Finite Group Databasebeta

← #4416068

Subgroups — #4416068

Order 512.

2310 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C22×#387C211yesmaximal, characteristic
3#17044C214yesmaximal, complemented
4#17044C214yesmaximal, complemented
5#16979C212yesmaximal, complemented
6#16804C212yesmaximal, complemented
7#15092C212yesmaximal
8#15026C212yesmaximal
9#15015C211yesmaximal, characteristic
10#14919C212yesmaximal
11#14919C212yesmaximal
12#14914C211yesmaximal, characteristic
13#14822C211yesmaximal, characteristic
14#14720C211yesmaximal, characteristic
15#14684C211yesmaximal, characteristic
16#14629C212yesmaximal, complemented
17#14626C212yesmaximal, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2855—28no—
23C2×#400C2212yes—
24C2×#395C2212yes—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#387—28no—
30C2×#387C2212yes—
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1446C2212yes—
47#1445C2214yes—
48#1445C2212yes—
49#1445C2214yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2214yes—
71#1441C2214yes—
72#1434C2212yes—
73#1434C2212yes—
74#1434C2212yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2211yescharacteristic
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2211yescharacteristic
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—44no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C42—44no—
116C22×C42D418yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#400—28no—
122#395—28no—
123#391—24no—
124#391—24no—
125#389—24no—
126#389—24no—
127#387—28no—
128#387—28no—
129#387—28no—
130#387—28no—
131#387—28no—
132#387—28no—
133#385—28no—
134#385—44no—
135#385—44no—
136#381—28no—
137#381—44no—
138#380—24no—
139C2×#95—24no—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95—24no—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C25C2411yescharacteristic
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—42no—
184C22×Q8—42no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C22⋊Q8—416no—
233C2×C4⋊C4—232no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4#4118yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C42—416no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42C2×D414yes—
281C2×C42C2×D414yes—
282C2×C42C2×D414yes—
283C2×C42C2×D414yes—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24C22×D412yes—
326C24H(5,𝔽2)14yes—
327C242−1+414yes—
328C242−1+412yes—
329C242−1+412yes—
330C24C2511yescharacteristic
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—88no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—416no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—416no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—88no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—88no—
457C4⋊C4—88no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C42—44no—
463C42—44no—
464C42—44no—
465C42—44no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—24no—
478C23#53414yes—
479C23#53614yes—
480C23#53614yes—
481C23#56318yes—
482C23#56318yes—
483C23C23×D411yescharacteristic
484C23C22×#4112yes—
485C23C2×H(5,𝔽2)12yes—
486C23C2×2−1+411yescharacteristic
487C23C2×2−1+411yescharacteristic
488C23C2×2−1+412yes—
489C23#58512yes—
490C23#58512yes—
491C23#58512yes—
492Q8—88no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
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—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—416no—
517C2×C4—416no—
518C2×C4—416no—
519C2×C4—416no—
520C2×C4—416no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
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—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—48no—
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—48no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—88no—
578C2×C4—88no—
579C2×C4#52514yes—
580C22—216no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—216no—
587C22—24no—
588C22#281718yes—
589C22C2×#52511yescharacteristic
590C22C2×#53411yescharacteristic
591C22C2×#53611yescharacteristic
592C22C2×#53611yescharacteristic
593C22#344911yescharacteristic
594C22#344911yescharacteristic
595C22#344911yescharacteristic
596C22#349512yes—
597C22#349512yes—
598C22#351314yes—
599C22#351614yes—
600C22C2×#56312yes—
601C22C2×#56312yes—
602C22#353314yes—
603C22#355314yes—
604C22#356412yes—
605C22#356412yes—
606C4—24no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—44no—
613C4—44no—
614C4—84no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—84no—
622C2—28no—
623C2#3506211yescharacteristic
624C2#3524611yescharacteristic
625C2#3532511yescharacteristic
626C2C2×#281711yescharacteristic
627C2#3744311yescharacteristic
628C2#3754811yescharacteristic
629C2#3760711yescharacteristic
630C2#4555914yes—
631C2#4558714yes—
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. #17044C2semidirect
4. #17044C2semidirect
5. #16979C2semidirect
6. #16804C2semidirect
16. #14629C2semidirect
17. #14626C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C22×#3872
#170442
#169792
#168042
#150922
#150262
#150152
#149192
#149142
#148222
#147202
#146842
#146292
#146262