Finite Group Databasebeta

← #719943

Subgroups — #719943

Order 512.

2734 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C212yesmaximal
2C2×#2551C211yesmaximal, characteristic
3C2×#2331C211yesmaximal, characteristic
4#16993C212yesmaximal
5#16507C212yesmaximal
6#15360C214yesmaximal
7#15318C214yesmaximal
8#15221C212yesmaximal
9#15190C212yesmaximal
10#15137C212yesmaximal
11#15012C212yesmaximal
12#14277C211yesmaximal, characteristic
13#14150C211yesmaximal, characteristic
14#14053C212yesmaximal
15#13910C211yesmaximal, characteristic
16#13848C211yesmaximal, characteristic
17#13804C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2212yes—
21C22×C4⋊Q8C2214yes—
22#2551—28no—
23#2551—216no—
24C2×#398C2214yes—
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#395C2214yes—
28#2331—28no—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2214yes—
32C2×#389C2214yes—
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#384C2212yes—
43C2×#384C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×#376C2211yescharacteristic
53#1446C2214yes—
54#1445C2214yes—
55#1444C2214yes—
56#1444C2211yescharacteristic
57#1444C2211yescharacteristic
58#1444C2212yes—
59#1444C2212yes—
60#1444C2212yes—
61#1444C2212yes—
62#1444C2214yes—
63#1444C2212yes—
64#1442C2212yes—
65#1442C2214yes—
66#1442C2214yes—
67#1442C2211yescharacteristic
68#1442C2211yescharacteristic
69#1442C2212yes—
70#1442C2214yes—
71#1442C2212yes—
72#1441C2212yes—
73#1441C2214yes—
74#1441C2212yes—
75#1441C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2314yes—
87C23×Q8C2312yes—
88C2×C4⋊Q8—216no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—48no—
96C2×C4⋊Q8—48no—
97C2×C4⋊Q8—48no—
98C2×C4⋊Q8D418yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#398—48no—
130#395—216no—
131#395—216no—
132#395—216no—
133#391—28no—
134#391—28no—
135#391—48no—
136#389—24no—
137#389—24no—
138#389—216no—
139#389—28no—
140#389—28no—
141#389—28no—
142#389—28no—
143#389—48no—
144#387—28no—
145#387—28no—
146#384—28no—
147#384—24no—
148#382—24no—
149#382Q818yescomplemented
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4D418yes—
155C4×C4⋊C4Q818yescomplemented
156#376—24no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2311yescharacteristic
189C2×#95C2312yes—
190C22×Q8—24no—
191C22×Q8—28no—
192C22×Q8—28no—
193C22×Q8—24no—
194C22×Q8—28no—
195C22×Q8—28no—
196C22×Q8—44no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8—44no—
201C22×Q8C2×D418yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C4⋊Q8—232no—
220C4⋊Q8—432no—
221C4⋊Q8—464no—
222C4⋊Q8—432no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×Q818yes—
281C2×C4⋊C4C2×Q818yes—
282C2×C4⋊C4#4114yes—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42C2×D414yes—
300C2×C42C2×D418yes—
301C2×C42C2×D414yes—
302C2×C42C2×Q818yes—
303C2×C42C2×Q814yes—
304C2×C42C2×Q814yes—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95C2×D418yes—
334#95C2×Q814yes—
335#95C2×Q814yes—
336#95C2×Q818yes—
337#95C2×Q818yes—
338C24C2511yescharacteristic
339C2×Q8—216no—
340C2×Q8—416no—
341C2×Q8—416no—
342C2×Q8—416no—
343C2×Q8—416no—
344C2×Q8—416no—
345C2×Q8—84no—
346C2×Q8—88no—
347C2×Q8—84no—
348C2×Q8—88no—
349C2×Q8—88no—
350C2×Q8—88no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
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—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4C22×D412yes—
424C22×C4C22×D414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×Q818yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q812yes—
437C22×C4C22×Q812yes—
438C22×C4C22×Q812yes—
439C22×C4C22×Q814yes—
440C22×C4C22×Q812yes—
441C22×C4C22×Q812yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4H(5,𝔽2)14yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C42−1+414yes—
447C22×C42−1+414yes—
448C22×C42−1+412yes—
449C22×C42−1+412yes—
450C4⋊C4—28no—
451C4⋊C4—216no—
452C4⋊C4—28no—
453C4⋊C4—216no—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—416no—
460C4⋊C4—416no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—416no—
464C4⋊C4—416no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—88no—
471C4⋊C4#121116yes—
472C42—28no—
473C42—216no—
474C42—48no—
475C42—416no—
476C42—48no—
477C42—44no—
478C42—84no—
479C42C4⋊Q8116yescomplemented
480C23C23×D411yescharacteristic
481C23C23×D411yescharacteristic
482C23C23×Q811yescharacteristic
483C23C23×Q811yescharacteristic
484C23C22×#4111yescharacteristic
485C23C2×H(5,𝔽2)12yes—
486C23C2×H(5,𝔽2)12yes—
487C23C2×2−1+411yescharacteristic
488C23C2×2−1+411yescharacteristic
489C23C2×2−1+412yes—
490C23C2×2−1+412yes—
491Q8—816no—
492Q8—832no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4C2×#12118yes—
582C2×C4C2×#12114yes—
583C2×C4C2×C4⋊Q814yes—
584C2×C4C2×C4⋊Q814yes—
585C2×C4C2×C4⋊Q814yes—
586C2×C4#53614yes—
587C2×C4#54418yes—
588C2×C4D4×Q814yes—
589C2×C4D4×Q814yes—
590C2×C4#55714yes—
591C2×C4#55714yes—
592C22C22×#12111yescharacteristic
593C22C22×C4⋊Q811yescharacteristic
594C22C2×#53611yescharacteristic
595C22C2×#54412yes—
596C22C2×D4×Q811yescharacteristic
597C22C2×D4×Q811yescharacteristic
598C22C2×#55711yescharacteristic
599C22C2×#55711yescharacteristic
600C22#348412yes—
601C22#348411yescharacteristic
602C22#348611yescharacteristic
603C22#348612yes—
604C22#349811yescharacteristic
605C22#349911yescharacteristic
606C22#350312yes—
607C22#350312yes—
608C22#350312yes—
609C22#351912yes—
610C22#352312yes—
611C22#354412yes—
612C22#355012yes—
613C22#355012yes—
614C22#355212yes—
615C4—28no—
616C4—28no—
617C4—24no—
618C4—24no—
619C4—44no—
620C4—44no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—44no—
625C4—48no—
626C4—44no—
627C4—81no—
628C4—82no—
629C4—84no—
630C4—81no—
631C4#254618yes—
632C2C2×#254611yescharacteristic
633C2#3527511yescharacteristic
634C2#3557611yescharacteristic
635C2#3569111yescharacteristic
636C2#3570011yescharacteristic
637C2#3768612yes—
638C2#3918112yes—
639C2#4242711yescharacteristic
640C2#4278911yescharacteristic
641C2#4493412yes—
642C2#4504412yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
149. #382Q8semidirect
155. C4×C4⋊C4Q8semidirect
479. C42C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#23312
#169932
#165072
#153602
#153182
#152212
#151902
#151372
#150122
#142772
#141502
#140532
#139102
#138482
#138042