Finite Group Databasebeta

← #6497132

Subgroups — #6497132

Order 512.

1810 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic, complemented
2D4×#117C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#33068C211yesmaximal, characteristic, complemented
5#32953C212yesmaximal
6#29998C211yesmaximal, characteristic, complemented
7#29937C212yesmaximal, complemented
8#29864C211yesmaximal, characteristic, complemented
9#27539C214yesmaximal
10#27475C214yesmaximal, complemented
11#27379C214yesmaximal, complemented
12#27315C214yesmaximal, complemented
13#22116C212yesmaximal, complemented
14#21926C211yesmaximal, characteristic, complemented
15#21613C212yesmaximal, complemented
16C2×#541—21no—
17C22×C4×D4C2211yescharacteristic
18C22×#117—21no—
19C2×#474C2212yes—
20C2×#474C2212yes—
21#2868C2211yescharacteristic, complemented
22#2859C2211yescharacteristic, complemented
23#2857C2212yes—
24#2642—21no—
25#2621—21no—
26#2506—22no—
27#2506C2214yescomplemented
28#2496—22no—
29#2496C2214yes—
30#2354C2212yes—
31#2352—22no—
32D4×C4⋊C4C2212yes—
33#2344C2212yes—
34#2342—22no—
35D4×C22⋊C4C2212yes—
36#2327—21no—
37C4×C4⋊Q8C2212yescomplemented
38#2293C2211yescharacteristic
39#2293C2211yescharacteristic
40#2290—21no—
41#2290—21no—
42#2285C2211yescharacteristic
43#2280—21no—
44C42×D4C2211yescharacteristic
45C42×D4C2211yescharacteristic
46C4×#117C2211yescharacteristic
47#2100C2214yescomplemented
48#2088C2214yescomplemented
49#2084C2214yescomplemented
50#2083C2214yescomplemented
51#2077C2214yes—
52#2077C2214yes—
53#2071C2214yes—
54#2065C2214yes—
55#1988C2212yescomplemented
56#1988C2212yescomplemented
57#1948C2212yescomplemented
58#1948C2212yescomplemented
59#1947C2214yescomplemented
60#1943C2214yescomplemented
61#1936C2214yes—
62#1926C2214yes—
63#1924C2214yes—
64#1921C2214yescomplemented
65#1921C2214yescomplemented
66#1920C2214yescomplemented
67#1917C2214yescomplemented
68#1864C2214yes—
69#1864C2214yes—
70#1862C2214yes—
71#1862C2214yes—
72#1852C2214yes—
73#1842C2211yescharacteristic, complemented
74#1831C2211yescharacteristic, complemented
75#1802C2212yes—
76#1802C2212yes—
77#1802C2214yes—
78#1797C2212yes—
79#1786C2214yes—
80#1776C2214yescomplemented
81#1767C2212yescomplemented
82#1762C2214yes—
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85#541—48no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2312yes—
89C2×#126—24no—
90C2×#125—24no—
91C2×#125—21no—
92C2×C22⋊Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4C2311yescharacteristic
99C2×C4×D4C2311yescharacteristic
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×C4×D4C2311yescharacteristic
103C2×C4×D4C2311yescharacteristic
104C2×#117—21no—
105C2×#117—22no—
106C2×#117—22no—
107C2×#117—21no—
108C2×#117—21no—
109C2×#117—48no—
110C2×#117D411yescharacteristic
111C2×#117D411yescharacteristic
112C22×C4⋊C4—22no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—22no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119#474—28no—
120#474—28no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×C22⋊C8C2314yes—
133C2×#97C2314yes—
134#400—24no—
135#400—24no—
136#398—24no—
137#397—24no—
138#391—22no—
139#391—22no—
140#390C2312yes—
141#390C2312yes—
142#389—22no—
143#389—22no—
144#389—22no—
145#389C2314yes—
146#389C2314yes—
147#389C2312yes—
148#389C2314yes—
149#389C2312yes—
150#388—22no—
151#388—22no—
152#388—22no—
153#388—22no—
154#387—24no—
155#387—24no—
156#387—22no—
157#387—22no—
158#386—22no—
159#386—22no—
160#385—22no—
161#385—22no—
162#384—22no—
163#384C2312yes—
164#384C2314yes—
165#384C2312yes—
166#382—21no—
167#382—21no—
168#381—22no—
169#380—21no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380C2312yes—
174#380C2314yes—
175#379—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4D412yes—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2311yescharacteristic
183C4×C4⋊C4C2311yescharacteristic
184C4×C4⋊C4C2312yes—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—21no—
191C4×C22⋊C4D412yes—
192C4×C22⋊C4C2312yes—
193#376D411yescharacteristic
194#376D411yescharacteristic
195C43C2311yescharacteristic
196C43C2311yescharacteristic
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25—21no—
209C22×Q8—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—42no—
215C22×D4C2411yescharacteristic
216C22×D4C2411yescharacteristic
217C23×C4—28no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2411yescharacteristic
230C22×C8C2414yes—
