Finite Group Databasebeta

← #6644708

Subgroups — #6644708

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3#33010C212yesmaximal
4#32451C214yesmaximal, complemented
5#32369C214yesmaximal, complemented
6#31659C214yesmaximal, complemented
7#31545C214yesmaximal, complemented
8#29641C212yesmaximal
9#29111C211yesmaximal, characteristic
10#28313C211yesmaximal, characteristic
11#28124C212yesmaximal
12#27969C211yesmaximal, characteristic
13#27919C211yesmaximal, characteristic
14#27653C212yesmaximal
15#21916C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#484C2212yes—
19C2×#483C2212yes—
20C2×#483C2212yes—
21C2×#451C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2646—21no—
26#2627—21no—
27#2590—22no—
28#2588—22no—
29#2568—22no—
30#2568—22no—
31#2568C2214yes—
32#2568C2214yes—
33#2511C2212yes—
34#2507C2212yes—
35#2451C2212yes—
36#2446C2212yes—
37C2×#400—21no—
38C2×#400—21no—
39#2329—21no—
40#2313—21no—
41C4×C22⋊Q8C2212yes—
42C2×#391—21no—
43C2×#391—21no—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2101C2214yes—
50#2097C2214yes—
51#2096C2212yes—
52#2085C2214yes—
53#2074C2214yes—
54#2072C2212yes—
55#2066C2214yes—
56#2058C2214yes—
57#2058C2214yes—
58#2056C2214yes—
59#2024C2212yes—
60#2024C2214yes—
61#2024C2214yes—
62#2016C2212yes—
63#1993C2212yes—
64#1971C2214yes—
65#1960C2212yes—
66#1949C2214yes—
67#1935C2214yes—
68#1927C2214yes—
69#1917C2214yes—
70#1917C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1878C2214yes—
74#1863C2214yes—
75#1863C2214yes—
76#1847C2212yes—
77#1836C2212yes—
78#1824C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1805C2214yes—
83#1797C2212yes—
84#1787C2212yes—
85#1749C2211yescharacteristic
86#1749C2211yescharacteristic
87C23×Q8—21no—
88C23×Q8C2311yescharacteristic
89C22×Q16C2314yes—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—44no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8C2312yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—44no—
99C2×C4×Q8C2314yes—
100C2×C4×Q8C2312yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#484—44no—
113#483—44no—
114#483—44no—
115#451—44no—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×#97C2312yes—
129C2×#97C2312yes—
130#400—22no—
131#400—24no—
132#400—24no—
133#400—24no—
134#400—24no—
135#400—22no—
136#400—22no—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#395—22no—
142#391—24no—
143#391—24no—
144#391—24no—
145#391—24no—
146#391—22no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391C2312yes—
152#391C2314yes—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#385—22no—
176#385—22no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#382—22no—
180#382—21no—
181#382—21no—
182#382C2314yes—
183#382C2312yes—
184#382C2312yes—
185#381—22no—
186#380—21no—
187#380—21no—
188#380—21no—
189#380C2312yes—
190C4×C4⋊C4—22no—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95D411yescharacteristic
206C2×#95D411yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2312yes—
217#336C2312yes—
218C22×Q8—24no—
219C22×Q8—21no—
220C22×Q8—24no—
221C22×Q8—21no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×Q16—28no—
241C2×Q16—44no—
242C2×Q16—44no—
243C22×C8C2414yes—
244C22×C8C2414yes—
245C4⋊Q8—48no—
246C4⋊Q8—42no—
247C4⋊Q8—48no—
248C4⋊Q8—42no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—44no—
252C4×Q8—48no—
253C4×Q8—48no—
254C4×Q8—44no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2412yes—
304C2×C42C2412yes—
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307#107—44no—
308#106—44no—
309#106—44no—
310C4⋊C8—44no—
311#103—28no—
312#103—44no—
313#103—44no—
314#103—44no—
315#103—44no—
316#103—44no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319#97—42no—
320#97—42no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
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—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350#95C2×D412yes—
351C24C22×D411yescharacteristic
352C2×Q8—24no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—416no—
361C2×Q8—42no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—48no—
367C2×Q8—81no—
368C2×Q8—81no—
369C2×Q8—816no—
370C2×Q82−1+414yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4#13714yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C4C2511yescharacteristic
422Q16—88no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—82no—
451C4⋊C4—81no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C42—24no—
461C42—24no—
462C42—22no—
463C42—22no—
464C42—24no—
465C42—42no—
466C42—41no—
467C42—41no—
468C42—48no—
469C42—82no—
470C42—82no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23#53512yes—
476C23C2×#13712yes—
477C23#57512yes—
478C23C23×D411yescharacteristic
479Q8—28no—
480Q8—48no—
481Q8—44no—
482Q8—44no—
483Q8—44no—
484Q8—816no—
485Q8—162no—
486C2×C4—22no—
487C2×C4—28no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—28no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
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—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4#53512yes—
547C2×C4C2×#13714yes—
548C2×C4C2×H(5,𝔽2)12yes—
549C2×C4C2×2−1+412yes—
550C2×C4#58512yes—
551C8—44no—
552C8—44no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22#321514yes—
560C22C2×#53511yescharacteristic
561C22#348511yescharacteristic
562C22#348611yescharacteristic
563C22C22×#13711yescharacteristic
564C22C2×#57511yescharacteristic
565C22#358611yescharacteristic
566C22#358611yescharacteristic
567C4—24no—
568C4—22no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—42no—
576C4—44no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4#353514yes—
585C2—24no—
586C2#3711811yescharacteristic
587C2C2×#321511yescharacteristic
588C2#6080711yescharacteristic
589C2#6171211yescharacteristic
590C2#6172311yescharacteristic
591C2#6182911yescharacteristic
592C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32451C2semidirect
5. #32369C2semidirect
6. #31659C2semidirect
7. #31545C2semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#25682
#330102
#324512
#323692
#316592
#315452
#296412
#291112
#283132
#281242
#279692
#279192
#276532
#219162