Finite Group Databasebeta

← #6629635

Subgroups — #6629635

Order 512.

2037 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34635C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic, complemented
3#33105C214yesmaximal, complemented
4#33104C214yesmaximal, complemented
5#33103C214yesmaximal, complemented
6#33102C214yesmaximal, complemented
7#21059C212yesmaximal, complemented
8#21059C212yesmaximal, complemented
9#21033C212yesmaximal, complemented
10#21033C212yesmaximal, complemented
11#20910C211yesmaximal, characteristic
12#20909C211yesmaximal, characteristic
13#20845C211yesmaximal, characteristic, complemented
14#20844C211yesmaximal, characteristic, complemented
15#20683C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#428C2212yes—
21C2×#428C2212yes—
22C2×#417C2212yes—
23C2×#417C2212yes—
24C2×#407C2211yescharacteristic
25C2×#407C2211yescharacteristic
26#2625C2212yes—
27#2625C2212yes—
28#2624C2212yes—
29#2624C2212yes—
30#2623—22no—
31#2622—22no—
32#2395—22no—
33#2395—22no—
34#2297C2211yescharacteristic
35#2297C2211yescharacteristic
36#2296C2211yescharacteristic
37#2296C2211yescharacteristic
38#2280C2211yescharacteristic
39#2274—22no—
40#2274—22no—
41#2274C2214yes—
42#2274C2214yes—
43C2×C4×C4⋊C4—22no—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4—22no—
47#1928C2214yescomplemented
48#1928C2214yescomplemented
49#1927C2214yescomplemented
50#1927C2214yescomplemented
51#1926C2214yescomplemented
52#1926C2214yescomplemented
53#1925C2214yescomplemented
54#1925C2214yescomplemented
55#1921C2214yescomplemented
56#1921C2214yescomplemented
57#1920C2214yescomplemented
58#1920C2214yescomplemented
59#1919C2214yescomplemented
60#1919C2214yescomplemented
61#1918C2214yescomplemented
62#1918C2214yescomplemented
63#1875C2214yes—
64#1874C2214yes—
65#1873C2214yes—
66#1872C2214yes—
67#1854C2214yes—
68#1854C2214yes—
69#1853C2214yes—
70#1852C2214yes—
71#1818C2214yes—
72#1818C2214yes—
73#1736C2212yes—
74#1736C2212yes—
75#1736C2212yes—
76#1736C2212yes—
77#1728C2212yes—
78#1728C2212yes—
79C22×#41C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C22×M16C2312yes—
83C22×M16C2312yes—
84#525—24no—
85#525—24no—
86C2×#123—21no—
87C2×C22⋊Q8C2312yes—
88C2×#121C2312yes—
89C2×#120—21no—
90C2×C4×D4—22no—
91C2×#117—22no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—28no—
95C2×#117D412yes—
96C2×#117D412yes—
97C2×#117C2314yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2312yes—
102C2×#117C2312yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4D412yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—21no—
111C22×C22⋊C4D412yes—
112C22×C42—21no—
113C22×C42—22no—
114C22×C42—22no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#428—28no—
118#428—28no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123#417—28no—
124#417—28no—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129#407—24no—
130#407—24no—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2314yes—
133#400—22no—
134#400—22no—
135#399—22no—
136#399—22no—
137#398—22no—
138#398—22no—
139#397—22no—
140#397—22no—
141#397—22no—
142#397—22no—
143#397—22no—
144#397—22no—
145#394—22no—
146#394—22no—
147#394—22no—
148#394—22no—
149#391C2311yescharacteristic
150#391C2312yes—
151#391C2311yescharacteristic
152#390C2311yescharacteristic
153#390C2312yes—
154#390C2311yescharacteristic
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#386—22no—
160#386—22no—
161#385—22no—
162#385—22no—
163#381—21no—
164#380C2312yes—
165#380C2312yes—
166#379—21no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2314yes—
181C4×C4⋊C4C2314yes—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—24no—
184C4×C22⋊C4—24no—
185C4×C22⋊C4—24no—
186C4×C22⋊C4—24no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C2×#95—22no—
191C2×#95—22no—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2314yes—
203#336C2314yes—
204C25C2×D411yescharacteristic
205C2×#41—22no—
206C2×#41—22no—
207C2×#41—24no—
208C2×#41—24no—
209C22×Q8C2411yescharacteristic
210C22×D4—21no—
211C22×D4—22no—
212C22×D4C2411yescharacteristic
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—21no—
220C23×C4—84no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4#4114yes—
226C23×C4#4114yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C2×M16—24no—
230C2×M16—24no—
231C2×M16—24no—
232C2×M16—24no—
233C2×M16—24no—
234C2×M16—24no—
235C22×C8C2414yes—
236C22×C8C2414yes—
237#123—44no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240#121—24no—
241#121—24no—
242#120—44no—
243C4×D4—48no—
