Finite Group Databasebeta

← #6971555

Subgroups — #6971555

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36454C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic
3#32741C212yesmaximal, complemented
4#31760C214yesmaximal, complemented
5#31103C214yesmaximal, complemented
6#30205C211yesmaximal, characteristic, complemented
7#30175C212yesmaximal, complemented
8#29642C212yesmaximal, complemented
9#29355C211yesmaximal, characteristic, complemented
10#29213C211yesmaximal, characteristic, complemented
11#26462C212yesmaximal, complemented
12#26440C212yesmaximal, complemented
13#26298C212yesmaximal
14#26236C212yesmaximal
15#24301C212yesmaximal, complemented
16#23838C211yesmaximal, characteristic, complemented
17#22012C211yesmaximal, characteristic
18C2×D4×Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#484C2214yes—
21C2×#475C2214yes—
22#2868C2211yescharacteristic
23#2862C2211yescharacteristic
24#2857C2212yes—
25#2855C2212yes—
26#2575—22no—
27#2551—21no—
28#2551C2212yes—
29#2547C2211yescharacteristic
30#2546—21no—
31#2542C2211yescharacteristic
32#2517—22no—
33#2511—21no—
34#2489C2212yes—
35#2447—21no—
36#2440—22no—
37#2440C2214yes—
38#2434—22no—
39#2421C2212yes—
40#2367—21no—
41#2365C2211yescharacteristic
42#2365C2211yescharacteristic
43#2355—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#125C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49C42×D4C2211yescharacteristic
50#2086C2214yes—
51#2085C2212yes—
52#2084C2211yescharacteristic
53#2083C2211yescharacteristic
54#2077C2214yes—
55#2077C2214yes—
56#2072C2214yes—
57#2072C2212yes—
58#2066C2214yes—
59#2059C2212yes—
60#2058C2212yes—
61#2058C2211yescharacteristic
62#2057C2212yes—
63#2057C2211yescharacteristic
64#2057C2214yes—
65#2036C2214yes—
66#2036C2214yes—
67#2024C2214yes—
68#2024C2212yes—
69#2024C2212yes—
70#2024C2212yes—
71#1992C2212yes—
72#1990C2212yes—
73#1988C2214yes—
74#1988C2212yes—
75#1981C2212yes—
76#1956C2212yes—
77#1954C2212yes—
78#1948C2212yes—
79#1947C2212yes—
80#1939C2212yes—
81#1935C2214yes—
82#1924C2214yes—
83#1918C2214yes—
84#1872C2214yes—
85#1865C2214yes—
86#1853C2214yes—
87#1847C2212yes—
88#1824C2214yes—
89#1817C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92C4×#106C2212yes—
93#1767C2211yescharacteristic
94#1744C2211yescharacteristic
95C23×Q8—22no—
96D4×Q8—48no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—48no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2312yes—
108C2×#125C2311yescharacteristic
109C2×#124—21no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×D4—22no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C22×C4⋊C4—22no—
122C22×C4⋊C4C2314yes—
123C22×C42C2312yes—
124#484—48no—
125#475—48no—
126C2×#107C2312yes—
127C2×#107C2312yes—
128C2×#107C2314yes—
129C2×#106C2312yes—
130C2×#106C2314yes—
131C2×#106C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142#400—22no—
143#400—22no—
144#399—22no—
145#398—24no—
146#397—22no—
147#397—22no—
148#397—22no—
149#395—24no—
150#395—24no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391—21no—
156#391—21no—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#390—21no—
161#389—22no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389C2311yescharacteristic
167#389C2311yescharacteristic
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#388—22no—
172#388—22no—
173#387—21no—
174#387—21no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#385—22no—
182#385—22no—
183#385—24no—
184#385—24no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—21no—
192#382—21no—
193#382—22no—
194#382C2311yescharacteristic
195#381—22no—
196#380—22no—
197#380C2312yes—
198#380C2314yes—
199#380C2312yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2312yes—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—24no—
