Finite Group Databasebeta

← #6357390

Subgroups — #6357390

Order 512.

2378 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60860C212yesmaximal, complemented
2#60798C212yesmaximal, complemented
3#60673C211yesmaximal, characteristic, complemented
4C22×#492C211yesmaximal, characteristic, complemented
5#19697C214yesmaximal, complemented
6#19679C214yesmaximal, complemented
7#19671C214yesmaximal, complemented
8#19660C212yesmaximal, complemented
9#19651C212yesmaximal, complemented
10#19625C214yesmaximal, complemented
11#19621C214yesmaximal, complemented
12#17323C211yesmaximal, characteristic, complemented
13C23×SD16—22no—
14#3444C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C22×#127C2211yescharacteristic
17#3239—24no—
18#3238—24no—
19#3230C2214yescomplemented
20#3229C2214yescomplemented
21#3227C2214yescomplemented
22#3226C2214yescomplemented
23#3217—24no—
24#3217—24no—
25#3208C2214yes—
26#3208C2214yes—
27#3205C2214yes—
28#3205C2214yes—
29#3170C2212yescomplemented
30#3169C2212yescomplemented
31#3168—24no—
32#3167C2212yescomplemented
33#3167C2212yescomplemented
34#3165C2214yescomplemented
35#3160C2211yescharacteristic, complemented
36#3151C2214yes—
37#3147C2211yescharacteristic, complemented
38C2×#492—24no—
39C2×#492—22no—
40C2×#492C2214yes—
41C2×#492C2212yes—
42C2×#492C2214yes—
43C2×#492C2212yes—
44#3141C2214yescomplemented
45#3138C2214yes—
46#3131C2212yescomplemented
47#3092—24no—
48#3079—24no—
49#3050—22no—
50#3038—24no—
51#3003—24no—
52C4×#135—22no—
53#2920C2212yes—
54C22×C4×C8C2211yescharacteristic
55#1722C2214yescomplemented
56#1721C2214yescomplemented
57#1719C2218yescomplemented
58#1717C2218yescomplemented
59#1715C2218yescomplemented
60#1714C2218yescomplemented
61#1714C2218yescomplemented
62#1708C2214yescomplemented
63#1706C2218yescomplemented
64#1705C2214yescomplemented
65#1452C2212yescomplemented
66#1451C2214yescomplemented
67#1450C2212yescomplemented
68C8×M16C2214yescomplemented
69C23×Q8—21no—
70C23×D4—21no—
71C23×D4—22no—
72C2×AGL(1,ℤ/8)—28no—
73C2×#135—22no—
74C22×SD16—24no—
75C22×SD16—28no—
76C22×SD16—28no—
77C22×SD16—24no—
78C22×SD16—416no—
79C23×C8D412yes—
80#533C2312yes—
81#533C2312yes—
82#533C2312yes—
83#532C2312yes—
84#532C2312yes—
85#532C2312yes—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8C2314yes—
92C2×C4⋊Q8C2312yes—
93C2×#127—21no—
94C2×#127—22no—
95C2×#127—22no—
96C2×#127—22no—
97C2×#127—22no—
98C2×#127C2314yes—
99C2×#127C2312yes—
100#529—22no—
101#529—22no—
102#529—22no—
103#529—22no—
104#529—22no—
105#529—22no—
106#525—22no—
107#525—22no—
108#525—22no—
109C4×#41—22no—
110C4×#41—22no—
111C4×#41—22no—
112C22×C4⋊C4—22no—
113C22×C42C2311yescharacteristic
114#501—28no—
115#500C2312yescomplemented
116#500C2318yescomplemented
117#499—22no—
118#498C2311yescharacteristic, complemented
119#498C2311yescharacteristic, complemented
120#498C2318yescomplemented
121#496—28no—
122#495—22no—
123#494C2311yescharacteristic, complemented
124#493C2318yes—
125#493C2311yescharacteristic, complemented
126#492—28no—
127#492—28no—
128#492—24no—
129#492—24no—
130#492C2314yes—
131#492C2314yes—
132#492C2318yes—
133#492C2318yes—
134#492C2312yescomplemented
135#490—28no—
136#489—28no—
137#488—24no—
138#487C2312yescomplemented
139#487C2312yescomplemented
140#487C2318yescomplemented
141#486C2318yes—
142#486C2318yes—
143#486C2312yescomplemented
144#486C2312yescomplemented
145#486C2318yescomplemented
146#484—28no—
147#483—28no—
148#481—28no—
149#480—28no—
150#477—28no—
151#476—28no—
152#475—28no—
153#474—28no—
154#467—24no—
155#466—24no—
156#465—24no—
157#465—24no—
158#460—28no—
159#460—28no—
