Finite Group Databasebeta

← #518826

Subgroups — #518826

Order 512.

3334 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C211yesmaximal, characteristic
2C2×#2347C211yesmaximal, characteristic, complemented
3C2×#2273C211yesmaximal, characteristic
4#14313C212yesmaximal, complemented
5#14295C211yesmaximal, characteristic, complemented
6#14240C212yesmaximal, complemented
7#14229C212yesmaximal, complemented
8#13990C214yesmaximal, complemented
9#13983C214yesmaximal, complemented
10#13971C214yesmaximal
11#13903C211yesmaximal, characteristic
12C4×#399C212yesmaximal, complemented
13#13853C211yesmaximal, characteristic, complemented
14#13823C214yesmaximal, complemented
15#13807C211yesmaximal, characteristic, complemented
16C22×#117C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C23×C22⋊C4C2211yescharacteristic
19#2482C4116yescomplemented
20C2×#399C2212yes—
21C2×#397C2212yes—
22#2347—28no—
23C2×#389C2212yescomplemented
24C2×#388C2214yes—
25C2×#388C2214yes—
26C2×#388C2214yes—
27C2×#385C2214yes—
28C2×#385C2212yes—
29C2×#383C2212yescomplemented
30C2×#381C2212yes—
31C2×#381C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#379C2212yes—
34C2×#379C2212yes—
35#2273—28no—
36C2×C4×C4⋊C4C2212yescomplemented
37C2×C4×C4⋊C4C2212yescomplemented
38C2×C4×C4⋊C4C2214yescomplemented
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2214yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48#1445C2211yescharacteristic, complemented
49#1445C2211yescharacteristic, complemented
50#1445C2211yescharacteristic, complemented
51#1445C2211yescharacteristic, complemented
52#1445C2214yescomplemented
53#1444C2212yescomplemented
54#1442C2212yescomplemented
55#1442C2214yescomplemented
56#1440C2214yes—
57#1440C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2212yes—
61#1439C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69#1436C2214yescomplemented
70#1434C2214yescomplemented
71#1434C2212yescomplemented
72#1434C2212yescomplemented
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2214yescomplemented
76C4×#95C2212yescomplemented
77C4×#95C2212yescomplemented
78C26C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C2×#117—24no—
84C2×#117—24no—
85C2×#117C2×C418yescomplemented
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4C2×C418yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C42C2314yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2314yes—
119C22×C42C2312yes—
120#399C2×C4116yescomplemented
121#397C2×C4116yescomplemented
122#389—28no—
123#388—216no—
124#388—216no—
125#388C2×C4132yescomplemented
126#385—216no—
127#385C2×C4116yescomplemented
128#383—28no—
129#381—44no—
130#381C2×C418yescomplemented
131#380—24no—
132#379—44no—
133#379C2×C4116yes—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—48no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—24no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—28no—
141C4×C22⋊C4—28no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—28no—
144C4×C22⋊C4—48no—
145C4×C22⋊C4D418yes—
146C2×#95C2311yescharacteristic
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C25—22no—
173C25—24no—
174C25—24no—
175C25—22no—
176C25—44no—
177C25—42no—
178C25—42no—
179C25C22×C414yes—
180C25C2411yescharacteristic
181C25C2412yes—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C22×C414yes—
197C23×C4C22×C414yes—
198C23×C4C2×D414yes—
199C23×C4#4118yes—
200C23×C4#4114yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214#117—232no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C22×C4116yes—
232C2×C22⋊C4—264no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—232no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C418yes—
270C2×C22⋊C4C22×C4116yes—
271C2×C22⋊C4C22×C418yes—
272C2×C22⋊C4C22×C4116yes—
273C2×C22⋊C4C2×D418yes—
274C2×C22⋊C4#4118yes—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—28no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—24no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42C22×C418yes—
293C2×C42C2×D418yes—
294#95—24no—
295#95—22no—
296#95—22no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—22no—
