Finite Group Databasebeta

← #6477785

Subgroups — #6477785

Order 512.

1558 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#35033C211yesmaximal, characteristic
2C2×#2347C211yesmaximal, characteristic
3#33150C212yesmaximal, complemented
4#33149C211yesmaximal, characteristic, complemented
5#33147C211yesmaximal, characteristic, complemented
6#29972C212yesmaximal, complemented
7#29968C211yesmaximal, characteristic, complemented
8#29966C211yesmaximal, characteristic, complemented
9#25999C214yesmaximal, complemented
10#25997C214yesmaximal, complemented
11#25991C214yesmaximal, complemented
12#25989C214yesmaximal, complemented
13#21941C212yesmaximal
14#21930C211yesmaximal, characteristic
15#20699C212yesmaximal
16C22×#117C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2851C2212yes—
21#2626—21no—
22#2626—21no—
23#2608—22no—
24#2607—22no—
25#2441C2212yes—
26#2441C2212yes—
27#2439C2212yes—
28#2439C2212yes—
29#2347—22no—
30#2347—22no—
31#2347C2214yes—
32#2347C2214yes—
33#2329—21no—
34#2329—21no—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#388—21no—
38C2×#388—21no—
39C2×#383—21no—
40#2280C2212yes—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4—21no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#2101C2214yescomplemented
47#2100C2212yescomplemented
48#2099C2212yescomplemented
49#2088C2214yescomplemented
50#2088C2214yescomplemented
51#2087C2214yescomplemented
52#2086C2214yescomplemented
53#2059C2214yescomplemented
54#2058C2212yescomplemented
55#2057C2212yescomplemented
56#1989C2214yescomplemented
57#1988C2212yescomplemented
58#1987C2212yescomplemented
59#1949C2214yescomplemented
60#1948C2212yescomplemented
61#1947C2212yescomplemented
62#1921C2214yescomplemented
63#1920C2214yescomplemented
64#1919C2214yescomplemented
65#1918C2214yescomplemented
66#1854C2214yes—
67#1854C2214yes—
68#1853C2212yes—
69#1853C2212yes—
70#1852C2212yes—
71#1852C2212yes—
72#1850C2214yes—
73#1850C2214yes—
74#1825C2214yes—
75#1825C2214yes—
76#1824C2212yes—
77#1824C2212yes—
78#1823C2212yes—
79#1823C2212yes—
80#1808C2214yes—
81#1807C2214yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1797C2212yes—
85#1735C2214yes—
86#1735C2214yes—
87C24×C4C2311yescharacteristic
88C23×C8C2314yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#125C2312yes—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×#117—22no—
95C2×#117—24no—
96C2×#117Q818yescomplemented
97C2×#117C2314yes—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4D411yescharacteristic
109C22×C22⋊C4D411yescharacteristic
110C22×C42—21no—
111C22×C42—21no—
112C22×C42—21no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2314yes—
121C2×C4⋊C8C2312yes—
122C2×C22⋊C8—28no—
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2314yes—
126#399—22no—
127#399—22no—
128#396—22no—
129#396—22no—
130#396—22no—
131#396—22no—
132#393—22no—
133#393—22no—
134#391—21no—
135#391—21no—
136#391—22no—
137#391—22no—
138#389—21no—
139#389—21no—
140#389—22no—
141#389—21no—
142#389—21no—
143#389—22no—
144#389—22no—
145#389C2312yes—
146#389C2312yes—
147#389C2314yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#389C2312yes—
152#388—24no—
153#388—24no—
154#388—24no—
155#388—24no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#385—22no—
163#385—22no—
164#384C2311yescharacteristic
165#384C2311yescharacteristic
166#383—24no—
167#383—24no—
168#382C2312yes—
169#381—22no—
170#380—21no—
171#380—21no—
172#380—22no—
173#380C2312yes—
174#380C2314yes—
175#380C2312yes—
176#380C2312yes—
177#380C2312yes—
178#379—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—24no—
183C4×C4⋊C4—24no—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4D412yes—
190C4×C22⋊C4D412yes—
191C4×C22⋊C4C2312yes—
192C4×C22⋊C4C2312yes—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215C25C2×D411yescharacteristic
216C22×Q8—21no—
217C22×Q8—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×Q814yes—
231C23×C4#4114yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C22×C8—28no—
235C22×C8—44no—
236C22×C8—44no—
237C22×C8C2414yes—
238C22×C8C2414yes—
239C4⋊Q8—42no—
240C4⋊Q8—42no—
241#125—44no—
242C22⋊Q8—44no—
243C22⋊Q8—44no—
