Finite Group Databasebeta

← #6631327

Subgroups — #6631327

Order 512.

1615 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34635C211yesmaximal, characteristic
2C2×#2629C211yesmaximal, characteristic, complemented
3#32757C214yesmaximal, complemented
4#32756C214yesmaximal, complemented
5#32755C214yesmaximal, complemented
6#32754C214yesmaximal, complemented
7#23670C212yesmaximal, complemented
8#23658C211yesmaximal, characteristic, complemented
9#23642C212yesmaximal, complemented
10#23630C211yesmaximal, characteristic, complemented
11#23617C212yesmaximal, complemented
12#23553C211yesmaximal, characteristic, complemented
13#23482C212yesmaximal
14#23453C211yesmaximal, characteristic, complemented
15#20627C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#423C2212yes—
22C2×#417C2212yes—
23C2×#417C2212yes—
24C2×#407C2212yes—
25#2629—22no—
26#2629—22no—
27#2629C2214yes—
28#2629C2214yes—
29#2625C2212yes—
30#2625C2212yes—
31#2624C2212yes—
32#2624C2212yes—
33#2623—22no—
34#2622—22no—
35#2395—22no—
36#2395—22no—
37C2×#398—22no—
38C2×#396—22no—
39#2297C2211yescharacteristic
40#2297C2211yescharacteristic
41#2296C2211yescharacteristic
42#2296C2211yescharacteristic
43C2×#389C2212yes—
44C2×#384C2211yescharacteristic
45#2280C2211yescharacteristic
46C2×#380C2212yes—
47#2088C2214yescomplemented
48#2088C2214yescomplemented
49#2087C2214yescomplemented
50#2086C2214yescomplemented
51#2084C2214yescomplemented
52#2083C2214yescomplemented
53#2077C2214yes—
54#2077C2214yes—
55#2076C2214yes—
56#2075C2214yes—
57#2072C2214yescomplemented
58#2071C2214yescomplemented
59#2068C2214yescomplemented
60#2067C2214yescomplemented
61#2066C2214yescomplemented
62#2065C2214yescomplemented
63#2059C2214yescomplemented
64#2059C2214yescomplemented
65#2036C2214yes—
66#2035C2214yes—
67#2034C2214yes—
68#2033C2214yes—
69#2026C2214yescomplemented
70#2025C2214yescomplemented
71#1798C2214yes—
72#1798C2214yes—
73#1736C2212yes—
74#1736C2212yes—
75#1736C2212yes—
76#1736C2212yes—
77#1728C2211yescharacteristic
78#1728C2211yescharacteristic
79C22×#41C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C22×M16C2312yes—
82C22×M16C2312yes—
83C2×#125—21no—
84C2×#125—21no—
85C2×#125—44no—
86C2×#125C2314yes—
87#525—44no—
88C2×#123—21no—
89C2×C22⋊Q8C2312yes—
90C2×#121C2312yes—
91C2×#120—21no—
92C2×C4×D4—22no—
93C2×#117—44no—
94C2×#117D412yes—
95C2×#117D412yes—
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C2×#117C2312yes—
100C2×#117C2312yes—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4D412yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—22no—
108C22×C22⋊C4D412yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111#428—44no—
112#428—44no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117#423—44no—
118C2×C4⋊C8C2314yes—
119C2×C4⋊C8C2314yes—
120#417—44no—
121#417—44no—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126#407—44no—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129#400—22no—
130#400—22no—
131#399—22no—
132#399—22no—
133#398—22no—
134#398—22no—
135#398—24no—
136#398—24no—
137#398—24no—
138#398—24no—
139#397—22no—
140#397—22no—
141#397—22no—
142#397—22no—
143#397—22no—
144#397—22no—
145#396—24no—
146#396—24no—
147#396—24no—
148#396—24no—
149#394—22no—
150#394—22no—
151#394—22no—
152#394—22no—
153#391C2311yescharacteristic
154#391C2311yescharacteristic
155#391C2312yes—
156#390C2311yescharacteristic
157#390C2311yescharacteristic
158#390C2312yes—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#389C2314yes—
164#389C2314yes—
165#389C2312yes—
166#389C2312yes—
167#386—22no—
168#386—22no—
169#385—22no—
170#385—22no—
171#384—21no—
172#384—21no—
173#384C2314yes—
174#381—21no—
175#380—22no—
176#380—22no—
177#380C2312yes—
178#380C2314yes—
179#380C2314yes—
180#380C2312yes—
181#379—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—22no—
189C2×#95—22no—
190C2×#95—22no—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25C2×D411yescharacteristic
202C2×#41—24no—
203C2×#41—22no—
204C2×#41—24no—
205C2×#41—22no—
206C22×Q8C2411yescharacteristic
207C22×D4—21no—
208C22×D4—22no—
209C22×D4C2411yescharacteristic
