Finite Group Databasebeta

← #193283

Subgroups — #193283

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34235C211yesmaximal, characteristic
2#21963C211yesmaximal, characteristic
3#21835C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic
5C2×#1561C211yesmaximal, characteristic
6#8926C211yesmaximal, characteristic, complemented
7#8926C211yesmaximal, characteristic, complemented
8#8925C211yesmaximal, characteristic, complemented
9#8925C211yesmaximal, characteristic, complemented
10#8923C211yesmaximal, characteristic, complemented
11#8923C211yesmaximal, characteristic, complemented
12#8922C211yesmaximal, characteristic, complemented
13#8922C211yesmaximal, characteristic, complemented
14#7901C211yesmaximal, characteristic
15#7899C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#428—21no—
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21#2365C2211yescharacteristic
22#2365C2211yescharacteristic
23#2365C2211yescharacteristic
24#2365C2211yescharacteristic
25#2360—21no—
26#2360—21no—
27#2356—21no—
28#2351—21no—
29#2343—22no—
30#2336—22no—
31#2320—21no—
32C4×C22⋊Q8—21no—
33C4×C22⋊Q8—21no—
34#2285C2211yescharacteristic
35#2283C2211yescharacteristic
36#2276—21no—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1990C2211yescharacteristic
41#1990C2211yescharacteristic
42#1989—22no—
43#1949—21no—
44#1949—21no—
45#1949—21no—
46#1949—21no—
47#1948C2211yescharacteristic
48#1948C2211yescharacteristic
49#1948C2211yescharacteristic
50#1948C2211yescharacteristic
51#1947C2211yescharacteristic
52#1947C2211yescharacteristic
53#1947C2211yescharacteristic
54#1947C2211yescharacteristic
55#1825—22no—
56#1809—21no—
57#1809—21no—
58#1736—21no—
59#1561C418yescomplemented
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72C22×M16—21no—
73C22×M16—21no—
74#533—22no—
75#533—22no—
76#533C2×C414yes—
77#533Q814yes—
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×#125D412yes—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8—21no—
83C2×C4×Q8—21no—
84C2×C4×Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×#117—21no—
90C2×#117D412yes—
91C22×C4⋊C4D411yescharacteristic
92C22×C4⋊C4D411yescharacteristic
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95#431—22no—
96#431—22no—
97#428—44no—
98C2×#107—21no—
99C2×#107—21no—
100C2×#107—21no—
101C2×#107—21no—
102C2×#106—21no—
103C2×#106—21no—
104C2×#106—21no—
105C2×#106—21no—
106#423—22no—
107#423—22no—
108#423C2×C414yes—
109#423C2×C414yes—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C22⋊C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#391—21no—
121#391—21no—
122#391—21no—
123#391—21no—
124#389—22no—
125#389—22no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#387—22no—
131#387—22no—
132#387—22no—
133#387—22no—
134#387—22no—
135#387—22no—
136#387—22no—
137#387—22no—
138#385—22no—
139#385—22no—
140#384C2311yescharacteristic
141#384C2311yescharacteristic
142#383—22no—
143#383—21no—
144#382—22no—
145#382—21no—
146#382D411yescharacteristic
147#382D411yescharacteristic
148#381—21no—
149#380—21no—
150#380—21no—
151#380D411yescharacteristic
152#380D411yescharacteristic
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C4⋊C4C2311yescharacteristic
167C4×C4⋊C4C2311yescharacteristic
168C4×C4⋊C4C2311yescharacteristic
169C4×C4⋊C4C2311yescharacteristic
170C4×C22⋊C4—22no—
171C4×C22⋊C4—22no—
172C4×C22⋊C4—21no—
173#376—22no—
174#376—21no—
175C2×#95—21no—
176#340—21no—
177#340—21no—
178#340—21no—
179#340—21no—
180#340—22no—
181#340—22no—
182#340—22no—
183#340—22no—
184#340—21no—
185#340—21no—
186#340—21no—
187#340—21no—
188#332C2×C414yescomplemented
189#332C2×C414yescomplemented
190#332C2×C414yescomplemented
191#332C2×C414yescomplemented
192C22×Q8—21no—
193C22×Q8—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C2×M16—24no—
200C2×M16—24no—
201C2×M16—42no—
202C2×M16—42no—
203C2×M16—81no—
204C2×M16—81no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8C22×C412yes—
216C4⋊Q8C22×C412yes—
217C4⋊Q8C2×Q812yes—
218C4⋊Q8C2×Q812yes—
219#125—24no—
220#125C4⋊C418yes—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—24no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8—42no—
