Finite Group Databasebeta

← #6806331

Subgroups — #6806331

Order 512.

1392 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic, complemented
2#36555C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#31472C214yesmaximal, complemented
5#31470C214yesmaximal
6#30233C212yesmaximal, complemented
7#29212C211yesmaximal, characteristic, complemented
8#29182C212yesmaximal, complemented
9#28726C211yesmaximal, characteristic
10#28726C212yesmaximal
11#26112C212yesmaximal, complemented
12#26110C212yesmaximal
13#24994C212yesmaximal, complemented
14#24990C212yesmaximal
15#22224C212yesmaximal
16C4×#489C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic, complemented
18C2×#557—21no—
19C2×#489C2211yescharacteristic
20C2×#489C2212yes—
21#2868C2211yescharacteristic
22#2868C2212yes—
23#2864C2212yes—
24#2859C2211yescharacteristic
25#2587—22no—
26#2582C2211yescharacteristic
27#2552—21no—
28#2552—22no—
29#2552C2212yes—
30#2552C2214yes—
31#2546—22no—
32#2546—21no—
33#2538C2211yescharacteristic
34#2522—22no—
35#2497—21no—
36#2454C2212yes—
37#2453C2212yes—
38#2423—21no—
39#2360—21no—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2212yes—
42Q8×C4⋊C4C2211yescharacteristic
43D4×C4⋊C4C2211yescharacteristic
44#2345—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×#124C2211yescharacteristic
48C4×#124C2212yes—
49C42×Q8C2211yescharacteristic
50#2100C2214yes—
51#2100C2214yes—
52#2099C2214yes—
53#2099C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2072C2214yes—
57#2072C2212yes—
58#2071C2214yes—
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#1920C2214yes—
77#1918C2214yes—
78#1917C2214yes—
79#1864C2214yes—
80#1863C2214yes—
81#1847C2212yes—
82#1836C2211yescharacteristic
83#1834C2212yes—
84#1834C2212yes—
85#1803C2212yes—
86#1803C2214yes—
87#1802C2212yes—
88#1802C2214yes—
89#1767C2211yescharacteristic
90#1767C2212yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95#557—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125—22no—
103C2×#125—24no—
104C2×#125C2312yes—
105C2×#124C2311yescharacteristic
106C2×C22⋊Q8—21no—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118#489—24no—
119#489—28no—
120C2×#107C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2314yes—
124C2×#103C2314yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#102C2314yes—
128C2×#102C2312yes—
129C2×#102C2312yes—
130C2×#97C2311yescharacteristic
131C2×#97C2312yes—
132C2×#97C2311yescharacteristic
133C2×#97C2312yes—
134C2×C4×C8C2312yes—
135#400—22no—
136#399—22no—
137#398—22no—
138#398—22no—
139#398—24no—
140#398—24no—
141#398—24no—
142#396—22no—
143#396—22no—
144#391—21no—
145#391—22no—
146#391—21no—
147#391C2312yes—
148#391C2311yescharacteristic
149#390C2312yes—
150#390C2311yescharacteristic
151#389—21no—
152#389—22no—
153#389—24no—
154#389—22no—
155#389—22no—
156#389—24no—
157#389—22no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389C2312yes—
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#388—21no—
171#388—22no—
172#388—22no—
173#388—21no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#385—22no—
180#385—22no—
181#384—21no—
182#384—22no—
183#384—21no—
184#384—22no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2314yes—
189#384C2311yescharacteristic
190#384C2314yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#382C2312yes—
194#382C2311yescharacteristic
195#380—21no—
196#380—21no—
197#380—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D412yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C43C2311yescharacteristic
214C43C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2312yes—
228#336C2314yes—
229#336C2314yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8C2×D411yescharacteristic
