Finite Group Databasebeta

← #6805647

Subgroups — #6805647

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2Q8×#124C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#32455C214yesmaximal
5#32451C214yesmaximal, complemented
6#30215C212yesmaximal, complemented
7#29211C212yesmaximal, complemented
8#28726C211yesmaximal, characteristic
9#28620C212yesmaximal
10#28522C211yesmaximal, characteristic, complemented
11#24683C212yesmaximal, complemented
12#24679C212yesmaximal
13#24333C212yesmaximal, complemented
14#24330C212yesmaximal
15#22224C212yesmaximal
16C4×#497C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic, complemented
18C2×Q82C2211yescharacteristic
19C2×D4×Q8C2211yescharacteristic
20C2×#497C2211yescharacteristic
21C2×#489C2212yes—
22C2×#477C2212yes—
23C2×#475C2212yes—
24C2×C4×Q16C2212yes—
25C2×C4×SD16C2212yes—
26#2868C2211yescharacteristic
27#2862C2212yes—
28#2862C2212yes—
29#2862C2211yescharacteristic
30#2582C2212yes—
31#2571—21no—
32#2570—22no—
33#2568—21no—
34#2568C2211yescharacteristic
35#2568C2212yes—
36#2563C2211yescharacteristic
37#2563C2212yes—
38#2511C2214yes—
39#2492—22no—
40#2492—21no—
41#2492—21no—
42#2492—22no—
43#2446—22no—
44#2365—21no—
45#2355—21no—
46#2355C2212yes—
47Q8×C22⋊C4—21no—
48Q8×C22⋊C4—21no—
49C4×C4⋊Q8C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#124C2211yescharacteristic
52C4×#124C2212yes—
53C42×Q8C2211yescharacteristic
54#2097C2214yes—
55#2074C2214yes—
56#2074C2214yes—
57#2072C2212yes—
58#2071C2212yes—
59#2058C2214yes—
60#2057C2214yes—
61#2056C2214yes—
62#2055C2214yes—
63#2024C2214yes—
64#2024C2214yes—
65#1993C2212yes—
66#1993C2212yes—
67#1993C2212yes—
68#1990C2212yes—
69#1990C2212yes—
70#1984C2212yes—
71#1973C2212yes—
72#1971C2212yes—
73#1960C2211yescharacteristic
74#1960C2212yes—
75#1959C2211yescharacteristic
76#1949C2214yes—
77#1949C2214yes—
78#1939C2212yes—
79#1927C2214yes—
80#1925C2214yes—
81#1917C2214yes—
82#1878C2214yes—
83#1877C2214yes—
84#1864C2214yes—
85#1863C2214yes—
86#1847C2211yescharacteristic
87#1847C2212yes—
88#1836C2214yes—
89#1803C2212yes—
90#1802C2212yes—
91#1767C2212yes—
92#1767C2211yescharacteristic
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C23×Q8—21no—
96C22×Q16C2312yes—
97C22×SD16C2312yes—
98Q82—44no—
99D4×Q8—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#124C2311yescharacteristic
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123#497—24no—
124#489—28no—
125#477—44no—
126#475—44no—
127C4×Q16—44no—
128C4×SD16—44no—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#97C2312yes—
144C2×#97C2312yes—
145C2×#97C2311yescharacteristic
146C2×#97C2311yescharacteristic
147C2×C4×C8C2312yes—
148#400—24no—
149#400—22no—
150#400—22no—
151#400—22no—
152#400—24no—
153#398—24no—
154#393—22no—
155#393—22no—
156#393—22no—
157#391—22no—
158#391—21no—
159#391—21no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—21no—
165#391—22no—
166#391—21no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—21no—
171#391—22no—
172#391—21no—
173#391C2312yes—
174#391C2314yes—
175#391C2312yes—
176#391C2311yescharacteristic
177#391C2314yes—
178#390C2312yes—
179#390C2311yescharacteristic
180#389—22no—
181#389—21no—
182#389—22no—
183#389—21no—
184#389C2314yes—
185#388—22no—
186#388—21no—
187#388—22no—
188#388—21no—
189#387—24no—
190#387—24no—
191#387—24no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#382—22no—
198#382—22no—
199#382—22no—
200#382C2312yes—
201#382C2312yes—
202#382C2312yes—
203#382C2311yescharacteristic
204#382C2312yes—
205#380—21no—
206#380—21no—
207#380—22no—
