Finite Group Databasebeta

← #6643201

Subgroups — #6643201

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2497C211yesmaximal, characteristic
3#33150C212yesmaximal, complemented
4#31689C214yesmaximal, complemented
5#31687C214yesmaximal, complemented
6#31429C214yesmaximal, complemented
7#31427C214yesmaximal, complemented
8#29973C212yesmaximal, complemented
9#29297C211yesmaximal, characteristic, complemented
10#29296C211yesmaximal, characteristic, complemented
11#27961C211yesmaximal, characteristic, complemented
12#27961C211yesmaximal, characteristic, complemented
13#27604C212yesmaximal
14#27604C212yesmaximal
15#21902C211yesmaximal, characteristic
16#2857C2211yescharacteristic
17#2857C2211yescharacteristic
18#2851C2212yes—
19#2626—21no—
20#2626—21no—
21#2608—22no—
22#2607—22no—
23#2497—22no—
24#2497—22no—
25#2497C2214yes—
26#2497C2214yes—
27#2441C2212yes—
28#2441C2212yes—
29#2439C2212yes—
30#2439C2212yes—
31C2×#400—21no—
32C2×#396—21no—
33#2329—21no—
34#2329—21no—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#388—21no—
38C2×#387—21no—
39C2×#384C2211yescharacteristic
40#2280C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#2101C2214yescomplemented
46#2100C2214yescomplemented
47#2100C2214yescomplemented
48#2088C2214yescomplemented
49#2086C2214yescomplemented
50#2084C2212yescomplemented
51#2083C2212yescomplemented
52#2077C2214yes—
53#2077C2214yes—
54#2075C2214yes—
55#2075C2214yes—
56#2059C2214yescomplemented
57#2058C2212yescomplemented
58#2057C2212yescomplemented
59#1990C2212yescomplemented
60#1990C2212yescomplemented
61#1989C2214yescomplemented
62#1949C2214yescomplemented
63#1942C2214yescomplemented
64#1941C2214yescomplemented
65#1923C2214yes—
66#1923C2214yes—
67#1920C2214yescomplemented
68#1918C2214yescomplemented
69#1854C2214yes—
70#1854C2214yes—
71#1853C2212yes—
72#1853C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1834C2212yescomplemented
76#1834C2212yescomplemented
77#1825C2214yes—
78#1825C2214yes—
79#1824C2212yes—
80#1824C2212yes—
81#1824C2212yes—
82#1824C2212yes—
83#1797C2212yes—
84#1779C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×#125C2312yes—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×#117C2312yes—
94C2×#117C2312yes—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4D411yescharacteristic
106C22×C22⋊C4D411yescharacteristic
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2314yes—
115C2×C22⋊C8C2312yes—
116C2×C22⋊C8C2312yes—
117C2×C22⋊C8C2314yes—
118C2×C22⋊C8C2314yes—
119C2×C4×C8C2312yes—
120C2×C4×C8C2312yes—
121#400—24no—
122#400—24no—
123#399—22no—
124#399—22no—
125#396—22no—
126#396—22no—
127#396—22no—
128#396—22no—
129#396—24no—
130#396—24no—
131#393—22no—
132#393—22no—
133#391—21no—
134#391—21no—
135#391—22no—
136#391—22no—
137#389—21no—
138#389—21no—
139#389—22no—
140#389—21no—
141#389—21no—
142#389—22no—
143#389—22no—
144#389C2312yes—
145#389C2312yes—
146#389C2314yes—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#388—24no—
152#388—24no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—24no—
160#387—24no—
161#385—22no—
162#385—22no—
163#384—21no—
164#384—21no—
165#384C2311yescharacteristic
166#384C2311yescharacteristic
167#384C2312yes—
168#384C2312yes—
169#382C2312yes—
170#381—22no—
171#380—21no—
172#380—21no—
173#380—22no—
174#380—22no—
175#380C2312yes—
176#380C2312yes—
177#380C2314yes—
178#380C2314yes—
179#380C2312yes—
180#380C2312yes—
181#379—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2314yes—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4D412yes—
188C4×C22⋊C4C2312yes—
189C4×C22⋊C4C2312yes—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95D411yescharacteristic
