Finite Group Databasebeta

← #6528425

Subgroups — #6528425

Order 512.

1354 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2C2×#2570C211yesmaximal, characteristic
3#29298C212yesmaximal
4#29297C211yesmaximal, characteristic
5#29296C211yesmaximal, characteristic
6#28832C214yesmaximal, complemented
7#28831C214yesmaximal, complemented
8#28794C214yesmaximal, complemented
9#28793C214yesmaximal, complemented
10#27952C212yesmaximal
11#27950C211yesmaximal, characteristic
12#27948C211yesmaximal, characteristic
13#21983C212yesmaximal
14#21361C212yesmaximal
15#20533C211yesmaximal, characteristic
16C2×#431C2212yes—
17#2857C2211yescharacteristic
18#2857C2211yescharacteristic
19#2851C2212yes—
20#2627—21no—
21#2627—21no—
22#2606—22no—
23#2605—22no—
24#2570—22no—
25#2570—22no—
26#2570C2212yes—
27#2570C2212yes—
28#2570C2214yes—
29#2570C2214yes—
30#2550C2212yes—
31#2550C2212yes—
32C2×#400—21no—
33C2×#398—21no—
34#2329—21no—
35#2329—21no—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#387—21no—
39C2×#387—21no—
40C2×#384C2211yescharacteristic
41#2283C2212yes—
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2085C2214yes—
47#2085C2214yes—
48#2085C2214yes—
49#2084C2214yes—
50#2084C2212yes—
51#2083C2214yes—
52#2083C2212yes—
53#2059C2214yes—
54#2058C2212yes—
55#2057C2212yes—
56#1991C2214yes—
57#1990C2212yes—
58#1990C2212yes—
59#1949C2214yes—
60#1949C2214yes—
61#1949C2214yes—
62#1949C2214yes—
63#1942C2214yes—
64#1942C2214yes—
65#1941C2214yes—
66#1941C2214yes—
67#1853C2212yes—
68#1853C2212yes—
69#1853C2212yes—
70#1853C2212yes—
71#1852C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1846C2214yes—
76#1846C2214yes—
77#1845C2212yes—
78#1844C2212yes—
79#1835C2214yes—
80C4×#107C2214yes—
81C4×#106C2214yes—
82C8×C4⋊C4C2214yes—
83C8×C4⋊C4C2214yes—
84C4×C22⋊C8C2211yescharacteristic
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2212yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×#125C2312yes—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4D411yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C42—21no—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106#431Q8116yescomplemented
107C2×#107C2314yes—
108C2×#107C2314yes—
109C2×#107C2314yes—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2314yes—
116C2×C22⋊C8C2312yes—
117C2×C22⋊C8C2312yes—
118C2×C4×C8C2312yes—
119C2×C4×C8C2312yes—
120C2×C4×C8C2314yes—
121C2×C4×C8C2314yes—
122#400—24no—
123#400—24no—
124#400—22no—
125#400—22no—
126#398—24no—
127#398—24no—
128#398—22no—
129#398—22no—
130#398—22no—
131#398—22no—
132#395—22no—
133#395—22no—
134#391—21no—
135#391—21no—
136#391—22no—
137#391—22no—
138#389—22no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—21no—
143#389—22no—
144#389—22no—
145#389C2312yes—
146#389C2312yes—
147#389C2314yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#389C2312yes—
152#387—22no—
153#387—22no—
154#387—24no—
155#387—24no—
156#387—24no—
157#387—24no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#385—22no—
163#385—22no—
164#384—21no—
165#384—21no—
166#384C2311yescharacteristic
167#384C2312yes—
168#384C2311yescharacteristic
169#384C2312yes—
170#383—22no—
171#382—22no—
172#382—22no—
173#382C2312yes—
174#382C2314yes—
175#382C2314yes—
176#382C2312yes—
177#382C2312yes—
178#381—22no—
179#380—21no—
180#380—21no—
181#380C2312yes—
182#380C2312yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
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×#95—21no—
199C2×#95—21no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#336C2312yes—
211#336C2312yes—
212#336C2314yes—
213C22×Q8—21no—
214C22×Q8—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2411yescharacteristic
226C22×C8C2414yes—
227C22×C8C2414yes—
228C4⋊Q8—42no—
229C4⋊Q8—42no—
230#125—44no—
231C22⋊Q8—44no—
232C22⋊Q8—44no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—82no—
265C2×C4⋊C4—82no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—82no—
286C2×C42C2×D412yes—
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D412yes—
290C2×C42C2×Q818yes—
291C2×C42C2412yes—
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2412yes—
295#107—44no—
296#107—44no—
297#107—44no—
298#107—44no—
299#106—44no—
300#106—44no—
301#106—44no—
302#106—44no—
303C4⋊C8C2×Q8116yes—
304C22⋊C8C2×Q818yes—
305C22⋊C8C2×Q818yes—
306C4×C8—44no—
307C4×C8—42no—
308C4×C8—42no—
309C4×C8C2×Q8116yes—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
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#95C2×D412yes—
333#95C2×D412yes—
334#95C2×D412yes—
335#95C2×D412yes—
336C24C22×D411yescharacteristic
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—81no—
340C2×Q8—81no—
341C2×Q8—81no—
342C2×Q8—81no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4C4⋊Q814yes—
380C22×C4C4⋊Q814yes—
381C22×C4#13514yes—
382C22×C4#13514yes—
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×Q814yes—
393C22×C4C2511yescharacteristic
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8C22×Q818yes—
399C2×C8C22×Q818yes—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—84no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C42—24no—
439C42—22no—
440C42—22no—
441C42—41no—
442C42—41no—
443C42—42no—
444C42—82no—
445C42—82no—
446C42—84no—
447C42C4⋊Q818yes—
448C42C22×Q814yes—
449C42C22×Q814yes—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23C2×C4⋊Q812yes—
455C23C2×#13512yes—
456C23C2×#13512yes—
457C23C23×D411yescharacteristic
458Q8—162no—
459Q8—162no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—28no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—28no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
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—82no—
510C2×C4—82no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4C2×C4⋊Q812yes—
516C2×C4C2×C4⋊Q814yes—
517C2×C4C2×C4⋊Q814yes—
518C2×C4C2×#13512yes—
519C2×C4C2×#13512yes—
520C2×C4C2×#13512yes—
521C2×C4C2×#13512yes—
522C2×C4C23×Q812yes—
523C2×C4#58512yes—
524C2×C4#58512yes—
525C8—28no—
526C8—28no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22#316014yes—
534C22C22×C4⋊Q811yescharacteristic
535C22#348511yescharacteristic
536C22#348711yescharacteristic
537C22C22×#13511yescharacteristic
538C22C22×#13511yescharacteristic
539C22#358511yescharacteristic
540C22#358611yescharacteristic
541C4—22no—
542C4—22no—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—24no—
547C4—24no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4#350414yes—
559C2—24no—
560C2#3564011yescharacteristic
561C2C2×#316011yescharacteristic
562C2#6066411yescharacteristic
563C2#6169611yescharacteristic
564C2#6169911yescharacteristic
565C2#6188211yescharacteristic
566C2#6188511yescharacteristic
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. #28832C2semidirect
7. #28831C2semidirect
8. #28794C2semidirect
9. #28793C2semidirect
106. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#25702
#292982
#292972
#292962
#288322
#288312
#287942
#287932
#279522
#279502
#279482
#219832
#213612
#205332