Finite Group Databasebeta

← #396504

Subgroups — #396504

Order 512.

2868 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
1C23×#102C211yesmaximal, characteristic
2C2×#2093C212yesmaximal, complemented
3C2×#1852C211yesmaximal, characteristic, complemented
4C2×#1808C211yesmaximal, characteristic, complemented
5C2×#1786C211yesmaximal, characteristic
6#13902C211yesmaximal, characteristic
7#13880C211yesmaximal, characteristic, complemented
8#11123C212yesmaximal, complemented
9#11123C212yesmaximal, complemented
10#11049C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic, complemented
16C22×#107C2212yescomplemented
17C22×#107C2211yescharacteristic, complemented
18C22×#102—28no—
19C22×#102—24no—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#385—22no—
28C2×#384C2212yescomplemented
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#379—22no—
32C2×C4×C22⋊C4C2211yescharacteristic
33#2093—28no—
34#1852—24no—
35#1808—24no—
36#1786—24no—
37C2×#340C2211yescharacteristic, complemented
38C2×#340C2212yescomplemented
39C2×#340C2211yescharacteristic, complemented
40C2×#336C2212yescomplemented
41C2×#336C2212yescomplemented
42#1444—22no—
43#1438—22no—
44C4⋊#95C2212yescomplemented
45C26—21no—
46C23×D4—24no—
47C23×D4—24no—
48C23×D4—22no—
49C23×D4—22no—
50C23×D4—21no—
51C23×D4—22no—
52C23×D4—22no—
53C23×D4—22no—
54C23×D4—42no—
55C23×D4—42no—
56C23×D4Q814yes—
57C23×D4C2312yes—
58C24×C4—21no—
59C24×C4C2311yescharacteristic
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—24no—
63C23×C8—42no—
64C23×C8D414yes—
65C23×C8C2312yes—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—24no—
83C22×C22⋊C4D412yes—
84C22×C42C2312yes—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—24no—
88C2×#107—24no—
89C2×#107—24no—
90C2×#107—42no—
91C2×#107—42no—
92C2×#102—232no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—42no—
103C2×#102—42no—
104C2×#102D418yes—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8D414yes—
108#390—24no—
109#390—24no—
110#390—42no—
111#390—42no—
112#385—48no—
113#384—24no—
114#384—42no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380—22no—
119#379—48no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C2×#95—24no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95D412yes—
126#340—22no—
127#340—22no—
128#340—24no—
129#336—24no—
130#336—42no—
131C25—22no—
132C25—24no—
133C25—44no—
134C25—44no—
135C25—42no—
136C25—42no—
137C25—44no—
138C25—44no—
139C25—82no—
140C25—82no—
141C25C2×D411yescharacteristic
142C22×D4—28no—
143C22×D4—216no—
144C22×D4—28no—
145C22×D4—216no—
146C22×D4—28no—
147C22×D4—28no—
148C22×D4—24no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—48no—
161C22×D4—48no—
162C22×D4—48no—
163C22×D4—48no—
164C22×D4—48no—
165C22×D4—48no—
166C22×D4—48no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—48no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4C2×D414yes—
177C22×D4C2×Q814yes—
178C22×D4#4114yes—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—44no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—82no—
192C23×C4C2×D414yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×Q812yes—
197C23×C4#4112yes—
198C23×C4#4112yes—
199C23×C4#4112yes—
200C23×C4C2411yescharacteristic
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—28no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—28no—
216C22×C8—44no—
217C22×C8—48no—
218C22×C8—48no—
219C22×C8C2×D412yes—
220C22×C8C2×D414yes—
221C22×C8C2×D414yes—
222C22×C8C2×D412yes—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—84no—
252C2×C4⋊C4C2×D414yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—84no—
264C2×C22⋊C4—84no—
265C2×C22⋊C4D818yes—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272#107—48no—
273#107—44no—
274#107—44no—
275#102—216no—
276#102—216no—
277#102—48no—
278#102—48no—
279C22⋊C8—28no—
280#95—24no—
281#95—42no—
282#95—42no—
283#95—48no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—416no—
292C24—416no—
293C24—48no—
294C24—48no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—44no—
300C24—44no—
301C24—44no—
302C24—44no—
303C24—48no—
304C24—416no—
305C24—88no—
306C24—816no—
307C24—816no—
308C24—816no—
309C24—816no—
310C24—88no—
311C24—84no—
312C24—84no—
313C24—88no—
314C24—84no—
315C24—84no—
316C24C22⋊Q812yes—
317C24#12312yes—
318C24#12712yes—
319C24C2×D814yes—
320C24#13714yes—
321C24C22×D411yescharacteristic
322C2×D4—232no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—432no—
339C2×D4—416no—
