Finite Group Databasebeta

← #704320

Subgroups — #704320

Order 512.

3142 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic
3C2×Q8×C22⋊C4C211yesmaximal, characteristic
4C2×#2310C211yesmaximal, characteristic
5#16396C214yesmaximal
6#15503C214yesmaximal, complemented
7#15500C214yesmaximal, complemented
8#15188C212yesmaximal, complemented
9#15186C212yesmaximal, complemented
10#14837C214yesmaximal
11#14138C211yesmaximal, characteristic
12#14129C211yesmaximal, characteristic
13#14024C212yesmaximal, complemented
14#13935C212yesmaximal, complemented
15#13853C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23#2625—28no—
24#2434—28no—
25C2×#400C2212yes—
26C2×#398C2212yes—
27C2×#397C2214yes—
28C2×#395C2214yes—
29Q8×C22⋊C4—28no—
30#2310—28no—
31C2×#391C2212yes—
32C2×#391C2214yes—
33C2×#391C2212yes—
34C2×#391C2211yescharacteristic
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#385C2214yes—
44C2×#381C2214yes—
45C2×#380C2212yes—
46C2×#380C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2212yes—
65#1440C2212yes—
66#1440C2212yes—
67#1440C2212yes—
68#1439C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2214yes—
76#1434C2214yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×Q8—28no—
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—44no—
83C23×Q8D418yes—
84C23×Q8C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—44no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C2×C4×Q8—44no—
100C2×#117—24no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117—44no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2312yes—
129#400—28no—
130#398—28no—
131#397—216no—
132#395—216no—
133#391—28no—
134#391—28no—
135#391—28no—
136#391—28no—
137#391—24no—
138#391—24no—
139#391—28no—
140#391—28no—
141#391—216no—
142#391—44no—
143#389—28no—
144#389—44no—
145#385—216no—
146#381—216no—
147#380—28no—
148#380D418yes—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—28no—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
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×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C25C2411yescharacteristic
182C22×Q8—232no—
183C22×Q8—232no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—28no—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—44no—
200C22×Q8—416no—
201C22×Q8—432no—
202C22×Q8—432no—
203C22×Q8C2×D414yes—
204C22×Q8C2×D418yes—
205C22×Q8C2×D414yes—
206C22×Q8#4118yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—44no—
213C23×C4C2×D418yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C4⋊Q8—416no—
232C4⋊Q8—416no—
233C22⋊Q8—416no—
234C4×Q8—432no—
235#117—416no—
236#117—416no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D418yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×D418yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—44no—
301C2×C42—44no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—44no—
324#95—42no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95C2×D418yes—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C2×#4112yes—
340C24C2511yescharacteristic
341C2×Q8—28no—
342C2×Q8—216no—
343C2×Q8—28no—
344C2×Q8—216no—
345C2×Q8—28no—
346C2×Q8—28no—
347C2×Q8—28no—
348C2×Q8—216no—
349C2×Q8—216no—
350C2×Q8—48no—
351C2×Q8—416no—
352C2×Q8—416no—
353C2×Q8—4128no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8#123116yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—216no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4#121116yes—
443C22×C4C22×D414yes—
444C22×C4C22×D418yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4118yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C42−1+414yes—
456C22×C42−1+414yes—
457C22×C42−1+414yes—
458C4⋊C4—216no—
459C4⋊C4—416no—
460C4⋊C4—416no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C22⋊C4—28no—
470C22⋊C4—216no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C42—416no—
477C42—48no—
478C42—48no—
479C42—44no—
480C42—44no—
481C42—48no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23C2×#12114yes—
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C22×#4111yescharacteristic
506C23C22×#4112yes—
507C23C2×H(5,𝔽2)12yes—
508C23C2×2−1+412yes—
509C23C2×2−1+412yes—
510C23C2×2−1+412yes—
511C23#58512yes—
512Q8—232no—
513Q8—432no—
514Q8—416no—
515Q8—88no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—232no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—432no—
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—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4C2×#12114yes—
598C2×C4C2×#12114yes—
599C2×C4C2×#12318yes—
600C2×C4C2×#12314yes—
601C2×C4#53414yes—
602C2×C4#54618yes—
603C2×C4#55118yes—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22C22×#12111yescharacteristic
614C22C22×#12311yescharacteristic
615C22C2×#53411yescharacteristic
616C22#346112yes—
617C22C2×#54612yes—
618C22C2×#55112yes—
619C22#348511yescharacteristic
620C22#348611yescharacteristic
621C22#348612yes—
622C22#349712yes—
623C22#349912yes—
624C22#350911yescharacteristic
625C22#350911yescharacteristic
626C22#351612yes—
627C22#351812yes—
628C22#352312yes—
629C22#352312yes—
630C22#355212yes—
631C22#355212yes—
632C22#355412yes—
633C22#356112yes—
634C4—24no—
635C4—28no—
636C4—44no—
637C4—44no—
638C4—48no—
639C4—44no—
640C4—44no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4#241218yes—
648C2—44no—
649C2C2×#241211yescharacteristic
650C2#3515311yescharacteristic
651C2#3566711yescharacteristic
652C2#3572312yes—
653C2#3825112yes—
654C2#3906712yes—
655C2#4278712yes—
656C2#4509512yes—
657C2#4532212yes—
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. #15503C2semidirect
7. #15500C2semidirect
8. #15188C2semidirect
9. #15186C2semidirect
13. #14024C2semidirect
14. #13935C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C2×#26252
C2×Q8×C22⋊C42
C2×#23102
#163962
#155032
#155002
#151882
#151862
#148372
#141382
#141292
#140242
#139352
#138532