Finite Group Databasebeta

← #6806332

Subgroups — #6806332

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#36555C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#31473C214yesmaximal, complemented
5#31471C214yesmaximal
6#30234C212yesmaximal, complemented
7#29212C211yesmaximal, characteristic, complemented
8#29181C212yesmaximal, complemented
9#28727C212yesmaximal
10#28726C211yesmaximal, characteristic
11#26109C212yesmaximal, complemented
12#26107C212yesmaximal
13#24997C212yesmaximal, complemented
14#24993C212yesmaximal
15#22224C212yesmaximal
16C4×#489C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic, complemented
18C2×#489C2212yes—
19C2×#489C2211yescharacteristic
20#2868C2211yescharacteristic
21#2864C2212yes—
22#2862C2212yes—
23#2859C2211yescharacteristic
24#2587—22no—
25#2582—21no—
26#2582C2211yescharacteristic
27#2571C2212yes—
28#2570C2214yes—
29#2568—21no—
30#2538C2211yescharacteristic
31#2522—22no—
32#2511—22no—
33#2497—21no—
34#2454C2212yes—
35#2453C2212yes—
36#2446—22no—
37#2423—21no—
38#2365—21no—
39#2360—21no—
40#2355—21no—
41#2355C2212yes—
42Q8×C4⋊C4C2211yescharacteristic
43D4×C4⋊C4C2211yescharacteristic
44#2345—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×#124C2211yescharacteristic
48C4×#124C2212yes—
49C42×Q8C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2101C2214yes—
53#2101C2214yes—
54#2088C2214yes—
55#2086C2214yes—
56#2074C2214yes—
57#2073C2214yes—
58#2072C2212yes—
59#2071C2212yes—
60#2059C2214yes—
61#1993C2212yes—
62#1993C2212yes—
63#1993C2211yescharacteristic
64#1993C2212yes—
65#1993C2212yes—
66#1992C2214yes—
67#1984C2212yes—
68#1959C2211yescharacteristic
69#1948C2212yes—
70#1947C2212yes—
71#1939C2212yes—
72#1936C2212yes—
73#1935C2212yes—
74#1927C2214yes—
75#1925C2214yes—
76#1921C2214yes—
77#1919C2214yes—
78#1917C2214yes—
79#1864C2214yes—
80#1863C2214yes—
81#1847C2212yes—
82#1836C2211yescharacteristic
83#1834C2212yes—
84#1834C2212yes—
85#1805C2214yes—
86#1804C2214yes—
87#1803C2212yes—
88#1802C2212yes—
89#1767C2211yescharacteristic
90#1767C2212yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#125C2312yes—
100C2×#124C2311yescharacteristic
101C2×C22⋊Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—22no—
104C2×C4×Q8D411yescharacteristic
105C2×C4×Q8D411yescharacteristic
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×D4C2311yescharacteristic
110C2×C4×D4C2312yes—
111C22×C4⋊C4—21no—
112#489—28no—
113#489—24no—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2314yes—
118C2×#103C2314yes—
119C2×#103C2312yes—
120C2×#103C2312yes—
121C2×#102C2314yes—
122C2×#102C2312yes—
123C2×#102C2312yes—
124C2×#97C2311yescharacteristic
125C2×#97C2312yes—
126C2×#97C2311yescharacteristic
127C2×#97C2312yes—
128C2×C4×C8C2312yes—
129#400—24no—
130#400—24no—
131#400—22no—
132#399—22no—
133#398—24no—
134#398—22no—
135#398—22no—
136#396—22no—
137#396—22no—
138#391—22no—
139#391—21no—
140#391—22no—
141#391—21no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391C2312yes—
147#391C2311yescharacteristic
148#390C2311yescharacteristic
149#390C2312yes—
150#389—22no—
151#389—21no—
152#389—22no—
153#389—22no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—22no—
160#389C2312yes—
161#389C2314yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#388—21no—
167#388—22no—
168#388—22no—
169#388—21no—
170#387—24no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—24no—
176#387—24no—
177#387—22no—
178#385—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#382—22no—
189#382C2314yes—
190#382C2314yes—
191#382C2312yes—
192#382C2311yescharacteristic
193#380—21no—
194#380—21no—
195#380—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4—22no—
209C4×C22⋊C4—21no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—22no—