232C4×Q8—42no—
233C4×Q8—81no—
234C4×Q8—81no—
235C4×Q8—81no—
236C4×Q8—81no—
237#117—24no—
238#117—42no—
239#117—42no—
240#117C4⋊C418yes—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C22⋊C4—22no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—41no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—41no—
287C2×C42—81no—
288C2×C42—81no—
289C2×C42C22×C412yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×Q812yes—
294C2×C42#4112yes—
295C2×C42#4112yes—
296C2×C42C2411yescharacteristic
297#107—42no—
298#107—42no—
299#107—81no—
300#107—81no—
301#106—42no—
302#106—42no—
303#106—81no—
304#106—81no—
305C4⋊C8—24no—
306C4⋊C8—24no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—42no—
312C4⋊C8C22×C412yes—
313C4⋊C8C22×C412yes—
314C4⋊C8C22×C412yes—
315C4⋊C8C22×C412yes—
316C22⋊C8—42no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—41no—
342#95—41no—
343C24#12011yescharacteristic
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4C2×C4⋊C412yes—
385C22×C4C4×D412yes—
386C22×C4C4×D412yes—
387C22×C4#12011yescharacteristic
388C22×C4#12011yescharacteristic
389C22×C4#12011yescharacteristic
390C22×C4C22⋊Q812yes—
391C22×C4C22⋊Q812yes—
392C22×C4#12312yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396M16—48no—
397M16—84no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4C2×C4⋊C414yes—
455C4⋊C4C2×C4⋊C414yes—
456C4⋊C4AGL(1,ℤ/8)12yes—
457C4⋊C4#13712yes—
458C22⋊C4—28no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—82no—
462C42—28no—
463C42—44no—
464C42—44no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—81no—
474C42—81no—
475C42—82no—
476C42—82no—
477C42—82no—
478C42C2×C4⋊C412yes—
479C42C4×D412yes—
480C42C4×D412yes—
481C42C22⋊Q812yes—
482C42C22⋊Q812yes—
483C42#12312yes—
484C42C23×C411yescharacteristic
485C42C22×Q811yescharacteristic
486C42C2×#4111yescharacteristic
487C42C2×#4111yescharacteristic
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23#38512yes—
495C23C2×#12011yescharacteristic
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501Q8—42no—
502Q8—42no—
503Q8—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4#38512yes—
561C2×C4#38512yes—
562C2×C4#38512yes—
563C2×C4C22×C4⋊C411yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×C22⋊Q811yescharacteristic
567C2×C4C2×C22⋊Q811yescharacteristic
568C2×C4C2×#12311yescharacteristic
569C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
570C2×C4C2×#13711yescharacteristic
571C2×C4#57511yescharacteristic
572C2×C4#57511yescharacteristic
573C2×C4#57611yescharacteristic
574C2×C4#57711yescharacteristic
575C2×C4#57711yescharacteristic
576C2×C4#57811yescharacteristic
577C8—42no—
578C8—42no—
579C8—42no—
580C8—42no—
581C8—81no—
582C8—81no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22C2×#38511yescharacteristic
591C22#300611yescharacteristic
592C22#300711yescharacteristic
593C22#301211yescharacteristic
594C22#301311yescharacteristic
595C22#302011yescharacteristic
596C22#302011yescharacteristic
597C4—21no—
598C4—21no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—42no—
604C4—44no—
605C4—44no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—161no—
613C4—161no—
614C4C4×AGL(1,ℤ/8)11yescharacteristic
615C4C4×#13711yescharacteristic
616C4#294611yescharacteristic
617C4#294711yescharacteristic
618C4#307911yescharacteristic
619C4#308011yescharacteristic
620C4#308311yescharacteristic
621C4#308411yescharacteristic
622C2—42no—
623C2#1882211yescharacteristic
624C2#1882311yescharacteristic
625C2#1882411yescharacteristic
626C2#1882511yescharacteristic
627C2#2085711yescharacteristic
628C2#2087511yescharacteristic
629C2#2087811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8926C2semidirect
7. #8926C2semidirect
8. #8925C2semidirect
9. #8925C2semidirect
10. #8923C2semidirect
11. #8923C2semidirect
12. #8922C2semidirect
13. #8922C2semidirect
59. #1561C4semidirect
188. #332C2×C4semidirect
189. #332C2×C4semidirect
190. #332C2×C4semidirect
191. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342352
#219632
#218352
#211702
C2×#15612
#89262
#89252
#89232
#89222
#79012
#78992