Finite Group Databasebeta

← #4309185

Subgroups — #4309185

Order 512.

2548 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2621C211yesmaximal, characteristic, complemented
2C2×#2290C211yesmaximal, characteristic
3#16624C214yesmaximal, complemented
4#16582C214yesmaximal, complemented
5#16402C214yesmaximal, complemented
6#16380C214yesmaximal
7#15235C211yesmaximal, characteristic, complemented
8#15223C211yesmaximal, characteristic, complemented
9#15180C211yesmaximal, characteristic, complemented
10#15179C212yesmaximal, complemented
11#15178C211yesmaximal, characteristic, complemented
12#15177C212yesmaximal
13#14702C212yesmaximal, complemented
14#14343C211yesmaximal, characteristic, complemented
15#14261C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2621—28no—
23C2×#398C2214yes—
24C2×#397C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30#2290—28no—
31C2×#388C2212yes—
32C2×#387C2212yes—
33C2×#386C2212yes—
34C2×#385C2212yes—
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×#376C2211yescharacteristic
42#1445C2214yes—
43#1445C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C23×Q8C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—44no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—44no—
83C2×C22⋊Q8—44no—
84C2×C22⋊Q8—44no—
85C2×C22⋊Q8—44no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×#117—44no—
90C2×#117D418yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C42—24no—
111C22×C42—44no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118#398—216no—
119#397—216no—
120#391—28no—
121#391—28no—
122#391—44no—
123#391—44no—
124#389—44no—
125#388—28no—
126#387—28no—
127#386—28no—
128#385—28no—
129#382—24no—
130#380—44no—
131#380—44no—
132C4×C4⋊C4—24no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—24no—
135#376—24no—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C25C2411yescharacteristic
162C25C2411yescharacteristic
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—24no—
173C22×D4—22no—
174C22×D4—42no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4C2×D414yes—
191C23×C4#4114yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C4⋊Q8—416no—
206C22⋊Q8—432no—
207C22⋊Q8—432no—
208C4×D4—232no—
209#117—416no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D418yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4C2×D418yes—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—416no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42C2×D414yes—
280C2×C42C2×D414yes—
281C2×C42#4114yes—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—21no—
315C24—21no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24C22×D412yes—
323C24C2×#4112yes—
324C242−1+414yes—
325C24C2511yescharacteristic
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×D4—28no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—44no—
337C2×D4—44no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4H(5,𝔽2)14yes—
434C22×C4H(5,𝔽2)14yes—
435C22×C42−1+414yes—
436C4⋊C4—28no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—416no—
440C4⋊C4—416no—
441C4⋊C4—416no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—88no—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C42—28no—
456C42—28no—
457C42—28no—
458C42—48no—
459C42—44no—
460C42—44no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23#54114yes—
503C23C23×D411yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4111yescharacteristic
506C23C2×H(5,𝔽2)12yes—
507C23C2×H(5,𝔽2)12yes—
508C23C2×2−1+412yes—
509C23C2×2−1+412yes—
510C23#58512yes—
511C23#58512yes—
512Q8—816no—
513D4—416no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—432no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
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—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—416no—
557C2×C4—416no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
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—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4#52514yes—
594C2×C4#53414yes—
595C2×C4#53514yes—
596C2×C4#53818yes—
597C2×C4#54018yes—
598C22—28no—
599C22—24no—
600C22—42no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—44no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C2×#52511yescharacteristic
617C22C2×#53411yescharacteristic
618C22C2×#53511yescharacteristic
619C22C2×#53812yes—
620C22C2×#54012yes—
621C22C2×#54111yescharacteristic
622C22#345511yescharacteristic
623C22#346112yes—
624C22#348912yes—
625C22#348912yes—
626C22#349512yes—
627C22#350911yescharacteristic
628C22#351211yescharacteristic
629C22#353812yes—
630C22#354012yes—
631C22#354112yes—
632C22#354112yes—
633C22#355212yes—
634C22#355412yes—
635C22#356112yes—
636C22#356212yes—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—84no—
645C4—82no—
646C4—84no—
647C4—84no—
648C4—84no—
649C4#263518yes—
650C2—44no—
651C2—82no—
652C2C2×#263511yescharacteristic
653C2#3530911yescharacteristic
654C2#3531011yescharacteristic
655C2#3737412yes—
656C2#3737912yes—
657C2#3739912yes—
658C2#3741712yes—
659C2#3749512yes—
660C2#3749912yes—
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×#2621C2semidirect
3. #16624C2semidirect
4. #16582C2semidirect
5. #16402C2semidirect
7. #15235C2semidirect
8. #15223C2semidirect
9. #15180C2semidirect
10. #15179C2semidirect
11. #15178C2semidirect
13. #14702C2semidirect
14. #14343C2semidirect
15. #14261C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
C2×#22902
#166242
#165822
#164022
#163802
#152352
#152232
#151802
#151792
#151782
#151772
#147022
#143432
#142612