196C2×#95D411yescharacteristic
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2314yes—
210#336C2312yes—
211#336C2312yes—
212C25C2×D411yescharacteristic
213C22×Q8—21no—
214C22×Q8—21no—
215C23×C4—24no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C22×C8C2414yes—
231C22×C8C2414yes—
232C4⋊Q8—42no—
233C4⋊Q8—42no—
234#125—44no—
235C22⋊Q8—44no—
236C22⋊Q8—44no—
237#117—44no—
238#117—44no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—82no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—82no—
289C2×C22⋊C4—82no—
290C2×C22⋊C4C2×D412yes—
291C2×C22⋊C4C2×D412yes—
292C2×C22⋊C4C2×D412yes—
293C2×C22⋊C4C2×D412yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—82no—
300C2×C42C2×D412yes—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2412yes—
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306#107—44no—
307#107—44no—
308#106—44no—
309#106—44no—
310C4⋊C8—44no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—44no—
314C22⋊C8—44no—
315C4×C8—42no—
316C4×C8—42no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95C2×D412yes—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—41no—
341C24—41no—
342C24—41no—
343C24—41no—
344C24C22×D412yes—
345C24C22×D411yescharacteristic
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—81no—
349C2×Q8—81no—
350C2×Q8—81no—
351C2×Q8—81no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—88no—
394C22×C4#13514yes—
395C22×C4#13714yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C42−1+414yes—
405C22×C4C2511yescharacteristic
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C42—22no—
457C42—22no—
458C42—24no—
459C42—41no—
460C42—41no—
461C42—42no—
462C42—84no—
463C42—82no—
464C42—82no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23#53612yes—
484C23C2×#13512yes—
485C23C2×#13712yes—
486C23C23×D411yescharacteristic
487Q8—162no—
488Q8—162no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—84no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4#53612yes—
545C2×C4C2×#13512yes—
546C2×C4C2×#13512yes—
547C2×C4C2×#13712yes—
548C2×C4C2×#13712yes—
549C2×C4C2×2−1+412yes—
550C2×C4#58512yes—
551C2×C4#58512yes—
552C8—44no—
553C8—44no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—88no—
568C22#323114yes—
569C22C2×#53611yescharacteristic
570C22#348511yescharacteristic
571C22#348511yescharacteristic
572C22C22×#13511yescharacteristic
573C22C22×#13711yescharacteristic
574C22#358511yescharacteristic
575C22#358511yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—22no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4#356414yes—
593C2—24no—
594C2—82no—
595C2#3697011yescharacteristic
596C2C2×#323111yescharacteristic
597C2#6086811yescharacteristic
598C2#6169911yescharacteristic
599C2#6170011yescharacteristic
600C2#6170911yescharacteristic
601C2#6171911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33150C2semidirect
4. #31689C2semidirect
5. #31687C2semidirect
6. #31429C2semidirect
7. #31427C2semidirect
8. #29973C2semidirect
9. #29297C2semidirect
10. #29296C2semidirect
11. #27961C2semidirect
12. #27961C2semidirect
45. #2101C22semidirect
46. #2100C22semidirect
47. #2100C22semidirect
48. #2088C22semidirect
49. #2086C22semidirect
50. #2084C22semidirect
51. #2083C22semidirect
56. #2059C22semidirect
57. #2058C22semidirect
58. #2057C22semidirect
59. #1990C22semidirect
60. #1990C22semidirect
61. #1989C22semidirect
62. #1949C22semidirect
63. #1942C22semidirect
64. #1941C22semidirect
67. #1920C22semidirect
68. #1918C22semidirect
75. #1834C22semidirect
76. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24972
#331502
#316892
#316872
#314292
#314272
#299732
#292972
#292962
#279612
#276042
#219022