Finite Group Databasebeta

← #6528522

Subgroups — #6528522

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37642C211yesmaximal, characteristic, complemented
2#35172C211yesmaximal, characteristic
3#29116C214yesmaximal
4#27971C214yesmaximal, complemented
5#27825C214yesmaximal, complemented
6#27821C212yesmaximal, complemented
7#27813C212yesmaximal, complemented
8#27743C214yesmaximal, complemented
9#27736C212yesmaximal, complemented
10#27728C212yesmaximal, complemented
11#21839C211yesmaximal, characteristic, complemented
12#21835C211yesmaximal, characteristic, complemented
13#20738C211yesmaximal, characteristic
14#20732C211yesmaximal, characteristic
15#20537C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2848—22no—
21#2844—22no—
22#2628—22no—
23#2628—22no—
24#2628C2214yes—
25#2626—22no—
26#2580—22no—
27#2571—21no—
28#2571C2212yescomplemented
29#2571C2212yescomplemented
30#2566C2214yes—
31#2563C2212yes—
32#2551—21no—
33#2551C2212yescomplemented
34#2551C2212yescomplemented
35#2542C2212yes—
36#2538—21no—
37#2538—21no—
38#2317C2212yes—
39#2309C2212yes—
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42#2285C2211yescharacteristic
43#2101C2214yescomplemented
44#2101C2214yescomplemented
45#2100C2214yescomplemented
46#2099C2214yescomplemented
47#2073C2214yes—
48#2073C2214yes—
49#2023C2214yes—
50#2023C2214yes—
51#1993C2214yescomplemented
52#1992C2212yescomplemented
53#1992C2212yescomplemented
54#1990C2214yescomplemented
55#1990C2211yescharacteristic, complemented
56#1990C2211yescharacteristic, complemented
57#1990C2211yescharacteristic, complemented
58#1990C2211yescharacteristic, complemented
59#1989C2214yescomplemented
60#1988C2212yescomplemented
61#1987C2212yescomplemented
62#1936C2214yes—
63#1936C2214yes—
64#1934C2214yes—
65#1934C2214yes—
66#1836C2214yescomplemented
67#1835C2214yescomplemented
68#1825C2214yes—
69#1825C2214yes—
70#1824C2212yes—
71#1823C2212yes—
72#1802C2212yes—
73#1802C2212yes—
74#1802C2212yes—
75#1802C2212yes—
76#1786C2212yes—
77#1786C2212yes—
78#1771C2214yescomplemented
79#1770C2212yescomplemented
80C8×C4⋊C4C2212yescomplemented
81#1762C2214yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84#1750C2214yes—
85C23×D4C2311yescharacteristic
86C24×C4—21no—
87#533—24no—
88#533—24no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2314yes—
94C2×#125C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×#117—21no—
108C2×#117—21no—
109C2×#117C2312yes—
110C2×#117C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C42C2311yescharacteristic
114#432—24no—
115#431—24no—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118C2×#102C2312yes—
119C2×#102C2314yes—
120C2×#102C2312yes—
121C2×#102C2312yes—
122C2×#102C2314yes—
123C2×#102C2312yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126C2×#97C2312yes—
127C2×#97C2314yes—
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2314yes—
131#400—24no—
132#400—24no—
133#400—24no—
134#399—24no—
135#399—24no—
136#396—24no—
137#396—24no—
138#395—24no—
139#395—24no—
140#395—24no—
141#393—24no—
142#393—24no—
143#390—22no—
144#390—22no—
145#390C2314yes—
146#390C2314yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389C2314yes—
154#389C2314yes—
155#385—24no—
156#385—24no—
157#385—24no—
158#384D411yescharacteristic
159#384D411yescharacteristic
160#384C2314yes—
161#384C2311yescharacteristic
162#384C2311yescharacteristic
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#382—22no—
166#382—22no—
167#382—21no—
168#382—21no—
169#382D412yes—
170#382C2314yes—
171#382C2314yes—
172#380—21no—
173#380—21no—
174#380—22no—
175#380—22no—
176#380D412yes—
177#380D412yes—
178#380C2312yes—
179#380C2312yes—
180#379—24no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187#340C2312yes—
188#340C2312yes—
189#340C2312yes—
190#340C2312yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25—21no—
204C22×Q8—21no—
205C22×Q8—21no—
206C22×Q8—24no—
207C22×D4—24no—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—42no—
212C22×D4C2412yes—
213C23×C4—21no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—44no—
217C23×C4—82no—
218C23×C4—82no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2411yescharacteristic
222C22×C8C2412yes—
223C22×C8C2412yes—
