Finite Group Databasebeta

← #712174

Subgroups — #712174

Order 512.

2686 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic
3C2×#2344C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic
5#16675C214yesmaximal
6#15738C214yesmaximal, complemented
7#15629C214yesmaximal, complemented
8#15189C212yesmaximal, complemented
9#15137C212yesmaximal, complemented
10#14907C214yesmaximal
11#14142C212yesmaximal, complemented
12#14138C211yesmaximal, characteristic
13#14129C211yesmaximal, characteristic
14#14005C212yesmaximal, complemented
15#13853C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21#2625—28no—
22#2502—28no—
23C2×#400C2212yes—
24C2×#398C2212yes—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#397C2214yes—
28#2344—28no—
29#2317—28no—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#387C2214yes—
41C2×#385C2214yes—
42C2×#385C2214yes—
43C2×#384C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#382C2212yes—
47C2×#381C2212yes—
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1445C2212yes—
53#1445C2212yes—
54#1444C2214yes—
55#1444C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2212yes—
66#1440C2212yes—
67#1440C2212yes—
68#1440C2212yes—
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76#1434C2214yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×Q8C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—28no—
90C2×#117—28no—
91C2×#117—48no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4D418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124#400—28no—
125#398—216no—
126#398—28no—
127#398—48no—
128#397—216no—
129#391—28no—
130#391—24no—
131#391—24no—
132#391—28no—
133#389—28no—
134#389—28no—
135#389—28no—
136#389—28no—
137#389—28no—
138#389—48no—
139#387—216no—
140#385—216no—
141#385—216no—
142#384—28no—
143#384—24no—
144#384—24no—
145#382—28no—
146#381—28no—
147#380D418yes—
148C4×C4⋊C4—28no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C25C2411yescharacteristic
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4C2×D414yes—
191C23×C4C2×D414yes—
192C23×C4C2×D418yes—
193C23×C4#4114yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C22⋊Q8—416no—
209#117—232no—
210#117—432no—
211#117—416no—
212C2×C4⋊C4—232no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D418yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4#4118yes—
261C2×C22⋊C4—22no—
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—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C2×D418yes—
278C2×C42—24no—
279C2×C42—28no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—44no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—28no—
309#95—24no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—42no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95C2×D418yes—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24C22×D412yes—
327C24C22×D412yes—
328C24C2×#4112yes—
329C24C2511yescharacteristic
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—48no—
334C2×Q8—48no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—216no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—416no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—48no—
415C22×C4#121116yes—
416C22×C4C22×D418yes—
417C22×C4C22×D414yes—
418C22×C4C22×D414yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C2×#4118yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4114yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C42−1+414yes—
429C22×C42−1+414yes—
430C22×C42−1+414yes—
431C4⋊C4—216no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—416no—
437C4⋊C4—416no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C4⋊C4—88no—
447C22⋊C4—216no—
448C22⋊C4—216no—
449C22⋊C4—28no—
450C22⋊C4—48no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C42—216no—
455C42—416no—
456C42—48no—
457C42—44no—
458C42—44no—
459C42—48no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23C2×#12114yes—
481C23C23×D411yescharacteristic
482C23C23×D411yescharacteristic
483C23C22×#4111yescharacteristic
484C23C22×#4112yes—
485C23C2×H(5,𝔽2)12yes—
486C23C2×2−1+412yes—
487C23C2×2−1+412yes—
488C23C2×2−1+412yes—
489C23#58512yes—
490Q8—88no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—232no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—432no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
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—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—44no—
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—42no—
573C2×C4C2×#12114yes—
574C2×C4C2×#12114yes—
575C2×C4#52514yes—
576C2×C4#53614yes—
577C2×C4#54718yes—
578C2×C4#55218yes—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22C22×#12111yescharacteristic
589C22C2×#52511yescharacteristic
590C22C2×#53611yescharacteristic
591C22#346012yes—
592C22C2×#54712yes—
593C22C2×#55212yes—
594C22#348411yescharacteristic
595C22#348612yes—
596C22#348711yescharacteristic
597C22#349612yes—
598C22#349912yes—
599C22#350911yescharacteristic
600C22#350911yescharacteristic
601C22#351512yes—
602C22#351612yes—
603C22#352012yes—
604C22#352212yes—
605C22#354412yes—
606C22#355212yes—
607C22#355512yes—
608C22#356312yes—
609C4—24no—
610C4—48no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—48no—
615C4—44no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#241318yes—
623C2—44no—
624C2C2×#241311yescharacteristic
625C2#3516811yescharacteristic
626C2#3566711yescharacteristic
627C2#3572412yes—
628C2#3826112yes—
629C2#3907712yes—
630C2#4243612yes—
631C2#4487512yes—
632C2#4523812yes—
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
6. #15738C2semidirect
7. #15629C2semidirect
8. #15189C2semidirect
9. #15137C2semidirect
11. #14142C2semidirect
14. #14005C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#26252
C2×#23442
C2×#23172
#166752
#157382
#156292
#151892
#151372
#149072
#141422
#141382
#141292
#140052
#138532