244#117—216no—
245#117—28no—
246#117—28no—
247#117—24no—
248#117—24no—
249#117—24no—
250#117—24no—
251#117—216no—
252#117—22no—
253#117—22no—
254#117—24no—
255#117—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—216no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4#4118yes—
275C2×C4⋊C4#4118yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2411yescharacteristic
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2411yescharacteristic
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2414yes—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
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—22no—
305C2×C42—22no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—84no—
314C2×C42—84no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D411yescharacteristic
318C2×C42C2412yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321#107—28no—
322#107—28no—
323#106—28no—
324#106—28no—
325#103—28no—
326#103—28no—
327#102—28no—
328#102—28no—
329C22⋊C8—28no—
330C22⋊C8—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
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#95—22no—
345C24—22no—
346C24—22no—
347C24—21no—
348C24—42no—
349C24—42no—
350C24#12314yes—
351C24C22×D412yes—
352C24C22×D411yescharacteristic
353#41—48no—
354#41—48no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—44no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—82no—
368C2×D4—82no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—216no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—816no—
403C22×C4#12314yes—
404C22×C4#12318yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4114yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4114yes—
417C22×C4C2511yescharacteristic
418M16—48no—
419M16—48no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C4⋊C4—216no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—44no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4C2×#4118yes—
447C4⋊C4C2×#4118yes—
448C22⋊C4—216no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C42—216no—
465C42—22no—
466C42—24no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—84no—
478C42—84no—
479C42—84no—
480C42—84no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—28no—
488C23—22no—
489C23—22no—
490C23—48no—
491C23—44no—
492C23—48no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23C2×#12314yes—
500C23C2×#12312yes—
501C23#52512yes—
502C23#52512yes—
503C23C23×D411yescharacteristic
504Q8—42no—
505Q8—42no—
506D4—42no—
507D4—42no—
508D4—88no—
509D4—84no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—216no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—416no—
533C2×C4—416no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—816no—
552C2×C4—84no—
553C2×C4C2×#12314yes—
554C2×C4C2×#12312yes—
555C2×C4#52512yes—
556C2×C4#52512yes—
557C2×C4#52918yes—
558C2×C4#52918yes—
559C2×C4C22×#4111yescharacteristic
560C2×C4C22×#4111yescharacteristic
561C2×C4C22×#4112yes—
562C2×C4C22×#4112yes—
563C8—44no—
564C8—44no—
565C22—24no—
566C22—22no—
567C22—24no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—416no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—82no—
579C22—81no—
580C22—82no—
581C22—81no—
582C22#285314yes—
583C22C22×#12311yescharacteristic
584C22C2×#52511yescharacteristic
585C22C2×#52511yescharacteristic
586C22#358512yes—
587C22#358612yes—
588C4—24no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—48no—
594C4—48no—
595C4—81no—
596C4—82no—
597C4—81no—
598C4—82no—
599C4—81no—
600C4#285314yes—
601C4C2×#52912yes—
602C4C2×#52912yes—
603C2—24no—
604C2—44no—
605C2—161no—
606C2C2×#285311yescharacteristic
607C2#6051712yes—
608C2#6056411yescharacteristic
609C2#6056512yes—
610C2#6056611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2274C2semidirect
3. #33105C2semidirect
4. #33104C2semidirect
5. #33103C2semidirect
6. #33102C2semidirect
7. #21059C2semidirect
8. #21059C2semidirect
9. #21033C2semidirect
10. #21033C2semidirect
13. #20845C2semidirect
14. #20844C2semidirect
15. #20683C2semidirect
47. #1928C22semidirect
48. #1928C22semidirect
49. #1927C22semidirect
50. #1927C22semidirect
51. #1926C22semidirect
52. #1926C22semidirect
53. #1925C22semidirect
54. #1925C22semidirect
55. #1921C22semidirect
56. #1921C22semidirect
57. #1920C22semidirect
58. #1920C22semidirect
59. #1919C22semidirect
60. #1919C22semidirect
61. #1918C22semidirect
62. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#346352
C2×#22742
#331052
#331042
#331032
#331022
#210592
#210332
#209102
#209092
#208452
#208442
#206832