Finite Group Databasebeta

← #6493470

Subgroups — #6493470

Order 512.

1621 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2#34123C211yesmaximal, characteristic
3#33162C211yesmaximal, characteristic, complemented
4#33066C211yesmaximal, characteristic, complemented
5#32907C212yesmaximal
6#29996C211yesmaximal, characteristic, complemented
7#29926C212yesmaximal, complemented
8#29799C211yesmaximal, characteristic, complemented
9#27405C214yesmaximal, complemented
10#27205C214yesmaximal, complemented
11#26170C214yesmaximal, complemented
12#25154C214yesmaximal
13#22148C212yesmaximal, complemented
14#21922C211yesmaximal, characteristic, complemented
15C4×#461C212yesmaximal, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#462C2212yes—
18C2×#461C2212yes—
19C2×#461C2214yes—
20C2×#452C2214yes—
21C2×C4×SD16C2214yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2647—21no—
26#2629—21no—
27#2507—22no—
28#2497—22no—
29#2446C2214yes—
30#2429C2214yes—
31#2356C2212yes—
32#2353—22no—
33#2349C2212yes—
34#2346C2212yes—
35#2343—22no—
36#2339C2212yes—
37#2330—21no—
38#2314—21no—
39C4×C4⋊Q8C2212yes—
40#2295C2211yescharacteristic
41#2295C2211yescharacteristic
42#2290—21no—
43#2290—21no—
44#2283—21no—
45#2283C2211yescharacteristic
46C42×D4C2211yescharacteristic
47C42×D4C2211yescharacteristic
48C4×#117C2211yescharacteristic
49C2×#376—21no—
50#2097C2214yes—
51#2088C2214yes—
52#2074C2214yes—
53#2066C2214yes—
54#2043C2214yes—
55#2036C2214yes—
56#2035C2214yes—
57#2016C2214yes—
58#1981C2214yes—
59#1981C2212yes—
60#1981C2212yes—
61#1969C2214yes—
62#1948C2212yes—
63#1947C2212yes—
64#1933C2214yes—
65#1927C2214yes—
66#1923C2214yes—
67#1918C2214yes—
68#1916C2214yes—
69#1878C2214yes—
70#1878C2214yes—
71#1877C2214yes—
72#1877C2214yes—
73#1873C2214yes—
74#1863C2214yes—
75#1862C2214yes—
76#1842C2211yescharacteristic
77#1831C2211yescharacteristic
78#1803C2212yes—
79#1802C2212yes—
80#1797C2212yes—
81C4×C4⋊C8C2212yes—
82C4×#103C2214yes—
83C22×#41C2311yescharacteristic
84C22×Q16C2314yes—
85C22×SD16C2314yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2314yes—
88C2×#125—21no—
89C2×C22⋊Q8C2314yes—
90C4×#41—24no—
91C4×#41—24no—
92C2×C4×Q8C2311yescharacteristic
93C2×C4×Q8C2311yescharacteristic
94C2×C4×Q8C2314yes—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4C2311yescharacteristic
98C2×C4×D4C2312yes—
99C2×C4×D4C2312yes—
100C2×C4×D4C2311yescharacteristic
101C2×#117—21no—
102C2×#117D411yescharacteristic
103C2×#117D411yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C42—21no—
106C22×C42—21no—
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109#462—28no—
110#461—28no—
111#461—48no—
112#452—48no—
113C4×SD16—48no—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#102C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C4×C8C2314yes—
129#400—24no—
130#398—24no—
131#398—24no—
132#396—24no—
133#391C2314yes—
134#391C2312yes—
135#391C2314yes—
136#391C2312yes—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—24no—
143#389C2314yes—
144#389C2312yes—
145#389C2312yes—
146#388—24no—
147#388—22no—
148#388—22no—
149#387—24no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—24no—
155#386—22no—
156#386—22no—
157#385—22no—
158#385—22no—
159#384—21no—
160#384—21no—
161#384—22no—
162#384C2312yes—
163#383—21no—
164#383—22no—
165#382—21no—
166#382—22no—
167#382C2312yes—
168#381—21no—
169#381—22no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380C2312yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4D412yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2311yescharacteristic
182C4×C4⋊C4C2311yescharacteristic
183C4×C4⋊C4C2312yes—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4D412yes—
190C4×C22⋊C4C2312yes—
191#376—21no—
192#376—21no—
193#376—21no—
194#376—22no—
195#376—22no—
196#376D411yescharacteristic
197#376D411yescharacteristic
198C2×#95—22no—
199C2×#95—22no—
200C43C2311yescharacteristic
201C43C2311yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208#336C2314yes—
209C2×#41—22no—
210C2×#41—22no—
211C2×#41—24no—
212C2×#41—42no—
213C22×Q8C2414yes—
