Finite Group Databasebeta

← #6566107

Subgroups — #6566107

Order 512.

1787 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3356C212yesmaximal
2#36188C211yesmaximal, characteristic
3#34332C211yesmaximal, characteristic
4#29866C211yesmaximal, characteristic, complemented
5#29756C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28581C212yesmaximal, complemented
8#28206C211yesmaximal, characteristic, complemented
9#27652C212yesmaximal, complemented
10#25454C212yesmaximal, complemented
11#25226C214yesmaximal, complemented
12#24818C212yesmaximal, complemented
13#24590C214yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15#23657C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic
17#22504C212yesmaximal
18C2×#556C2212yes—
19C2×#554C2211yescharacteristic
20#3356—216no—
21C2×#498C2212yes—
22C2×#485C2214yes—
23C2×#483C2214yes—
24C2×#477C2212yes—
25C2×#475C2212yes—
26C2×#470C2214yes—
27C2×C4×Q16C2212yes—
28C2×#436C2212yes—
29C2×#428C2214yes—
30#2868C2211yescharacteristic
31#2857C2212yes—
32#2568C2212yes—
33#2547—21no—
34#2547—21no—
35#2499C2214yes—
36#2496—22no—
37#2496—22no—
38#2370—21no—
39#2369C2211yescharacteristic
40#2365—21no—
41#2365—21no—
42#2360—22no—
43#2360C2212yes—
44Q8×C4⋊C4C2211yescharacteristic
45D4×C4⋊C4—21no—
46D4×C4⋊C4C2211yescharacteristic
47#2345C2212yes—
48#2330—21no—
49#2329—22no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2295C2211yescharacteristic
54C2×#389C2212yes—
55#2285C2212yes—
56#2275—21no—
57C42×D4C2211yescharacteristic
58#2085C2214yes—
59#2085C2214yes—
60#2084C2212yes—
61#2084C2212yes—
62#2084C2212yes—
63#2084C2212yes—
64#2083C2212yes—
65#2083C2211yescharacteristic
66#2072C2211yescharacteristic
67#2072C2212yes—
68#2072C2212yes—
69#2068C2212yes—
70#2066C2214yes—
71#2066C2214yes—
72#2056C2212yes—
73#2043C2212yes—
74#2043C2211yescharacteristic
75#2016C2211yescharacteristic
76#1990C2212yes—
77#1988C2212yes—
78#1949C2214yes—
79#1948C2214yes—
80#1939C2212yes—
81#1928C2214yes—
82#1927C2214yes—
83#1894C2214yes—
84#1878C2214yes—
85#1872C2214yes—
86#1851C2214yes—
87#1831C2211yescharacteristic
88#1825C2214yes—
89#1805C2214yes—
90#1803C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C22×Q16C2312yes—
97C22×M16C2314yes—
98#556—28no—
99#556—28no—
100#554—24no—
101#554D418yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#126—24no—
108C2×#125—21no—
109C2×#124—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117C2314yes—
124C22×C4⋊C4—22no—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42C2312yes—
128#498—28no—
129#485—216no—
130#483—216no—
131#477—28no—
132#475—28no—
133#470—216no—
134C4×Q16—28no—
135#436—28no—
136#428—216no—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C4×C8C2312yes—
154#400—24no—
155#397—24no—
156#397—24no—
157#391—22no—
158#391—22no—
159#391—21no—
160#391—22no—
161#391C2312yes—
162#391C2311yescharacteristic
163#391C2314yes—
164#391C2311yescharacteristic
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2311yescharacteristic
180#389C2311yescharacteristic
181#387—21no—
182#387—22no—
183#387—21no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—24no—
188#387—24no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#382C2314yes—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2312yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227#376—21no—
228C2×#95—22no—
229C43C2311yescharacteristic
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2312yes—
240#336C2311yescharacteristic
241#336C2314yes—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8C2412yes—
245C22×Q8C2411yescharacteristic
246C22×D4C2×D411yescharacteristic
247C23×C4—22no—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C2×Q16—24no—
253C2×Q16—24no—
254C2×Q16—42no—
255C2×M16—28no—
256C2×M16—28no—
257C2×M16—44no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—24no—
264C4⋊Q8—42no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8C2×D414yes—
268#126—416no—
269#125—28no—
270#124—48no—
271C22⋊Q8—24no—
272C22⋊Q8—216no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8C2×D418yes—
276C4×Q8—22no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×Q8—82no—
282C4×Q8C2×D414yes—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—42no—
286C4×D4—44no—
287C4×D4C2×D414yes—
288C4×D4#4114yes—
289#117—28no—
290#117—28no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—84no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D418yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—21no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—44no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—24no—
355#107—28no—
356#106—28no—
357#106—28no—
358#106—28no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361C4⋊C8—44no—
362#103—24no—
363#103—28no—
364#103—24no—
365#103—24no—
366#103—28no—
367#103—42no—
368#103—44no—
369C22⋊C8—28no—
370C4×C8—24no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—24no—
389#95—24no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24C22×D412yes—
394C2×Q8—24no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—81no—
408C2×Q8—81no—
409C2×Q8C22×D412yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4#12114yes—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D414yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4114yes—
447C22×C4C2511yescharacteristic
448Q16—48no—
449M16—416no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C82−1+414yes—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—81no—
494C4⋊C4#12114yes—
495C4⋊C4C22×D412yes—
496C4⋊C4C22×D414yes—
497C22⋊C4—28no—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4#12118yes—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—42no—
511C42—42no—
512C42—41no—
513C42—48no—
514C42—44no—
515C42—42no—
516C42—82no—
517C42—82no—
518C42—82no—
519C42#13714yes—
520C42C22×D412yes—
521C42C2×#4112yes—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23C2×#12114yes—
529C23C23×D411yescharacteristic
530Q8—24no—
531Q8—42no—
532Q8—42no—
533Q8—42no—
534Q8—42no—
535Q8—42no—
536Q8—84no—
537Q8—88no—
538D4—28no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—88no—
590C2×C4—164no—
591C2×C4C2×#12114yes—
592C2×C4C2×#12112yes—
593C2×C4C2×#12112yes—
594C2×C4C2×#13712yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4#57512yes—
598C2×C4#57712yes—
599C2×C4C23×D411yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C2×2−1+412yes—
602C2×C4#58512yes—
603C8—24no—
604C8—42no—
605C8—44no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22C22×#12111yescharacteristic
614C22#348511yescharacteristic
615C22#349911yescharacteristic
616C22C22×#13711yescharacteristic
617C22C2×#57511yescharacteristic
618C22C2×#57711yescharacteristic
619C22#358611yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—82no—
632C4—82no—
633C4—81no—
634C4—81no—
635C4—82no—
636C4—84no—
637C4#303714yes—
638C4#348912yes—
639C4#351512yes—
640C2—44no—
641C2#3602011yescharacteristic
642C2C2×#303711yescharacteristic
643C2#6042911yescharacteristic
644C2#6172311yescharacteristic
645C2#6173911yescharacteristic
646C2#6247811yescharacteristic
647C2#6251111yescharacteristic
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
4. #29866C2semidirect
5. #29756C2semidirect
6. #28687C2semidirect
7. #28581C2semidirect
8. #28206C2semidirect
9. #27652C2semidirect
10. #25454C2semidirect
11. #25226C2semidirect
12. #24818C2semidirect
13. #24590C2semidirect
14. #23679C2semidirect
15. #23657C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33562
#361882
#343322
#298662
#297562
#286872
#285812
#282062
#276522
#254542
#252262
#248182
#245902
#236792
#236572
#227002
#225042