Finite Group Databasebeta

← #6490794

Subgroups — #6490794

Order 512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic, complemented
3#33121C212yesmaximal, complemented
4#33120C212yesmaximal, complemented
5#30068C214yesmaximal, complemented
6#29267C214yesmaximal, complemented
7#29213C212yesmaximal, complemented
8#28757C212yesmaximal, complemented
9#27919C211yesmaximal, characteristic, complemented
10#27918C211yesmaximal, characteristic
11#25911C211yesmaximal, characteristic
12#25910C211yesmaximal, characteristic, complemented
13#24074C212yesmaximal, complemented
14#24073C212yesmaximal, complemented
15#22226C212yesmaximal, complemented
16#22225C212yesmaximal, complemented
17#20783C211yesmaximal, characteristic, complemented
18C2×#542C2212yes—
19C2×#533C2211yescharacteristic
20C2×#491C2212yes—
21C2×#490C2212yes—
22C2×#484C2212yes—
23C2×#481C2212yes—
24C2×#483C2212yes—
25C2×#480C2212yes—
26#2868C2211yescharacteristic
27#2858C2211yescharacteristic
28#2857C2211yescharacteristic
29#2856C2211yescharacteristic
30#2640C2212yes—
31#2620C2212yes—
32#2512—22no—
33#2491—22no—
34#2452—22no—
35#2422—22no—
36#2368—22no—
37#2368—22no—
38#2355—22no—
39#2355—22no—
40#2331C2212yes—
41#2331C2212yes—
42#2326C2212yes—
43#2317C2211yescharacteristic
44#2317C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#125C2212yes—
48C4×#125C2212yes—
49C2×#382C2211yescharacteristic
50#2274C2211yescharacteristic
51#2274C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2097C2214yes—
54#2092C2214yes—
55#2058C2214yes—
56#2057C2214yes—
57#2042C2214yes—
58#2042C2214yes—
59#2024C2214yes—
60#2023C2214yes—
61#1992C2212yes—
62#1992C2212yes—
63#1983C2214yes—
64#1983C2214yes—
65#1961C2212yes—
66#1960C2212yes—
67#1948C2214yes—
68#1947C2214yes—
69#1928C2214yes—
70#1927C2214yes—
71#1926C2214yes—
72#1925C2214yes—
73#1875C2212yes—
74#1875C2212yes—
75#1873C2212yes—
76#1873C2212yes—
77#1865C2214yes—
78#1864C2214yes—
79#1863C2214yes—
80#1862C2214yes—
81#1847C2212yes—
82#1847C2212yes—
83#1840C2212yes—
84#1832C2212yes—
85#1818C2212yes—
86#1798C2212yes—
87#1767C2212yes—
88#1749C2211yescharacteristic
89#1749C2211yescharacteristic
90#1745C2212yes—
91#542—44no—
92#542—44no—
93#533—28no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2312yes—
98C2×#127C2311yescharacteristic
99C2×#126—22no—
100C2×#125C2312yes—
101C2×#124—22no—
102C2×#124C2312yes—
103C2×#124C2312yes—
104C2×#123—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×#121—22no—
109C2×#121C2312yes—
110C2×C4×Q8—22no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—22no—
113C2×C4×D4C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#491—28no—
122#490—28no—
123#484—44no—
124#483—44no—
125#481—44no—
126#480—44no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×#97C2312yes—
143C2×#97C2312yes—
144#397—24no—
145#396—24no—
146#394—24no—
147#393—24no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391C2312yes—
153#391C2312yes—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#388—24no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#385—21no—
176#385—21no—
177#385—21no—
178#385—21no—
179#385—24no—
180#384—21no—
181#384—21no—
182#384—22no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#382—21no—
187#382—21no—
188#382—22no—
189#382C2312yes—
190#382C2312yes—
191#380—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4D412yes—
208C4×C22⋊C4D412yes—
209#376C2311yescharacteristic
210#376C2311yescharacteristic
211C2×#95—22no—
212C43C2311yescharacteristic
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2314yes—
220#336C2314yes—
221#336C2312yes—
