Finite Group Databasebeta

← #6497127

Subgroups — #6497127

Order 512.

1471 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2#34125C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#33069C211yesmaximal, characteristic, complemented
5#32958C212yesmaximal
6#29998C211yesmaximal, characteristic, complemented
7#29942C212yesmaximal, complemented
8#29868C211yesmaximal, characteristic, complemented
9#27548C214yesmaximal
10#27484C214yesmaximal, complemented
11#27388C214yesmaximal, complemented
12#27324C214yesmaximal, complemented
13#22117C212yesmaximal, complemented
14#21926C211yesmaximal, characteristic, complemented
15#21616C212yesmaximal, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#477C2212yes—
18C2×#476C2212yes—
19#2868C2211yescharacteristic
20#2859C2211yescharacteristic
21#2857C2212yes—
22#2647—21no—
23#2629—21no—
24#2507C2214yes—
25#2497C2214yes—
26#2446—22no—
27#2429—22no—
28#2357—22no—
29#2357C2212yes—
30#2351C2212yes—
31#2347—22no—
32#2347C2212yes—
33#2341C2212yes—
34#2330—21no—
35#2314—21no—
36C4×C4⋊Q8C2212yes—
37#2293—21no—
38#2293—21no—
39#2293C2211yescharacteristic
40#2293C2211yescharacteristic
41#2283—21no—
42#2283C2211yescharacteristic
43C42×Q8C2211yescharacteristic
44C42×Q8C2211yescharacteristic
45C4×#117—21no—
46C4×#117C2211yescharacteristic
47#2101C2214yes—
48#2088C2214yes—
49#2085C2214yes—
50#2084C2214yes—
51#2077C2214yes—
52#2075C2214yes—
53#2071C2214yes—
54#2065C2214yes—
55#1987C2212yes—
56#1987C2212yes—
57#1948C2212yes—
58#1947C2212yes—
59#1947C2214yes—
60#1944C2214yes—
61#1936C2214yes—
62#1925C2214yes—
63#1924C2214yes—
64#1921C2214yes—
65#1920C2214yes—
66#1918C2214yes—
67#1917C2214yes—
68#1865C2214yes—
69#1865C2214yes—
70#1863C2214yes—
71#1862C2214yes—
72#1854C2214yes—
73#1842C2211yescharacteristic
74#1831C2211yescharacteristic
75#1803C2212yes—
76#1803C2214yes—
77#1802C2212yes—
78#1797C2212yes—
79#1788C2214yes—
80#1776C2214yes—
81#1767C2212yes—
82#1763C2214yes—
83C22×#41C2311yescharacteristic
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×#125—21no—
87C2×C22⋊Q8—22no—
88C4×#41—24no—
89C4×#41—24no—
90C2×C4×Q8—22no—
91C2×C4×Q8C2311yescharacteristic
92C2×C4×Q8C2312yes—
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2311yescharacteristic
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4C2311yescharacteristic
98C2×C4×D4C2311yescharacteristic
99C2×#117—21no—
100C2×#117—21no—
101C2×#117—21no—
102C2×#117D411yescharacteristic
103C2×#117D411yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C42—21no—
106C22×C42C2311yescharacteristic
107#477—28no—
108#476—28no—
109C2×#107C2314yes—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2314yes—
122#400—24no—
123#398—24no—
124#398—24no—
125#396—24no—
126#391—22no—
127#391—22no—
128#390—22no—
129#390—22no—
130#390C2312yes—
131#390C2312yes—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—24no—
138#389C2314yes—
139#389C2314yes—
140#389C2312yes—
141#389C2314yes—
142#389C2312yes—
143#388—22no—
144#388—22no—
145#388—24no—
146#388—22no—
147#388—22no—
148#388—22no—
149#388—22no—
150#387—24no—
151#387—24no—
152#384—21no—
153#384—21no—
154#384C2312yes—
155#384C2314yes—
156#383—21no—
157#383—22no—
158#382—21no—
159#382—22no—
160#382C2312yes—
161#381—21no—
162#381—22no—
163#380—22no—
164#380—22no—
165#380C2312yes—
166#380C2314yes—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2311yescharacteristic
179C4×C4⋊C4C2311yescharacteristic
180C4×C4⋊C4C2312yes—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4C2312yes—
188#376—21no—
189#376D411yescharacteristic
190#376D411yescharacteristic
191C43—21no—
192C43—21no—
193C43C2311yescharacteristic
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2314yes—
205#336C2314yes—
206C2×#41—22no—
207C2×#41—22no—
208C2×#41—24no—
209C2×#41—42no—
210C22×Q8—22no—
