Finite Group Databasebeta

← #396505

Subgroups — #396505

Order 512.

2796 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#102C211yesmaximal, characteristic
2C2×#2092C212yesmaximal, complemented
3C2×#1853C211yesmaximal, characteristic, complemented
4C2×#1808C211yesmaximal, characteristic, complemented
5C2×#1786C211yesmaximal, characteristic
6#13902C211yesmaximal, characteristic
7#13881C211yesmaximal, characteristic, complemented
8#11123C212yesmaximal, complemented
9#11121C212yesmaximal, complemented
10#11051C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic, complemented
16C22×#107C2211yescharacteristic, complemented
17C22×#106C2212yescomplemented
18C22×#102—28no—
19C22×#102—24no—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#382C2212yescomplemented
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#379—22no—
31C2×C4×C22⋊C4C2211yescharacteristic
32#2092—28no—
33#1853—24no—
34#1808—24no—
35#1786—24no—
36C2×#340C2211yescharacteristic, complemented
37C2×#340C2212yescomplemented
38C2×#340C2211yescharacteristic, complemented
39C2×#336C2212yescomplemented
40C2×#336C2212yescomplemented
41#1442—22no—
42#1438—22no—
43C4⋊#95C2212yescomplemented
44C22⋊#95—22no—
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—42no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—24no—
81C22×C22⋊C4—24no—
82C22×C22⋊C4D412yes—
83C22×C42C2312yes—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—42no—
91C2×#102—232no—
92C2×#102—24no—
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—42no—
102C2×#102—42no—
103C2×#102D418yes—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8D414yes—
107#390—24no—
108#390—24no—
109#390—42no—
110#390—42no—
111#382—24no—
112#382—42no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117#379—48no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C2×#95—24no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—24no—
124C2×#95D412yes—
125#340—22no—
126#340—22no—
127#340—24no—
128#336—24no—
129#336—42no—
130C25—22no—
131C25—24no—
132C25—44no—
133C25—44no—
134C25—42no—
135C25—42no—
136C25—44no—
137C25—44no—
138C25—82no—
139C25—82no—
140C25C2×D411yescharacteristic
141C22×D4—28no—
142C22×D4—216no—
143C22×D4—28no—
144C22×D4—216no—
145C22×D4—28no—
146C22×D4—28no—
147C22×D4—24no—
148C22×D4—22no—
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—48no—
160C22×D4—48no—
161C22×D4—48no—
162C22×D4—48no—
163C22×D4—48no—
164C22×D4—48no—
165C22×D4—48no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—48no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4C2×D414yes—
176C22×D4C2×Q814yes—
177C22×D4#4114yes—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—44no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—82no—
191C23×C4C2×D414yes—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×Q812yes—
196C23×C4#4112yes—
197C23×C4#4112yes—
198C23×C4#4112yes—
199C23×C4C2411yescharacteristic
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—28no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—28no—
215C22×C8—44no—
216C22×C8—48no—
217C22×C8—48no—
218C22×C8C2×D412yes—
219C22×C8C2×D414yes—
220C22×C8C2×D414yes—
221C22×C8C2×D412yes—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
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—22no—
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—48no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4C2×D414yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—84no—
261C2×C22⋊C4—84no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268#107—44no—
269#107—44no—
270#106—48no—
271#102—216no—
272#102—216no—
273#102—48no—
274#102—48no—
275C22⋊C8—28no—
276#95—24no—
277#95—48no—
278#95—42no—
279#95—42no—
280#95—84no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—416no—
289C24—416no—
290C24—48no—
291C24—48no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—44no—
297C24—44no—
298C24—44no—
299C24—44no—
300C24—48no—
301C24—416no—
302C24—88no—
303C24—816no—
304C24—816no—
305C24—816no—
306C24—816no—
307C24—88no—
308C24—84no—
309C24—84no—
310C24—88no—
311C24—84no—
312C24—84no—
313C24C22⋊Q812yes—
314C24#12312yes—
315C24#12712yes—
316C24#13514yes—
317C24AGL(1,ℤ/8)14yes—
318C24C22×D411yescharacteristic
319C2×D4—232no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
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—432no—
336C2×D4—416no—
337C2×D4—416no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—416no—
341C2×D4—416no—
342C2×D4—432no—
343C2×D4—416no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
