Finite Group Databasebeta

← #6492057

Subgroups — #6492057

Order 512.

1778 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
1#35292C211yesmaximal, characteristic
2#34127C211yesmaximal, characteristic, complemented
3#33109C212yesmaximal, complemented
4#33106C212yesmaximal, complemented
5#29926C212yesmaximal, complemented
6#29925C212yesmaximal, complemented
7#26329C214yesmaximal, complemented
8#26244C214yesmaximal, complemented
9#24866C214yesmaximal, complemented
10#24864C214yesmaximal, complemented
11#21961C211yesmaximal, characteristic, complemented
12#21851C211yesmaximal, characteristic, complemented
13#21809C211yesmaximal, characteristic, complemented
14#21364C211yesmaximal, characteristic, complemented
15#21257C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#484C2214yes—
19C2×#480C2214yes—
20C2×#432C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#416C2214yes—
23#2857C2212yes—
24#2856C2212yes—
25#2624—21no—
26#2615—21no—
27#2592C2212yes—
28#2580—21no—
29#2540C2212yes—
30#2535—21no—
31#2414—22no—
32#2407C2214yes—
33#2355C2212yes—
34#2352C2212yes—
35#2352C2212yes—
36D4×C4⋊C4C2212yes—
37#2347—21no—
38#2343—22no—
39#2339—21no—
40#2324—22no—
41#2295—21no—
42#2290—22no—
43#2285C2211yescharacteristic
44#2285C2211yescharacteristic
45#2283C2212yes—
46C42×D4—21no—
47C4×#117C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49#2097C2214yes—
50#2092C2214yes—
51#2074C2214yes—
52#2073C2214yes—
53#2036C2214yes—
54#2035C2214yes—
55#2034C2214yes—
56#2033C2214yes—
57#1985C2212yes—
58#1983C2214yes—
59#1982C2211yescharacteristic
60#1982C2211yescharacteristic
61#1982C2212yes—
62#1981C2211yescharacteristic
63#1981C2211yescharacteristic
64#1981C2212yes—
65#1959C2214yes—
66#1955C2212yes—
67#1948C2212yes—
68#1947C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1947C2214yes—
72#1928C2214yes—
73#1928C2214yes—
74#1925C2214yes—
75#1925C2214yes—
76#1874C2214yes—
77#1872C2214yes—
78#1869C2214yes—
79#1854C2214yes—
80#1808C2212yes—
81#1808C2212yes—
82#1797C2212yes—
83#1797C2212yes—
84#1788C2212yes—
85#1788C2212yes—
86#1776C2214yes—
87C22×#41D411yescharacteristic
88C22×#41C2312yes—
89C23×C8C2312yes—
90#532—24no—
91#532—24no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#124C2312yes—
95C2×C22⋊Q8C2314yes—
96C2×#121—22no—
97C2×#121—21no—
98C2×#121—21no—
99C2×#121—22no—
100C2×#121C2314yes—
101C4×#41—24no—
102C4×#41—24no—
103C2×C4×Q8D411yescharacteristic
104C2×C4×Q8C2311yescharacteristic
105C2×C4×D4—22no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4D412yes—
109C2×C4×D4D411yescharacteristic
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2312yes—
112C2×#117—21no—
113C2×#117—21no—
114C2×#117D412yes—
115C2×#117D411yescharacteristic
116C2×#117C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C42—22no—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#484—48no—
123#480—48no—
124#432—24no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8—24no—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134#416—48no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2312yes—
147#399—22no—
148#397—22no—
149#394—24no—
150#393—22no—
151#392—24no—
152#392—22no—
153#391—22no—
154#391C2314yes—
155#390—24no—
156#390—22no—
157#390—22no—
158#390C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#388—24no—
163#388—22no—
164#388—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#386—22no—
170#386—22no—
171#386—24no—
172#385—24no—
173#385—22no—
174#385—22no—
175#384C2312yes—
176#384C2312yes—
177#383—22no—
178#382—21no—
179#382—21no—
180#382C2312yes—
181#382C2312yes—
182#381—22no—
183#380—22no—
184#380—21no—
185#380—22no—
186#380—21no—
187#380C2311yescharacteristic
188#380C2311yescharacteristic
189#380C2312yes—
190#380C2312yes—
191#380C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C22⋊C4—21no—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209#376—21no—
210#376—22no—
211C2×#95—22no—
212C43—21no—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219C2×#41—24no—
220C2×#41—24no—
221C2×#41—24no—
222C2×#41—44no—
223C2×#41—44no—
224C2×#41—44no—
225C22×Q8C2×D411yescharacteristic
226C22×Q8C2412yes—
