Finite Group Databasebeta

← #6528428

Subgroups — #6528428

Order 512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic
3#29302C211yesmaximal, characteristic
4#29301C211yesmaximal, characteristic
5#29298C212yesmaximal
6#28840C214yesmaximal, complemented
7#28839C214yesmaximal, complemented
8#28808C214yesmaximal, complemented
9#28806C214yesmaximal, complemented
10#27956C211yesmaximal, characteristic
11#27956C211yesmaximal, characteristic
12#27952C212yesmaximal
13#21987C212yesmaximal
14#21363C212yesmaximal
15#20534C211yesmaximal, characteristic
16C2×#432C2212yes—
17#2857C2211yescharacteristic
18#2857C2211yescharacteristic
19#2851C2212yes—
20#2627—21no—
21#2627—21no—
22#2606—22no—
23#2605—22no—
24#2571—22no—
25#2571—22no—
26#2571C2214yes—
27#2571C2214yes—
28#2570C2212yes—
29#2570C2212yes—
30#2550C2212yes—
31#2550C2212yes—
32C2×#400—21no—
33C2×#400—21no—
34#2329—21no—
35#2329—21no—
36C2×#389—21no—
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#384C2211yescharacteristic
40#2283C2212yes—
41C2×#382—21no—
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2085C2214yes—
47#2085C2214yes—
48#2085C2214yes—
49#2085C2214yes—
50#2085C2214yes—
51#2084C2212yes—
52#2083C2212yes—
53#2059C2214yes—
54#2058C2212yes—
55#2057C2212yes—
56#1993C2212yes—
57#1993C2212yes—
58#1991C2214yes—
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#1847C2212yes—
76#1847C2212yes—
77#1847C2214yes—
78#1846C2214yes—
79#1836C2214yes—
80#1776C2214yes—
81#1776C2214yes—
82#1771C2214yes—
83#1771C2214yes—
84#1749C2211yescharacteristic
85#1749C2211yescharacteristic
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#432Q8116yescomplemented
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×#97C2312yes—
119C2×#97C2312yes—
120C2×#97C2314yes—
121C2×C4×C8C2314yes—
122#400—22no—
123#400—22no—
124#400—24no—
125#400—24no—
126#400—24no—
127#400—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—24no—
139#389—24no—
140#389—22no—
141#389—21no—
142#389—21no—
143#389—21no—
144#389—21no—
145#389—22no—
146#389—22no—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#389C2314yes—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#385—22no—
161#385—22no—
162#384—22no—
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#384C2314yes—
166#383—22no—
167#382—24no—
168#382—24no—
169#382—22no—
170#382—21no—
171#382—21no—
172#382C2312yes—
173#382C2312yes—
174#382C2314yes—
175#382C2312yes—
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⋊C4C2312yes—
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×#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—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42C2×D412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D412yes—
288C2×C42C2×Q818yes—
289C2×C42C2412yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2412yes—
293#107—44no—
294#107—44no—
295#107—44no—
296#107—44no—
297#106—44no—
298#106—44no—
299#106—44no—
300#106—44no—
301C4⋊C8C2×Q8116yes—
302C22⋊C8C2×Q818yes—
303C22⋊C8C2×Q818yes—
304#97—42no—
305#97—42no—
306#97C2×Q8116yes—
307C4×C8—44no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—22no—
317#95—22no—
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—22no—
358C22×C4—22no—
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—41no—
368C22×C4—41no—
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×C4—42no—
380C22×C4—42no—
381C22×C4C4⋊Q814yes—
382C22×C4C4⋊Q814yes—
383C22×C4C22×D412yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
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—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—81no—
423C4⋊C4—82no—
424C4⋊C4—81no—
425C4⋊C4—82no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C42—24no—
437C42—22no—
438C42—22no—
439C42—42no—
440C42—41no—
441C42—41no—
442C42—48no—
443C42—82no—
444C42—82no—
445C42C4⋊Q818yes—
446C42C22×Q814yes—
447C42C22×Q814yes—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23C2×C4⋊Q812yes—
453C23#57512yes—
454C23#57512yes—
455C23C23×D411yescharacteristic
456Q8—162no—
457Q8—162no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—28no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—28no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
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—82no—
512C2×C4—82no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4C2×C4⋊Q812yes—
518C2×C4C2×C4⋊Q814yes—
519C2×C4C2×C4⋊Q814yes—
520C2×C4C23×Q812yes—
521C2×C4#58512yes—
522C2×C4#58512yes—
523C8—28no—
524C8—28no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22#316614yes—
532C22C22×C4⋊Q811yescharacteristic
533C22#348511yescharacteristic
534C22#348711yescharacteristic
535C22C2×#57511yescharacteristic
536C22C2×#57511yescharacteristic
537C22#358511yescharacteristic
538C22#358611yescharacteristic
539C4—22no—
540C4—24no—
541C4—24no—
542C4—41no—
543C4—42no—
544C4—41no—
545C4—41no—
546C4—42no—
547C4—41no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4#350414yes—
557C2—24no—
558C2#3564011yescharacteristic
559C2C2×#316611yescharacteristic
560C2#6066411yescharacteristic
561C2#6171011yescharacteristic
562C2#6171211yescharacteristic
563C2#6189411yescharacteristic
564C2#6189511yescharacteristic
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. #28840C2semidirect
7. #28839C2semidirect
8. #28808C2semidirect
9. #28806C2semidirect
106. #432Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#25712
#293022
#293012
#292982
#288402
#288392
#288082
#288062
#279562
#279522
#219872
#213632
#205342