Finite Group Databasebeta

← #6643205

Subgroups — #6643205

Order 512.

1324 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#35033C211yesmaximal, characteristic
2C2×#2495C211yesmaximal, characteristic
3#33151C212yesmaximal, complemented
4#31709C214yesmaximal, complemented
5#31707C214yesmaximal, complemented
6#31453C214yesmaximal, complemented
7#31451C214yesmaximal, complemented
8#29973C212yesmaximal, complemented
9#29302C211yesmaximal, characteristic, complemented
10#29301C211yesmaximal, characteristic, complemented
11#27969C211yesmaximal, characteristic, complemented
12#27969C211yesmaximal, characteristic, complemented
13#27604C212yesmaximal
14#27602C212yesmaximal
15#21906C211yesmaximal, characteristic
16#2857C2211yescharacteristic
17#2857C2211yescharacteristic
18#2851C2212yes—
19#2626—21no—
20#2626—21no—
21#2608—22no—
22#2607—22no—
23#2495—22no—
24#2495—22no—
25#2495C2214yes—
26#2495C2214yes—
27#2441C2212yes—
28#2441C2212yes—
29#2439C2212yes—
30#2439C2212yes—
31C2×#400—21no—
32C2×#397—21no—
33#2329—21no—
34#2329—21no—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#388—21no—
38C2×#383—21no—
39C2×#382C2211yescharacteristic
40#2280C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#2101C2214yescomplemented
46#2101C2214yescomplemented
47#2101C2214yescomplemented
48#2088C2214yescomplemented
49#2086C2214yescomplemented
50#2084C2212yescomplemented
51#2083C2212yescomplemented
52#2077C2214yes—
53#2077C2214yes—
54#2076C2214yes—
55#2075C2214yes—
56#2059C2214yescomplemented
57#2058C2212yescomplemented
58#2057C2212yescomplemented
59#1993C2212yescomplemented
60#1993C2212yescomplemented
61#1989C2214yescomplemented
62#1949C2214yescomplemented
63#1944C2214yescomplemented
64#1943C2214yescomplemented
65#1924C2214yes—
66#1924C2214yes—
67#1921C2214yescomplemented
68#1919C2214yescomplemented
69#1854C2214yes—
70#1854C2214yes—
71#1853C2212yes—
72#1853C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1836C2212yescomplemented
76#1836C2212yescomplemented
77#1825C2214yes—
78#1825C2214yes—
79#1824C2212yes—
80#1824C2212yes—
81#1823C2212yes—
82#1823C2212yes—
83#1797C2212yes—
84#1779C2212yes—
85#1749C2211yescharacteristic
86#1749C2211yescharacteristic
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⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4D411yescharacteristic
104C22×C22⋊C4D411yescharacteristic
105C22×C42—21no—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C2×#107C2314yes—
109C2×#107C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×C4⋊C8C2314yes—
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2312yes—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117C2×#97C2312yes—
118C2×#97C2312yes—
119#400—24no—
120#400—24no—
121#399—22no—
122#399—22no—
123#397—24no—
124#397—24no—
125#396—22no—
126#396—22no—
127#396—22no—
128#396—22no—
129#393—22no—
130#393—22no—
131#391—21no—
132#391—21no—
133#391—22no—
134#391—22no—
135#389—21no—
136#389—21no—
137#389—22no—
138#389—21no—
139#389—21no—
140#389—22no—
141#389—22no—
142#389C2312yes—
143#389C2312yes—
144#389C2314yes—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#389C2312yes—
149#388—24no—
150#388—24no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#385—22no—
158#385—22no—
159#384C2311yescharacteristic
160#384C2311yescharacteristic
161#383—24no—
162#383—24no—
163#382—21no—
164#382—21no—
165#382C2312yes—
166#382C2312yes—
167#382C2312yes—
168#381—22no—
169#380—21no—
170#380—21no—
171#380—22no—
172#380—22no—
173#380C2312yes—
174#380C2314yes—
175#380C2314yes—
176#380C2312yes—
177#380C2312yes—
178#380C2312yes—
179#379—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4C2314yes—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4D412yes—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4C2312yes—
187C4×C22⋊C4C2312yes—
188C2×#95—21no—
189C2×#95—21no—
