Finite Group Databasebeta

← #6795518

Subgroups — #6795518

Order 512.

1477 conjugacy classes of subgroups.

The subgroup lattice has 651 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#32291C214yesmaximal, complemented
5#31940C214yesmaximal, complemented
6#29926C212yesmaximal, complemented
7#28689C211yesmaximal, characteristic, complemented
8#28532C212yesmaximal, complemented
9#28283C211yesmaximal, characteristic, complemented
10#27730C212yesmaximal, complemented
11#27682C212yesmaximal, complemented
12#25705C212yesmaximal
13#24652C212yesmaximal, complemented
14#24123C212yesmaximal
15#22795C211yesmaximal, characteristic
16#21985C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#556C2212yes—
19C2×#500C2212yes—
20C2×#475C2212yes—
21C2×#475C2212yes—
22C2×#462C2211yescharacteristic
23C2×C4×Q16C2212yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2573—22no—
28#2571C2211yescharacteristic
29#2568C2212yes—
30#2568C2211yescharacteristic
31#2547—21no—
32#2547—21no—
33#2547C2211yescharacteristic
34#2538C2211yescharacteristic
35#2514—21no—
36#2499C2214yes—
37#2496—22no—
38#2496—22no—
39#2494C2212yes—
40#2447—21no—
41#2421C2212yes—
42#2401—22no—
43#2365—21no—
44#2365—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124—21no—
48C4×C22⋊Q8C2212yes—
49#2295—21no—
50#2285C2212yes—
51#2283C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2101C2214yes—
55#2097C2214yes—
56#2097C2214yes—
57#2083C2214yes—
58#2074C2214yes—
59#2066C2214yes—
60#2036C2214yes—
61#2036C2214yes—
62#2035C2214yes—
63#2001C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1981C2212yes—
67#1981C2211yescharacteristic
68#1969C2214yes—
69#1954C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2211yescharacteristic
76#1933C2214yes—
77#1928C2212yes—
78#1923C2214yes—
79#1878C2212yes—
80#1873C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1854C2214yes—
84#1846C2214yes—
85#1842C2211yescharacteristic
86#1835C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96#556—44no—
97#556—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#126—24no—
104C2×#124—21no—
105C2×#124—22no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2314yes—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×#117C2314yes—
118C22×C42C2312yes—
119#500—44no—
120#475—44no—
121#475—44no—
122#462—24no—
123C4×Q16—44no—
124#431—28no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#107C2312yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C4×C8C2314yes—
145C2×C4×C8C2312yes—
146#401—22no—
147#400—22no—
148#400—24no—
149#400—22no—
150#400—22no—
151#397—22no—
152#397—22no—
153#397—22no—
154#397—24no—
155#397—24no—
156#396—22no—
157#391—22no—
158#391—22no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391C2312yes—
164#391C2314yes—
165#391C2312yes—
166#390—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—24no—
182#387—24no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#383—22no—
191#382—21no—
192#382C2314yes—
193#382C2312yes—
194#382C2312yes—
195#381—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380C2312yes—
201#380C2312yes—
202#380C2312yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224#376—22no—
225C43—21no—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C2×Q16—24no—
245C2×Q16—24no—
246C2×Q16—42no—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258#126—416no—
259#124—44no—
260#124—48no—
261C22⋊Q8—24no—
262C22⋊Q8—42no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—41no—
273C4×Q8—48no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—42no—
279#117—44no—
280#117—44no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—42no—
342#107—44no—
343#107—44no—
344#106—44no—
345#106—44no—
346C4⋊C8—24no—
347C4⋊C8—24no—
348C4⋊C8—22no—
349C4⋊C8—42no—
350C4⋊C8#4114yes—
351#103—24no—
352#103—42no—
353#103—44no—
354#103—42no—
355#103—42no—
356#103—44no—
357#103—44no—
358C22⋊C8—28no—
359C22⋊C8—44no—
360C4×C8—28no—
361C4×C8—24no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24C22×D412yes—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q82−1+412yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2511yescharacteristic
439Q16—48no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8C2×#4114yes—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—82no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—82no—
495C42—24no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—21no—
502C42—21no—
503C42—41no—
504C42—42no—
505C42—48no—
506C42—42no—
507C42—44no—
508C42—42no—
509C42—44no—
510C42—42no—
511C42—42no—
512C42—81no—
513C42#13514yes—
514C42C2×#4112yes—
515C42H(5,𝔽2)12yes—
516C42H(5,𝔽2)12yes—
517C422−1+412yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23C23×D411yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—42no—
531Q8—42no—
532Q8—42no—
533Q8—42no—
534Q8—88no—
535Q8—84no—
536D4—44no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—84no—
588C2×C4#53512yes—
589C2×C4#53512yes—
590C2×C4#54712yes—
591C2×C4#54712yes—
592C2×C4C2×#13512yes—
593C2×C4C2×#13512yes—
594C2×C4C2×#13512yes—
595C2×C4#57512yes—
596C2×C4#57812yes—
597C2×C4C22×#4111yescharacteristic
598C2×C4C2×H(5,𝔽2)11yescharacteristic
599C2×C4C2×H(5,𝔽2)11yescharacteristic
600C2×C4C2×2−1+411yescharacteristic
601C2×C4C2×2−1+411yescharacteristic
602C2×C4#58511yescharacteristic
603C8—24no—
604C8—44no—
605C8—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22C2×#53511yescharacteristic
614C22C2×#54711yescharacteristic
615C22#348911yescharacteristic
616C22C22×#13511yescharacteristic
617C22C2×#57511yescharacteristic
618C22C2×#57811yescharacteristic
619C22#358611yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—41no—
628C4—82no—
629C4—81no—
630C4—84no—
631C4—82no—
632C4—82no—
633C4—81no—
634C4—82no—
635C4—81no—
636C4#321214yes—
637C4#334214yes—
638C4#351711yescharacteristic
639C4#352011yescharacteristic
640C4#354711yescharacteristic
641C4#355211yescharacteristic
642C2—82no—
643C2#3892011yescharacteristic
644C2C2×#321211yescharacteristic
645C2#6083711yescharacteristic
646C2C2×#334211yescharacteristic
647C2#6125711yescharacteristic
648C2#6210111yescharacteristic
649C2#6214611yescharacteristic
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. #32291C2semidirect
5. #31940C2semidirect
6. #29926C2semidirect
7. #28689C2semidirect
8. #28532C2semidirect
9. #28283C2semidirect
10. #27730C2semidirect
11. #27682C2semidirect
13. #24652C2semidirect
16. #21985C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#349452
#331962
#322912
#319402
#299262
#286892
#285322
#282832
#277302
#276822
#257052
#246522
#241232
#227952
#219852
C4×#4622