214C22×Q8C2411yescharacteristic
215C22×D4—21no—
216C22×D4C2411yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2411yescharacteristic
225C2×Q16—28no—
226C2×Q16—44no—
227C2×Q16—44no—
228C2×SD16—28no—
229C2×SD16—28no—
230C2×SD16—28no—
231C22×C8C2414yes—
232C22×C8C2414yes—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8—44no—
236C4⋊Q8—44no—
237#125—44no—
238C22⋊Q8—44no—
239C22⋊Q8—44no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—44no—
244C4×Q8—44no—
245C4×Q8#4114yes—
246C4×D4—28no—
247C4×D4—28no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—22no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4#4114yes—
256#117—28no—
257#117—22no—
258#117—22no—
259#117—24no—
260#117—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2411yescharacteristic
280C2×C4⋊C4C2411yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C42—28no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2412yes—
312C2×C42C2411yescharacteristic
313C2×C42C2412yes—
314C2×C42C2411yescharacteristic
315#107—44no—
316#106—44no—
317C4⋊C8—24no—
318C4⋊C8—24no—
319C4⋊C8—44no—
320#103—28no—
321#103—28no—
322#103—44no—
323#103—44no—
324#103—44no—
325#103—44no—
326#103—44no—
327#102—28no—
328C22⋊C8—44no—
329C4×C8—44no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—21no—
337#95—21no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95C2×D412yes—
349#95C2×D412yes—
350#95C2×D412yes—
351#95C2×D412yes—
352C24—21no—
353C24C22×D411yescharacteristic
354#41—48no—
355#41—44no—
356#41—44no—
357C2×Q8—21no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8C2×#4112yes—
367C2×D4—24no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—44no—
371C2×D4C2×#4112yes—
372C2×D4H(5,𝔽2)14yes—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C2511yescharacteristic
412Q16—88no—
413SD16—416no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4C2×#4112yes—
444C4⋊C4C2×#4112yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—82no—
453C22⋊C4—84no—
454C42—24no—
455C42—24no—
456C42—22no—
457C42—24no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—21no—
462C42—21no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—42no—
468C42—42no—
469C42—44no—
470C42—42no—
471C42—44no—
472C42—42no—
473C42—84no—
474C42#13514yes—
475C42#13714yes—
476C42C2×#4112yes—
477C42C2×#4112yes—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—21no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23C23×D411yescharacteristic
489Q8—24no—
490Q8—41no—
491Q8—41no—
492Q8—44no—
493Q8—44no—
494Q8—44no—
495Q8—44no—
496D4—22no—
497D4—24no—
498D4—48no—
499D4#54014yes—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—48no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—82no—
544C2×C4—88no—
545C2×C4—88no—
546C2×C4—84no—
547C2×C4#52512yes—
548C2×C4#52512yes—
549C2×C4C2×#13514yes—
550C2×C4C2×#13512yes—
551C2×C4C2×#13714yes—
552C2×C4C2×#13712yes—
553C2×C4C22×#4111yescharacteristic
554C2×C4C22×#4111yescharacteristic
555C2×C4C2×H(5,𝔽2)12yes—
556C2×C4#58512yes—
557C8—44no—
558C8—44no—
559C22—24no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22C2×#52511yescharacteristic
570C22#348812yes—
571C22C22×#13511yescharacteristic
572C22C22×#13711yescharacteristic
573C22#358612yes—
574C4—24no—
575C4—22no—
576C4—24no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—82no—
586C4—84no—
587C4—84no—
588C4—82no—
589C4#310714yes—
590C4C2×#54011yescharacteristic
591C4#345511yescharacteristic
592C4#349412yes—
593C2—44no—
594C2—81no—
595C2#3526111yescharacteristic
596C2C2×#310711yescharacteristic
597C2#6056911yescharacteristic
598C2#6195012yes—
599C2#6196012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35284C2semidirect
3. #33162C2semidirect
4. #33066C2semidirect
6. #29996C2semidirect
7. #29926C2semidirect
8. #29799C2semidirect
9. #27405C2semidirect
10. #27205C2semidirect
11. #26170C2semidirect
13. #22148C2semidirect
14. #21922C2semidirect
15. C4×#461C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#341232
#331622
#330662
#329072
#299962
#299262
#297992
#274052
#272052
#261702
#251542
#221482
#219222
C4×#4612