Finite Group Databasebeta

← #6493045

Subgroups — #6493045

Order 512.

1796 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35278C211yesmaximal, characteristic
2#34145C211yesmaximal, characteristic, complemented
3#33161C211yesmaximal, characteristic, complemented
4#33106C212yesmaximal, complemented
5#33069C211yesmaximal, characteristic, complemented
6#29999C211yesmaximal, characteristic, complemented
7#29796C211yesmaximal, characteristic, complemented
8#29661C212yesmaximal, complemented
9#26947C214yesmaximal, complemented
10#26709C214yesmaximal, complemented
11#26644C214yesmaximal, complemented
12#26605C214yesmaximal, complemented
13#22153C212yesmaximal, complemented
14#21926C211yesmaximal, characteristic, complemented
15#21505C212yesmaximal, complemented
16C2×#540—21no—
17C2×#533C2211yescharacteristic
18C2×#480C2214yes—
19C2×#460C2212yes—
20C2×#460C2212yes—
21C2×#460C2214yes—
22C2×#440C2214yes—
23#2867C2211yescharacteristic
24#2859C2211yescharacteristic
25#2856C2212yes—
26#2641—21no—
27#2619—21no—
28#2445—22no—
29#2445C2214yes—
30#2428—22no—
31#2428C2214yes—
32#2370—22no—
33#2369C2212yes—
34#2368—22no—
35#2365C2212yes—
36#2360C2212yes—
37#2360C2212yes—
38#2328—21no—
39C4×#127C2212yes—
40#2296—21no—
41#2295—21no—
42#2295—21no—
43#2295C2211yescharacteristic
44#2295C2211yescharacteristic
45C42×Q8C2211yescharacteristic
46C42×Q8C2211yescharacteristic
47C4×#117—21no—
48C4×#117C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2087C2214yes—
51#2068C2214yes—
52#2041C2214yes—
53#2023C2214yes—
54#2014C2214yes—
55#2013C2214yes—
56#2000C2214yes—
57#2000C2214yes—
58#1982C2212yes—
59#1982C2214yes—
60#1982C2212yes—
61#1973C2214yes—
62#1947C2212yes—
63#1947C2212yes—
64#1936C2214yes—
65#1925C2214yes—
66#1925C2214yes—
67#1925C2214yes—
68#1924C2214yes—
69#1921C2214yes—
70#1917C2214yes—
71#1880C2214yes—
72#1877C2214yes—
73#1874C2214yes—
74#1863C2214yes—
75#1862C2214yes—
76#1842C2211yescharacteristic
77#1831C2211yescharacteristic
78#1803C2212yes—
79#1803C2212yes—
80#1797C2212yes—
81#1767C2212yes—
82#1762C2214yes—
83C22×SD16C2314yes—
84C22×SD16C2314yes—
85#540—48no—
86#533—24no—
87#533—24no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127—21no—
91C2×#127—21no—
92C2×#127—22no—
93C2×#127C2312yes—
94C2×#127C2314yes—
95C2×#125—21no—
96C2×#125—21no—
97C2×#123—24no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2311yescharacteristic
100C2×C22⋊Q8C2311yescharacteristic
101C2×#121—22no—
102C2×#121—22no—
103C2×#121—24no—
104C2×#121C2314yes—
105C2×C4×Q8—21no—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4C2314yes—
113C2×#117—21no—
114C2×#117D411yescharacteristic
115C2×#117D411yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119#480—48no—
120#460—28no—
121#460—28no—
122#460—48no—
123#440—48no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2314yes—
139#394—24no—
140#394—24no—
141#394—24no—
142#392—24no—
143#391—22no—
144#391—22no—
145#391C2312yes—
146#391C2312yes—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390C2314yes—
152#390C2314yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389C2314yes—
159#389C2312yes—
160#389C2312yes—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#386—24no—
168#386—24no—
169#385—22no—
170#385—22no—
171#384C2312yes—
172#382—21no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2311yescharacteristic
188C4×C4⋊C4C2311yescharacteristic
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4D412yes—
193C4×C22⋊C4C2312yes—
194#376—21no—
195#376—21no—
196#376D411yescharacteristic
197#376D411yescharacteristic
198C2×#95—21no—
199C43—21no—
200C43—21no—
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210C22×Q8—21no—
211C22×Q8C2411yescharacteristic
212C22×Q8C2411yescharacteristic
213C22×D4—24no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—24no—
217C22×D4—24no—
218C22×D4C2414yes—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2411yescharacteristic
224C2×SD16—28no—
225C2×SD16—28no—
226C2×SD16—44no—
227C2×SD16—44no—
228C2×SD16—44no—
229C2×SD16—44no—
230C22×C8C2414yes—
231C22×C8C2414yes—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236#127—24no—
237#127—24no—
