Finite Group Databasebeta

← #4397654

Subgroups — #4397654

Order 512.

2708 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2820C211yesmaximal, characteristic, complemented
2C2×#2562C211yesmaximal, characteristic, complemented
3C2×#2544C211yesmaximal, characteristic
4C2×#2336C211yesmaximal, characteristic, complemented
5#16804C212yesmaximal, complemented
6#16503C212yesmaximal
7#15795C214yesmaximal, complemented
8#15619C214yesmaximal, complemented
9#15607C214yesmaximal, complemented
10#15380C214yesmaximal, complemented
11#15107C212yesmaximal, complemented
12#15070C212yesmaximal, complemented
13#14345C211yesmaximal, characteristic, complemented
14#14288C211yesmaximal, characteristic, complemented
15#13955C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2820—28no—
22#2562—28no—
23#2544—28no—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#395C2212yes—
29C2×#394C2212yes—
30C2×#394C2214yes—
31C2×#394C2212yes—
32C2×#394C2214yes—
33#2336—28no—
34C2×#391C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#387C2212yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#386C2214yes—
44C2×#383C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#1446C2212yes—
52#1445C2212yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2212yes—
66#1438C2214yes—
67#1438C2212yes—
68#1438C2214yes—
69#1438C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1437C2212yes—
73#1437C2214yes—
74#1437C2212yes—
75C4⋊#95C2214yes—
76#1434C2212yes—
77#1434C2212yes—
78C22⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2314yes—
83C24×C4C2312yes—
84C2×#127—24no—
85C2×#127—44no—
86C2×#125—24no—
87C2×#125—24no—
88C2×#125D418yescomplemented
89C2×#124—24no—
90C2×#124—24no—
91C2×#124—24no—
92C2×#124—24no—
93C2×#124—44no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#400—28no—
128#399—28no—
129#397—28no—
130#396—216no—
131#395—28no—
132#394—28no—
133#394—28no—
134#394—216no—
135#394—48no—
136#391—24no—
137#390—24no—
138#389—28no—
139#388—216no—
140#388—48no—
141#387—28no—
142#387—28no—
143#387—28no—
144#387—28no—
145#386—216no—
146#383—24no—
147#382—24no—
148#380—28no—
149#380—28no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4D418yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25C2412yes—
178C25C2412yes—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—28no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219#127—416no—
220#125—232no—
221#124—416no—
222#124—416no—
223C4×D4—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
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—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D418yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D418yes—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
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—
306C2×C42#4114yes—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—24no—
316#95—22no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
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×D418yes—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24H(5,𝔽2)14yes—
351C24C2511yescharacteristic
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—84no—
356C2×Q8—84no—
357C2×D4—28no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—44no—
361C2×D4—416no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—416no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4—88no—
373C2×D4—88no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
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—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
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—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C4⋊C4—216no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—88no—
460C22⋊C4—416no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C42—44no—
474C42—44no—
475C42—48no—
476C42—84no—
477C42—84no—
478C42#121116yes—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23C23×D411yescharacteristic
524C23C23×D411yescharacteristic
525C23C22×#4111yescharacteristic
526C23C2×H(5,𝔽2)12yes—
527C23#58512yes—
528C23#58512yes—
529C23#58512yes—
530C23#58514yes—
531Q8—88no—
532D4—416no—
533D4—88no—
534D4—832no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
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—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
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—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—816no—
603C2×C4C2×#12114yes—
604C2×C4C2×#12114yes—
605C2×C4C2×#12114yes—
606C22—44no—
607C22—48no—
608C22—44no—
609C22—48no—
610C22—44no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—84no—
616C22—84no—
617C22—82no—
618C22—84no—
619C22—84no—
620C22—82no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—82no—
626C22C22×#12111yescharacteristic
627C22#344912yes—
628C22#348411yescharacteristic
629C22#348712yes—
630C22#348711yescharacteristic
631C22#348812yes—
632C22#349412yes—
633C22#350011yescharacteristic
634C22#350011yescharacteristic
635C22#350012yes—
636C22#351012yes—
637C22#351212yes—
638C22#353314yes—
639C22#356014yes—
640C22#356614yes—
641C22#356614yes—
642C4—24no—
643C4—24no—
644C4—24no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—84no—
650C4—82no—
651C4—82no—
652C4—84no—
653C4—82no—
654C4—82no—
655C2—84no—
656C2—162no—
657C2#3532811yescharacteristic
658C2#3556311yescharacteristic
659C2#3567711yescharacteristic
660C2#4050212yes—
661C2#4238812yes—
662C2#4774814yes—
663C2#5069512yes—
664C2#5107012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2820C2semidirect
2. C2×#2562C2semidirect
4. C2×#2336C2semidirect
5. #16804C2semidirect
7. #15795C2semidirect
8. #15619C2semidirect
9. #15607C2semidirect
10. #15380C2semidirect
11. #15107C2semidirect
12. #15070C2semidirect
13. #14345C2semidirect
14. #14288C2semidirect
15. #13955C2semidirect
88. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28202
C2×#25622
C2×#25442
C2×#23362
#168042
#165032
#157952
#156192
#156072
#153802
#151072
#150702
#143452
#142882
#139552