Finite Group Databasebeta

← #4389758

Subgroups — #4389758

Order 512.

2702 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2658C211yesmaximal, characteristic
3C2×#2632C211yesmaximal, characteristic
4C2×#2337C211yesmaximal, characteristic
5#17062C212yesmaximal, complemented
6#16803C212yesmaximal, complemented
7#15351C214yesmaximal, complemented
8#15318C214yesmaximal, complemented
9#15311C214yesmaximal
10#15294C214yesmaximal, complemented
11#15161C212yesmaximal, complemented
12#15135C212yesmaximal, complemented
13#14663C211yesmaximal, characteristic, complemented
14#13964C211yesmaximal, characteristic, complemented
15#13961C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×#124C2211yescharacteristic
19C22×#124C2212yes—
20C22×C4×Q8C2211yescharacteristic
21#2658—28no—
22#2632—28no—
23#2567—28no—
24C2×#399C2212yes—
25C2×#399C2212yes—
26C2×#399C2212yes—
27C2×#399C2214yes—
28C2×#398C2214yes—
29C2×#397C2214yes—
30C2×#395C2212yes—
31C2×#395C2214yes—
32C2×#394C2212yes—
33C2×#393C2212yes—
34#2337—28no—
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#389C2214yes—
39C2×#389C2214yes—
40C2×#388C2212yes—
41C2×#388C2212yes—
42C2×#388C2214yes—
43C2×#386C2212yes—
44C2×#385C2212yes—
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#383C2211yescharacteristic
48C2×#383C2211yescharacteristic
49C2×#380C2214yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53#1446C2212yes—
54#1446C2214yes—
55#1445C2212yes—
56#1445C2212yes—
57#1445C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1444C2214yes—
61#1444C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75#1434C2212yes—
76#1434C2212yes—
77#1434C2212yes—
78C22⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2312yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4C2312yes—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—48no—
89C2×#124—28no—
90C2×#124—24no—
91C2×#124—24no—
92C2×#124—48no—
93C2×#124D418yes—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#399—28no—
126#399—216no—
127#399—28no—
128#399—28no—
129#398—48no—
130#397—48no—
131#395—28no—
132#395—216no—
133#394—28no—
134#393—28no—
135#391—24no—
136#391—24no—
137#391—28no—
138#389—216no—
139#389—48no—
140#388—28no—
141#388—216no—
142#388—28no—
143#386—28no—
144#385—28no—
145#384—24no—
146#384—24no—
147#383—24no—
148#383—24no—
149#380—48no—
150C4×C4⋊C4D418yes—
151C4×C22⋊C4—28no—
152#376—24no—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C25C2412yes—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8—44no—
193C22×Q8C2×D414yes—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—44no—
197C22×D4C2×D414yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C4⋊Q8—432no—
218C4⋊Q8—416no—
219#124—232no—
220#124—432no—
221C4×Q8—416no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
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⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D418yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
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⋊C4C2×D418yes—
284C2×C22⋊C4C2×D418yes—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298C2×C42#4114yes—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—24no—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×D418yes—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24C22×D414yes—
336C24C2511yescharacteristic
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—44no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—44no—
346C2×Q8—48no—
347C2×Q8—84no—
348C2×Q8—88no—
349C2×Q8—84no—
350C2×Q8—88no—
351C2×D4—28no—
352C2×D4—48no—
353C2×D4—84no—
354C2×D4—84no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
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—24no—
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—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—416no—
415C22×C4—48no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—88no—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D414yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4H(5,𝔽2)18yes—
432C22×C42−1+414yes—
433C22×C42−1+414yes—
434C22×C42−1+414yes—
435C4⋊C4—216no—
436C4⋊C4—416no—
437C4⋊C4—416no—
438C4⋊C4—416no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C4⋊C4—84no—
447C4⋊C4—88no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C22⋊C4—88no—
457C42—216no—
458C42—44no—
459C42—44no—
460C42—48no—
461C42—84no—
462C42#121116yes—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—24no—
467C23—24no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23C23×D411yescharacteristic
487C23C23×D411yescharacteristic
488C23C22×#4111yescharacteristic
489C23C2×H(5,𝔽2)14yes—
490C23C2×2−1+412yes—
491C23C2×2−1+412yes—
492C23C2×2−1+412yes—
493C23#58512yes—
494Q8—416no—
495Q8—88no—
496Q8—816no—
497D4—88no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
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—816no—
570C2×C4C2×#12114yes—
571C2×C4C2×#12114yes—
572C2×C4C2×#12114yes—
573C2×C4#53618yes—
574C22—44no—
575C22—48no—
576C22—48no—
577C22—44no—
578C22—44no—
579C22—84no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—84no—
585C22C22×#12111yescharacteristic
586C22C2×#53612yes—
587C22#348412yes—
588C22#348511yescharacteristic
589C22#348611yescharacteristic
590C22#348912yes—
591C22#349512yes—
592C22#349812yes—
593C22#349912yes—
594C22#350812yes—
595C22#350911yescharacteristic
596C22#350911yescharacteristic
597C22#353514yes—
598C22#354114yes—
599C22#354414yes—
600C22#355014yes—
601C4—24no—
602C4—28no—
603C4—24no—
604C4—24no—
605C4—44no—
606C4—48no—
607C4—44no—
608C4—44no—
609C4—82no—
610C4—84no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—82no—
615C2—84no—
616C2#3531611yescharacteristic
617C2#3552811yescharacteristic
618C2#3563311yescharacteristic
619C2#4137012yes—
620C2#4278912yes—
621C2#4776614yes—
622C2#5091712yes—
623C2#5122512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2567C2semidirect
5. #17062C2semidirect
6. #16803C2semidirect
7. #15351C2semidirect
8. #15318C2semidirect
10. #15294C2semidirect
11. #15161C2semidirect
12. #15135C2semidirect
13. #14663C2semidirect
14. #13964C2semidirect
15. #13961C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#26582
C2×#26322
C2×#23372
#170622
#168032
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