Finite Group Databasebeta

← #6564402

Subgroups — #6564402

Order 512.

1831 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3328C212yesmaximal
2#36188C211yesmaximal, characteristic
3#34334C211yesmaximal, characteristic
4#29866C211yesmaximal, characteristic, complemented
5#29715C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28581C212yesmaximal, complemented
8#28161C211yesmaximal, characteristic, complemented
9#27652C212yesmaximal, complemented
10#25323C212yesmaximal, complemented
11#25220C214yesmaximal, complemented
12#24683C212yesmaximal, complemented
13#24540C214yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15#23629C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic
17#22504C212yesmaximal
18C2×#556C2212yes—
19C2×D4×Q8C2211yescharacteristic
20#3328—216no—
21C2×#487C2212yes—
22C2×#485C2214yes—
23C2×#475C2212yes—
24C2×#470C2214yes—
25C2×#462C2212yes—
26C2×#451C2214yes—
27C2×C4×Q16C2212yes—
28C2×#436C2212yes—
29C2×#419C2214yes—
30#2868C2211yescharacteristic
31#2857C2212yes—
32#2568C2212yes—
33#2547—21no—
34#2547—21no—
35#2499C2214yes—
36#2496—22no—
37#2496—22no—
38#2369—21no—
39#2368—22no—
40#2365—21no—
41#2365—21no—
42#2365C2211yescharacteristic
43#2360C2212yes—
44#2355C2211yescharacteristic
45#2349—21no—
46D4×C4⋊C4C2211yescharacteristic
47Q8×C22⋊C4C2212yes—
48#2331—21no—
49#2329—22no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2295C2211yescharacteristic
54C2×#391C2212yes—
55#2285C2212yes—
56#2275—21no—
57C42×D4C2211yescharacteristic
58#2084C2212yes—
59#2084C2212yes—
60#2083C2211yescharacteristic
61#2074C2214yes—
62#2074C2214yes—
63#2072C2212yes—
64#2072C2212yes—
65#2072C2212yes—
66#2072C2212yes—
67#2072C2211yescharacteristic
68#2072C2212yes—
69#2068C2212yes—
70#2056C2214yes—
71#2056C2214yes—
72#2056C2212yes—
73#2043C2211yescharacteristic
74#2016C2212yes—
75#2016C2211yescharacteristic
76#1990C2212yes—
77#1988C2212yes—
78#1949C2214yes—
79#1939C2212yes—
80#1927C2214yes—
81#1894C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1872C2214yes—
85#1851C2214yes—
86#1831C2211yescharacteristic
87#1825C2214yes—
88#1805C2214yes—
89#1803C2212yes—
90#1803C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C23×Q8C2312yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C22×M16C2314yes—
100#556—28no—
101#556—28no—
102D4×Q8—24no—
103D4×Q8D418yes—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×#126—24no—
111C2×#124—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2312yes—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C42C2312yes—
128#487—28no—
129#485—216no—
130#475—28no—
131#470—216no—
132#462—28no—
133#451—216no—
134C4×Q16—28no—
135#436—28no—
136C2×#107C2312yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2312yes—
141#419—216no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2312yes—
153#400—24no—
154#397—24no—
155#397—24no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2312yes—
163#391C2312yes—
164#391C2314yes—
165#391C2311yescharacteristic
166#391C2312yes—
167#391C2312yes—
168#391C2311yescharacteristic
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#387—21no—
181#387—22no—
182#387—21no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—24no—
187#387—24no—
188#385—22no—
189#385—22no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#382—21no—
198#382C2314yes—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205C4×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226#376—21no—
227C2×#95—22no—
228C2×#95—22no—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2314yes—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8C2×D418yes—
248C22×Q8C2412yes—
249C22×Q8C2412yes—
250C22×Q8C2411yescharacteristic
251C22×D4C2×D411yescharacteristic
252C23×C4—22no—
253C23×C4—22no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C2×Q16—24no—
258C2×Q16—28no—
259C2×Q16—42no—
260C2×Q16—42no—
261C2×Q16—44no—
262C2×Q16—44no—
263C2×M16—28no—
264C2×M16—44no—
265C2×M16—44no—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C4⋊Q8—28no—
270C4⋊Q8—24no—
271C4⋊Q8—22no—
272C4⋊Q8—24no—
273C4⋊Q8—42no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C4⋊Q8C2×D414yes—
277#126—416no—
278#124—48no—
279C22⋊Q8—24no—
280C22⋊Q8—216no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8C2×D418yes—
284C4×Q8—22no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—48no—
290C4×Q8C2×D414yes—
291C4×D4—28no—
292C4×D4—24no—
293C4×D4—42no—
294C4×D4—44no—
295C4×D4C2×D414yes—
296C4×D4#4114yes—
297#117—28no—
298#117—28no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—24no—
357#106—28no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360C4⋊C8—42no—
361#103—24no—
362#103—28no—
363#103—28no—
364#103—28no—
365#103—28no—
366#103—24no—
367#103—42no—
368#103—44no—
369#103—42no—
370C22⋊C8—28no—
371C4×C8—24no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—24no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24C22×D412yes—
397C2×Q8—28no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—416no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8—88no—
417C2×Q8C22×D412yes—
418C2×Q8C22×D414yes—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4#12114yes—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D414yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4114yes—
456C22×C4C2511yescharacteristic
457Q16—48no—
458Q16—88no—
459M16—88no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4#12114yes—
502C4⋊C4C22×D412yes—
503C22⋊C4—28no—
504C22⋊C4—24no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4#12118yes—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—42no—
517C42—42no—
518C42—41no—
519C42—48no—
520C42—44no—
521C42—42no—
522C42—82no—
523C42—82no—
524C42—82no—
525C42#13714yes—
526C42C22×D412yes—
527C42C2×#4112yes—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23C2×#12114yes—
535C23C23×D411yescharacteristic
536Q8—24no—
537Q8—28no—
538Q8—44no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—88no—
545Q8—88no—
546D4—28no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—88no—
598C2×C4—164no—
599C2×C4C2×#12114yes—
600C2×C4C2×#12112yes—
601C2×C4C2×#12112yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4#57512yes—
606C2×C4#57812yes—
607C2×C4C23×D411yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4#58512yes—
610C2×C4#58512yes—
611C8—42no—
612C8—44no—
613C8—42no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C22×#12111yescharacteristic
622C22#348511yescharacteristic
623C22#350011yescharacteristic
624C22C22×#13711yescharacteristic
625C22C2×#57511yescharacteristic
626C22C2×#57811yescharacteristic
627C22#358611yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—44no—
639C4—82no—
640C4—81no—
641C4—82no—
642C4—81no—
643C4—82no—
644C4—84no—
645C4#303714yes—
646C4#348512yes—
647C4#352112yes—
648C2—44no—
649C2#3600911yescharacteristic
650C2C2×#303711yescharacteristic
651C2#6043211yescharacteristic
652C2#6172311yescharacteristic
653C2#6173011yescharacteristic
654C2#6253611yescharacteristic
655C2#6259311yescharacteristic
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. #29715C2semidirect
6. #28717C2semidirect
7. #28581C2semidirect
8. #28161C2semidirect
9. #27652C2semidirect
10. #25323C2semidirect
11. #25220C2semidirect
12. #24683C2semidirect
13. #24540C2semidirect
14. #23679C2semidirect
15. #23629C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33282
#361882
#343342
#298662
#297152
#287172
#285812
#281612
#276522
#253232
#252202
#246832
#245402
#236792
#236292
#227002
#225042