Finite Group Databasebeta

← #412561

Subgroups — #412561

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23611C211yesmaximal, characteristic
2#23610C211yesmaximal, characteristic
3C2×#1946C211yesmaximal, characteristic, complemented
4#21158C211yesmaximal, characteristic, complemented
5C2×#1847C211yesmaximal, characteristic, complemented
6C2×#1816C211yesmaximal, characteristic
7#20662C211yesmaximal, characteristic
8C2×#1765C211yesmaximal, characteristic
9#11239C211yesmaximal, characteristic, complemented
10#11238C211yesmaximal, characteristic, complemented
11#11211C211yesmaximal, characteristic, complemented
12#11210C211yesmaximal, characteristic, complemented
13#11079C211yesmaximal, characteristic
14#11078C211yesmaximal, characteristic
15#10869C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C2×#424C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C22×#104C2211yescharacteristic
26C2×#417C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#416C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2316C2211yescharacteristic
32#2316C2211yescharacteristic
33#2311C2211yescharacteristic
34#2311C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#382C2211yescharacteristic
37#2068—22no—
38#2067—22no—
39#2066—22no—
40#2065—22no—
41#1946—24no—
42#1943—22no—
43#1942—22no—
44#1847—24no—
45#1816—24no—
46#1809—22no—
47#1807—22no—
48#1765—24no—
49C2×#343C2211yescharacteristic
50C2×#343C2211yescharacteristic
51C2×#343C2211yescharacteristic
52C2×#343C2211yescharacteristic
53C2×#341C2211yescharacteristic
54#1736C2211yescharacteristic
55#1736C2211yescharacteristic
56#1736C2211yescharacteristic
57#1736C2211yescharacteristic
58#1736C2211yescharacteristic
59#1736C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62C2×#339C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C2×#131—22no—
65C2×#131—22no—
66C2×#131—22no—
67C2×#131—22no—
68C2×#131—22no—
69C2×#131Q814yes—
70C22×M16—21no—
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C23×C8—21no—
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4—41no—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#108—22no—
88C2×#108—22no—
89C2×#108—22no—
90#427—22no—
91#427—22no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107—41no—
98C2×#107D412yes—
99C2×#107D414yes—
100C2×#106—21no—
101C2×#106—21no—
102C2×#106—22no—
103C2×#106—22no—
104C2×#106—22no—
105C2×#106—22no—
106C2×#106—41no—
107C2×#106D412yes—
108#424—22no—
109#424—22no—
110#424—22no—
111#424—22no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8—22no—
114C2×#104—22no—
115C2×#104—22no—
116C2×#104Q814yes—
117#417—22no—
118#417—22no—
119#417—41no—
120#417—41no—
121#416—22no—
122#416—22no—
123C2×#103—21no—
124C2×#103—21no—
125C2×#103D412yes—
126C2×#102—21no—
127C2×#102—21no—
128C2×#102D412yes—
129#405—22no—
130#405—22no—
131C2×#97—22no—
132C2×#97—22no—
133C2×#97—21no—
134C2×#97D414yes—
135C2×C4×C8—21no—
136#391—21no—
137#391—21no—
138#390—21no—
139#390—21no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#386—41no—
145#386—41no—
146#384—22no—
147#384—22no—
148#382—22no—
149#382—22no—
150#380—22no—
151#380—22no—
152#380—41no—
153#380—41no—
154C2×#95—41no—
155#343—22no—
156#343—22no—
157#343—41no—
158#343—41no—
159#341—22no—
160#340—22no—
161#340—22no—
162#340—22no—
163#340—22no—
164#340—22no—
165#340—22no—
166#339—22no—
167#339—22no—
168#339—22no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41C2×Q812yes—
180C2×#41C2×Q812yes—
181C22×Q8C2×D411yescharacteristic
182C22×D4—21no—
183C22×D4C2×D411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C2411yescharacteristic
191#131—28no—
192#131—28no—
193#131—44no—
194#131—44no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—24no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—42no—
219C2×M16—42no—
220C2×M16C2×Q812yes—
221C2×M16C2×Q812yes—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—24no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—42no—
232C22×C8—42no—
233C22×C8C2×D412yes—
234C22×C8C2×D412yes—
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D412yes—
238C22×C8C2×D412yes—
239C22×C8C2×Q812yes—
240C22×C8#4112yes—
241C22×C8#4112yes—
242C22×C8#4112yes—
243C22×C8#4112yes—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—84no—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42C2×D412yes—
299C2×C42C2×Q812yes—
300C2×C42#4112yes—
301#108—44no—
302#107—24no—
303#107—24no—
