Finite Group Databasebeta

← #6482289

Subgroups — #6482289

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2C2×#2370C211yesmaximal, characteristic
3#33150C212yesmaximal
4#33148C211yesmaximal, characteristic
5#33146C211yesmaximal, characteristic
6#29972C212yesmaximal
7#29969C211yesmaximal, characteristic
8#29967C211yesmaximal, characteristic
9#26517C214yesmaximal, complemented
10#26515C214yesmaximal, complemented
11#26451C214yesmaximal, complemented
12#26449C214yesmaximal, complemented
13#21951C211yesmaximal, characteristic
14#21836C212yesmaximal
15#21361C212yesmaximal
16C22×#125C2211yescharacteristic
17C2×#431C2212yes—
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2851C2212yes—
21#2627—21no—
22#2627—21no—
23#2606—22no—
24#2605—22no—
25#2570C2212yes—
26#2570C2212yes—
27#2550C2212yes—
28#2550C2212yes—
29#2370—22no—
30#2370—22no—
31#2370C2214yes—
32#2370C2214yes—
33#2329—21no—
34#2329—21no—
35C2×#389—21no—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#387—21no—
39C2×#387—21no—
40#2283C2212yes—
41C2×C4×C4⋊C4—21no—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2101C2214yes—
47#2100C2212yes—
48#2099C2212yes—
49#2088C2214yes—
50#2088C2214yes—
51#2087C2214yes—
52#2087C2214yes—
53#2059C2214yes—
54#2058C2212yes—
55#2057C2212yes—
56#1991C2214yes—
57#1991C2214yes—
58#1989C2214yes—
59#1989C2214yes—
60#1989C2214yes—
61#1988C2212yes—
62#1987C2212yes—
63#1949C2214yes—
64#1949C2214yes—
65#1949C2214yes—
66#1948C2212yes—
67#1947C2212yes—
68#1939C2214yes—
69#1939C2214yes—
70#1920C2214yes—
71#1920C2214yes—
72#1918C2214yes—
73#1918C2214yes—
74#1853C2212yes—
75#1853C2212yes—
76#1852C2212yes—
77#1852C2212yes—
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1797C2212yes—
85C4×#107C2214yes—
86C4×#106C2214yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×#125—22no—
90C2×#125—24no—
91C2×#125Q818yescomplemented
92C2×#125C2314yes—
93C2×#125C2312yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C42—21no—
107C22×C42—21no—
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#431—28no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2314yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2312yes—
123C2×C4×C8C2314yes—
124#400—22no—
125#400—22no—
126#398—22no—
127#398—22no—
128#398—22no—
129#398—22no—
130#395—22no—
131#395—22no—
132#391—21no—
133#391—21no—
134#391—22no—
135#391—22no—
136#389—24no—
137#389—24no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#389C2314yes—
152#387—24no—
153#387—24no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—24no—
161#387—24no—
162#385—22no—
163#385—22no—
164#384C2311yescharacteristic
165#384C2311yescharacteristic
166#383—22no—
167#382C2312yes—
168#382C2312yes—
169#382C2312yes—
170#381—22no—
171#380—21no—
172#380—21no—
173#380C2312yes—
174#380C2312yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—24no—
180C4×C4⋊C4—24no—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192#376—22no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215C22×Q8—21no—
216C22×Q8—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2411yescharacteristic
228C22×C8C2414yes—
229C22×C8C2414yes—
230C4⋊Q8—42no—
231C4⋊Q8—42no—
232#125—216no—
233#125—28no—
234#125—44no—
235#125C2×Q8116yes—
236C22⋊Q8—44no—
237C22⋊Q8—44no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
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—44no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×Q814yes—
274C2×C4⋊C4C2×Q814yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42—82no—
295C2×C42—82no—
296C2×C42C2×D412yes—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D412yes—
300C2×C42C2×Q814yes—
301C2×C42#4114yes—
302C2×C42C2412yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2412yes—
306#107—28no—
307#107—44no—
308#106—28no—
309#106—44no—
310C4⋊C8—28no—
311C4⋊C8—44no—
312C4⋊C8—42no—
313C4⋊C8—42no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316C4×C8—28no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—24no—
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#95C2×D412yes—
339#95C2×D412yes—
340#95C2×D412yes—
341#95C2×D412yes—
342C24C22×D411yescharacteristic
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—81no—
346C2×Q8—81no—
347C2×Q8—81no—
348C2×Q8—81no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
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—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
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×C4C22⋊Q814yes—
387C22×C4C22⋊Q814yes—
388C22×C4#13514yes—
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×Q812yes—
399C22×C4C2×#4112yes—
400C22×C4C2511yescharacteristic
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C4⋊C4—24no—
408C4⋊C4—216no—
409C4⋊C4—216no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—48no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—81no—
431C4⋊C4—82no—
432C4⋊C4—81no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4C22×Q814yes—
441C4⋊C4C22×Q814yes—
442C4⋊C42−1+414yes—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C42—24no—
446C42—22no—
447C42—22no—
448C42—41no—
449C42—41no—
450C42—42no—
451C42—44no—
452C42—44no—
453C42—82no—
454C42—82no—
455C42—84no—
456C42—84no—
457C42C22⋊Q818yes—
458C42C22×Q814yes—
459C42C2×#4114yes—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23C2×C22⋊Q812yes—
465C23C2×#13512yes—
466C23#57512yes—
467C23C23×D411yescharacteristic
468Q8—162no—
469Q8—162no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—28no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—48no—
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—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—84no—
528C2×C4C2×C22⋊Q812yes—
529C2×C4C2×C22⋊Q814yes—
530C2×C4C2×C22⋊Q814yes—
531C2×C4C2×#13512yes—
532C2×C4C2×#13512yes—
533C2×C4C23×Q811yescharacteristic
534C2×C4C22×#4111yescharacteristic
535C2×C4C2×2−1+412yes—
536C2×C4#58512yes—
537C8—44no—
538C8—44no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22#307614yes—
546C22C22×C22⋊Q811yescharacteristic
547C22#348511yescharacteristic
548C22#348611yescharacteristic
549C22C22×#13511yescharacteristic
550C22C2×#57511yescharacteristic
551C22#358511yescharacteristic
552C22#358611yescharacteristic
553C4—22no—
554C4—22no—
555C4—24no—
556C4—24no—
557C4—22no—
558C4—42no—
559C4—42no—
560C4—41no—
561C4—41no—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4#346112yes—
571C4#350212yes—
572C2—24no—
573C2#3515411yescharacteristic
574C2C2×#307611yescharacteristic
575C2#6049211yescharacteristic
576C2#6169611yescharacteristic
577C2#6171811yescharacteristic
578C2#6181911yescharacteristic
579C2#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. #26517C2semidirect
10. #26515C2semidirect
11. #26451C2semidirect
12. #26449C2semidirect
91. C2×#125Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#23702
#331502
#331482
#331462
#299722
#299692
#299672
#265172
#265152
#264512
#264492
#219512
#218362
#213612