Finite Group Databasebeta

← #4311714

Subgroups — #4311714

Order 512.

2136 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic
2C2×C4×#125C211yesmaximal, characteristic, complemented
3#16981C212yesmaximal, complemented
4#16964C211yesmaximal, characteristic, complemented
5#16896C214yesmaximal, complemented
6#16802C212yesmaximal
7#16692C211yesmaximal, characteristic
8#16674C212yesmaximal, complemented
9#16673C212yesmaximal, complemented
10#16634C211yesmaximal, characteristic, complemented
11#16633C211yesmaximal, characteristic
12#15103C212yesmaximal, complemented
13#15081C212yesmaximal, complemented
14#15077C214yesmaximal, complemented
15#14688C211yesmaximal, characteristic, complemented
16#14318C212yesmaximal, complemented
17#14312C212yesmaximal
18C22×#126C2212yes—
19C22×#125C2211yescharacteristic
20#2860—28no—
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#399C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#394C2212yes—
27C4×#125—28no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#388C2214yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#383C2212yes—
43C2×#382C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C43C2211yescharacteristic
49#1445C2214yes—
50#1445C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2212yes—
54#1444C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2212yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1438C2212yes—
75#1437C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2211yescharacteristic
82#1434C2211yescharacteristic
83#1434C2212yes—
84#1434C2214yes—
85C23×D4C2311yescharacteristic
86C2×#126—28no—
87C2×#126—44no—
88C2×#126—44no—
89C2×#125—28no—
90C2×#125—24no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#400—28no—
128#400—44no—
129#399—44no—
130#398—44no—
131#397—44no—
132#394—28no—
133#390—24no—
134#390—24no—
135#390—24no—
136#389—24no—
137#389—24no—
138#389—24no—
139#389—24no—
140#389—24no—
141#388—28no—
142#388—28no—
143#388—28no—
144#388—48no—
145#387—28no—
146#387—28no—
147#383—44no—
148#382—24no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C43D418yes—
182C25C2412yes—
183C22×D4—22no—
184C22×D4—24no—
185C22×D4—42no—
186C22×D4—42no—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203#126—432no—
204#125—232no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4#4118yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42C2×D414yes—
290C2×C42C2×D414yes—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324C24—22no—
325C24—24no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24C2511yescharacteristic
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—88no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
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—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
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—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4114yes—
419C22×C4C2×#4114yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C42−1+414yes—
423C22×C42−1+412yes—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—216no—
427C4⋊C4—216no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C4⋊C4—88no—
441C22⋊C4—48no—
442C22⋊C4—416no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C42—28no—
452C42—28no—
453C42—28no—
454C42—28no—
455C42—44no—
456C42—44no—
457C42—44no—
458C42—48no—
459C42—48no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23C23×D411yescharacteristic
484C23C22×#4112yes—
485C23C2×H(5,𝔽2)12yes—
486C23C2×H(5,𝔽2)11yescharacteristic
487C23C2×2−1+412yes—
488C23C2×2−1+411yescharacteristic
489C23#58512yes—
490C23#58512yes—
491C23#58512yes—
492D4—88no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
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—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
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—44no—
572C2×C4—44no—
573C2×C4#52514yes—
574C2×C4#53514yes—
575C2×C4#53614yes—
576C2×C4#53714yes—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—82no—
588C22C2×#52511yescharacteristic
589C22C2×#53511yescharacteristic
590C22C2×#53611yescharacteristic
591C22C2×#53711yescharacteristic
592C22#344911yescharacteristic
593C22#344911yescharacteristic
594C22#344911yescharacteristic
595C22#346012yes—
596C22#346112yes—
597C22#352114yes—
598C22#352214yes—
599C22#354612yes—
600C22#354712yes—
601C22#355912yes—
602C22#356214yes—
603C22#356312yes—
604C22#356414yes—
605C4—24no—
606C4—24no—
607C4—24no—
608C4—24no—
609C4—42no—
610C4—44no—
611C4—44no—
612C4—42no—
613C4—42no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—84no—
618C4—84no—
619C4—82no—
620C4—82no—
621C2—162no—
622C2#3461411yescharacteristic
623C2#3514311yescharacteristic
624C2#3524911yescharacteristic
625C2#3708514yes—
626C2#3710414yes—
627C2#3754411yescharacteristic
628C2#3754911yescharacteristic
629C2#3755711yescharacteristic
630C2#3762111yescharacteristic
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
2. C2×C4×#125C2semidirect
3. #16981C2semidirect
4. #16964C2semidirect
5. #16896C2semidirect
8. #16674C2semidirect
9. #16673C2semidirect
10. #16634C2semidirect
12. #15103C2semidirect
13. #15081C2semidirect
14. #15077C2semidirect
15. #14688C2semidirect
16. #14318C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28602
C2×C4×#1252
#169812
#169642
#168962
#168022
#166922
#166742
#166732
#166342
#166332
#151032
#150812
#150772
#146882
#143182
#143122