236C22×Q8—48no—
237C22×Q8—48no—
238C22×Q8—84no—
239C22×Q8C2412yes—
240C22×D4C2×D411yescharacteristic
241C23×C4—24no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C4⋊Q8—24no—
249C4⋊Q8—28no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—48no—
256C4⋊Q8—48no—
257C4⋊Q8—48no—
258C4⋊Q8—82no—
259#125—22no—
260#125—41no—
261#124—44no—
262C22⋊Q8—24no—
263C22⋊Q8—416no—
264C22⋊Q8—42no—
265C22⋊Q8—48no—
266C22⋊Q8—44no—
267C4×Q8—44no—
268C4×Q8—44no—
269C4×Q8—44no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—48no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—44no—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—24no—
336#107—24no—
337#107—44no—
338#106—24no—
339#106—42no—
340#106—44no—
341C4⋊C8—24no—
342C4⋊C8—44no—
343#103—28no—
344#103—44no—
345#103—44no—
346#103—44no—
347C22⋊C8—44no—
348C22⋊C8—44no—
349#97—42no—
350C4×C8—24no—
351#95—21no—
352#95—22no—
353#95—22no—
354#95—21no—
355#95—22no—
356#95—22no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—24no—
365#95—22no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24C22×D412yes—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—48no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—48no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—84no—
391C2×Q8—832no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4#13714yes—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—84no—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2511yescharacteristic
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4AGL(1,ℤ/8)14yes—
473C4⋊C4H(5,𝔽2)14yes—
474C4⋊C42−1+412yes—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—82no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—22no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—42no—
493C42—44no—
494C42—42no—
495C42—44no—
496C42—42no—
497C42—82no—
498C422−1+414yes—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23C2×#13714yes—
506C23C23×D411yescharacteristic
507Q8—48no—
508Q8—44no—
509Q8—44no—
510Q8—816no—
511Q8—88no—
512Q8—84no—
513D4—28no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—816no—
566C2×C4#53612yes—
567C2×C4#53612yes—
568C2×C4C2×AGL(1,ℤ/8)14yes—
569C2×C4C2×AGL(1,ℤ/8)12yes—
570C2×C4C2×#13712yes—
571C2×C4#57712yes—
572C2×C4#57812yes—
573C2×C4C2×H(5,𝔽2)12yes—
574C2×C4C2×2−1+411yescharacteristic
575C2×C4C2×2−1+412yes—
576C2×C4#58511yescharacteristic
577C8—42no—
578C8—44no—
579C8—42no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22C2×#53611yescharacteristic
588C22#348411yescharacteristic
589C22#348911yescharacteristic
590C22C22×AGL(1,ℤ/8)11yescharacteristic
591C22C22×#13711yescharacteristic
592C22C2×#57711yescharacteristic
593C22C2×#57811yescharacteristic
594C4—22no—
595C4—24no—
596C4—22no—
597C4—22no—
598C4—44no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—81no—
605C4—84no—
606C4—81no—
607C4—82no—
608C4—81no—
609C4—82no—
610C4—82no—
611C4#323914yes—
612C4#355012yes—
613C4#355412yes—
614C2—44no—
615C2#4248111yescharacteristic
616C2C2×#323911yescharacteristic
617C2#6087811yescharacteristic
618C2#6160011yescharacteristic
619C2#6160211yescharacteristic
620C2#6213111yescharacteristic
621C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32741C2semidirect
4. #31760C2semidirect
5. #31103C2semidirect
6. #30205C2semidirect
7. #30175C2semidirect
8. #29642C2semidirect
9. #29355C2semidirect
10. #29213C2semidirect
11. #26462C2semidirect
12. #26440C2semidirect
15. #24301C2semidirect
16. #23838C2semidirect

Maximal subgroups

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

subgroupindex
#364542
D4×C4⋊Q82
#327412
#317602
#311032
#302052
#301752
#296422
#293552
#292132
#264622
#264402
#262982
#262362
#243012
#238382
#220122