Finite Group Databasebeta

← #6475511

Subgroups — #6475511

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#33049C212yesmaximal, complemented
4#33043C212yesmaximal, complemented
5#30193C214yesmaximal, complemented
6#29344C214yesmaximal, complemented
7#29118C214yesmaximal, complemented
8#27975C212yesmaximal, complemented
9#27962C212yesmaximal, complemented
10#25902C212yesmaximal, complemented
11#25856C212yesmaximal, complemented
12#21771C214yesmaximal
13C4×#408C211yesmaximal, characteristic
14C2×#536C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#487C2212yes—
19C2×#487C2212yes—
20C2×#408C2212yes—
21C2×C4×M16C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2857C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2628C2212yes—
28#2628C2212yes—
29#2621C2212yes—
30#2621C2212yes—
31#2568—21no—
32#2567—21no—
33#2563—21no—
34#2547—21no—
35#2547—21no—
36#2546—21no—
37#2546—21no—
38#2542—21no—
39#2487—21no—
40#2487—21no—
41#2486—21no—
42#2486—21no—
43#2486—21no—
44#2486—21no—
45#2409—21no—
46#2409—21no—
47#2317C2212yes—
48#2311C2212yes—
49#2310C2212yes—
50#2285C2211yescharacteristic
51#2285C2211yescharacteristic
52#2101C2218yes—
53#2074C2218yes—
54#2043C2218yes—
55#2024C2218yes—
56#1990C2212yes—
57#1990C2212yes—
58#1956C2212yes—
59#1956C2212yes—
60#1935C2218yes—
61#1933C2218yes—
62#1894C2218yes—
63#1892C2218yes—
64#1873C2218yes—
65#1872C2218yes—
66#1834C2212yes—
67#1834C2212yes—
68#1825C2218yes—
69#1787C2218yes—
70#1750C2212yes—
71#1750C2212yes—
72C4×C22⋊C8C2212yes—
73C4×C22⋊C8C2212yes—
74#1729C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8—21no—
77C23×Q8C2312yes—
78C22×M16C2312yes—
79C23×C8C2312yes—
80#536—216no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8—42no—
88C2×C4⋊Q8—42no—
89C2×C4⋊Q8—42no—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×#124—22no—
95C2×#123—22no—
96C2×#123—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8C2314yes—
100C2×C22⋊Q8C2314yes—
101C4×#41—42no—
102C4×#41—42no—
103C2×C4×Q8D411yescharacteristic
104C2×C4×Q8D411yescharacteristic
105C2×C4×D4D411yescharacteristic
106C2×C4×D4D411yescharacteristic
107C2×C4×D4D411yescharacteristic
108C2×C4×D4D411yescharacteristic
109C2×C4×D4D411yescharacteristic
110C2×C4×D4D411yescharacteristic
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#487—44no—
124#487—44no—
125C2×#103C2318yes—
126C2×#103C2318yes—
127C2×#103C2318yes—
128C2×#103C2318yes—
129#408—28no—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C4×M16—42no—
133C2×#97C2312yes—
134C2×C4×C8—42no—
135C2×C4×C8C2312yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2312yes—
138#400—22no—
139#400—22no—
140#399—22no—
141#399—22no—
142#398—22no—
143#398—22no—
144#398—22no—
145#398—22no—
146#397—22no—
147#397—22no—
148#397—22no—
149#397—22no—
150#395—22no—
151#395—22no—
152#393—22no—
153#393—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391C2314yes—
160#391C2314yes—
161#390—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2314yes—
169#389C2314yes—
170#386—22no—
171#386—22no—
172#385—22no—
173#385—22no—
174#384—22no—
175#384C2312yes—
176#384C2312yes—
177#382—22no—
178#382C2312yes—
179#382C2312yes—
180#381—22no—
181#381—22no—
182#380—22no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380—22no—
187#380—22no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—21no—
192#380C2314yes—
193#380C2314yes—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198#340C2318yes—
199#340C2318yes—
200#340C2318yes—
201#340C2318yes—
202#336C2314yes—
203#336C2314yes—
204C2×#41—41no—
205C2×#41—41no—
206C2×#41—42no—
207C2×#41#4118yes—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—28no—
212C22×Q8—42no—
213C22×Q8—42no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×Q8—84no—
217C22×Q8C2×D411yescharacteristic
218C22×Q8C2414yes—
219C22×D4C2×D411yescharacteristic
220C22×D4C2×D411yescharacteristic
221C22×D4C2×D411yescharacteristic