238#127—44no—
239#127—44no—
240#127—44no—
241#127—44no—
242#127—84no—
243#125—28no—
244#125—44no—
245#123—416no—
246C22⋊Q8—28no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251#121—416no—
252#121—48no—
253#121—44no—
254#121—44no—
255#121—84no—
256C4×Q8—28no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—42no—
262C4×Q8—42no—
263C4×Q8#4114yes—
264C4×Q8#4114yes—
265C4×D4—48no—
266C4×D4—44no—
267C4×D4—44no—
268C4×D4—84no—
269#117—28no—
270#117—24no—
271#117—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
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—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D412yes—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—44no—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2412yes—
327C2×C42C2412yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330#107—28no—
331#107—44no—
332C4⋊C8—24no—
333C4⋊C8—24no—
334C4⋊C8—44no—
335#103—44no—
336#103—44no—
337#102—28no—
338#102—28no—
339#102—28no—
340#102—28no—
341#102—44no—
342#102—44no—
343C22⋊C8—44no—
344#97—44no—
345#95—21no—
346#95—21no—
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#95—24no—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361C24—24no—
362C24C22×D411yescharacteristic
363C2×Q8—24no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—82no—
371C2×Q8C2×#4112yes—
372C2×Q8C2×#4112yes—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—48no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4—84no—
392C2×D4—84no—
393C2×D4—84no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—82no—
415C22×C4AGL(1,ℤ/8)14yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C2511yescharacteristic
426SD16—416no—
427SD16—88no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4C2×#4112yes—
459C4⋊C4C2×#4112yes—
460C4⋊C4H(5,𝔽2)14yes—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—24no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—21no—
479C42—21no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—42no—
485C42—42no—
486C42—44no—
487C42—44no—
488C42—42no—
489C42—44no—
490C42—42no—
491C42—82no—
492C42—82no—
493C42#13714yes—
494C42C2×#4112yes—
495C42C2×#4112yes—
496C23—22no—
497C23—21no—
498C23—22no—
499C23—21no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23C2×AGL(1,ℤ/8)12yes—
508C23C23×D411yescharacteristic
509Q8—24no—
510Q8—22no—
511Q8—22no—
512Q8—42no—
513Q8—41no—
514Q8—41no—
515Q8—84no—
516D4—44no—
517D4—44no—
518D4—44no—
519D4—44no—
520D4—88no—
521D4—88no—
522D4—168no—
523D4—168no—
524D4—168no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—82no—
565C2×C4—88no—
566C2×C4—168no—
567C2×C4#52512yes—
568C2×C4#52512yes—
569C2×C4C2×AGL(1,ℤ/8)12yes—
570C2×C4C2×AGL(1,ℤ/8)12yes—
571C2×C4C2×#13714yes—
572C2×C4C2×#13712yes—
573C2×C4C22×#4111yescharacteristic
574C2×C4C22×#4111yescharacteristic
575C2×C4C2×H(5,𝔽2)12yes—
576C2×C4#58512yes—
577C8—44no—
578C8—44no—
579C22—24no—
580C22—22no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22C2×#52511yescharacteristic
592C22#348812yes—
593C22C22×AGL(1,ℤ/8)11yescharacteristic
594C22C22×#13711yescharacteristic
595C22#358512yes—
596C4—24no—
597C4—22no—
598C4—22no—
599C4—42no—
600C4—42no—
601C4—44no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—42no—
608C4—42no—
609C4—82no—
610C4—84no—
611C4#311214yes—
612C4#346012yes—
613C4#350811yescharacteristic
614C4#351211yescharacteristic
615C2—42no—
616C2—164no—
617C2#3525911yescharacteristic
618C2C2×#311211yescharacteristic
619C2#6056811yescharacteristic
620C2#6195912yes—
621C2#6196512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35278C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34145C2semidirect
3. #33161C2semidirect
4. #33106C2semidirect
5. #33069C2semidirect
6. #29999C2semidirect
7. #29796C2semidirect
8. #29661C2semidirect
9. #26947C2semidirect
10. #26709C2semidirect
11. #26644C2semidirect
12. #26605C2semidirect
13. #22153C2semidirect
14. #21926C2semidirect
15. #21505C2semidirect

Maximal subgroups

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

subgroupindex
#352782
#341452
#331612
#331062
#330692
#299992
#297962
#296612
#269472
#267092
#266442
#266052
#221532
#219262
#215052