Finite Group Databasebeta

← #6487852

Subgroups — #6487852

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic
3#32960C212yesmaximal
4#32957C211yesmaximal, characteristic
5#32954C211yesmaximal, characteristic
6#29972C212yesmaximal
7#29968C211yesmaximal, characteristic
8#29966C211yesmaximal, characteristic
9#26116C214yesmaximal, complemented
10#26114C214yesmaximal, complemented
11#26050C214yesmaximal, complemented
12#26048C214yesmaximal, complemented
13#21983C212yesmaximal
14#21951C211yesmaximal, characteristic
15#20735C212yesmaximal
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#479C2214yes—
20C2×#477C2212yes—
21C2×#475C2212yes—
22C2×#431C2212yes—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2851C2212yes—
26#2627—21no—
27#2627—21no—
28#2606—22no—
29#2605—22no—
30#2570C2212yes—
31#2570C2212yes—
32#2550C2212yes—
33#2550C2212yes—
34#2351—22no—
35#2351—22no—
36#2351C2214yes—
37#2351C2214yes—
38#2329—21no—
39#2329—21no—
40C2×#389—21no—
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44#2283C2212yes—
45C2×#382—21no—
46C2×#382C2211yescharacteristic
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2101C2214yes—
50#2100C2212yes—
51#2099C2212yes—
52#2088C2214yes—
53#2088C2214yes—
54#2087C2214yes—
55#2086C2214yes—
56#2059C2214yes—
57#2058C2212yes—
58#2057C2212yes—
59#1949C2214yes—
60#1949C2214yes—
61#1939C2214yes—
62#1939C2214yes—
63#1865C2214yes—
64#1865C2214yes—
65#1863C2214yes—
66#1863C2214yes—
67#1853C2212yes—
68#1853C2212yes—
69#1852C2212yes—
70#1852C2212yes—
71#1846C2214yes—
72#1835C2214yes—
73#1823C2212yes—
74#1823C2212yes—
75#1805C2214yes—
76#1805C2214yes—
77#1805C2214yes—
78#1803C2212yes—
79#1803C2212yes—
80#1797C2211yescharacteristic
81#1797C2211yescharacteristic
82#1797C2212yes—
83#1787C2212yes—
84#1787C2212yes—
85C4×#103C2214yes—
86C4×#103C2214yes—
87C23×Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×#125C2312yes—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×Q8—24no—
98C2×C4×Q8—44no—
99C2×C4×Q8—44no—
100C2×C4×Q8Q818yescomplemented
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2314yes—
104C2×C4×Q8C2314yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C42—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#479—48no—
122#477—44no—
123#475—44no—
124#431—28no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C4×C8C2314yes—
140#400—22no—
141#400—22no—
142#398—22no—
143#398—22no—
144#398—22no—
145#398—22no—
146#395—22no—
147#395—22no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391—22no—
152#389—24no—
153#389—24no—
154#389—24no—
155#389—24no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#385—22no—
177#385—22no—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#383—22no—
181#382—22no—
182#382—24no—
183#382—24no—
184#382C2312yes—
185#382C2312yes—
186#382C2312yes—
187#382C2314yes—
188#381—22no—
189#380—21no—
190#380—21no—
191#380C2312yes—
192#380C2312yes—
193C4×C4⋊C4—24no—
194C4×C4⋊C4—24no—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
202#376—22no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95D411yescharacteristic
208C2×#95D411yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8C2×Q814yes—
227C22×Q8C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—42no—
242C4⋊Q8—42no—
243#125—44no—
244C22⋊Q8—44no—
245C22⋊Q8—44no—
246C4×Q8—216no—
247C4×Q8—28no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—28no—
251C4×Q8—44no—
252C4×Q8—42no—
253C4×Q8—42no—
254C4×Q8—44no—
255C4×Q8—44no—
256C4×Q8—48no—
257C4×Q8C2×Q8116yes—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×Q814yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×Q814yes—
320C2×C42#4114yes—
321C2×C42C2412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2412yes—
325#107—44no—
326#107—44no—
327#106—44no—
328#106—44no—
329C4⋊C8—28no—
330C4⋊C8—42no—
331C4⋊C8—42no—
332C4⋊C8—44no—
333#103—28no—
334#103—28no—
335#103—28no—
336#103—28no—
337C22⋊C8—24no—
338C22⋊C8—24no—
339C4×C8—28no—
340#95—24no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363C24C22×D411yescharacteristic
364C2×Q8—24no—
365C2×Q8—216no—
366C2×Q8—24no—
367C2×Q8—216no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—81no—
375C2×Q8—81no—
376C2×Q8—81no—
377C2×Q8—81no—
378C2×Q8C22×Q814yes—
379C2×Q8H(5,𝔽2)14yes—
380C2×Q82−1+414yes—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4C22⋊Q814yes—
419C22×C4C22⋊Q814yes—
420C22×C4#13514yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×Q812yes—
431C22×C4C2×#4112yes—
432C22×C4C2511yescharacteristic
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C4⋊C4—216no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4C22×Q814yes—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C42—216no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—41no—
481C42—41no—
482C42—42no—
483C42—44no—
484C42—44no—
485C42—48no—
486C42—82no—
487C42—82no—
488C42C22⋊Q818yes—
489C42C22×Q814yes—
490C42C2×#4114yes—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C2×C22⋊Q812yes—
496C23C2×#13512yes—
497C23#57512yes—
498C23C23×D411yescharacteristic
499Q8—216no—
500Q8—24no—
501Q8—24no—
502Q8—24no—
503Q8—162no—
504Q8—162no—
505Q8#54418yes—
506C2×C4—216no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—24no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4C2×C22⋊Q814yes—
565C2×C4C2×C22⋊Q812yes—
566C2×C4C2×C22⋊Q814yes—
567C2×C4C2×#13512yes—
568C2×C4C2×#13512yes—
569C2×C4C23×Q811yescharacteristic
570C2×C4C22×#4111yescharacteristic
571C2×C4C2×H(5,𝔽2)12yes—
572C2×C4C2×2−1+412yes—
573C8—44no—
574C8—44no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22#307614yes—
582C22C22×C22⋊Q811yescharacteristic
583C22#348411yescharacteristic
584C22#348611yescharacteristic
585C22C22×#13511yescharacteristic
586C22C2×#57511yescharacteristic
587C22#358611yescharacteristic
588C22#358611yescharacteristic
589C4—24no—
590C4—24no—
591C4—22no—
592C4—22no—
593C4—28no—
594C4—22no—
595C4—44no—
596C4—42no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4C2×#54412yes—
607C4#349212yes—
608C2—24no—
609C2#3522411yescharacteristic
610C2C2×#307611yescharacteristic
611C2#6049111yescharacteristic
612C2#6156411yescharacteristic
613C2#6158311yescharacteristic
614C2#6182211yescharacteristic
615C2#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
9. #26116C2semidirect
10. #26114C2semidirect
11. #26050C2semidirect
12. #26048C2semidirect
100. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#23512
#329602
#329572
#329542
#299722
#299682
#299662
#261162
#261142
#260502
#260482
#219832
#219512
#207352