212C43C2311yescharacteristic
213C43C2312yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#336C2312yes—
227#336C2314yes—
228#336C2314yes—
229#336C2314yes—
230C22×Q8—22no—
231C22×Q8—21no—
232C22×Q8C2×D411yescharacteristic
233C22×Q8C2411yescharacteristic
234C22×D4C2411yescharacteristic
235C23×C4—21no—
236C23×C4—22no—
237C22×C8C2412yes—
238C22×C8C2414yes—
239C22×C8C2412yes—
240C4⋊Q8—22no—
241C4⋊Q8—24no—
242C4⋊Q8—22no—
243C4⋊Q8—24no—
244C4⋊Q8—44no—
245C4⋊Q8—82no—
246#125—44no—
247#124—22no—
248#124#4114yes—
249C22⋊Q8—44no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—42no—
253C4×Q8—42no—
254C4×Q8—44no—
255C4×Q8—44no—
256C4×Q8—84no—
257C4×D4—24no—
258C4×D4—44no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C42—22no—
304C2×C42—21no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2412yes—
316C2×C42C2412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320#107—44no—
321#106—44no—
322C4⋊C8—24no—
323C4⋊C8—44no—
324#103—28no—
325#103—24no—
326#103—42no—
327#102—28no—
328#102—24no—
329#102—42no—
330#97—22no—
331#97—24no—
332#97—22no—
333#97—42no—
334C4×C8—24no—
335#95—21no—
336#95—21no—
337#95—22no—
338#95—22no—
339#95—21no—
340#95—22no—
341#95—21no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95C2×D412yes—
356#95C2×D412yes—
357#95C2×D412yes—
358C24—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—42no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—44no—
371C2×Q8C2×#4112yes—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4C2×#4112yes—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—82no—
389C22×C4—82no—
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×D411yescharacteristic
400C22×C4C2511yescharacteristic
401C2×C8—24no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—84no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—44no—
451C22⋊C4—42no—
452C22⋊C4—82no—
453C22⋊C4—81no—
454C22⋊C4—81no—
455C22⋊C4—82no—
456C42—22no—
457C42—22no—
458C42—24no—
459C42—22no—
460C42—24no—
461C42—41no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—41no—
467C42—42no—
468C42—44no—
469C42—82no—
470C42—84no—
471C42C2×#4112yes—
472C422−1+414yes—
473C422−1+412yes—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23C23×D411yescharacteristic
482Q8—24no—
483Q8—44no—
484Q8—42no—
485Q8—84no—
486Q8—88no—
487D4—24no—
488D4—42no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—84no—
538C2×C4—164no—
539C2×C4#53612yes—
540C2×C4#53612yes—
541C2×C4#54812yes—
542C2×C4#54812yes—
543C2×C4#57512yes—
544C2×C4#57512yes—
545C2×C4C22×#4111yescharacteristic
546C2×C4C2×2−1+412yes—
547C2×C4C2×2−1+411yescharacteristic
548C2×C4#58512yes—
549C8—42no—
550C8—44no—
551C8—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×#53611yescharacteristic
560C22C2×#54811yescharacteristic
561C22#350011yescharacteristic
562C22C2×#57511yescharacteristic
563C22C2×#57511yescharacteristic
564C22#358511yescharacteristic
565C22#358611yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—42no—
570C4—41no—
571C4—42no—
572C4—41no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—84no—
581C4—81no—
582C4—81no—
583C4#324014yes—
584C4#351512yes—
585C4#355512yes—
586C2—82no—
587C2#3917311yescharacteristic
588C2C2×#324011yescharacteristic
589C2#6088711yescharacteristic
590C2#6132311yescharacteristic
591C2#6132411yescharacteristic
592C2#6258611yescharacteristic
593C2#6259311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36598C2semidirect
3. #33297C2semidirect
4. #31473C2semidirect
6. #30234C2semidirect
7. #29212C2semidirect
8. #29181C2semidirect
11. #26109C2semidirect
13. #24997C2semidirect
17. #21345C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#365552
#332972
#314732
#314712
#302342
#292122
#291812
#287272
#287262
#261092
#261072
#249972
#249932
#222242
C4×#4892
#213452