Finite Group Databasebeta

← #6796022

Subgroups — #6796022

Order 512.

1479 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic
2#34945C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#32414C214yesmaximal, complemented
5#31612C214yesmaximal, complemented
6#29942C212yesmaximal, complemented
7#29287C211yesmaximal, characteristic, complemented
8#28704C212yesmaximal
9#28689C211yesmaximal, characteristic, complemented
10#27631C212yesmaximal, complemented
11#26513C212yesmaximal, complemented
12#25456C212yesmaximal, complemented
13#25341C212yesmaximal
14#24837C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic
16#21962C212yesmaximal, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#500C2212yes—
20C2×#477C2211yescharacteristic
21C2×#475C2212yes—
22C2×C8×D4C2212yes—
23C22×C4⋊C8C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2573—22no—
27#2571C2211yescharacteristic
28#2568—21no—
29#2568C2211yescharacteristic
30#2547—21no—
31#2547C2211yescharacteristic
32#2547C2212yes—
33#2538C2211yescharacteristic
34#2514—21no—
35#2499—22no—
36#2496—22no—
37#2496C2214yes—
38#2494C2212yes—
39#2447—21no—
40#2421C2212yes—
41#2401—22no—
42#2365—21no—
43#2365C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#124—21no—
47C4×C22⋊Q8C2212yes—
48#2295—21no—
49#2285C2212yes—
50#2283C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2101C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2085C2214yes—
57#2083C2214yes—
58#2077C2214yes—
59#2075C2214yes—
60#2075C2214yes—
61#2068C2214yes—
62#2035C2214yes—
63#1990C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1987C2212yes—
67#1987C2211yescharacteristic
68#1949C2214yes—
69#1949C2214yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1947C2211yescharacteristic
77#1939C2212yes—
78#1928C2214yes—
79#1921C2212yes—
80#1919C2214yes—
81#1892C2214yes—
82#1863C2212yes—
83#1850C2214yes—
84#1842C2211yescharacteristic
85#1808C2214yes—
86#1808C2214yes—
87#1807C2214yes—
88#1803C2212yes—
89#1803C2211yescharacteristic
90#1803C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C23×C8C2314yes—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#126—24no—
103C2×#124—21no—
104C2×#124—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2312yes—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C22×C42C2312yes—
118#500—44no—
119#477—24no—
120#475—44no—
121C8×D4—44no—
122C2×#107C2312yes—
123C2×#107C2314yes—
124C2×#107C2312yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8—28no—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×C22⋊C8C2314yes—
139C2×C4×C8C2312yes—
140#401—22no—
141#400—24no—
142#400—22no—
143#400—22no—
144#400—22no—
145#397—22no—
146#397—22no—
147#397—22no—
148#397—24no—
149#397—24no—
150#396—22no—
151#391—22no—
152#391—22no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391—22no—
159#391C2312yes—
160#390—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#388—22no—
171#388—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—24no—
176#387—24no—
177#385—22no—
178#384—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#383—22no—
185#382—21no—
186#382—22no—
187#382C2312yes—
188#382C2312yes—
189#381—22no—
190#380—22no—
191#380—22no—
192#380C2312yes—
193#380C2314yes—
194#380C2312yes—
195#380C2312yes—
196#380C2314yes—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—22no—
219C43—21no—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8C2411yescharacteristic
238C22×D4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C22×C8—28no—
243C22×C8—44no—
244C22×C8—44no—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C4⋊Q8—22no—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256#126—416no—
257#124—44no—
258#124—48no—
259C22⋊Q8—24no—
260C22⋊Q8—42no—
261C22⋊Q8—48no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—41no—
268C4×Q8—44no—
269C4×Q8—48no—
270C4×Q8—44no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—22no—
274C4×D4—22no—
275#117—48no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—24no—
337#107—42no—
338#107—44no—
339#107—44no—
340#106—44no—
341#106—44no—
342C4⋊C8—28no—
343C4⋊C8—24no—
344C4⋊C8—24no—
345C4⋊C8—22no—
346C4⋊C8—42no—
347C4⋊C8#4114yes—
348#103—24no—
349#103—24no—
350#103—44no—
351C22⋊C8—44no—
352C4×C8—24no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24C22×D412yes—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q82−1+412yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4AGL(1,ℤ/8)14yes—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—42no—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2511yescharacteristic
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—88no—
436C2×C8C2×#4114yes—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—82no—
484C42—24no—
485C42—24no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—21no—
491C42—21no—
492C42—41no—
493C42—42no—
494C42—48no—
495C42—42no—
496C42—42no—
497C42—44no—
498C42—44no—
499C42—42no—
500C42—42no—
501C42—81no—
502C42#13514yes—
503C42C2×#4112yes—
504C42H(5,𝔽2)12yes—
505C422−1+412yes—
506C422−1+412yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23C2×AGL(1,ℤ/8)14yes—
514C23C23×D411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—41no—
518Q8—41no—
519Q8—84no—
520Q8—84no—
521Q8—88no—
522D4—28no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—48no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4#53712yes—
575C2×C4#53712yes—
576C2×C4#54712yes—
577C2×C4#54712yes—
578C2×C4C2×#13512yes—
579C2×C4C2×#13512yes—
580C2×C4C2×#13512yes—
581C2×C4C2×AGL(1,ℤ/8)12yes—
582C2×C4#57812yes—
583C2×C4C22×#4111yescharacteristic
584C2×C4C2×H(5,𝔽2)11yescharacteristic
585C2×C4C2×2−1+411yescharacteristic
586C2×C4C2×2−1+411yescharacteristic
587C2×C4C2×2−1+411yescharacteristic
588C2×C4#58511yescharacteristic
589C8—24no—
590C8—44no—
591C8—42no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C2×#53711yescharacteristic
600C22C2×#54711yescharacteristic
601C22#348911yescharacteristic
602C22C22×#13511yescharacteristic
603C22C22×AGL(1,ℤ/8)11yescharacteristic
604C22C2×#57811yescharacteristic
605C22#358611yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—41no—
614C4—81no—
615C4—82no—
616C4—82no—
617C4—81no—
618C4—82no—
619C4—82no—
620C4—84no—
621C4—81no—
622C4#324714yes—
623C4#334214yes—
624C4#351411yescharacteristic
625C4#352311yescharacteristic
626C4#355211yescharacteristic
627C4#355511yescharacteristic
628C2—44no—
629C2#3892311yescharacteristic
630C2C2×#324711yescharacteristic
631C2#6092511yescharacteristic
632C2C2×#334211yescharacteristic
633C2#6125211yescharacteristic
634C2#6210111yescharacteristic
635C2#6213211yescharacteristic
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. #33196C2semidirect
4. #32414C2semidirect
5. #31612C2semidirect
6. #29942C2semidirect
7. #29287C2semidirect
9. #28689C2semidirect
10. #27631C2semidirect
11. #26513C2semidirect
12. #25456C2semidirect
14. #24837C2semidirect
16. #21962C2semidirect
17. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#349452
#331962
#324142
#316122
#299422
#292872
#287042
#286892
#276312
#265132
#254562
#253412
#248372
D4×C4⋊C82
#219622
C4×#4772