224C22×C8C2414yes—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—44no—
230C4⋊Q8—44no—
231C4⋊Q8—84no—
232C4⋊Q8—84no—
233#125—24no—
234#125—24no—
235C22⋊Q8—44no—
236C22⋊Q8—44no—
237C22⋊Q8—84no—
238C22⋊Q8—84no—
239C4×Q8—44no—
240C4×Q8—44no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—44no—
246C4×D4—44no—
247#117—24no—
248#117—24no—
249#117—44no—
250#117—82no—
251#117—82no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4C2×D414yes—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42#4114yes—
302C2×C42#4114yes—
303C2×C42C2414yes—
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306C4⋊C8—24no—
307C4⋊C8—24no—
308#102—24no—
309#102—28no—
310#102—24no—
311#102—42no—
312#102—44no—
313#102—42no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316#97—24no—
317#97—44no—
318C4×C8—22no—
319C4×C8—22no—
320C4×C8—44no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95C2×D414yes—
332#95C2×D414yes—
333C24—24no—
334C24—22no—
335C24—42no—
336C24—42no—
337C24—81no—
338C24—81no—
339C24—81no—
340C24—81no—
341C24C22×D411yescharacteristic
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—81no—
347C2×Q8—81no—
348C2×Q8—84no—
349C2×Q8—84no—
350C2×Q8—84no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D42−1+414yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—816no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4#12414yes—
386C22×C4#12414yes—
387C22×C4C22×D414yes—
388C22×C4C22×D414yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2511yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C42—24no—
453C42—24no—
454C42—44no—
455C42—44no—
456C42—44no—
457C42#12414yes—
458C42#12414yes—
459C42C2×#4112yes—
460C42C2×#4112yes—
461C42C2×#4112yes—
462C42C2×#4112yes—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—24no—
470C23—48no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—84no—
478C23—84no—
479C23—84no—
480C23—84no—
481C23—84no—
482C23—84no—
483C23C2×#12412yes—
484C23C23×D411yescharacteristic
485Q8—84no—
486Q8—84no—
487Q8—168no—
488D4—24no—
489D4—24no—
490D4—48no—
491D4—42no—
492D4—42no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
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—88no—
533C2×C4—84no—
534C2×C4—88no—
535C2×C4C2×#12412yes—
536C2×C4C2×#12412yes—
537C2×C4C2×#12412yes—
538C2×C4C2×#12412yes—
539C2×C4C2×#12412yes—
540C2×C4#57612yes—
541C2×C4#57712yes—
542C2×C4C22×#4111yescharacteristic
543C2×C4C22×#4111yescharacteristic
544C2×C4C2×2−1+412yes—
545C2×C4#58512yes—
546C8—42no—
547C8—42no—
548C8—44no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—816no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22C22×#12411yescharacteristic
565C22#348511yescharacteristic
566C22#348611yescharacteristic
567C22C2×#57611yescharacteristic
568C22C2×#57711yescharacteristic
569C22#358511yescharacteristic
570C22#358611yescharacteristic
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—44no—
576C4—44no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4#313614yes—
585C4#349512yes—
586C4#349512yes—
587C2—42no—
588C2—84no—
589C2#3564411yescharacteristic
590C2C2×#313611yescharacteristic
591C2#6062111yescharacteristic
592C2#6180211yescharacteristic
593C2#6180311yescharacteristic
594C2#6184911yescharacteristic
595C2#6185011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37642C2semidirect
4. #27971C2semidirect
5. #27825C2semidirect
6. #27821C2semidirect
7. #27813C2semidirect
8. #27743C2semidirect
9. #27736C2semidirect
10. #27728C2semidirect
11. #21839C2semidirect
12. #21835C2semidirect
15. #20537C2semidirect
28. #2571C22semidirect
29. #2571C22semidirect
33. #2551C22semidirect
34. #2551C22semidirect
43. #2101C22semidirect
44. #2101C22semidirect
45. #2100C22semidirect
46. #2099C22semidirect
51. #1993C22semidirect
52. #1992C22semidirect
53. #1992C22semidirect
54. #1990C22semidirect
55. #1990C22semidirect
56. #1990C22semidirect
57. #1990C22semidirect
58. #1990C22semidirect
59. #1989C22semidirect
60. #1988C22semidirect
61. #1987C22semidirect
66. #1836C22semidirect
67. #1835C22semidirect
78. #1771C22semidirect
79. #1770C22semidirect
80. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#376422
#351722
#291162
#279712
#278252
#278212
#278132
#277432
#277362
#277282
#218392
#218352
#207382
#207322
#205372