234C22×Q8C2411yescharacteristic
235C22×D4C2411yescharacteristic
236C23×C4—21no—
237C23×C4—22no—
238C22×C8C2412yes—
239C22×C8C2414yes—
240C22×C8C2412yes—
241C4⋊Q8—22no—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—24no—
245C4⋊Q8—48no—
246C4⋊Q8—44no—
247C4⋊Q8—82no—
248C4⋊Q8—84no—
249#125—44no—
250#125—416no—
251#125—48no—
252#124—22no—
253#124#4114yes—
254C22⋊Q8—44no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—48no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—48no—
263C4×D4—24no—
264C4×D4—44no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D412yes—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2412yes—
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330#107—44no—
331#106—44no—
332C4⋊C8—24no—
333C4⋊C8—44no—
334#103—28no—
335#103—24no—
336#103—42no—
337#102—28no—
338#102—24no—
339#102—42no—
340#97—22no—
341#97—24no—
342#97—22no—
343#97—42no—
344C4×C8—24no—
345#95—21no—
346#95—21no—
347#95—22no—
348#95—22no—
349#95—21no—
350#95—22no—
351#95—21no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D412yes—
364C24—21no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—44no—
371C2×Q8—42no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8AGL(1,ℤ/8)14yes—
377C2×Q8C2×#4112yes—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4C2×#4112yes—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C2511yescharacteristic
407C2×C8—24no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—84no—
455C4⋊C4—82no—
456C4⋊C4—84no—
457C4⋊C4#13714yes—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—81no—
463C22⋊C4—82no—
464C22⋊C4—81no—
465C22⋊C4—82no—
466C42—24no—
467C42—22no—
468C42—24no—
469C42—22no—
470C42—24no—
471C42—22no—
472C42—24no—
473C42—41no—
474C42—42no—
475C42—41no—
476C42—42no—
477C42—42no—
478C42—48no—
479C42—44no—
480C42—82no—
481C42C2×#4112yes—
482C422−1+414yes—
483C422−1+412yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C23×D411yescharacteristic
492Q8—28no—
493Q8—24no—
494Q8—42no—
495Q8—88no—
496Q8—84no—
497D4—24no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—164no—
547C2×C4#53612yes—
548C2×C4#53612yes—
549C2×C4#54812yes—
550C2×C4#54812yes—
551C2×C4C2×AGL(1,ℤ/8)14yes—
552C2×C4C2×AGL(1,ℤ/8)12yes—
553C2×C4C2×#13714yes—
554C2×C4C2×#13712yes—
555C2×C4C22×#4111yescharacteristic
556C2×C4C2×2−1+412yes—
557C2×C4C2×2−1+411yescharacteristic
558C2×C4#58512yes—
559C8—42no—
560C8—44no—
561C8—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#53611yescharacteristic
570C22C2×#54811yescharacteristic
571C22#350011yescharacteristic
572C22C22×AGL(1,ℤ/8)11yescharacteristic
573C22C22×#13711yescharacteristic
574C22#358511yescharacteristic
575C22#358611yescharacteristic
576C4—24no—
577C4—22no—
578C4—22no—
579C4—24no—
580C4—22no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—42no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4—84no—
590C4—82no—
591C4—81no—
592C4—81no—
593C4#323814yes—
594C4#351512yes—
595C4#355512yes—
596C2—82no—
597C2#3917311yescharacteristic
598C2C2×#323811yescharacteristic
599C2#6088711yescharacteristic
600C2#6132111yescharacteristic
601C2#6132211yescharacteristic
602C2#6258211yescharacteristic
603C2#6259511yescharacteristic
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. #36593C2semidirect
3. #33297C2semidirect
4. #31472C2semidirect
6. #30233C2semidirect
7. #29212C2semidirect
8. #29182C2semidirect
11. #26112C2semidirect
13. #24994C2semidirect
17. #21345C2semidirect

Maximal subgroups

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

subgroupindex
#365932
#365552
#332972
#314722
#314702
#302332
#292122
#291822
#287262
#261122
#261102
#249942
#249902
#222242
C4×#4892
#213452