Finite Group Databasebeta

← #6644701

Subgroups — #6644701

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2C2×#2565C211yesmaximal, characteristic, complemented
3#33003C212yesmaximal, complemented
4#32456C214yesmaximal, complemented
5#32372C214yesmaximal, complemented
6#31662C214yesmaximal, complemented
7#31546C214yesmaximal, complemented
8#29640C212yesmaximal, complemented
9#29111C211yesmaximal, characteristic, complemented
10#28314C211yesmaximal, characteristic
11#28115C212yesmaximal, complemented
12#27969C211yesmaximal, characteristic, complemented
13#27918C211yesmaximal, characteristic
14#27644C212yesmaximal, complemented
15#21916C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×#125C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#481C2212yes—
20C2×#480C2212yes—
21C2×#480C2212yes—
22C2×#448C2212yes—
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2854C2212yes—
26#2639—21no—
27#2618—21no—
28#2565—22no—
29#2565—22no—
30#2565C2214yes—
31#2565C2214yes—
32#2560—22no—
33#2558—22no—
34#2512C2212yes—
35#2506C2212yes—
36#2452C2212yes—
37#2445C2212yes—
38C2×#398—21no—
39C2×#398—21no—
40#2325—21no—
41C4×#121C2212yes—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#387—21no—
45C2×#387—21no—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2101C2214yes—
50#2095C2212yes—
51#2092C2214yes—
52#2083C2214yes—
53#2072C2212yes—
54#2071C2214yes—
55#2066C2214yes—
56#2059C2214yes—
57#2059C2214yes—
58#2055C2214yes—
59#2025C2214yes—
60#2025C2214yes—
61#2024C2212yes—
62#2014C2212yes—
63#1993C2212yes—
64#1973C2214yes—
65#1961C2212yes—
66#1947C2214yes—
67#1935C2214yes—
68#1927C2214yes—
69#1917C2214yes—
70#1917C2214yes—
71#1880C2214yes—
72#1880C2214yes—
73#1877C2214yes—
74#1863C2214yes—
75#1863C2214yes—
76#1847C2212yes—
77#1836C2212yes—
78#1824C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1802C2214yes—
83#1797C2212yes—
84#1787C2212yes—
85#1749C2211yescharacteristic
86#1749C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C22×SD16C2314yes—
89#539—48no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#126—22no—
93C2×#125—22no—
94C2×#125—22no—
95C2×#125—44no—
96C2×#125C2314yes—
97C2×#123—22no—
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—22no—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×Q8—22no—
105C2×C4×Q8—44no—
106C2×C4×Q8C2314yes—
107C2×C4×D4—22no—
108C2×C4×D4C2312yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#481—44no—
124#480—44no—
125#480—44no—
126#448—44no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×#97C2312yes—
140C2×#97C2312yes—
141#400—22no—
142#398—22no—
143#398—24no—
144#398—24no—
145#398—24no—
146#398—24no—
147#397—22no—
148#396—22no—
149#396—22no—
150#394—22no—
151#394—22no—
152#393—22no—
153#391—21no—
154#391—22no—
155#391—21no—
156#391—21no—
157#391C2314yes—
158#391C2312yes—
159#391C2312yes—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#389—21no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—24no—
177#387—24no—
178#387—24no—
179#387—24no—
180#386—22no—
181#386—22no—
182#386—22no—
183#385—22no—
184#384—22no—
185#384C2314yes—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#382—21no—
189#382—21no—
190#382C2312yes—
191#382C2312yes—
192#380—21no—
193#380C2312yes—
194C4×C4⋊C4D412yes—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8C2412yes—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×SD16—28no—
242C2×SD16—44no—
243C2×SD16—44no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—42no—
247#127—42no—
248#126—48no—
249#125—48no—
250#125—48no—
251#125—48no—
252#123—48no—
253#123—48no—
254C22⋊Q8—48no—
255C22⋊Q8—44no—
256#121—24no—
257#121—24no—
258#121—44no—
259#121—44no—
260C4×Q8—48no—
261C4×Q8—48no—
262C4×D4—44no—
263C4×D4—48no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2412yes—
323C2×C42C2412yes—
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326#107—44no—
327#107—44no—
328#106—44no—
329C4⋊C8—44no—
330#103—28no—
331#103—44no—
332#103—44no—
333#102—44no—
334#102—44no—
335#102—44no—
336C22⋊C8—24no—
337C22⋊C8—24no—
338#97—42no—
339#97—42no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360C24—22no—
361C24C22×D411yescharacteristic
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—416no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—81no—
371C2×Q8—81no—
372C2×Q82−1+414yes—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—81no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4—81no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
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—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4AGL(1,ℤ/8)14yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4H(5,𝔽2)14yes—
438C22×C4C2511yescharacteristic
439SD16—88no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C42—24no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—24no—
490C42—42no—
491C42—41no—
492C42—41no—
493C42—48no—
494C42—44no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23#53512yes—
507C23C2×AGL(1,ℤ/8)12yes—
508C23#57512yes—
509C23C23×D411yescharacteristic
510Q8—28no—
511Q8—48no—
512Q8—44no—
513Q8—162no—
514D4—44no—
515D4—44no—
516D4—88no—
517D4—162no—
518D4—162no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—164no—
577C2×C4#53512yes—
578C2×C4C2×AGL(1,ℤ/8)14yes—
579C2×C4C2×H(5,𝔽2)12yes—
580C2×C4C2×2−1+412yes—
581C2×C4#58512yes—
582C8—44no—
583C8—44no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22#321414yes—
598C22C2×#53511yescharacteristic
599C22#348511yescharacteristic
600C22#348611yescharacteristic
601C22C22×AGL(1,ℤ/8)11yescharacteristic
602C22C2×#57511yescharacteristic
603C22#358611yescharacteristic
604C22#358611yescharacteristic
605C4—24no—
606C4—22no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—41no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4#353514yes—
622C2—24no—
623C2—162no—
624C2#3711811yescharacteristic
625C2C2×#321411yescharacteristic
626C2#6080711yescharacteristic
627C2#6170411yescharacteristic
628C2#6171611yescharacteristic
629C2#6182811yescharacteristic
630C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2565C2semidirect
3. #33003C2semidirect
4. #32456C2semidirect
5. #32372C2semidirect
6. #31662C2semidirect
7. #31546C2semidirect
8. #29640C2semidirect
9. #29111C2semidirect
11. #28115C2semidirect
12. #27969C2semidirect
14. #27644C2semidirect
15. #21916C2semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#25652
#330032
#324562
#323722
#316622
#315462
#296402
#291112
#283142
#281152
#279692
#279182
#276442
#219162