208#380—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C43C2312yes—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2412yes—
246C22×Q8C2411yescharacteristic
247C22×D4C2411yescharacteristic
248C23×C4—22no—
249C23×C4—21no—
250C2×Q16—24no—
251C2×Q16—24no—
252C2×Q16—42no—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—42no—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C4⋊Q8—44no—
270C4⋊Q8—82no—
271#124—22no—
272#124#4114yes—
273C22⋊Q8—44no—
274C22⋊Q8—48no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—42no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—84no—
284C4×D4—24no—
285C4×D4—44no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—42no—
341#107—44no—
342#106—42no—
343#106—44no—
344C4⋊C8—28no—
345C4⋊C8—24no—
346#103—28no—
347#103—42no—
348#103—42no—
349#103—42no—
350#103—44no—
351#103—44no—
352#102—28no—
353#102—42no—
354#97—24no—
355#97—24no—
356#97—22no—
357#97—22no—
358C4×C8—24no—
359#95—22no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—21no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—81no—
408C2×Q8—81no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—816no—
412C2×Q8#13714yes—
413C2×Q8C2×#4112yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4C2×#4112yes—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C2511yescharacteristic
443Q16—48no—
444SD16—48no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C82−1+414yes—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—84no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—82no—
490C22⋊C4—81no—
491C22⋊C4—82no—
492C22⋊C4—81no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—24no—
497C42—22no—
498C42—24no—
499C42—41no—
500C42—42no—
501C42—42no—
502C42—42no—
503C42—41no—
504C42—44no—
505C42—44no—
506C42—42no—
507C42—84no—
508C42C2×#4112yes—
509C42H(5,𝔽2)14yes—
510C422−1+412yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C23×D411yescharacteristic
519Q8—28no—
520Q8—24no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—88no—
527Q8—88no—
528Q8—84no—
529Q8—162no—
530Q8—164no—
531D4—24no—
532D4—42no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—88no—
581C2×C4—84no—
582C2×C4#53512yes—
583C2×C4#53512yes—
584C2×C4#54812yes—
585C2×C4#54812yes—
586C2×C4C2×#13714yes—
587C2×C4C2×#13712yes—
588C2×C4#57512yes—
589C2×C4C22×#4111yescharacteristic
590C2×C4C2×H(5,𝔽2)12yes—
591C2×C4C2×2−1+412yes—
592C2×C4C2×2−1+411yescharacteristic
593C8—24no—
594C8—44no—
595C8—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22C2×#53511yescharacteristic
604C22C2×#54811yescharacteristic
605C22#349911yescharacteristic
606C22C22×#13711yescharacteristic
607C22C2×#57511yescharacteristic
608C22#358511yescharacteristic
609C22#358611yescharacteristic
610C4—24no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4—81no—
622C4—84no—
623C4—82no—
624C4—82no—
625C4—81no—
626C4—82no—
627C4#321514yes—
628C4#352012yes—
629C4#355012yes—
630C2—82no—
631C2#3916711yescharacteristic
632C2C2×#321511yescharacteristic
633C2#6082911yescharacteristic
634C2#6132211yescharacteristic
635C2#6132411yescharacteristic
636C2#6250711yescharacteristic
637C2#6251011yescharacteristic
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. #36598C2semidirect
3. #33293C2semidirect
5. #32451C2semidirect
6. #30215C2semidirect
7. #29211C2semidirect
10. #28522C2semidirect
11. #24683C2semidirect
13. #24333C2semidirect
17. #21345C2semidirect

Maximal subgroups

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

subgroupindex
#365982
Q8×#1242
#332932
#324552
#324512
#302152
#292112
#287262
#286202
#285222
#246832
#246792
#243332
#243302
#222242
C4×#4972
#213452