160#459—28no—
161#459—28no—
162#450—28no—
163#450—28no—
164#447—28no—
165#447—28no—
166C4×Q16—24no—
167C4×SD16—24no—
168C4×SD16—24no—
169C4×D8—24no—
170#432D414yes—
171#432D414yes—
172#431C2314yes—
173#431C2314yes—
174#428—28no—
175#427—22no—
176#423D414yes—
177#418—28no—
178#417—28no—
179#416—24no—
180#407D414yes—
181C4×M16C2314yes—
182C2×C4×C8D414yes—
183C2×C4×C8D412yes—
184C2×C4×C8C2311yescharacteristic
185C2×C4×C8C2311yescharacteristic
186C2×C4×C8C2312yes—
187C2×C4×C8C2312yes—
188#322C2318yescomplemented
189#322C2314yescomplemented
190C82C2314yescomplemented
191C25—41no—
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—22no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—48no—
198C22×Q8—42no—
199C22×Q8—41no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—48no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—41no—
211C22×D4—88no—
212C23×C4—41no—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215AGL(1,ℤ/8)—816no—
216#135—48no—
217C2×Q16—22no—
218C2×SD16—216no—
219C2×SD16—28no—
220C2×SD16—28no—
221C2×SD16—24no—
222C2×SD16—216no—
223C2×SD16—416no—
224C2×SD16—432no—
225C2×SD16—432no—
226C2×SD16—416no—
227C2×D8—22no—
228C2×D8—216no—
229C2×M16C2×D418yes—
230C22×C8—28no—
231C22×C8C2×D412yes—
232C22×C8C2×D414yes—
233C22×C8C2×D414yes—
234C22×C8C2×D412yes—
235C4⋊Q8—24no—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238C4⋊Q8—24no—
239C4⋊Q8—22no—
240C4⋊Q8—41no—
241C4⋊Q8—42no—
242C4⋊Q8—41no—
243C4⋊Q8—42no—
244C4⋊Q8C2414yes—
245C4⋊Q8C2412yes—
246C4⋊Q8C2412yes—
247C4⋊Q8C2412yes—
248#127—24no—
249#127—24no—
250#127—24no—
251#127—24no—
252#127—22no—
253#127—41no—
254#127—41no—
255#127—42no—
256#127—42no—
257#127C2414yes—
258#127C2412yes—
259#126—42no—
260#126—42no—
261#125—22no—
262#125—22no—
263#125—22no—
264#124—22no—
265#124—22no—
266#124—22no—
267#123—42no—
268#123—42no—
269#123—42no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275C22⋊Q8—22no—
276#121—22no—
277#121—22no—
278#121—22no—
279#121—22no—
280#121—22no—
281#121—22no—
282#120—42no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—42no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—42no—
299C4×D4—42no—
300C4×D4—42no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—42no—
305#117—42no—
306#117—42no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C42—42no—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2412yes—
320C2×C42C2411yescharacteristic
321#107—24no—
322#107—216no—
323#106—22no—
324#106—22no—
325#106—216no—
326C4⋊C8C2×D418yes—
327C4⋊C8C2×D418yes—
328#103—24no—
329#103—24no—
330#103—216no—
331#102—24no—
332#102—24no—
333#102—216no—
334#102—216no—
335#102—216no—
336C22⋊C8C2×D418yes—
337C22⋊C8C2×D418yes—
338#97C2×D418yes—
339C4×C8C2×D414yes—
340C4×C8C2×D414yes—
341C4×C8C2×D412yes—
342C4×C8C2×D412yes—
343C4×C8C2411yescharacteristic
344C4×C8C2418yes—
345C4×C8C2412yes—
346C4×C8C2411yescharacteristic
347C24—44no—
348C24—42no—
349C24—84no—
350C24—88no—
351C24#12711yescharacteristic
352#41—48no—
353#41—48no—
354#41—48no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—24no—
359C2×Q8—416no—
360C2×Q8—44no—
361C2×Q8—42no—
362C2×Q8—48no—
363C2×Q8—84no—
364C2×Q8—88no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—28no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—416no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—416no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—84no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—88no—
387C2×D4—88no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—84no—
396C22×C4#12712yes—
397C22×C4#12712yes—
398C22×C4#12712yes—
