Finite Group Databasebeta

← #6690752

Subgroups — #6690752

Order 512.

2051 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39378C211yesmaximal, characteristic, complemented
2#36179C211yesmaximal, characteristic, complemented
3#33066C211yesmaximal, characteristic, complemented
4#32039C214yesmaximal, complemented
5#30723C214yesmaximal, complemented
6#30220C211yesmaximal, characteristic, complemented
7#29798C211yesmaximal, characteristic, complemented
8#26236C212yesmaximal, complemented
9#25393C212yesmaximal, complemented
10#24454C214yesmaximal, complemented
11#24237C212yesmaximal, complemented
12#24054C214yesmaximal, complemented
13#22431C212yesmaximal, complemented
14#22264C211yesmaximal, characteristic, complemented
15#21113C211yesmaximal, characteristic, complemented
16C2×#547C2212yes—
17C2×#547C2214yes—
18C2×#540—21no—
19C2×#484C2214yes—
20C2×#468C2214yes—
21C2×#461C2212yes—
22C2×#449C2214yes—
23C2×C4×D8C2211yescharacteristic
24C2×#436C2212yes—
25C2×#417C2214yes—
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2856—21no—
29#2701—24no—
30#2629—21no—
31#2542C2214yes—
32#2542C2212yes—
33#2537C2212yes—
34#2501—22no—
35#2501—24no—
36#2486C2214yes—
37#2420—22no—
38#2416C2214yes—
39D4×C4⋊C4C2212yes—
40C4×#127C2211yescharacteristic
41#2293C2211yescharacteristic
42C2×#384—21no—
43C42×D4C2211yescharacteristic
44#2043C2214yes—
45#2043C2214yes—
46#2043C2214yes—
47#2036C2214yes—
48#2036C2214yes—
49#2036C2214yes—
50#2036C2214yes—
51#2013C2214yes—
52#2000C2214yes—
53#2000C2214yes—
54#1981C2212yes—
55#1956C2212yes—
56#1956C2212yes—
57#1954C2212yes—
58#1947C2212yes—
59#1925C2212yes—
60#1918C2214yes—
61#1918C2212yes—
62#1894C2214yes—
63#1894C2214yes—
64#1879C2212yes—
65#1872C2214yes—
66#1872C2214yes—
67#1870C2214yes—
68#1870C2214yes—
69#1862C2214yes—
70#1862C2212yes—
71#1831C2211yescharacteristic
72#1802C2212yes—
73#1788C2214yes—
74#1788C2214yes—
75C4×#107C2211yescharacteristic
76C4×#102C2211yescharacteristic
77C4×#102C2211yescharacteristic
78#1744C2211yescharacteristic
79C22×#41C2314yes—
80C22×SD16C2314yes—
81C22×D8C2311yescharacteristic
82C22×M16C2314yes—
83#547—28no—
84#547—44no—
85#547—48no—
86#547—48no—
87#540—48no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2314yes—
90C2×#127D411yescharacteristic
91C2×#125—21no—
92C2×#123—24no—
93C2×#123—24no—
94C2×#123—24no—
95C2×#123—24no—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2314yes—
99C2×#121—22no—
100C2×#121—24no—
101C2×#121—24no—
102C2×#121—22no—
103C2×#121C2314yes—
104C2×C4×D4—22no—
105C2×C4×D4C2312yes—
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2311yescharacteristic
108C2×C4×D4C2312yes—
109C2×C4×D4C2314yes—
110C2×#117—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—24no—
113C22×C4⋊C4C2314yes—
114C22×C42—21no—
115#484—48no—
116#468—48no—
117#461—44no—
118#449—48no—
119C4×D8—24no—
120#436—44no—
121C2×#107C2312yes—
122C2×#107C2314yes—
123C2×#107C2311yescharacteristic
124C2×C4⋊C8C2312yes—
125#417—48no—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2311yescharacteristic
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2311yescharacteristic
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143#399—24no—
144#399—24no—
145#399—24no—
146#398—24no—
147#397—24no—
148#396—24no—
149#394—24no—
150#394—24no—
151#390—22no—
152#390—24no—
153#390D412yes—
154#390D414yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389C2314yes—
159#389C2312yes—
160#388—22no—
161#386—24no—
162#385—24no—
163#384—22no—
164#384—24no—
165#384—21no—
166#384—21no—
167#384—22no—
168#384D411yescharacteristic
169#384C2312yes—
170#382—21no—
171#381—24no—
172#380—22no—
173#380C2314yes—
174#380C2314yes—
175C4×C4⋊C4C2311yescharacteristic
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2311yescharacteristic
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—22no—
181C43C2311yescharacteristic
182#340C2314yes—
183#340C2314yes—
184#340C2312yes—
185#340C2312yes—
186#340C2312yes—
187#336C2314yes—
188#336C2312yes—
189#336C2312yes—
190C2×#41—28no—
191C2×#41—28no—
192C2×#41—28no—
193C2×#41—44no—
194C2×#41—44no—
195C2×#41—44no—
196C22×Q8C2414yes—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4C2×D412yes—
200C22×D4C2×D414yes—
201C22×D4C2411yescharacteristic
202C22×D4C2411yescharacteristic
203C23×C4—22no—
204C23×C4—21no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—21no—
208C23×C4C2414yes—
209C2×SD16—44no—
210C2×SD16—44no—
211C2×SD16—44no—
212C2×D8—22no—
213C2×D8—24no—
214C2×M16—44no—
215C2×M16—44no—
216C2×M16—44no—
217C22×C8C2412yes—
218C22×C8C2414yes—
219C22×C8C2411yescharacteristic