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—48no—
357C2×D4C22⋊Q818yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
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—48no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
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—816no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4C22⋊Q814yes—
407C22×C4C22⋊Q814yes—
408C22×C4C22⋊Q812yes—
409C22×C4#12314yes—
410C22×C4#12314yes—
411C22×C4#12312yes—
412C22×C4#12712yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×Q811yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4112yes—
420C22×C4C2×#4111yescharacteristic
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)12yes—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—416no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8#12014yes—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—48no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C22⋊C4—48no—
459C22⋊C4—816no—
460C22⋊C4—816no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—42no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—22no—
473C23—24no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—22no—
478C23—28no—
479C23—28no—
480C23—432no—
481C23—48no—
482C23—48no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—48no—
489C23—48no—
490C23—48no—
491C23—44no—
492C23—48no—
493C23—864no—
494C23—864no—
495C23—816no—
496C23—816no—
497C23—816no—
498C23—816no—
499C23—816no—
500C23—84no—
501C23—84no—
502C23—816no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23#39914yes—
512C23#47818yes—
513C23#48218yes—
514C23#49514yes—
515C23#49614yes—
516C23C2×C22⋊Q811yescharacteristic
517C23C2×#12311yescharacteristic
518C23C2×#12711yescharacteristic
519C23D4212yes—
520C23#54612yes—
521C23C2×#13512yes—
522C23C2×AGL(1,ℤ/8)12yes—
523C23#57612yes—
524C23#57712yes—
525D4—28no—
526D4—28no—
527D4—432no—
528D4—44no—
529D4—44no—
530D4—44no—
531D4—44no—
532D4—44no—
533D4—44no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—816no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—84no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4#39914yes—
574C2×C4C2×#12011yescharacteristic
575C2×C4C2×C22⋊Q811yescharacteristic
576C2×C4C2×C22⋊Q812yes—
577C2×C4C2×C22⋊Q811yescharacteristic
578C2×C4C2×#12311yescharacteristic
579C2×C4C2×#12311yescharacteristic
580C2×C4#52512yes—
581C2×C4#52511yescharacteristic
582C2×C4#53411yescharacteristic
583C2×C4#53511yescharacteristic
584C2×C4#53812yes—
585C2×C4#54312yes—
586C2×C4D4212yes—
587C2×C4#54612yes—
588C2×C4#54612yes—
589C2×C4#54612yes—
590C2×C4#55112yes—
591C2×C4#55112yes—
592C8—24no—
593C8—42no—
594C8—44no—
595C22—216no—
596C22—216no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—416no—
604C22—48no—
605C22—864no—
606C22—88no—
607C22—88no—
608C22—88no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22#209218yescomplemented
616C22C2×#39911yescharacteristic
617C22#241812yes—
618C22#243311yescharacteristic
619C22#243712yes—
620C22#253911yescharacteristic
621C22C2×#47812yes—
622C22#308112yes—
623C22C2×#48212yes—
624C22#309612yes—
625C22C2×#49511yescharacteristic
626C22C2×#49611yescharacteristic
627C22#315411yescharacteristic
628C22#315511yescharacteristic
629C22#326514yes—
630C22#328014yes—
631C22#329614yes—
632C22#331814yes—
633C4—24no—
634C4—84no—
635C4—84no—
636C4—161no—
637C4—161no—
638C4—161no—
639C4—161no—
640C4#240511yescharacteristic
641C4#241812yes—
642C4#243712yes—
643C4#248211yescharacteristic
644C4#262611yescharacteristic
645C4#284811yescharacteristic
646C2—28no—
647C2—88no—
648C2#1473411yescharacteristic
649C2C2×#209211yescharacteristic
650C2#2373511yescharacteristic
651C2#2486914yes—
652C2#2578111yescharacteristic
653C2#2578311yescharacteristic
654C2#2595214yes—
655C2#2833011yescharacteristic
656C2#2834111yescharacteristic
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×#2092C2semidirect
3. C2×#1853C2semidirect
4. C2×#1808C2semidirect
7. #13881C2semidirect
8. #11123C2semidirect
9. #11121C2semidirect
10. #11051C2semidirect
11. #10714C2semidirect
15. C22×#107C22semidirect
16. C22×#107C22semidirect
17. C22×#106C22semidirect
27. C2×#382C22semidirect
36. C2×#340C22semidirect
37. C2×#340C22semidirect
38. C2×#340C22semidirect
39. C2×#336C22semidirect
40. C2×#336C22semidirect
43. C4⋊#95C22semidirect
615. C22#2092semidirect

Maximal subgroups

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

subgroupindex
C23×#1022
C2×#20922
C2×#18532
C2×#18082
C2×#17862
#139022
#138812
#111232
#111212
#110512
#107142