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×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
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—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—82no—
284C2×C22⋊C4C2×D412yes—
285C2×C22⋊C4C2×D412yes—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42C2×D412yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2412yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300#107—44no—
301#107—44no—
302#106—44no—
303#106—44no—
304C4⋊C8—44no—
305C22⋊C8—24no—
306C22⋊C8—24no—
307C22⋊C8—44no—
308C22⋊C8—44no—
309#97—42no—
310#97—42no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—22no—
320#95—22no—
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#95C2×D412yes—
332#95C2×D412yes—
333#95C2×D412yes—
334#95C2×D412yes—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—41no—
339C24—41no—
340C24—41no—
341C24—41no—
342C24C22×D412yes—
343C24C22×D411yescharacteristic
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—81no—
347C2×Q8—81no—
348C2×Q8—81no—
349C2×Q8—81no—
350C22×C4—22no—
351C22×C4—22no—
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—42no—
377C22×C4—42no—
378C22×C4—41no—
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×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C42−1+414yes—
403C22×C4C2511yescharacteristic
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—81no—
428C4⋊C4—82no—
429C4⋊C4—81no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—84no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—48no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—82no—
446C22⋊C4—82no—
447C22⋊C4—84no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C42—22no—
451C42—22no—
452C42—24no—
453C42—42no—
454C42—41no—
455C42—41no—
456C42—48no—
457C42—82no—
458C42—82no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23#53612yes—
478C23#57512yes—
479C23#57512yes—
480C23C23×D411yescharacteristic
481Q8—162no—
482Q8—162no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—28no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—44no—
508C2×C4—48no—
509C2×C4—42no—
510C2×C4—42no—
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—42no—
533C2×C4—84no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4#53612yes—
543C2×C4C2×2−1+412yes—
544C2×C4#58512yes—
545C2×C4#58512yes—
546C8—44no—
547C8—44no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—88no—
562C22#324014yes—
563C22C2×#53611yescharacteristic
564C22#348511yescharacteristic
565C22#348511yescharacteristic
566C22C2×#57511yescharacteristic
567C22C2×#57511yescharacteristic
568C22#358511yescharacteristic
569C22#358611yescharacteristic
570C4—22no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—42no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4#356414yes—
587C2—24no—
588C2—82no—
589C2#3697011yescharacteristic
590C2C2×#324011yescharacteristic
591C2#6087011yescharacteristic
592C2#6171011yescharacteristic
593C2#6171111yescharacteristic
594C2#6171311yescharacteristic
595C2#6171611yescharacteristic
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. #33151C2semidirect
4. #31709C2semidirect
5. #31707C2semidirect
6. #31453C2semidirect
7. #31451C2semidirect
8. #29973C2semidirect
9. #29302C2semidirect
10. #29301C2semidirect
11. #27969C2semidirect
12. #27969C2semidirect
45. #2101C22semidirect
46. #2101C22semidirect
47. #2101C22semidirect
48. #2088C22semidirect
49. #2086C22semidirect
50. #2084C22semidirect
51. #2083C22semidirect
56. #2059C22semidirect
57. #2058C22semidirect
58. #2057C22semidirect
59. #1993C22semidirect
60. #1993C22semidirect
61. #1989C22semidirect
62. #1949C22semidirect
63. #1944C22semidirect
64. #1943C22semidirect
67. #1921C22semidirect
68. #1919C22semidirect
75. #1836C22semidirect
76. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24952
#331512
#317092
#317072
#314532
#314512
#299732
#293022
#293012
#279692
#276042
#276022
#219062