340C2×D4—416no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—416no—
344C2×D4—416no—
345C2×D4—432no—
346C2×D4—416no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—48no—
360C2×D4C22⋊Q818yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—48no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—816no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4C22⋊Q814yes—
409C22×C4C22⋊Q814yes—
410C22×C4C22⋊Q812yes—
411C22×C4#12314yes—
412C22×C4#12314yes—
413C22×C4#12312yes—
414C22×C4#12712yes—
415C22×C4C2×D818yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×Q811yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4112yes—
423C22×C4C2×#4111yescharacteristic
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)12yes—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—416no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8#12014yes—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—48no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—816no—
462C22⋊C4—216no—
463C22⋊C4—816no—
464C22⋊C4—816no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—22no—
477C23—24no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—22no—
482C23—28no—
483C23—28no—
484C23—432no—
485C23—48no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—48no—
492C23—48no—
493C23—48no—
494C23—44no—
495C23—48no—
496C23—48no—
497C23—864no—
498C23—864no—
499C23—816no—
500C23—816no—
501C23—816no—
502C23—816no—
503C23—816no—
504C23—84no—
505C23—84no—
506C23—816no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23#39914yes—
516C23#47418yes—
517C23#48018yes—
518C23#49314yes—
519C23#49714yes—
520C23C2×C22⋊Q811yescharacteristic
521C23C2×#12311yescharacteristic
522C23C2×#12711yescharacteristic
523C23D4212yes—
524C23#54612yes—
525C23C22×D812yes—
526C23C2×#13712yes—
527C23#57612yes—
528C23#57712yes—
529D4—28no—
530D4—28no—
531D4—432no—
532D4—44no—
533D4—44no—
534D4—44no—
535D4—44no—
536D4—44no—
537D4—44no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—816no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—84no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4#39914yes—
578C2×C4C2×#12011yescharacteristic
579C2×C4C2×C22⋊Q811yescharacteristic
580C2×C4C2×C22⋊Q812yes—
581C2×C4C2×C22⋊Q811yescharacteristic
582C2×C4C2×#12311yescharacteristic
583C2×C4C2×#12311yescharacteristic
584C2×C4#52512yes—
585C2×C4#52511yescharacteristic
586C2×C4#53411yescharacteristic
587C2×C4#53511yescharacteristic
588C2×C4#53812yes—
589C2×C4#54312yes—
590C2×C4D4212yes—
591C2×C4#54612yes—
592C2×C4#54612yes—
593C2×C4#54612yes—
594C2×C4#55112yes—
595C2×C4#55112yes—
596C8—24no—
597C8—42no—
598C8—44no—
599C22—216no—
600C22—216no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—416no—
608C22—48no—
609C22—864no—
610C22—88no—
611C22—88no—
612C22—88no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22#209318yescomplemented
620C22C2×#39911yescharacteristic
621C22#241812yes—
622C22#243311yescharacteristic
623C22#243712yes—
624C22#253911yescharacteristic
625C22C2×#47412yes—
626C22#308112yes—
627C22C2×#48012yes—
628C22#309612yes—
629C22C2×#49311yescharacteristic
630C22C2×#49711yescharacteristic
631C22#315411yescharacteristic
632C22#315511yescharacteristic
633C22#326614yes—
634C22D4×D814yes—
635C22#329414yes—
636C22#331414yes—
637C4—24no—
638C4—48no—
639C4—84no—
640C4—161no—
641C4—161no—
642C4—161no—
643C4—161no—
644C4#240511yescharacteristic
645C4#241812yes—
646C4#243712yes—
647C4#248211yescharacteristic
648C4#262611yescharacteristic
649C4#284811yescharacteristic
650C2—28no—
651C2—88no—
652C2#1473411yescharacteristic
653C2C2×#209311yescharacteristic
654C2#2373511yescharacteristic
655C2#2486114yes—
656C2#2577911yescharacteristic
657C2#2578411yescharacteristic
658C2#2595014yes—
659C2#2832911yescharacteristic
660C2#2834211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC24×D4C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2093C2semidirect
3. C2×#1852C2semidirect
4. C2×#1808C2semidirect
7. #13880C2semidirect
8. #11123C2semidirect
9. #11123C2semidirect
10. #11049C2semidirect
11. #10714C2semidirect
15. C22×#107C22semidirect
16. C22×#107C22semidirect
17. C22×#107C22semidirect
28. C2×#384C22semidirect
37. C2×#340C22semidirect
38. C2×#340C22semidirect
39. C2×#340C22semidirect
40. C2×#336C22semidirect
41. C2×#336C22semidirect
44. C4⋊#95C22semidirect
619. C22#2093semidirect

Maximal subgroups

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

subgroupindex
C23×#1022
C2×#20932
C2×#18522
C2×#18082
C2×#17862
#139022
#138802
#111232
#110492
#107142