Finite Group Databasebeta

← #4929643

Subgroups — #4929643

Order 512.

2700 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2584C211yesmaximal, characteristic, complemented
2C2×#2611C211yesmaximal, characteristic
3C2×#2599C211yesmaximal, characteristic
4C2×#2276C211yesmaximal, characteristic, complemented
5#15174C212yesmaximal
6#15149C212yesmaximal, complemented
7#14908C212yesmaximal, complemented
8#14894C212yesmaximal
9#14880C212yesmaximal, complemented
10#14877C212yesmaximal
11#14870C212yesmaximal
12#14818C212yesmaximal, complemented
13#14289C211yesmaximal, characteristic, complemented
14#14208C214yesmaximal, complemented
15#14204C214yesmaximal
16#13952C211yesmaximal, characteristic, complemented
17#13852C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2611—28no—
23#2599—28no—
24#2584—28no—
25C2×#400C2211yescharacteristic
26C2×#399C2211yescharacteristic
27C2×#398C2214yes—
28C2×#398C2211yescharacteristic
29C2×#398C2212yes—
30C2×#398C2211yescharacteristic
31C2×#397C2211yescharacteristic
32C2×#397C2211yescharacteristic
33C2×#396C2214yes—
34C2×#395C2212yes—
35C2×#395C2211yescharacteristic
36C2×#393C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#389C2214yes—
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#387C2212yes—
45C2×#387C2212yes—
46C2×#387C2212yes—
47C2×#385C2212yes—
48C2×#385C2212yes—
49C2×#383C2211yescharacteristic
50#2276—28no—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2214yes—
56C2×C4×C22⋊C4C2212yes—
57C2×#376C2212yes—
58C2×#376C2211yescharacteristic
59#1445C2212yes—
60#1445C2212yes—
61#1444C2212yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2212yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2212yes—
71#1441C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1439C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83C4⋊#95C2214yes—
84C4⋊#95C2212yes—
85#1434C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#125—24no—
90C2×#125—24no—
91C2×#125—24no—
92C2×C4×Q8—216no—
93C2×C4×Q8—28no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8D418yes—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2314yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133#400—24no—
134#399—24no—
135#398—28no—
136#398—24no—
137#398—216no—
138#398—24no—
139#397—24no—
140#397—24no—
141#396—216no—
142#395—28no—
143#395—24no—
144#393—24no—
145#389—216no—
146#389—28no—
147#389—28no—
148#389—28no—
149#389—48no—
150#388—28no—
151#388—28no—
152#387—28no—
153#387—28no—
154#387—28no—
155#385—28no—
156#385—28no—
157#383—24no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—216no—
162C4×C4⋊C4—28no—
163C4×C22⋊C4—44no—
164#376—28no—
165#376D418yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C2×#95C2312yes—
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—28no—
200C22×Q8C2×D414yes—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—41no—
207C23×C4—41no—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224#125—416no—
225C4×Q8—264no—
226C4×Q8—232no—
227#117—416no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
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—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4#41116yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42C2×D418yes—
316C2×C42C2×D418yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42#4114yes—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95C2×D414yes—
348#95C2×D418yes—
349#95C2×D414yes—
350C24—21no—
351C24—21no—
352C24—24no—
353C24—21no—
354C24—21no—
355C24—24no—
356C24—21no—
357C24—21no—
358C242−1+412yes—
359C24C2511yescharacteristic
360C2×Q8—216no—
361C2×Q8—216no—
362C2×Q8—28no—
363C2×Q8—216no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C2×#4118yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4118yes—
456C22×C4C2×#4118yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+418yes—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—232no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—48no—
476C4⋊C4—88no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C42—232no—
485C42—216no—
486C42—216no—
487C42—44no—
488C42—88no—
489C42—84no—
490C42—84no—
491C42#121116yes—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—28no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23C23×D411yescharacteristic
510C23C23×D411yescharacteristic
511C23C22×#4111yescharacteristic
512C23C22×#4114yes—
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×2−1+414yes—
515C23C2×2−1+411yescharacteristic
516C23#58511yescharacteristic
517C23#58511yescharacteristic
518Q8—232no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—416no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4C2×#12114yes—
604C2×C4C2×#12114yes—
605C2×C4C2×#12114yes—
606C2×C4#52518yes—
607C2×C4#53218yes—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22—81no—
613C22—82no—
614C22—81no—
615C22—82no—
616C22—82no—
617C22C22×#12111yescharacteristic
618C22C2×#52512yes—
619C22C2×#53212yes—
620C22#344412yes—
621C22#348612yes—
622C22#348811yescharacteristic
623C22#348811yescharacteristic
624C22#348911yescharacteristic
625C22#348911yescharacteristic
626C22#349211yescharacteristic
627C22#349311yescharacteristic
628C22#349912yes—
629C22#350912yes—
630C22#351514yes—
631C22#351614yes—
632C22#352014yes—
633C22#352314yes—
634C4—24no—
635C4—28no—
636C4—24no—
637C4—28no—
638C4—24no—
639C4—48no—
640C4—44no—
641C4—48no—
642C4—44no—
643C4—42no—
644C4—81no—
645C4—82no—
646C4—82no—
647C4—81no—
648C4—82no—
649C4—82no—
650C2—82no—
651C2#3543412yes—
652C2#3549412yes—
653C2#3551811yescharacteristic
654C2#3566711yescharacteristic
655C2#3590912yes—
656C2#3594912yes—
657C2#3599712yes—
658C2#3600112yes—
659C2#3662411yescharacteristic
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
1. C2×#2584C2semidirect
4. C2×#2276C2semidirect
6. #15149C2semidirect
7. #14908C2semidirect
9. #14880C2semidirect
12. #14818C2semidirect
13. #14289C2semidirect
14. #14208C2semidirect
16. #13952C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25842
C2×#26112
C2×#25992
C2×#22762
#151742
#151492
#149082
#148942
#148802
#148772
#148702
#148182
#142892
#142082
#142042
#139522
#138522