227C22×D4—24no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2×D411yescharacteristic
232C22×D4C2×D412yes—
233C22×D4C2412yes—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C22×C8—24no—
244C22×C8—44no—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250#127—22no—
251#127—22no—
252#124—24no—
253#124—24no—
254C22⋊Q8—44no—
255C22⋊Q8—44no—
256#121—48no—
257#121—44no—
258#121—44no—
259#121—44no—
260#121—44no—
261#121—84no—
262C4×Q8—24no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×D4—28no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—48no—
272C4×D4—44no—
273C4×D4—44no—
274#117—24no—
275#117—28no—
276#117—24no—
277#117—42no—
278#117—82no—
279#117—82no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—84no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42#4114yes—
333C2×C42#4114yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—28no—
338#107—44no—
339#107—44no—
340#106—44no—
341C4⋊C8—22no—
342C4⋊C8—22no—
343C4⋊C8—24no—
344C4⋊C8—24no—
345#103—28no—
346#103—44no—
347#103—44no—
348#103—44no—
349#102—28no—
350#102—44no—
351#102—44no—
352#102—44no—
353C22⋊C8—24no—
354C22⋊C8—44no—
355C22⋊C8—44no—
356#97—24no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—24no—
375C24—22no—
376C24C22×D411yescharacteristic
377C24C22×D412yes—
378#41—416no—
379#41—416no—
380#41—84no—
381#41—84no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—42no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—84no—
405C2×D4—84no—
406C2×D4—84no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—84no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4#12314yes—
439C22×C4#12314yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2511yescharacteristic
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—84no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4H(5,𝔽2)14yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C22⋊C4—82no—
496C22⋊C4—84no—
497C42—24no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—44no—
502C42—44no—
503C42—42no—
504C42—44no—
505C42—42no—
506C42—82no—
507C42—82no—
508C42—82no—
509C42#12314yes—
510C42#12314yes—
511C42C2×#4112yes—
512C42C2×#4112yes—
513C42C2×#4112yes—
514C42C2×#4112yes—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23—21no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23C2×#12312yes—
531C23#57514yes—
532C23C23×D411yescharacteristic
533Q8—44no—
534Q8—44no—
535Q8—42no—
536Q8—42no—
537D4—28no—
538D4—48no—
539D4—44no—
540D4—42no—
541D4—42no—
542D4—84no—
543D4—88no—
544D4—84no—
545D4—168no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—88no—
589C2×C4—88no—
590C2×C4—88no—
591C2×C4C2×#12312yes—
592C2×C4C2×#12312yes—
593C2×C4C2×#12312yes—
594C2×C4C2×#12312yes—
595C2×C4C2×#12312yes—
596C2×C4#57612yes—
597C2×C4#57812yes—
598C2×C4C22×#4111yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C2×H(5,𝔽2)12yes—
601C2×C4#58512yes—
602C8—44no—
603C8—42no—
604C8—42no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—84no—
618C22—84no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22C22×#12311yescharacteristic
623C22#348812yes—
624C22C2×#57512yes—
625C22C2×#57611yescharacteristic
626C22C2×#57811yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—44no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—82no—
637C4—84no—
638C4—82no—
639C4—82no—
640C4#309814yes—
641C4#346012yes—
642C4#350811yescharacteristic
643C4#351011yescharacteristic
644C2—44no—
645C2—42no—
646C2—162no—
647C2#3525411yescharacteristic
648C2#6051111yescharacteristic
649C2C2×#309811yescharacteristic
650C2#6197712yes—
651C2#6198212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35292C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34127C2semidirect
3. #33109C2semidirect
4. #33106C2semidirect
5. #29926C2semidirect
6. #29925C2semidirect
7. #26329C2semidirect
8. #26244C2semidirect
9. #24866C2semidirect
10. #24864C2semidirect
11. #21961C2semidirect
12. #21851C2semidirect
13. #21809C2semidirect
14. #21364C2semidirect
15. #21257C2semidirect

Maximal subgroups

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

subgroupindex
#352922
#341272
#331092
#331062
#299262
#299252
#263292
#262442
#248662
#248642
#219612
#218512
#218092
#213642
#212572