222#336C2312yes—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8C2412yes—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4C2×D412yes—
233C23×C4C2411yescharacteristic
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238C4⋊Q8—24no—
239C4⋊Q8—42no—
240C4⋊Q8—42no—
241C4⋊Q8—84no—
242#127—24no—
243#126—48no—
244#125—24no—
245#125#4118yes—
246#124—28no—
247#124—42no—
248#124—42no—
249#124—48no—
250#123—48no—
251C22⋊Q8—216no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—48no—
255#121—24no—
256#121—24no—
257#121—84no—
258C4×Q8—216no—
259C4×Q8—42no—
260C4×Q8—42no—
261C4×D4—42no—
262C4×D4—42no—
263C4×D4—84no—
264#117—24no—
265#117—24no—
266#117—24no—
267#117—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4C2×D411yescharacteristic
306C2×C22⋊C4C2×D411yescharacteristic
307C2×C22⋊C4C2×D412yes—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327#107—44no—
328#107—44no—
329#106—44no—
330#106—44no—
331C4⋊C8—28no—
332#103—28no—
333#103—28no—
334#103—44no—
335#103—44no—
336#102—28no—
337#102—28no—
338#102—44no—
339#102—44no—
340C22⋊C8—24no—
341C22⋊C8—24no—
342#97—24no—
343#97—24no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—24no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362C24—22no—
363C24C22×D411yescharacteristic
364C2×Q8—28no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—44no—
370C2×Q8—82no—
371C2×Q8—82no—
372C2×Q8—84no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—84no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—84no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4H(5,𝔽2)14yes—
410C22×C4C2511yescharacteristic
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—84no—
444C4⋊C4C2×#4114yes—
445C4⋊C4C2×#4114yes—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C42—24no—
460C42—24no—
461C42—24no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—44no—
471C42—82no—
472C42—82no—
473C42—82no—
474C42—82no—
475C42C2×#4114yes—
476C422−1+414yes—
477C23—22no—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23#53512yes—
487C23C23×D411yescharacteristic
488Q8—44no—
489Q8—44no—
490Q8—88no—
491Q8—164no—
492D4—44no—
493D4—44no—
494D4—88no—
495D4—164no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
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—42no—
535C2×C4—42no—
536C2×C4—84no—
537C2×C4—164no—
538C2×C4—162no—
539C2×C4—162no—
540C2×C4#52514yes—
541C2×C4#53512yes—
542C2×C4#53614yes—
543C2×C4C22×#4112yes—
544C2×C4C2×H(5,𝔽2)12yes—
545C2×C4C2×2−1+412yes—
546C8—44no—
547C8—44no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—42no—
552C22—42no—
553C22—84no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—84no—
558C22C2×#52511yescharacteristic
559C22C2×#53511yescharacteristic
560C22C2×#53611yescharacteristic
561C22#358512yes—
562C22#358612yes—
563C4—24no—
564C4—22no—
565C4—24no—
566C4—42no—
567C4—44no—
568C4—44no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4#352214yes—
580C2—42no—
581C2—162no—
582C2#3524911yescharacteristic
583C2#6056712yes—
584C2#6082711yescharacteristic
585C2#6082811yescharacteristic
586C2#6088612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34215C2semidirect
3. #33121C2semidirect
4. #33120C2semidirect
5. #30068C2semidirect
6. #29267C2semidirect
7. #29213C2semidirect
8. #28757C2semidirect
9. #27919C2semidirect
12. #25910C2semidirect
13. #24074C2semidirect
14. #24073C2semidirect
15. #22226C2semidirect
16. #22225C2semidirect
17. #20783C2semidirect

Maximal subgroups

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

subgroupindex
#352772
#342152
#331212
#331202
#300682
#292672
#292132
#287572
#279192
#279182
#259112
#259102
#240742
#240732
#222262
#222252
#207832