304#107—24no—
305#107—44no—
306#106—24no—
307#106—44no—
308#106—42no—
309#106—42no—
310C4⋊C8—44no—
311C4⋊C8—44no—
312#104—28no—
313#103—24no—
314#103—42no—
315#103—42no—
316#102—24no—
317#102—42no—
318#102—42no—
319#97—28no—
320#97—42no—
321C4×C8—42no—
322#95—44no—
323#95—42no—
324#95—42no—
325#95—42no—
326#95—42no—
327C24—21no—
328C24H(5,𝔽2)11yescharacteristic
329#41—22no—
330#41—22no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—22no—
336#41—22no—
337#41—41no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8C4⋊Q812yes—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4C4⋊Q812yes—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C22×Q811yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C42−1+411yescharacteristic
407C22×C42−1+411yescharacteristic
408M16—22no—
409M16—22no—
410M16—22no—
411M16—22no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—48no—
417M16—41no—
418M16—41no—
419M16—41no—
420M16—41no—
421M16—41no—
422M16—41no—
423M16—41no—
424M16—41no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—48no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8#12114yes—
445C2×C8#12112yes—
446C2×C8#12114yes—
447C2×C8#12112yes—
448C2×C8#12112yes—
449C2×C8#12112yes—
450C2×C8C22⋊Q812yes—
451C2×C8C22⋊Q812yes—
452C2×C8C22⋊Q814yes—
453C2×C8C22⋊Q814yes—
454C2×C8#12614yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—88no—
478C4⋊C4—162no—
479C4⋊C4—162no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—162no—
487C22⋊C4—162no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42C22⋊Q814yes—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23#53711yescharacteristic
506C23#54011yescharacteristic
507C23#54411yescharacteristic
508C23D4211yescharacteristic
509C23#54611yescharacteristic
510C23#55411yescharacteristic
511C23#55511yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516D4—22no—
517D4—22no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—41no—
523D4—41no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—161no—
558C2×C4—161no—
559C2×C4—161no—
560C2×C4—161no—
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×C22⋊Q811yescharacteristic
566C2×C4C2×C22⋊Q811yescharacteristic
567C2×C4C2×C22⋊Q811yescharacteristic
568C2×C4C2×C22⋊Q811yescharacteristic
569C2×C4C2×#12611yescharacteristic
570C2×C4C2×C4⋊Q811yescharacteristic
571C2×C4#53211yescharacteristic
572C2×C4#53311yescharacteristic
573C2×C4#53311yescharacteristic
574C2×C4#53711yescharacteristic
575C2×C4#54111yescharacteristic
576C2×C4#54411yescharacteristic
577C2×C4D4211yescharacteristic
578C2×C4#54711yescharacteristic
579C2×C4#54711yescharacteristic
580C2×C4#54711yescharacteristic
581C2×C4#54811yescharacteristic
582C2×C4D4×Q811yescharacteristic
583C2×C4D4×Q811yescharacteristic
584C2×C4#55011yescharacteristic
585C2×C4#55211yescharacteristic
586C2×C4#55311yescharacteristic
587C2×C4#55411yescharacteristic
588C2×C4#55611yescharacteristic
589C8—24no—
590C8—42no—
591C8—42no—
592C8—81no—
593C8—81no—
594C8#39614yes—
595C8#39614yes—
596C22—22no—
597C22—22no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—22no—
603C22—22no—
604C22—42no—
605C22—44no—
606C22—82no—
607C22—81no—
608C22—81no—
609C22#241711yescharacteristic
610C22#242811yescharacteristic
611C22#243911yescharacteristic
612C22#244011yescharacteristic
613C22#249711yescharacteristic
614C22#253711yescharacteristic
615C22#262911yescharacteristic
616C22#328812yes—
617C22#328912yes—
618C4—44no—
619C4—42no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—161no—
626C4—161no—
627C4C2×#39611yescharacteristic
628C4#241611yescharacteristic
629C4#242911yescharacteristic
630C4#244011yescharacteristic
631C4#244111yescharacteristic
632C4#249611yescharacteristic
633C4#253811yescharacteristic
634C4#285711yescharacteristic
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—82no—
640C2#1513111yescharacteristic
641C2#2543811yescharacteristic
642C2#2543911yescharacteristic
643C2#2608412yes—
644C2#2824812yes—
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1946C2semidirect
4. #21158C2semidirect
5. C2×#1847C2semidirect
9. #11239C2semidirect
10. #11238C2semidirect
11. #11211C2semidirect
12. #11210C2semidirect
15. #10869C2semidirect

Maximal subgroups

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

subgroupindex
#236112
#236102
C2×#19462
#211582
C2×#18472
C2×#18162
#206622
C2×#17652
#112392
#112382
#112112
#112102
#110792
#110782
#108692