231C22×C8C2414yes—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—44no—
235C4⋊Q8—44no—
236#126—416no—
237#125—44no—
238#125—416no—
239C22⋊Q8—48no—
240C4×Q8—48no—
241C4×D4—28no—
242C4×D4—28no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—42no—
252C4×D4—42no—
253C4×D4#4114yes—
254C4×D4#4114yes—
255#117—24no—
256#117—24no—
257#117—44no—
258#117—48no—
259#117—48no—
260#117—44no—
261#117—44no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2411yescharacteristic
285C2×C4⋊C4C2411yescharacteristic
286C2×C22⋊C4—28no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2412yes—
321C2×C42C2412yes—
322C2×C42C2411yescharacteristic
323#107—28no—
324#107—28no—
325#106—44no—
326#106—44no—
327C4⋊C8—24no—
328C4⋊C8—24no—
329C4⋊C8—44no—
330#102—28no—
331#102—28no—
332#102—28no—
333#102—44no—
334C22⋊C8—44no—
335#97—44no—
336#95—22no—
337#95—22no—
338#95—21no—
339#95—21no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95C2×D412yes—
349#95C2×D412yes—
350#95C2×D412yes—
351#95C2×D412yes—
352C24—21no—
353C24—24no—
354C24—21no—
355C24—44no—
356C24—42no—
357C24—42no—
358C24—82no—
359C24C22×D411yescharacteristic
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×D4—24no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4C2×#4112yes—
374C2×D4C2×#4112yes—
375C2×D42−1+414yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—432no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4AGL(1,ℤ/8)14yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C2511yescharacteristic
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—82no—
454C4⋊C4C2×#4112yes—
455C4⋊C4C2×#4112yes—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—21no—
476C42—21no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—42no—
482C42—42no—
483C42—44no—
484C42—48no—
485C42—44no—
486C42—42no—
487C42—42no—
488C42AGL(1,ℤ/8)14yes—
489C42C2×#4112yes—
490C42C2×#4112yes—
491C23—22no—
492C23—22no—
493C23—21no—
494C23—24no—
495C23—22no—
496C23—21no—
497C23—22no—
498C23—48no—
499C23—48no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—88no—
508C23—84no—
509C23—88no—
510C23C2×AGL(1,ℤ/8)12yes—
511C23C23×D411yescharacteristic
512Q8—88no—
513D4—22no—
514D4—24no—
515D4—24no—
516D4—48no—
517D4—41no—
518D4—41no—
519D4#54114yescomplemented
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—82no—
566C2×C4—88no—
567C2×C4—88no—
568C2×C4#52512yes—
569C2×C4#52512yes—
570C2×C4C2×AGL(1,ℤ/8)14yes—
571C2×C4C2×AGL(1,ℤ/8)12yes—
572C2×C4C2×AGL(1,ℤ/8)12yes—
573C2×C4C2×AGL(1,ℤ/8)12yes—
574C2×C4C22×#4111yescharacteristic
575C2×C4C22×#4111yescharacteristic
576C2×C4C2×2−1+412yes—
577C2×C4#58512yes—
578C8—44no—
579C8—44no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—42no—
584C22—41no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—816no—
594C22—82no—
595C22—82no—
596C22C2×#52511yescharacteristic
597C22#348912yes—
598C22C22×AGL(1,ℤ/8)11yescharacteristic
599C22C22×AGL(1,ℤ/8)11yescharacteristic
600C22#358612yes—
601C4—22no—
602C4—22no—
603C4—24no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—41no—
611C4—84no—
612C4—84no—
613C4—82no—
614C4—82no—
615C4#310914yes—
616C4C2×#54111yescharacteristic
617C4#345511yescharacteristic
618C4#349512yes—
619C2—42no—
620C2—44no—
621C2—82no—
622C2#3531011yescharacteristic
623C2C2×#310911yescharacteristic
624C2#6056911yescharacteristic
625C2#6210712yes—
626C2#6210912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #35319C2semidirect
3. #33195C2semidirect
4. #33068C2semidirect
6. #29998C2semidirect
7. #29937C2semidirect
8. #29864C2semidirect
10. #27475C2semidirect
11. #27379C2semidirect
12. #27315C2semidirect
13. #22116C2semidirect
14. #21926C2semidirect
15. #21613C2semidirect
21. #2868C22semidirect
22. #2859C22semidirect
27. #2506C22semidirect
37. C4×C4⋊Q8C22semidirect
47. #2100C22semidirect
48. #2088C22semidirect
49. #2084C22semidirect
50. #2083C22semidirect
55. #1988C22semidirect
56. #1988C22semidirect
57. #1948C22semidirect
58. #1948C22semidirect
59. #1947C22semidirect
60. #1943C22semidirect
64. #1921C22semidirect
65. #1921C22semidirect
66. #1920C22semidirect
67. #1917C22semidirect
73. #1842C22semidirect
74. #1831C22semidirect
80. #1776C22semidirect
81. #1767C22semidirect
519. D4#541semidirect

Maximal subgroups

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

subgroupindex
#353192
D4×#1172
#331952
#330682
#329532
#299982
#299372
#298642
#275392
#274752
#273792
#273152
#221162
#219262
#216132