Finite Group Databasebeta

← #6528427

Subgroups — #6528427

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic
3#29303C212yesmaximal
4#29297C211yesmaximal, characteristic
5#29296C211yesmaximal, characteristic
6#28838C214yesmaximal, complemented
7#28837C214yesmaximal, complemented
8#28804C214yesmaximal, complemented
9#28802C214yesmaximal, complemented
10#27953C212yesmaximal
11#27950C211yesmaximal, characteristic
12#27948C211yesmaximal, characteristic
13#21987C212yesmaximal
14#21363C212yesmaximal
15#20534C211yesmaximal, characteristic
16C22×#125—21no—
17C2×#432C2212yes—
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2851C2212yes—
21#2627—21no—
22#2627—21no—
23#2606—22no—
24#2605—22no—
25#2570C2212yes—
26#2570C2212yes—
27#2552—22no—
28#2552—22no—
29#2552C2214yes—
30#2552C2214yes—
31#2550C2212yes—
32#2550C2212yes—
33C2×#398—21no—
34C2×#398—21no—
35#2329—21no—
36#2329—21no—
37C2×#389—21no—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43#2283C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2085C2214yes—
47#2084C2214yes—
48#2084C2214yes—
49#2084C2212yes—
50#2083C2214yes—
51#2083C2214yes—
52#2083C2212yes—
53#2059C2214yes—
54#2058C2212yes—
55#2057C2212yes—
56#1992C2214yes—
57#1990C2212yes—
58#1990C2212yes—
59#1949C2214yes—
60#1949C2214yes—
61#1948C2214yes—
62#1947C2214yes—
63#1944C2214yes—
64#1943C2214yes—
65#1942C2214yes—
66#1941C2214yes—
67#1853C2212yes—
68#1853C2212yes—
69#1853C2212yes—
70#1853C2212yes—
71#1852C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1847C2214yes—
76#1847C2214yes—
77#1845C2212yes—
78#1844C2212yes—
79#1836C2214yes—
80#1776C2214yes—
81#1776C2214yes—
82#1771C2214yes—
83#1771C2214yes—
84#1749C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×#125—24no—
90C2×#125—24no—
91C2×#125—48no—
92C2×#125C2312yes—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C42—21no—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#432Q8116yescomplemented
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2314yes—
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2312yes—
125C2×#97C2314yes—
126C2×#97C2314yes—
127C2×C4×C8C2312yes—
128C2×C4×C8C2312yes—
129#400—22no—
130#400—22no—
131#398—22no—
132#398—22no—
133#398—22no—
134#398—22no—
135#398—24no—
136#398—24no—
137#398—24no—
138#398—24no—
139#395—22no—
140#395—22no—
141#391—21no—
142#391—21no—
143#391—22no—
144#391—22no—
145#389—24no—
146#389—24no—
147#389—21no—
148#389—21no—
149#389—22no—
150#389—21no—
151#389—21no—
152#389—22no—
153#389—22no—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2314yes—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#385—22no—
168#385—22no—
169#384—22no—
170#384—21no—
171#384—21no—
172#384—22no—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2312yes—
176#384C2314yes—
177#384C2312yes—
178#384C2314yes—
179#383—22no—
180#382C2312yes—
181#382C2312yes—
182#382C2312yes—
183#381—22no—
184#380—21no—
185#380—21no—
186#380C2312yes—
187#380C2312yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4D412yes—
190C4×C4⋊C4D412yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197#376—22no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95D411yescharacteristic
202C2×#95D411yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#336C2312yes—
212#336C2312yes—
213#336C2314yes—
214C22×Q8—21no—
215C22×Q8—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2411yescharacteristic
227C22×C8C2414yes—
228C22×C8C2414yes—
229C4⋊Q8—42no—
230C4⋊Q8—42no—
231#125—44no—
232#125—416no—
233#125—416no—
234C22⋊Q8—44no—
235C22⋊Q8—44no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D412yes—
301C2×C42C2×Q818yes—
302C2×C42C2412yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2412yes—
306#107—44no—
307#107—44no—
308#107—44no—
309#107—44no—
310#106—44no—
311#106—44no—
312#106—44no—
313#106—44no—
314C4⋊C8C2×Q8116yes—
315C22⋊C8C2×Q818yes—
316C22⋊C8C2×Q818yes—
317#97—44no—
318#97C2×Q8116yes—
319C4×C8—42no—
320C4×C8—42no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
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#95C2×D412yes—
340#95C2×D412yes—
341#95C2×D412yes—
342#95C2×D412yes—
343C24C22×D411yescharacteristic
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—81no—
347C2×Q8—81no—
348C2×Q8—81no—
349C2×Q8—81no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4C4⋊Q814yes—
387C22×C4C4⋊Q814yes—
388C22×C4AGL(1,ℤ/8)14yes—
389C22×C4#13714yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×Q814yes—
400C22×C4C2511yescharacteristic
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8C22×Q818yes—
406C2×C8C22×Q818yes—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—84no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C42—22no—
450C42—22no—
451C42—24no—
452C42—41no—
453C42—41no—
454C42—42no—
455C42—48no—
456C42—82no—
457C42—82no—
458C42C4⋊Q818yes—
459C42C22×Q814yes—
460C42C22×Q814yes—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23C2×C4⋊Q812yes—
466C23C2×AGL(1,ℤ/8)12yes—
467C23C2×#13712yes—
468C23C23×D411yescharacteristic
469Q8—162no—
470Q8—162no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—28no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4C2×C4⋊Q812yes—
527C2×C4C2×C4⋊Q814yes—
528C2×C4C2×C4⋊Q814yes—
529C2×C4C2×AGL(1,ℤ/8)12yes—
530C2×C4C2×AGL(1,ℤ/8)12yes—
531C2×C4C2×#13712yes—
532C2×C4C2×#13712yes—
533C2×C4C23×Q812yes—
534C2×C4#58512yes—
535C2×C4#58512yes—
536C8—28no—
537C8—28no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22#316514yes—
545C22C22×C4⋊Q811yescharacteristic
546C22#348511yescharacteristic
547C22#348711yescharacteristic
548C22C22×AGL(1,ℤ/8)11yescharacteristic
549C22C22×#13711yescharacteristic
550C22#358511yescharacteristic
551C22#358611yescharacteristic
552C4—22no—
553C4—22no—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—24no—
558C4—24no—
559C4—42no—
560C4—42no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4#350414yes—
570C2—24no—
571C2#3564011yescharacteristic
572C2C2×#316511yescharacteristic
573C2#6066411yescharacteristic
574C2#6170611yescharacteristic
575C2#6170911yescharacteristic
576C2#6189011yescharacteristic
577C2#6189311yescharacteristic
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
6. #28838C2semidirect
7. #28837C2semidirect
8. #28804C2semidirect
9. #28802C2semidirect
113. #432Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#25522
#293032
#292972
#292962
#288382
#288372
#288042
#288022
#279532
#279502
#279482
#219872
#213632
#205342