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C2×M16—24no—
224C2×M16—24no—
225C2×M16—24no—
226C2×M16—24no—
227C2×M16—24no—
228C2×M16—24no—
229C22×C8C2414yes—
230C22×C8C2414yes—
231#125—44no—
232#125—44no—
233#125—44no—
234#125—44no—
235#123—44no—
236C22⋊Q8—42no—
237C22⋊Q8—42no—
238#121—42no—
239#121—42no—
240#120—44no—
241C4×D4—48no—
242#117—24no—
243#117—22no—
244#117—24no—
245#117—22no—
246#117—44no—
247#117—44no—
248#117—42no—
249#117—42no—
250#117—42no—
251#117—42no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—84no—
272C2×C4⋊C4—84no—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2411yescharacteristic
282C2×C4⋊C4C2411yescharacteristic
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
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⋊C4—44no—
297C2×C22⋊C4—84no—
298C2×C22⋊C4C2×D412yes—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D412yes—
310C2×C42C2412yes—
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#107—44no—
314#107—44no—
315#106—44no—
316#106—44no—
317C4⋊C8—44no—
318C4⋊C8—44no—
319#103—44no—
320#103—44no—
321#102—44no—
322#102—44no—
323C22⋊C8—44no—
324C22⋊C8—44no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—42no—
344C24—42no—
345C24C22×D412yes—
346C24C22×D411yescharacteristic
347#41—48no—
348#41—48no—
349C2×Q8—21no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×D4—21no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—44no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—82no—
362C2×D4—82no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4H(5,𝔽2)14yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+414yes—
408C22×C42−1+412yes—
409C22×C4C2511yescharacteristic
410M16—48no—
411M16—48no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—42no—
471C42—42no—
472C42—42no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—48no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—84no—
490C23—82no—
491C23—82no—
492C23#53412yes—
493C23#53612yes—
494C23#53712yes—
495C23C23×D411yescharacteristic
496Q8—42no—
497Q8—42no—
498D4—42no—
499D4—42no—
500D4—88no—
501D4—84no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—48no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—84no—
544C2×C4—88no—
545C2×C4—88no—
546C2×C4—84no—
547C2×C4#53412yes—
548C2×C4#53612yes—
549C2×C4#53712yes—
550C2×C4#56218yes—
551C2×C4#56318yes—
552C2×C4C2×H(5,𝔽2)11yescharacteristic
553C2×C4C2×H(5,𝔽2)12yes—
554C2×C4C2×2−1+412yes—
555C2×C4C2×2−1+411yescharacteristic
556C8—44no—
557C8—44no—
558C22—24no—
559C22—24no—
560C22—22no—
561C22—22no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—88no—
570C22—81no—
571C22—82no—
572C22—82no—
573C22—81no—
574C22—81no—
575C22#281414yes—
576C22C2×#53411yescharacteristic
577C22C2×#53611yescharacteristic
578C22C2×#53711yescharacteristic
579C22#358512yes—
580C22#358612yes—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—84no—
588C4—82no—
589C4—82no—
590C4—81no—
591C4—84no—
592C4—81no—
593C4#281414yes—
594C4C2×#56212yes—
595C4C2×#56312yes—
596C2—24no—
597C2—82no—
598C2—161no—
599C2C2×#281411yescharacteristic
600C2#6075512yes—
601C2#6086712yes—
602C2#6092011yescharacteristic
603C2#6092111yescharacteristic
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×#2629C2semidirect
3. #32757C2semidirect
4. #32756C2semidirect
5. #32755C2semidirect
6. #32754C2semidirect
7. #23670C2semidirect
8. #23658C2semidirect
9. #23642C2semidirect
10. #23630C2semidirect
11. #23617C2semidirect
12. #23553C2semidirect
14. #23453C2semidirect
15. #20627C2semidirect
47. #2088C22semidirect
48. #2088C22semidirect
49. #2087C22semidirect
50. #2086C22semidirect
51. #2084C22semidirect
52. #2083C22semidirect
57. #2072C22semidirect
58. #2071C22semidirect
59. #2068C22semidirect
60. #2067C22semidirect
61. #2066C22semidirect
62. #2065C22semidirect
63. #2059C22semidirect
64. #2059C22semidirect
69. #2026C22semidirect
70. #2025C22semidirect

Maximal subgroups

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

subgroupindex
#346352
C2×#26292
#327572
#327562
#327552
#327542
#236702
#236582
#236422
#236302
#236172
#235532
#234822
#234532
#206272