Finite Group Databasebeta

← #6495533

Subgroups — #6495533

Order 512.

1738 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33297C211yesmaximal, characteristic, complemented
5#33230C211yesmaximal, characteristic, complemented
6#33230C211yesmaximal, characteristic, complemented
7#29854C214yesmaximal, complemented
8#29192C212yesmaximal, complemented
9#29191C212yesmaximal, complemented
10#28658C214yesmaximal, complemented
11#27892C212yesmaximal, complemented
12#27890C212yesmaximal
13#25678C214yesmaximal, complemented
14#22223C214yesmaximal, complemented
15C4×#432C211yesmaximal, characteristic, complemented
16C2×#533—21no—
17C2×#529C2212yes—
18C2×#501C2214yes—
19C2×#489C2214yes—
20C2×#489C2214yes—
21C2×#432C2212yes—
22#2868C2211yescharacteristic
23#2868C2211yescharacteristic
24#2862—21no—
25#2861C2212yes—
26#2860—21no—
27#2860—21no—
28#2859—21no—
29#2859C2212yes—
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2857—21no—
33#2857—21no—
34#2857C2212yes—
35#2857C2212yes—
36#2857C2211yescharacteristic
37#2856C2211yescharacteristic
38#2854—21no—
39#2854—21no—
40#2850C2212yes—
41C4×C4⋊Q8—21no—
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#124C2212yes—
48C4×#121—21no—
49C4×#121—21no—
50#2293—21no—
51#2293—21no—
52#2285—21no—
53#2285C2212yes—
54C2×C43C2211yescharacteristic
55#2067C2218yes—
56#2066C2218yes—
57#2036C2218yes—
58#2034C2218yes—
59#1993C2214yes—
60#1992C2214yes—
61#1959C2214yes—
62#1959C2214yes—
63#1957C2212yes—
64#1957C2212yes—
65#1956C2212yes—
66#1956C2212yes—
67#1928C2218yes—
68#1926C2218yes—
69#1865C2218yes—
70#1862C2218yes—
71#1847C2214yes—
72#1845C2212yes—
73#1844C2212yes—
74#1834C2212yes—
75#1834C2212yes—
76#1767C2212yes—
77C4×C4⋊C8C2212yes—
78#1749C2212yes—
79C4×C22⋊C8C2212yes—
80#1745C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83#533—48no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#127C2311yescharacteristic
92#529—48no—
93C2×#126—22no—
94C2×#125—22no—
95C2×#125C2312yes—
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×#123—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C2×#117—22no—
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#501—48no—
122#489—216no—
123#489—48no—
124#432—28no—
125C2×#107C2318yes—
126C2×#106C2318yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2318yes—
129C2×#103C2318yes—
130C2×#102C2318yes—
131C2×#102C2318yes—
132C2×C22⋊C8C2314yes—
133C2×#97C2312yes—
134C2×#97C2312yes—
135C2×#97C2312yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2312yes—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391C2312yes—
144#391C2312yes—
145#390—21no—
146#390—21no—
147#390—21no—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390C2312yes—
154#390C2312yes—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389C2314yes—
170#389C2314yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#386—22no—
180#386—22no—
181#384—21no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382—22no—
193#382C2312yes—
194#380—21no—
195#380—21no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380C2314yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4C2314yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C43D411yescharacteristic
209C43D411yescharacteristic
210C43D411yescharacteristic
211C43D411yescharacteristic
212C43C2311yescharacteristic
213C43C2311yescharacteristic
214C43C2311yescharacteristic
215C43C2311yescharacteristic
216#340C2318yes—
217#340C2318yes—
218#336C2314yes—
219#336C2314yes—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8C2412yes—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C2412yes—
227C23×C4—22no—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2411yescharacteristic
235C22×C8C2414yes—
236C22×C8C2414yes—
237C4⋊Q8—28no—
238C4⋊Q8—28no—
239C4⋊Q8—24no—
240C4⋊Q8—44no—
241C4⋊Q8—44no—
242C4⋊Q8—44no—
243C4⋊Q8—44no—
244#127—24no—
245#126—48no—
246#125—28no—
247#125—48no—
248#124—28no—
249#124—28no—
250#123—48no—
251C22⋊Q8—44no—
252C22⋊Q8—48no—
253C22⋊Q8—48no—
254#121—44no—
255#121—44no—
256#121—44no—
257C4×Q8—44no—
258C4×Q8—48no—
259C4×Q8—48no—
260C4×D4—44no—
261C4×D4—44no—
262C4×D4—48no—
263C4×D4—44no—
264#117—48no—
265#117—48no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2414yes—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—48no—
342#106—48no—
343C4⋊C8—28no—
344#103—216no—
345#103—48no—
346#102—216no—
347#102—48no—
348C22⋊C8—28no—
349#97—24no—
350#97—24no—
351#97—24no—
352#97#4118yes—
353C4×C8—24no—
354C4×C8—24no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—24no—
365C24—22no—
366C24C22×D411yescharacteristic
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4#13714yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2511yescharacteristic
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8C2×#4118yes—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—44no—
500C42—48no—
501C42AGL(1,ℤ/8)14yes—
502C42C2×#4114yes—
503C42H(5,𝔽2)14yes—
504C422−1+414yes—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23C2×#13712yes—
519C23C23×D411yescharacteristic
520Q8—48no—
521Q8—88no—
522Q8—88no—
523D4—48no—
524D4—88no—
525D4—84no—
526D4—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—28no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—88no—
576C2×C4#53212yes—
577C2×C4#53212yes—
578C2×C4#53512yes—
579C2×C4#53512yes—
580C2×C4#53612yes—
581C2×C4#53612yes—
582C2×C4C2×AGL(1,ℤ/8)12yes—
583C2×C4C2×AGL(1,ℤ/8)12yes—
584C2×C4C2×AGL(1,ℤ/8)12yes—
585C2×C4C2×#13712yes—
586C2×C4C2×#13712yes—
587C2×C4#57512yes—
588C2×C4#57512yes—
589C2×C4C22×#4112yes—
590C2×C4C2×H(5,𝔽2)12yes—
591C2×C4C2×2−1+412yes—
592C8—28no—
593C8—44no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22C2×#53211yescharacteristic
609C22C2×#53511yescharacteristic
610C22C2×#53611yescharacteristic
611C22C22×AGL(1,ℤ/8)11yescharacteristic
612C22C22×#13711yescharacteristic
613C22C2×#57511yescharacteristic
614C22C2×#57511yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—84no—
627C4—82no—
628C4—84no—
629C4—82no—
630C4—82no—
631C4#323914yes—
632C4#324014yes—
633C4#351114yes—
634C2—42no—
635C2—162no—
636C2#3529811yescharacteristic
637C2#6070011yescharacteristic
638C2#6070111yescharacteristic
639C2#6080311yescharacteristic
640C2#6080411yescharacteristic
641C2C2×#323911yescharacteristic
642C2C2×#324011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #33316C2semidirect
4. #33297C2semidirect
5. #33230C2semidirect
6. #33230C2semidirect
7. #29854C2semidirect
8. #29192C2semidirect
9. #29191C2semidirect
10. #28658C2semidirect
11. #27892C2semidirect
13. #25678C2semidirect
14. #22223C2semidirect
15. C4×#432C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#366312
#333162
#332972
#332302
#298542
#291922
#291912
#286582
#278922
#278902
#256782
#222232
C4×#4322