Finite Group Databasebeta

← #6566109

Subgroups — #6566109

Order 512.

1809 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#36180C211yesmaximal, characteristic
3#34324C211yesmaximal, characteristic
4#29869C211yesmaximal, characteristic, complemented
5#29759C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28581C212yesmaximal, complemented
8#28205C211yesmaximal, characteristic, complemented
9#27647C212yesmaximal, complemented
10#25453C212yesmaximal, complemented
11#25219C214yesmaximal, complemented
12#24817C212yesmaximal, complemented
13#24598C214yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15#23658C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic
17#22502C212yesmaximal
18C2×#556C2211yescharacteristic
19C2×#554C2212yes—
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#2567C2212yes—
33#2552—21no—
34#2552—21no—
35#2502C2214yes—
36#2440—22no—
37#2440—22no—
38#2373—22no—
39#2367—21no—
40#2365C2211yescharacteristic
41#2362C2212yes—
42#2355—21no—
43#2354C2211yescharacteristic
44Q8×C4⋊C4C2211yescharacteristic
45D4×C4⋊C4—21no—
46D4×C4⋊C4—21no—
47#2346C2212yes—
48#2334—22no—
49#2326—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2296C2212yes—
54#2295C2211yescharacteristic
55C2×#384C2212yes—
56#2276—21no—
57C42×D4C2211yescharacteristic
58#2084C2212yes—
59#2084C2212yes—
60#2084C2212yes—
61#2084C2214yes—
62#2083C2212yes—
63#2083C2214yes—
64#2083C2212yes—
65#2083C2211yescharacteristic
66#2072C2211yescharacteristic
67#2072C2212yes—
68#2072C2212yes—
69#2068C2212yes—
70#2068C2214yes—
71#2066C2214yes—
72#2056C2212yes—
73#2043C2212yes—
74#2043C2211yescharacteristic
75#2016C2211yescharacteristic
76#1990C2212yes—
77#1987C2212yes—
78#1948C2214yes—
79#1948C2214yes—
80#1939C2212yes—
81#1927C2214yes—
82#1927C2214yes—
83#1894C2214yes—
84#1878C2214yes—
85#1873C2214yes—
86#1851C2214yes—
87#1831C2211yescharacteristic
88#1823C2214yes—
89#1803C2212yes—
90#1803C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C22×Q16C2312yes—
97C22×M16C2314yes—
98#556—24no—
99#556D418yes—
100#554—28no—
101#554—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#126—22no—
108C2×#125—22no—
109C2×#124—21no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2312yes—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2314yes—
128C22×C42C2312yes—
129#498—28no—
130#485—216no—
131#483—216no—
132#477—28no—
133#475—28no—
134#470—216no—
135C4×Q16—28no—
136#436—28no—
137#428—216no—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C4×C8C2312yes—
155#398—24no—
156#398—24no—
157#395—24no—
158#391—21no—
159#391—21no—
160#391C2312yes—
161#391C2314yes—
162#391C2311yescharacteristic
163#391C2311yescharacteristic
164#390—21no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2311yescharacteristic
176#389C2312yes—
177#389C2311yescharacteristic
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—21no—
184#387—22no—
185#387—21no—
186#387—22no—
187#387—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—24no—
192#385—24no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384—22no—
197#384—22no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2314yes—
201#384C2312yes—
202#384C2312yes—
203#382—22no—
204#380—22no—
205#380—22no—
206#380C2312yes—
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—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227#376—21no—
228#376—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—24no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C2×Q16—24no—
252C2×Q16—24no—
253C2×Q16—42no—
254C2×M16—28no—
255C2×M16—28no—
256C2×M16—44no—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—24no—
263C4⋊Q8—42no—
264C4⋊Q8—82no—
265C4⋊Q8—82no—
266C4⋊Q8C2×D414yes—
267#126—216no—
268#125—48no—
269#124—28no—
270C22⋊Q8—24no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—416no—
274C22⋊Q8C2×D418yes—
275C4×Q8—28no—
276C4×Q8—22no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8C2×D414yes—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—48no—
286C4×D4C2×D414yes—
287C4×D4#4114yes—
288#117—24no—
289#117—44no—
290#117C2×D418yes—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—84no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—24no—
358#107—28no—
359#106—28no—
360#106—28no—
361#106—28no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#103—24no—
366#103—28no—
367#103—24no—
368#103—24no—
369#103—28no—
370#103—42no—
371#103—44no—
372C22⋊C8—28no—
373C4×C8—24no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—24no—
390#95—24no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24C22×D412yes—
395C2×Q8—24no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8C22×D412yes—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4#12114yes—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D414yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4114yes—
445C22×C4C2511yescharacteristic
446Q16—48no—
447M16—416no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C82−1+414yes—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—82no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4#12114yes—
493C4⋊C4#13714yes—
494C4⋊C4C22×D414yes—
495C4⋊C4C22×D412yes—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—48no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4#12118yes—
504C42—28no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—24no—
510C42—22no—
511C42—42no—
512C42—42no—
513C42—41no—
514C42—44no—
515C42—44no—
516C42—44no—
517C42—42no—
518C42—82no—
519C42—82no—
520C42#13714yes—
521C42C22×D412yes—
522C42C2×#4112yes—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23C2×#12114yes—
530C23C23×D411yescharacteristic
531Q8—24no—
532Q8—42no—
533Q8—48no—
534Q8—42no—
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—164no—
539D4—28no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—168no—
590C2×C4C2×#12114yes—
591C2×C4C2×#12112yes—
592C2×C4C2×#12112yes—
593C2×C4C2×#13712yes—
594C2×C4C2×#13714yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
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
617C22C22×#13711yescharacteristic
618C22C2×#57711yescharacteristic
619C22#358511yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—24no—
625C4—24no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—44no—
631C4—82no—
632C4—84no—
633C4—82no—
634C4—81no—
635C4—81no—
636C4—82no—
637C4#303914yes—
638C4#348912yes—
639C4#351512yes—
640C2—44no—
641C2#3602011yescharacteristic
642C2C2×#303911yescharacteristic
643C2#6043111yescharacteristic
644C2#6171911yescharacteristic
645C2#6173511yescharacteristic
646C2#6247811yescharacteristic
647C2#6251011yescharacteristic
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. #29869C2semidirect
5. #29759C2semidirect
6. #28687C2semidirect
7. #28581C2semidirect
8. #28205C2semidirect
9. #27647C2semidirect
10. #25453C2semidirect
11. #25219C2semidirect
12. #24817C2semidirect
13. #24598C2semidirect
14. #23679C2semidirect
15. #23658C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33562
#361802
#343242
#298692
#297592
#286872
#285812
#282052
#276472
#254532
#252192
#248172
#245982
#236792
#236582
#227002
#225022