211C22×Q8C2411yescharacteristic
212C22×D4—21no—
213C22×D4C2411yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2411yescharacteristic
220C22×C8C2414yes—
221C22×C8C2414yes—
222C4⋊Q8—24no—
223C4⋊Q8—24no—
224C4⋊Q8—84no—
225#125—44no—
226C22⋊Q8—84no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—42no—
233C4×Q8—42no—
234C4×Q8—84no—
235C4×Q8#4114yes—
236C4×D4—28no—
237C4×D4—28no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4#4114yes—
242#117—28no—
243#117—22no—
244#117—22no—
245#117—24no—
246#117—42no—
247#117—44no—
248#117—44no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2411yescharacteristic
270C2×C4⋊C4C2411yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C42—28no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2412yes—
305C2×C42C2412yes—
306C2×C42C2411yescharacteristic
307#107—28no—
308#107—44no—
309#106—28no—
310#106—44no—
311C4⋊C8—24no—
312C4⋊C8—24no—
313C4⋊C8—44no—
314#103—28no—
315#103—28no—
316#102—28no—
317#102—44no—
318C22⋊C8—44no—
319#97—44no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—21no—
328#95—21no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337#95C2×D412yes—
338C24—21no—
339C24C22×D411yescharacteristic
340#41—48no—
341#41—44no—
342#41—44no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8C2×#4112yes—
349C2×Q82−1+414yes—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—44no—
355C2×D4C2×#4112yes—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—82no—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C2511yescharacteristic
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—48no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4C2×#4112yes—
422C4⋊C4C2×#4112yes—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—84no—
434C42—24no—
435C42—24no—
436C42—22no—
437C42—24no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—21no—
443C42—21no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—42no—
449C42—42no—
450C42—44no—
451C42—44no—
452C42—42no—
453C42—42no—
454C42—82no—
455C42—82no—
456C42—84no—
457C42#13714yes—
458C42C2×#4112yes—
459C42C2×#4112yes—
460C23—22no—
461C23—21no—
462C23—22no—
463C23—22no—
464C23—21no—
465C23—21no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23C23×D411yescharacteristic
471Q8—22no—
472Q8—24no—
473Q8—88no—
474Q8#54114yes—
475D4—24no—
476D4—48no—
477D4—41no—
478D4—41no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—82no—
524C2×C4—84no—
525C2×C4#52512yes—
526C2×C4#52512yes—
527C2×C4C2×#13714yes—
528C2×C4C2×#13712yes—
529C2×C4#57512yes—
530C2×C4C22×#4111yescharacteristic
531C2×C4C22×#4111yescharacteristic
532C2×C4C2×2−1+412yes—
533C2×C4#58512yes—
534C8—44no—
535C8—44no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—41no—
544C22—82no—
545C22—82no—
546C22C2×#52511yescharacteristic
547C22#348912yes—
548C22C22×#13711yescharacteristic
549C22C2×#57511yescharacteristic
550C22#358512yes—
551C4—22no—
552C4—24no—
553C4—24no—
554C4—41no—
555C4—42no—
556C4—41no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—41no—
562C4—84no—
563C4—82no—
564C4—82no—
565C4—84no—
566C4#311614yes—
567C4C2×#54111yescharacteristic
568C4#345511yescharacteristic
569C4#349512yes—
570C2—81no—
571C2—82no—
572C2#3531011yescharacteristic
573C2C2×#311611yescharacteristic
574C2#6056811yescharacteristic
575C2#6211212yes—
576C2#6211712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
1. #35284C2semidirect
3. #33198C2semidirect
4. #33069C2semidirect
6. #29998C2semidirect
7. #29942C2semidirect
8. #29868C2semidirect
10. #27484C2semidirect
11. #27388C2semidirect
12. #27324C2semidirect
13. #22117C2semidirect
14. #21926C2semidirect
15. #21616C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#341252
#331982
#330692
#329582
#299982
#299422
#298682
#275482
#274842
#273882
#273242
#221172
#219262
#216162