302#95—22no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95C22×C418yes—
316#95C22×C418yes—
317#95C22×C4116yes—
318#95C2×D414yes—
319#95C2×D414yes—
320C24—216no—
321C24—28no—
322C24—28no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—22no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—28no—
337C24—216no—
338C24—48no—
339C24—48no—
340C24—84no—
341C24—84no—
342C24—84no—
343C24—82no—
344C24—82no—
345C24—88no—
346C24—88no—
347C24—88no—
348C24—88no—
349C24C23×C412yes—
350C24C23×C414yes—
351C24C22×D412yes—
352C24C2×#4112yes—
353C24C2×#4114yes—
354C24C2×#4114yes—
355C24C2511yescharacteristic
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—88no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—88no—
420C22×C4—84no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—88no—
424C22×C4C4×D4116yes—
425C22×C4C23×C418yes—
426C22×C4C23×C414yes—
427C22×C4C23×C414yes—
428C22×C4C23×C414yes—
429C22×C4C23×C412yes—
430C22×C4C23×C412yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4H(5,𝔽2)14yes—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—416no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—216no—
451C22⋊C4—232no—
452C22⋊C4—216no—
453C22⋊C4—216no—
454C22⋊C4—216no—
455C22⋊C4—216no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—416no—
460C22⋊C4—84no—
461C22⋊C4—88no—
462C42—216no—
463C42—48no—
464C42—416no—
465C42—84no—
466C42—84no—
467C42—88no—
468C42—84no—
469C42—84no—
470C23—232no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—28no—
476C23—28no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—832no—
509C23—88no—
510C23—88no—
511C23—816no—
512C23—816no—
513C23—816no—
514C23—816no—
515C23C2×C4×D414yes—
516C23C4×#4118yes—
517C23C24×C411yescharacteristic
518C23C23×D411yescharacteristic
519C23C22×#4111yescharacteristic
520C23C22×#4112yes—
521C23C22×#4112yes—
522C23C2×H(5,𝔽2)12yes—
523C23#58514yes—
524C23#58512yes—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—416no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—816no—
591C2×C4—88no—
592C2×C4—816no—
593C2×C4C2×C4×D414yes—
594C2×C4C2×C4×D414yes—
595C2×C4#52518yes—
596C22—216no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—28no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—48no—
605C22—48no—
606C22—44no—
607C22—48no—
608C22—44no—
609C22—48no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—832no—
616C22C22×C4×D411yescharacteristic
617C22C2×C4×#4112yes—
618C22#342812yes—
619C22C4×H(5,𝔽2)12yes—
620C22C2×#52512yes—
621C22#344411yescharacteristic
622C22#344411yescharacteristic
623C22#348512yes—
624C22#349112yes—
625C22#349412yes—
626C22#349811yescharacteristic
627C22#350011yescharacteristic
628C22#350514yes—
629C22#351014yes—
630C22#351314yes—
631C22#352114yes—
632C4—28no—
633C4—24no—
634C4—24no—
635C4—44no—
636C4—48no—
637C4—44no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—84no—
644C4—82no—
645C2—48no—
646C2—44no—
647C2C4×#52511yescharacteristic
648C2#3419911yescharacteristic
649C2#3427511yescharacteristic
650C2#3431514yes—
651C2#3584012yes—
652C2#3585112yes—
653C2#3595412yes—
654C2#3603312yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2347C2semidirect
4. #14313C2semidirect
5. #14295C2semidirect
6. #14240C2semidirect
7. #14229C2semidirect
8. #13990C2semidirect
9. #13983C2semidirect
12. C4×#399C2semidirect
13. #13853C2semidirect
14. #13823C2semidirect
15. #13807C2semidirect
19. #2482C4semidirect
23. C2×#389C22semidirect
29. C2×#383C22semidirect
36. C2×C4×C4⋊C4C22semidirect
37. C2×C4×C4⋊C4C22semidirect
38. C2×C4×C4⋊C4C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1445C22semidirect
53. #1444C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. #1434C22semidirect
71. #1434C22semidirect
72. #1434C22semidirect
75. C4×#95C22semidirect
76. C4×#95C22semidirect
77. C4×#95C22semidirect
85. C2×#117C2×C4semidirect
120. #399C2×C4semidirect
121. #397C2×C4semidirect
125. #388C2×C4semidirect
127. #385C2×C4semidirect
130. #381C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C2×#23472
C2×#22732
#143132
#142952
#142402
#142292
#139902
#139832
#139712
#139032
C4×#3992
#138532
#138232
#138072