399C22×C4#12712yes—
400C22×C4#12711yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406Q16—44no—
407SD16—432no—
408SD16—416no—
409SD16—416no—
410SD16—48no—
411SD16—816no—
412D8—44no—
413D8—816no—
414M16—216no—
415C2×C8—24no—
416C2×C8—28no—
417C2×C8—28no—
418C2×C8AGL(1,ℤ/8)18yescomplemented
419C2×C8C22×D414yes—
420C2×C8C22×D4116yes—
421C2×C8C22×D412yes—
422C2×C8C22×D412yes—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—28no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42#12711yescharacteristic
452C42#12711yescharacteristic
453C42#12712yes—
454C42#12712yes—
455C42C22×D412yes—
456C42C22×D411yescharacteristic
457C42C22×D412yes—
458C42C22×D412yes—
459C42C22×D412yes—
460C42C2511yescharacteristic
461C23—22no—
462C23—24no—
463C23—42no—
464C23—44no—
465C23—88no—
466C23—88no—
467C23—88no—
468C23—84no—
469C23—88no—
470C23—1616no—
471C23C2×#12711yescharacteristic
472C23C2×#12712yes—
473Q8—48no—
474Q8—44no—
475Q8—44no—
476Q8—44no—
477Q8—88no—
478Q8—84no—
479D4—44no—
480D4—48no—
481D4—44no—
482D4—416no—
483D4—44no—
484D4—88no—
485D4—88no—
486D4—88no—
487D4—84no—
488D4—84no—
489D4—84no—
490D4—84no—
491D4—88no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—88no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4C2×#12712yes—
510C2×C4C2×#12712yes—
511C2×C4C2×#12712yes—
512C2×C4C2×#12711yescharacteristic
513C2×C4C2×#12712yes—
514C2×C4C2×AGL(1,ℤ/8)14yes—
515C2×C4#57718yes—
516C2×C4C23×D411yescharacteristic
517C2×C4C23×D412yes—
518C8—24no—
519C8—216no—
520C8C2×AGL(1,ℤ/8)14yes—
521C8C2×AGL(1,ℤ/8)14yes—
522C22—28no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—82no—
527C22—84no—
528C22—82no—
529C22—82no—
530C22—88no—
531C22—168no—
532C22—1616no—
533C22#315112yes—
534C22#315414yes—
535C22C22×#12711yescharacteristic
536C4—22no—
537C4—22no—
538C4—22no—
539C4—84no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4#315112yes—
544C4#315414yes—
545C4C22×AGL(1,ℤ/8)12yes—
546C4C2×#57714yes—
547C2—22no—
548C2—24no—
549C2—162no—
550C2—164no—
551C2C2×#315111yescharacteristic
552C2C2×#315412yes—
subgroupG/H
center, socleC22C22×#127
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60860C2semidirect
2. #60798C2semidirect
3. #60673C2semidirect
4. C22×#492C2semidirect
5. #19697C2semidirect
6. #19679C2semidirect
7. #19671C2semidirect
8. #19660C2semidirect
9. #19651C2semidirect
10. #19625C2semidirect
11. #19621C2semidirect
12. #17323C2semidirect
19. #3230C22semidirect
20. #3229C22semidirect
21. #3227C22semidirect
22. #3226C22semidirect
29. #3170C22semidirect
30. #3169C22semidirect
32. #3167C22semidirect
33. #3167C22semidirect
34. #3165C22semidirect
35. #3160C22semidirect
37. #3147C22semidirect
44. #3141C22semidirect
46. #3131C22semidirect
55. #1722C22semidirect
56. #1721C22semidirect
57. #1719C22semidirect
58. #1717C22semidirect
59. #1715C22semidirect
60. #1714C22semidirect
61. #1714C22semidirect
62. #1708C22semidirect
63. #1706C22semidirect
64. #1705C22semidirect
65. #1452C22semidirect
66. #1451C22semidirect
67. #1450C22semidirect
68. C8×M16C22semidirect
115. #500C23semidirect
116. #500C23semidirect
118. #498C23semidirect
119. #498C23semidirect
120. #498C23semidirect
123. #494C23semidirect
125. #493C23semidirect
134. #492C23semidirect
138. #487C23semidirect
139. #487C23semidirect
140. #487C23semidirect
143. #486C23semidirect
144. #486C23semidirect
145. #486C23semidirect
188. #322C23semidirect
189. #322C23semidirect
190. C82C23semidirect
418. C2×C8AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#608602
#607982
#606732
C22×#4922
#196972
#196792
#196712
#196602
#196512
#196252
#196212
#173232