋊C4—216no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C42—28no—
470C42—28no—
471C42—44no—
472C42—44no—
473C42—48no—
474C42—48no—
475C42—88no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—28no—
483C23—24no—
484C23—28no—
485C23—24no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23C2×C4⋊Q814yes—
493C23C23×D411yescharacteristic
494C23C23×Q812yes—
495C23C22×#4112yes—
496C23C2×H(5,𝔽2)12yes—
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×2−1+411yescharacteristic
499C23#58512yes—
500C23#58512yes—
501C23#58512yes—
502Q8—416no—
503Q8—88no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—416no—
535C2×C4—416no—
536C2×C4—416no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
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—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4#52514yes—
587C2×C4C2×C4⋊Q814yes—
588C2×C4C2×C4⋊Q814yes—
589C2×C4#53414yes—
590C2×C4#53714yes—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—48no—
595C22—48no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22C2×#52511yescharacteristic
601C22C22×C4⋊Q811yescharacteristic
602C22C2×#53411yescharacteristic
603C22C2×#53711yescharacteristic
604C22#344911yescharacteristic
605C22#344911yescharacteristic
606C22#344911yescharacteristic
607C22#346012yes—
608C22#346112yes—
609C22Q8×#4114yes—
610C22#350112yes—
611C22#350114yes—
612C22#350212yes—
613C22#351314yes—
614C22#354112yes—
615C22#354212yes—
616C22#356614yes—
617C4—24no—
618C4—24no—
619C4—24no—
620C4—24no—
621C4—24no—
622C4—48no—
623C4—48no—
624C4—44no—
625C4—42no—
626C4—44no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C2—44no—
634C2#3461411yescharacteristic
635C2#3495311yescharacteristic
636C2#3501111yescharacteristic
637C2#3506211yescharacteristic
638C2#3517311yescharacteristic
639C2#3744411yescharacteristic
640C2#3755211yescharacteristic
641C2#3969614yes—
642C2#3969714yes—
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. #16981C2semidirect
5. #16806C2semidirect
10. #15101C2semidirect
11. #14879C2semidirect
12. #14872C2semidirect
13. #14818C2semidirect
15. #14241C2semidirect
16. #14226C2semidirect
147. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×C4×C22⋊Q82
#170572
#169812
#168062
#163652
#162542
#152042
#151012
#148792
#148722
#148182
#142972
#142412
#142262
C4×#3802