220C22×C8C2411yescharacteristic
221C4⋊Q8—24no—
222C4⋊Q8—24no—
223C4⋊Q8—44no—
224C4⋊Q8—44no—
225#127—24no—
226#125—44no—
227#123—416no—
228#123—416no—
229#123—416no—
230#123—416no—
231C22⋊Q8—28no—
232C22⋊Q8—28no—
233C22⋊Q8—28no—
234C22⋊Q8—44no—
235C22⋊Q8—44no—
236C22⋊Q8—44no—
237#121—48no—
238#121—416no—
239#121—416no—
240#121—48no—
241#121—44no—
242#121—44no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—24no—
249C4×D4—22no—
250C4×D4—44no—
251C4×D4—44no—
252C4×D4—48no—
253C4×D4#4114yes—
254#117—44no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2411yescharacteristic
276C2×C4⋊C4C2411yescharacteristic
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C22⋊C4C2×D414yes—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—21no—
295C2×C42—44no—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2412yes—
300C2×C42C2411yescharacteristic
301#107—22no—
302#107—28no—
303#107—24no—
304C4⋊C8—24no—
305#103—44no—
306#103—44no—
307#103—44no—
308#103—44no—
309#103—44no—
310#103—44no—
311#103—44no—
312#102—22no—
313#102—24no—
314#102—28no—
315#102—22no—
316#102—24no—
317C22⋊C8—44no—
318C22⋊C8—44no—
319C22⋊C8—44no—
320C4×C8—22no—
321C4×C8—22no—
322#95—22no—
323#95—21no—
324#95—22no—
325#95—21no—
326#95—24no—
327#95—24no—
328#95—24no—
329C24—21no—
330C24—21no—
331C24C22×D414yes—
332#41—416no—
333#41—88no—
334#41—88no—
335#41—88no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—28no—
345C2×D4—24no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—21no—
349C2×D4—22no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—84no—
359C2×D4—84no—
360C2×D4C2×#4112yes—
361C2×D4H(5,𝔽2)14yes—
362C22×C4—28no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—44no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—48no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4C22×D412yes—
390C22×C4C22×D414yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C2511yescharacteristic
395SD16—88no—
396D8—44no—
397M16—88no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—22no—
407C2×C8—22no—
408C4⋊C4—24no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—82no—
429C4⋊C4—84no—
430C4⋊C4C2×#4112yes—
431C4⋊C4H(5,𝔽2)14yes—
432C22⋊C4—28no—
433C22⋊C4—24no—
434C22⋊C4—28no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—42no—
443C22⋊C4—44no—
444C22⋊C4—82no—
445C22⋊C4—84no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—42no—
450C42—44no—
451C42—42no—
452C42—42no—
453C42AGL(1,ℤ/8)14yes—
454C42C2×#4112yes—
455C42H(5,𝔽2)12yes—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—41no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23C23×D411yescharacteristic
474Q8—44no—
475Q8—44no—
476Q8—44no—
477Q8—44no—
478D4—22no—
479D4—24no—
480D4—48no—
481D4—41no—
482D4—41no—
483D4—42no—
484D4—88no—
485D4—816no—
486D4—168no—
487D4#54014yescomplemented
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—416no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—44no—
520C2×C4—41no—
521C2×C4—816no—
522C2×C4—84no—
523C2×C4—168no—
524C2×C4#54612yes—
525C2×C4#54612yes—
526C2×C4C2×AGL(1,ℤ/8)14yes—
527C2×C4C2×AGL(1,ℤ/8)12yes—
528C2×C4#57812yes—
529C2×C4C22×#4111yescharacteristic
530C2×C4C2×H(5,𝔽2)12yes—
531C2×C4C2×H(5,𝔽2)12yes—
532C2×C4C2×H(5,𝔽2)11yescharacteristic
533C8—41no—
534C8—41no—
535C8—42no—
536C8—44no—
537C22—24no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—81no—
550C22—81no—
551C22—82no—
552C22—81no—
553C22—82no—
554C22C2×#54611yescharacteristic
555C22#348412yes—
556C22C22×AGL(1,ℤ/8)11yescharacteristic
557C22C2×#57811yescharacteristic
558C22#358612yes—
559C4—22no—
560C4—24no—
561C4—44no—
562C4—44no—
563C4—42no—
564C4—42no—
565C4—84no—
566C4—82no—
567C4—84no—
568C4—81no—
569C4#333014yes—
570C4C2×#54011yescharacteristic
571C4#350811yescharacteristic
572C4#354312yes—
573C2—44no—
574C2—84no—
575C2—161no—
576C2#3769411yescharacteristic
577C2C2×#333011yescharacteristic
578C2#6122411yescharacteristic
579C2#6158512yes—
580C2#6169312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #39378C2semidirect
2. #36179C2semidirect
3. #33066C2semidirect
4. #32039C2semidirect
5. #30723C2semidirect
6. #30220C2semidirect
7. #29798C2semidirect
8. #26236C2semidirect
9. #25393C2semidirect
10. #24454C2semidirect
11. #24237C2semidirect
12. #24054C2semidirect
13. #22431C2semidirect
14. #22264C2semidirect
15. #21113C2semidirect
487. D4#540semidirect

Maximal subgroups

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

subgroupindex
#393782
#361792
#330662
#320392
#307232
#302202
#297982
#262362
#253932
#244542
#242372
#240542
#224312
#222642
#211132