Finite Group Databasebeta

← #241312

Subgroups — #241312

Order 512.

930 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21365C211yesmaximal, characteristic
4#21362C211yesmaximal, characteristic
5#10582C211yesmaximal, characteristic, complemented
6#10581C211yesmaximal, characteristic, complemented
7#10580C211yesmaximal, characteristic, complemented
8#10579C211yesmaximal, characteristic, complemented
9#10575C211yesmaximal, characteristic, complemented
10#10575C211yesmaximal, characteristic, complemented
11#10574C211yesmaximal, characteristic, complemented
12#10573C211yesmaximal, characteristic, complemented
13#8046C211yesmaximal, characteristic
14#7902C211yesmaximal, characteristic
15#7892C211yesmaximal, characteristic
16C2×#533—21no—
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2552C2211yescharacteristic
21#2552C2211yescharacteristic
22#2551C2211yescharacteristic
23#2551C2211yescharacteristic
24#2550—21no—
25#2550—21no—
26#2549—22no—
27#2440—21no—
28#2440—21no—
29#2399—22no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32#2296—21no—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35C4×#117C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1949—22no—
38#1948C2211yescharacteristic
39#1948C2211yescharacteristic
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1947C2211yescharacteristic
43#1947C2211yescharacteristic
44#1947C2211yescharacteristic
45#1947C2211yescharacteristic
46#1854—22no—
47#1854—21no—
48#1854—21no—
49#1853—21no—
50#1852—21no—
51#1845C2211yescharacteristic
52#1844C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1776—21no—
56#1776—21no—
57C4×#107—21no—
58C4×#106—21no—
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#533—44no—
72#533—44no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×#125—21no—
78C2×#125—21no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C4×Q8—22no—
82C2×C4×Q8—22no—
83C2×#117—21no—
84C2×#117—21no—
85C2×#117D411yescharacteristic
86C2×#117D411yescharacteristic
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4D411yescharacteristic
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#432—22no—
92#432Q814yescomplemented
93#431—22no—
94#431—22no—
95#431Q814yescomplemented
96#431Q814yescomplemented
97C2×#107—21no—
98C2×#107—21no—
99C2×#107—21no—
100C2×#107—21no—
101C2×#107—22no—
102C2×#106—21no—
103C2×#106—21no—
104C2×#106—21no—
105C2×#106—21no—
106C2×#106—22no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×#97—21no—
116C2×C4×C8—21no—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#398—22no—
120#398—22no—
121#398—22no—
122#398—22no—
123#395—22no—
124#395—22no—
125#395—22no—
126#395—22no—
127#391—22no—
128#391—22no—
129#389—22no—
130#389—22no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#385—22no—
140#385—22no—
141#384—21no—
142#384—21no—
143#384C2311yescharacteristic
144#384C2311yescharacteristic
145#384C2311yescharacteristic
146#384C2311yescharacteristic
147#382—21no—
148#382—21no—
149#380—21no—
150#380—21no—
151#380—21no—
152#380D412yes—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4D411yescharacteristic
161C4×C4⋊C4D411yescharacteristic
162C4×C4⋊C4D412yes—
163C4×C4⋊C4D411yescharacteristic
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4C2311yescharacteristic
166C4×C4⋊C4C2311yescharacteristic
167C4×C4⋊C4C2311yescharacteristic
168C4×C4⋊C4C2311yescharacteristic
169C4×C22⋊C4—21no—
170C4×C22⋊C4—21no—
171#376—21no—
172C2×#95—21no—
173C43—21no—
174#340—22no—
175#340—22no—
176#340—21no—
177#340—21no—
178#340—21no—
179#340—21no—
180#340—21no—
181#340—21no—
182#340—21no—
183#340—21no—
184C22×Q8—22no—
185C22×Q8—22no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C4⋊Q8—42no—
198C4⋊Q8—42no—
199C4⋊Q8—42no—
200C4⋊Q8—42no—
201C4⋊Q8—42no—
202C4⋊Q8—42no—
203C4⋊Q8—42no—
204C4⋊Q8—42no—
205#125—42no—
206#125—42no—
207#125—42no—
208#125—42no—
209C22⋊Q8—48no—
210C22⋊Q8—48no—
211C4×Q8—48no—
212C4×Q8—48no—
213#117—22no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—42no—
218#117—42no—
219#117—82no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
