Finite Group Databasebeta

← #6643213

Subgroups — #6643213

Order 512.

1346 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2497C211yesmaximal, characteristic
3#33149C211yesmaximal, characteristic, complemented
4#33147C211yesmaximal, characteristic, complemented
5#31717C214yesmaximal, complemented
6#31715C214yesmaximal, complemented
7#31441C214yesmaximal, complemented
8#31439C214yesmaximal, complemented
9#29969C211yesmaximal, characteristic, complemented
10#29967C211yesmaximal, characteristic, complemented
11#29303C212yesmaximal, complemented
12#27965C212yesmaximal, complemented
13#27604C212yesmaximal
14#27604C212yesmaximal
15#21906C211yesmaximal, characteristic
16#2857C2211yescharacteristic
17#2857C2211yescharacteristic
18#2851C2212yes—
19#2626—21no—
20#2626—21no—
21#2608—22no—
22#2607—22no—
23#2497—22no—
24#2497—22no—
25#2497C2214yes—
26#2497C2214yes—
27#2441C2212yes—
28#2441C2212yes—
29#2439C2212yes—
30#2439C2212yes—
31C2×#400—21no—
32C2×#396—21no—
33#2329—21no—
34#2329—21no—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#388—21no—
38C2×#387—21no—
39C2×#384C2211yescharacteristic
40#2280C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#2100C2212yescomplemented
46#2099C2212yescomplemented
47#2099C2214yescomplemented
48#2099C2214yescomplemented
49#2088C2214yescomplemented
50#2087C2214yescomplemented
51#2085C2214yescomplemented
52#2077C2214yes—
53#2077C2214yes—
54#2075C2214yes—
55#2075C2214yes—
56#2059C2214yescomplemented
57#2058C2212yescomplemented
58#2057C2212yescomplemented
59#1992C2214yescomplemented
60#1988C2212yescomplemented
61#1987C2212yescomplemented
62#1948C2212yescomplemented
63#1947C2212yescomplemented
64#1944C2214yescomplemented
65#1943C2214yescomplemented
66#1924C2214yes—
67#1924C2214yes—
68#1921C2214yescomplemented
69#1919C2214yescomplemented
70#1854C2214yes—
71#1854C2214yes—
72#1853C2212yes—
73#1853C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1836C2214yescomplemented
77#1825C2214yes—
78#1825C2214yes—
79#1824C2212yes—
80#1824C2212yes—
81#1824C2212yes—
82#1824C2212yes—
83#1797C2211yescharacteristic
84#1797C2211yescharacteristic
85#1779C2212yes—
86#1749C2212yes—
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⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4D411yescharacteristic
106C22×C22⋊C4D411yescharacteristic
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×C22⋊C8C2312yes—
117C2×C22⋊C8C2312yes—
118C2×C22⋊C8C2314yes—
119C2×C22⋊C8C2314yes—
120C2×#97C2314yes—
121#400—24no—
122#400—24no—
123#399—22no—
124#399—22no—
125#396—22no—
126#396—22no—
127#396—22no—
128#396—22no—
129#396—24no—
130#396—24no—
131#393—22no—
132#393—22no—
133#391—21no—
134#391—21no—
135#391—22no—
136#391—22no—
137#389—21no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389C2312yes—
145#389C2312yes—
146#389C2314yes—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#388—24no—
152#388—24no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—24no—
160#387—24no—
161#385—22no—
162#385—22no—
163#384—22no—
164#384C2311yescharacteristic
165#384C2311yescharacteristic
166#384C2314yes—
167#382C2312yes—
168#381—22no—
169#380—21no—
170#380—21no—
171#380—22no—
172#380—22no—
173#380C2312yes—
174#380C2312yes—
175#380C2314yes—
176#380C2314yes—
177#380C2312yes—
178#380C2312yes—
179#379—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2312yes—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4D412yes—
188C4×C22⋊C4C2312yes—
189C4×C22⋊C4C2312yes—
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#336C2312yes—
210#336C2312yes—
211#336C2314yes—
212C25C2×D411yescharacteristic
213C22×Q8—21no—
214C22×Q8—21no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C22×C8C2414yes—
232C22×C8C2414yes—
233C4⋊Q8—42no—
234C4⋊Q8—42no—
235#125—44no—
236C22⋊Q8—44no—
237C22⋊Q8—44no—
238#117—44no—
239#117—44no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4—82no—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—82no—
290C2×C22⋊C4—82no—
291C2×C22⋊C4C2×D412yes—
292C2×C22⋊C4C2×D412yes—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D412yes—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—82no—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D412yes—
305C2×C42C2412yes—
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#107—44no—
309#107—44no—
310#106—44no—
311#106—44no—
312C4⋊C8—42no—
313C4⋊C8—42no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316C22⋊C8—44no—
317C22⋊C8—44no—
318#97—44no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95C2×D412yes—
337#95C2×D412yes—
338#95C2×D412yes—
339#95C2×D412yes—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—41no—
344C24—41no—
345C24—41no—
346C24—41no—
347C24C22×D412yes—
348C24C22×D411yescharacteristic
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—81no—
352C2×Q8—81no—
353C2×Q8—81no—
354C2×Q8—81no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4#13714yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C42−1+412yes—
410C22×C42−1+412yes—
411C22×C4C2511yescharacteristic
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C42—24no—
463C42—22no—
464C42—22no—
465C42—41no—
466C42—41no—
467C42—42no—
468C42—82no—
469C42—82no—
470C42—82no—
471C42—82no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23#53612yes—
491C23C2×#13712yes—
492C23#57512yes—
493C23C23×D411yescharacteristic
494Q8—162no—
495Q8—162no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
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—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
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—84no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4#53612yes—
556C2×C4C2×#13712yes—
557C2×C4C2×#13712yes—
558C2×C4C2×2−1+411yescharacteristic
559C2×C4C2×2−1+411yescharacteristic
560C2×C4#58512yes—
561C2×C4#58512yes—
562C8—44no—
563C8—44no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—88no—
578C22#323714yes—
579C22C2×#53611yescharacteristic
580C22#348511yescharacteristic
581C22#348511yescharacteristic
582C22C22×#13711yescharacteristic
583C22C2×#57511yescharacteristic
584C22#358511yescharacteristic
585C22#358511yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4#356412yes—
603C4#356412yes—
604C2—24no—
605C2—82no—
606C2#3697011yescharacteristic
607C2C2×#323711yescharacteristic
608C2#6086811yescharacteristic
609C2#6170911yescharacteristic
610C2#6171011yescharacteristic
611C2#6171311yescharacteristic
612C2#6171911yescharacteristic
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. #33149C2semidirect
4. #33147C2semidirect
5. #31717C2semidirect
6. #31715C2semidirect
7. #31441C2semidirect
8. #31439C2semidirect
9. #29969C2semidirect
10. #29967C2semidirect
11. #29303C2semidirect
12. #27965C2semidirect
45. #2100C22semidirect
46. #2099C22semidirect
47. #2099C22semidirect
48. #2099C22semidirect
49. #2088C22semidirect
50. #2087C22semidirect
51. #2085C22semidirect
56. #2059C22semidirect
57. #2058C22semidirect
58. #2057C22semidirect
59. #1992C22semidirect
60. #1988C22semidirect
61. #1987C22semidirect
62. #1948C22semidirect
63. #1947C22semidirect
64. #1944C22semidirect
65. #1943C22semidirect
68. #1921C22semidirect
69. #1919C22semidirect
76. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24972
#331492
#331472
#317172
#317152
#314412
#314392
#299692
#299672
#293032
#279652
#276042
#219062