222C23×C4—22no—
223C23×C4—22no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C2×M16—24no—
234C2×M16—42no—
235C2×M16—42no—
236C22×C8—24no—
237C22×C8—42no—
238C22×C8—42no—
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—24no—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247C4⋊Q8—42no—
248C4⋊Q8—42no—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253C4⋊Q8—48no—
254#124—48no—
255#123—48no—
256#123—48no—
257C22⋊Q8—216no—
258C22⋊Q8—216no—
259C22⋊Q8—84no—
260C22⋊Q8—84no—
261C4×Q8—42no—
262C4×Q8—42no—
263C4×D4—42no—
264C4×D4—42no—
265C4×D4—42no—
266C4×D4—42no—
267C4×D4—42no—
268C4×D4—42no—
269#117—28no—
270#117—41no—
271#117—41no—
272#117—41no—
273#117—41no—
274#117—41no—
275#117—41no—
276#117—44no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2414yes—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4C2×D411yescharacteristic
313C2×C22⋊C4C2×D411yescharacteristic
314C2×C22⋊C4C2×D411yescharacteristic
315C2×C22⋊C4C2×D411yescharacteristic
316C2×C22⋊C4C2×D411yescharacteristic
317C2×C22⋊C4C2×D411yescharacteristic
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#103—48no—
337#103—48no—
338#103—48no—
339#103—48no—
340C22⋊C8—28no—
341C22⋊C8—28no—
342#97—42no—
343C4×C8—24no—
344C4×C8—24no—
345C4×C8—42no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358C24C22×D411yescharacteristic
359C24C22×D411yescharacteristic
360C24C22×D411yescharacteristic
361#41—48no—
362#41—48no—
363C2×Q8—28no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—28no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—84no—
377C2×Q8—88no—
378C2×Q8—88no—
379C2×Q8—816no—
380C2×Q8#12414yes—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4#12414yes—
388C2×D4#12414yes—
389C2×D4#12414yes—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
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—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—84no—
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×C4C22×D411yescharacteristic
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4C2511yescharacteristic
435M16—48no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—48no—
443C4⋊C4—28no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—28no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—44no—
493C42—44no—
494C42H(5,𝔽2)12yes—
495C42H(5,𝔽2)12yes—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23C2×#12412yes—
509C23C2×#12412yes—
510C23C2×#12412yes—
511C23C23×D411yescharacteristic
512Q8—44no—
513Q8—48no—
514Q8—48no—
515Q8—88no—
516Q8—88no—
517Q8—88no—
518D4—44no—
519D4—44no—
520D4—44no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—88no—
564C2×C4—88no—
565C2×C4C2×#12412yes—
566C2×C4C2×#12412yes—
567C2×C4C2×#12412yes—
568C2×C4#53412yes—
569C2×C4#53412yes—
570C2×C4#57714yes—
571C2×C4#57814yes—
572C2×C4C22×#4112yes—
573C2×C4C2×H(5,𝔽2)11yescharacteristic
574C2×C4C2×H(5,𝔽2)11yescharacteristic
575C2×C4#58512yes—
576C8—44no—
577C8—44no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22C22×#12411yescharacteristic
594C22C2×#53411yescharacteristic
595C22#344911yescharacteristic
596C22C2×#57712yes—
597C22C2×#57812yes—
598C4—22no—
599C4—22no—
600C4—22no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—84no—
608C4—84no—
609C4—84no—
610C4—84no—
611C4#320014yes—
612C4#320214yes—
613C4#350512yes—
614C4#350512yes—
615C2—42no—
616C2—42no—
617C2—42no—
618C2#3501911yescharacteristic
619C2#6062312yes—
620C2C2×#320011yescharacteristic
621C2C2×#320211yescharacteristic
622C2#6100812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
3. #33049C2semidirect
4. #33043C2semidirect
5. #30193C2semidirect
6. #29344C2semidirect
7. #29118C2semidirect
8. #27975C2semidirect
9. #27962C2semidirect
10. #25902C2semidirect
11. #25856C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#352792
#330492
#330432
#301932
#293442
#291182
#279752
#279622
#259022
#258562
#217712
C4×#4082