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—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—41no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—41no—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×Q812yes—
278C2×C42C2×Q812yes—
279C2×C42C2×Q812yes—
280C2×C42#4112yes—
281C2×C42C2411yescharacteristic
282#107—42no—
283#107—42no—
284#107—42no—
285#107—42no—
286#107—82no—
287#106—42no—
288#106—42no—
289#106—42no—
290#106—42no—
291#106—82no—
292C4⋊C8—24no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8C2×Q812yes—
298C4⋊C8C2×Q812yes—
299C4⋊C8C2×Q812yes—
300C4⋊C8C2×Q812yes—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C22⋊C8—24no—
304#97—24no—
305C4×C8—24no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8C2×Q812yes—
309C4×C8C2×Q812yes—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—22no—
319#95—22no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—41no—
325#95—41no—
326C24#12011yescharacteristic
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—82no—
332C2×Q8—82no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4#12011yescharacteristic
362C22×C4#12011yescharacteristic
363C22×C4#12011yescharacteristic
364C22×C4C22⋊Q812yes—
365C22×C4C22⋊Q812yes—
366C22×C4C22⋊Q812yes—
367C22×C4C4⋊Q812yes—
368C22×C4C4⋊Q812yes—
369C22×C4C4⋊Q812yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4#13512yes—
434C4⋊C4#13512yes—
435C4⋊C4AGL(1,ℤ/8)12yes—
436C4⋊C4AGL(1,ℤ/8)12yes—
437C4⋊C4#13712yes—
438C4⋊C4#13712yes—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—44no—
442C22⋊C4—82no—
443C42—24no—
444C42—24no—
445C42—42no—
446C42—42no—
447C42—44no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—81no—
457C42—81no—
458C42—82no—
459C42C22⋊Q812yes—
460C42C22⋊Q812yes—
461C42C22⋊Q812yes—
462C42C4⋊Q812yes—
463C42C4⋊Q812yes—
464C42C4⋊Q812yes—
465C42C22×Q811yescharacteristic
466C42C22×Q811yescharacteristic
467C42C22×Q811yescharacteristic
468C42C2×#4111yescharacteristic
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23#39512yes—
476C23C2×#12011yescharacteristic
477Q8—88no—
478Q8—88no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4#39512yes—
536C2×C4#39512yes—
537C2×C4#39512yes—
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×C22⋊Q811yescharacteristic
541C2×C4C2×C4⋊Q811yescharacteristic
542C2×C4C2×C4⋊Q811yescharacteristic
543C2×C4C2×C4⋊Q811yescharacteristic
544C2×C4C2×#13511yescharacteristic
545C2×C4C2×#13511yescharacteristic
546C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
547C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
548C2×C4C2×#13711yescharacteristic
549C2×C4C2×#13711yescharacteristic
550C2×C4#57511yescharacteristic
551C2×C4#57511yescharacteristic
552C8—42no—
553C8—42no—
554C8—42no—
555C8—42no—
556C8—42no—
557C8—42no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22C2×#39511yescharacteristic
566C22#300211yescharacteristic
567C22#300211yescharacteristic
568C22#300311yescharacteristic
569C22#300411yescharacteristic
570C22#302111yescharacteristic
571C22#302211yescharacteristic
572C4—22no—
573C4—21no—
574C4—21no—
575C4—22no—
576C4—22no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4—82no—
585C4—81no—
586C4—82no—
587C4—161no—
588C4—161no—
589C4#307611yescharacteristic
590C4#307611yescharacteristic
591C4#307911yescharacteristic
592C4#307911yescharacteristic
593C4#316011yescharacteristic
594C4#316211yescharacteristic
595C4#316511yescharacteristic
596C4#316511yescharacteristic
597C2—42no—
598C2#1879411yescharacteristic
599C2#1879711yescharacteristic
600C2#1880611yescharacteristic
601C2#1880911yescharacteristic
602C2#2355111yescharacteristic
603C2#2355211yescharacteristic
604C2#2356211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
5. #10582C2semidirect
6. #10581C2semidirect
7. #10580C2semidirect
8. #10579C2semidirect
9. #10575C2semidirect
10. #10575C2semidirect
11. #10574C2semidirect
12. #10573C2semidirect
92. #432Q8semidirect
95. #431Q8semidirect
96. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#349382
#219842
#213652
#213622
#105822
#105812
#105802
#105792
#105752
#105742
#105732
#80462
#79022
#78922