244#117—216no—
245#117—28no—
246#117—44no—
247#117—44no—
248#117C2×Q8116yes—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×Q814yes—
275C2×C4⋊C4C2×Q814yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—82no—
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D412yes—
299C2×C22⋊C4C2×D412yes—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—82no—
310C2×C42—82no—
311C2×C42C2×D412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2412yes—
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317#107—44no—
318#107—44no—
319#106—44no—
320#106—44no—
321C4⋊C8—44no—
322C4⋊C8—42no—
323C4⋊C8—42no—
324C22⋊C8—24no—
325C22⋊C8—24no—
326C22⋊C8—28no—
327#95—24no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95C2×D412yes—
345#95C2×D412yes—
346#95C2×D412yes—
347#95C2×D412yes—
348C24—22no—
349C24—22no—
350C24—41no—
351C24—41no—
352C24—41no—
353C24—41no—
354C24C22⋊Q814yes—
355C24C22×D412yes—
356C24C22×D411yescharacteristic
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—81no—
360C2×Q8—81no—
361C2×Q8—81no—
362C2×Q8—81no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4C22⋊Q814yes—
405C22×C4C22⋊Q818yes—
406C22×C4#13514yes—
407C22×C4C22×D412yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×Q812yes—
416C22×C4C22×Q814yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4114yes—
419C22×C4C2511yescharacteristic
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—88no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—48no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—81no—
448C4⋊C4—82no—
449C4⋊C4—81no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4C22×Q814yes—
456C4⋊C4C22×Q814yes—
457C22⋊C4—216no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4—84no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C42—216no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—41no—
474C42—41no—
475C42—42no—
476C42—82no—
477C42—82no—
478C42—84no—
479C42—82no—
480C42—82no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—28no—
492C23—44no—
493C23—44no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23C2×C22⋊Q814yes—
499C23C2×C22⋊Q812yes—
500C23C2×#13512yes—
501C23#57512yes—
502C23C23×D411yescharacteristic
503Q8—162no—
504Q8—162no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4C2×C22⋊Q814yes—
564C2×C4C2×C22⋊Q812yes—
565C2×C4C2×#13512yes—
566C2×C4C2×#13512yes—
567C2×C4C23×Q811yescharacteristic
568C2×C4C22×#4111yescharacteristic
569C2×C4#58512yes—
570C2×C4#58512yes—
571C8—44no—
572C8—44no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—88no—
587C22#307614yes—
588C22C22×C22⋊Q811yescharacteristic
589C22#348511yescharacteristic
590C22#348511yescharacteristic
591C22C22×#13511yescharacteristic
592C22C2×#57511yescharacteristic
593C22#358511yescharacteristic
594C22#358611yescharacteristic
595C4—22no—
596C4—22no—
597C4—22no—
598C4—24no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—44no—
604C4—44no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4#345812yes—
612C4#349112yes—
613C2—24no—
614C2—44no—
615C2#3509011yescharacteristic
616C2C2×#307611yescharacteristic
617C2#6049211yescharacteristic
618C2#6169911yescharacteristic
619C2#6170111yescharacteristic
620C2#6171211yescharacteristic
621C2#6171311yescharacteristic
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. #33149C2semidirect
5. #33147C2semidirect
6. #29972C2semidirect
7. #29968C2semidirect
8. #29966C2semidirect
9. #25999C2semidirect
10. #25997C2semidirect
11. #25991C2semidirect
12. #25989C2semidirect
46. #2101C22semidirect
47. #2100C22semidirect
48. #2099C22semidirect
49. #2088C22semidirect
50. #2088C22semidirect
51. #2087C22semidirect
52. #2086C22semidirect
53. #2059C22semidirect
54. #2058C22semidirect
55. #2057C22semidirect
56. #1989C22semidirect
57. #1988C22semidirect
58. #1987C22semidirect
59. #1949C22semidirect
60. #1948C22semidirect
61. #1947C22semidirect
62. #1921C22semidirect
63. #1920C22semidirect
64. #1919C22semidirect
65. #1918C22semidirect
96. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#23472
#331502
#331492
#331472
#299722
#299682
#299662
#259992
#259972
#259912
#259892
#219412
#219302
#206992