Finite Group Databasebeta

← #544491

Subgroups — #544491

Order 512.

2552 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2395C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic, complemented
3C2×#2318C211yesmaximal, characteristic
4#14629C212yesmaximal, complemented
5#14540C214yesmaximal
6#14526C214yesmaximal, complemented
7#14344C211yesmaximal, characteristic
8#14322C212yesmaximal, complemented
9#14306C212yesmaximal, complemented
10#14244C212yesmaximal, complemented
11#13965C211yesmaximal, characteristic, complemented
12#13958C211yesmaximal, characteristic, complemented
13#13918C212yesmaximal, complemented
14#13915C212yesmaximal, complemented
15#13900C212yesmaximal, complemented
16#13895C212yesmaximal, complemented
17#13807C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2395C4116yescomplemented
21C2×#400C2211yescharacteristic
22C2×#399C2211yescharacteristic
23C2×#398C2211yescharacteristic
24C2×#397C2212yes—
25C2×#397C2211yescharacteristic
26C2×#394C2212yes—
27#2346—28no—
28#2318—28no—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#388C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#387C2212yes—
35C2×#386C2212yes—
36C2×#385C2212yes—
37C2×#385C2214yes—
38C2×#383C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2212yes—
41C2×#381C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×#376C2212yes—
51C22×#95C2211yescharacteristic
52#1445C2212yes—
53#1445C2214yes—
54#1445C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1444C2212yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2212yes—
63#1440C2212yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2212yes—
68#1437C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2214yes—
77#1434C2212yes—
78#1434C2212yes—
79#1434C2212yes—
80#1434C2212yes—
81#1434C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4C2×C418yes—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117C2×C418yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125#400C2×C418yescomplemented
126#399C2×C418yescomplemented
127#398C2×C418yescomplemented
128#397C2×C4116yes—
129#397C2×C418yescomplemented
130#394C2×C4116yes—
131#389—28no—
132#389—48no—
133#388—216no—
134#388—28no—
135#388—48no—
136#387—28no—
137#386—28no—
138#385—28no—
139#385—48no—
140#383—24no—
141#382—24no—
142#382—28no—
143#381—24no—
144#381—24no—
145#380—24no—
146#380—24no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4C2×C4116yescomplemented
149C4×C22⋊C4C2×C418yes—
150C4×C22⋊C4C2×C4116yescomplemented
151C4×C22⋊C4D418yes—
152#376—28no—
153C2×#95—24no—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C25C2411yescharacteristic
187C25C2411yescharacteristic
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4C22×C414yes—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C22×C414yes—
203C23×C4C22×C414yes—
204C23×C4C2×D414yes—
205C23×C4#4114yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C4×D4—232no—
222#117—232no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
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⋊C4C22×C414yes—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C22×C418yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
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×C22⋊C4C22×C418yes—
277C2×C22⋊C4C22×C414yes—
278C2×C22⋊C4C22×C414yes—
279C2×C22⋊C4C22×C418yes—
280C2×C22⋊C4C22×C418yes—
281C2×C22⋊C4C22×C418yes—
282C2×C22⋊C4C2×D418yes—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42C22×C418yes—
296C2×C42C22×C414yes—
297C2×C42C22×C418yes—
298C2×C42C2×D418yes—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C22×C414yes—
324#95C22×C414yes—
325#95C22×C418yes—
326#95C22×C418yes—
327#95C22×C418yes—
328#95C22×C418yes—
329#95C2×D414yes—
330#95C2×D414yes—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24C23×C412yes—
344C24C23×C412yes—
345C24C22×D412yes—
346C24C2×#4112yes—
347C24C2511yescharacteristic
348C2×D4—28no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—48no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4C4×D4116yes—
427C22×C4C23×C414yes—
428C22×C4C23×C412yes—
429C22×C4C23×C412yes—
430C22×C4C23×C414yes—
431C22×C4C23×C414yes—
432C22×C4C23×C412yes—
433C22×C4C23×C412yes—
434C22×C4C23×C412yes—
435C22×C4C23×C414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C42−1+414yes—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—216no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—88no—
451C4⋊C4—84no—
452C4⋊C4—88no—
453C4⋊C4—84no—
454C4⋊C4—88no—
455C22⋊C4—216no—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—216no—
459C22⋊C4—216no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C22⋊C4—84no—
465C22⋊C4—84no—
466C22⋊C4—88no—
467C42—216no—
468C42—28no—
469C42—48no—
470C42—48no—
471C42—416no—
472C42—88no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23C2×C4×D414yes—
505C23C24×C411yescharacteristic
506C23C23×D411yescharacteristic
507C23C22×#4111yescharacteristic
508C23C2×H(5,𝔽2)12yes—
509C23C2×2−1+412yes—
510C23#58512yes—
511C23#58512yes—
512C23#58514yes—
513D4—416no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—416no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—88no—
591C2×C4—816no—
592C2×C4C2×C4×D414yes—
593C2×C4C2×C4×D414yes—
594C2×C4#52018yes—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—48no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—82no—
609C22—82no—
610C22—84no—
611C22C22×C4×D411yescharacteristic
612C22C2×#52012yes—
613C22#342811yescharacteristic
614C22#342811yescharacteristic
615C22#342812yes—
616C22#344912yes—
617C22#345512yes—
618C22#348512yes—
619C22#349112yes—
620C22#349811yescharacteristic
621C22#349911yescharacteristic
622C22#351112yes—
623C22#354514yes—
624C22#355114yes—
625C22#355314yes—
626C22#355714yes—
627C4—28no—
628C4—24no—
629C4—24no—
630C4—44no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—44no—
637C4—44no—
638C4—84no—
639C4—82no—
640C4—82no—
641C4—84no—
642C2—44no—
643C2—82no—
644C2#3406511yescharacteristic
645C2#3419911yescharacteristic
646C2#3420711yescharacteristic
647C2#3439012yes—
648C2#3439712yes—
649C2#4138112yes—
650C2#4138312yes—
651C2#4140114yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
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×#2346C2semidirect
4. #14629C2semidirect
6. #14526C2semidirect
8. #14322C2semidirect
9. #14306C2semidirect
10. #14244C2semidirect
11. #13965C2semidirect
12. #13958C2semidirect
13. #13918C2semidirect
14. #13915C2semidirect
15. #13900C2semidirect
16. #13895C2semidirect
17. #13807C2semidirect
20. #2395C4semidirect
125. #400C2×C4semidirect
126. #399C2×C4semidirect
127. #398C2×C4semidirect
129. #397C2×C4semidirect
148. C4×C4⋊C4C2×C4semidirect
150. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23952
C2×#23462
C2×#23182
#146292
#145402
#145262
#143442
#143222
#143062
#142442
#139652
#139582
#139182
#139152
#139002
#138952
#138072