Finite Group Databasebeta

← #6564401

Subgroups — #6564401

Order 512.

1901 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3331C212yesmaximal
2#36180C211yesmaximal, characteristic, complemented
3#34316C211yesmaximal, characteristic
4#29868C211yesmaximal, characteristic, complemented
5#29717C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28581C212yesmaximal, complemented
8#28161C211yesmaximal, characteristic, complemented
9#27647C212yesmaximal, complemented
10#25324C212yesmaximal, complemented
11#25219C214yesmaximal, complemented
12#24680C212yesmaximal, complemented
13#24546C214yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23630C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22502C212yesmaximal, complemented
18C2×#554C2212yes—
19C2×#547C2211yescharacteristic
20#3331—216no—
21C2×#487C2212yes—
22C2×#483C2214yes—
23C2×#476C2212yes—
24C2×#470C2214yes—
25C2×#462C2212yes—
26C2×#452C2214yes—
27C2×C4×SD16C2212yes—
28C2×#436C2212yes—
29C2×#417C2214yes—
30#2868C2211yescharacteristic
31#2857C2212yes—
32#2567C2212yes—
33#2552—21no—
34#2552—21no—
35#2502C2214yes—
36#2440—22no—
37#2440—22no—
38#2374—22no—
39#2365C2211yescharacteristic
40#2363—21no—
41#2362C2212yes—
42#2354C2211yescharacteristic
43#2354C2211yescharacteristic
44D4×C4⋊C4—21no—
45D4×C4⋊C4—21no—
46#2345—21no—
47#2341C2212yes—
48#2335—22no—
49#2324—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2297C2212yes—
54#2293C2211yescharacteristic
55C2×#384C2212yes—
56#2274—21no—
57C42×D4C2211yescharacteristic
58#2084C2211yescharacteristic
59#2084C2212yes—
60#2084C2212yes—
61#2072C2212yes—
62#2072C2212yes—
63#2072C2212yes—
64#2072C2214yes—
65#2071C2212yes—
66#2071C2211yescharacteristic
67#2071C2214yes—
68#2071C2212yes—
69#2066C2212yes—
70#2056C2214yes—
71#2055C2214yes—
72#2055C2212yes—
73#2043C2211yescharacteristic
74#2016C2212yes—
75#2013C2211yescharacteristic
76#1990C2212yes—
77#1987C2212yes—
78#1948C2214yes—
79#1939C2212yes—
80#1927C2214yes—
81#1894C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1873C2214yes—
85#1851C2214yes—
86#1831C2211yescharacteristic
87#1823C2214yes—
88#1803C2212yes—
89#1803C2214yes—
90#1803C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C22×#41C2312yes—
97C22×Q16C2314yes—
98C22×SD16C2312yes—
99C22×M16C2314yes—
100#554—28no—
101#554—28no—
102#547—24no—
103#547D418yes—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#125—22no—
110C2×#123—22no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2312yes—
114C2×#121—21no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×D4—22no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117—21no—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4C2314yes—
130C22×C42C2312yes—
131#487—28no—
132#483—216no—
133#476—28no—
134#470—216no—
135#462—28no—
136#452—216no—
137C4×SD16—28no—
138#436—28no—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2312yes—
144#417—216no—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156#398—24no—
157#398—24no—
158#395—24no—
159#391C2312yes—
160#391C2312yes—
161#391C2314yes—
162#391C2312yes—
163#390—21no—
164#390—22no—
165#390—22no—
166#390C2311yescharacteristic
167#390C2312yes—
168#390C2311yescharacteristic
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#388—22no—
181#388—22no—
182#388—22no—
183#388—21no—
184#388—21no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#386—22no—
190#386—22no—
191#385—22no—
192#385—24no—
193#385—24no—
194#384—21no—
195#384—21no—
196#384—22no—
197#384—22no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2314yes—
201#384C2312yes—
202#384C2312yes—
203#380—22no—
204#380—22no—
205#380—21no—
206#380—22no—
207#380C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C43C2311yescharacteristic
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2311yescharacteristic
240#336C2314yes—
241#336C2312yes—
242C2×#41—24no—
243C2×#41—44no—
244C2×#41—44no—
245C2×#41C2×D418yes—
246C22×Q8—22no—
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C22×D4—22no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—24no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C2×Q16—28no—
259C2×Q16—44no—
260C2×Q16—44no—
261C2×SD16—24no—
262C2×SD16—42no—
263C2×SD16—42no—
264C2×M16—28no—
265C2×M16—44no—
266C2×M16—44no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—22no—
272C4⋊Q8—24no—
273C4⋊Q8—42no—
274C4⋊Q8—82no—
275C4⋊Q8—82no—
276C4⋊Q8C2×D414yes—
277#125—48no—
278#123—216no—
279C22⋊Q8—24no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8—416no—
283C22⋊Q8C2×D418yes—
284#121—28no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—22no—
292C4×D4—48no—
293C4×D4C2×D414yes—
294C4×D4C2×D414yes—
295C4×D4#4114yes—
296#117—44no—
297#117—44no—
298#117—44no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—84no—
326C2×C4⋊C4—84no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#106—24no—
368#106—28no—
369C4⋊C8—24no—
370C4⋊C8—44no—
371C4⋊C8—42no—
372#103—28no—
373#103—28no—
374#103—28no—
375#103—24no—
376#103—42no—
377#103—44no—
378#103—42no—
379#102—24no—
380#102—28no—
381C22⋊C8—28no—
382C4×C8—24no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95—24no—
395#95—24no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—21no—
400C24C22×D412yes—
401#41—416no—
402#41—88no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—44no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8C22×D414yes—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4#12114yes—
425C2×D4C22×D412yes—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—48no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4114yes—
462C22×C4C2511yescharacteristic
463Q16—88no—
464SD16—48no—
465M16—88no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C4⋊C4—24no—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—81no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4#12114yes—
508C4⋊C4#13714yes—
509C4⋊C4C22×D412yes—
510C22⋊C4—28no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—81no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4#12118yes—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—42no—
530C42—42no—
531C42—41no—
532C42—44no—
533C42—44no—
534C42—42no—
535C42—82no—
536C42—82no—
537C42—82no—
538C42#13714yes—
539C42C22×D412yes—
540C42C2×#4112yes—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C2×#12114yes—
555C23C23×D411yescharacteristic
556Q8—28no—
557Q8—44no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—164no—
563D4—24no—
564D4—28no—
565D4—42no—
566D4—88no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—48no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—88no—
615C2×C4—168no—
616C2×C4C2×#12112yes—
617C2×C4C2×#12114yes—
618C2×C4C2×#12112yes—
619C2×C4C2×#13714yes—
620C2×C4C2×#13712yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4#57712yes—
625C2×C4C23×D411yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4#58512yes—
628C2×C4#58512yes—
629C8—42no—
630C8—44no—
631C8—42no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22C22×#12111yescharacteristic
647C22#348511yescharacteristic
648C22#350011yescharacteristic
649C22C22×#13711yescharacteristic
650C22C22×#13711yescharacteristic
651C22C2×#57711yescharacteristic
652C22#358511yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—24no—
657C4—22no—
658C4—24no—
659C4—42no—
660C4—41no—
661C4—44no—
662C4—42no—
663C4—82no—
664C4—84no—
665C4—81no—
666C4—81no—
667C4—82no—
668C4—82no—
669C4#303914yes—
670C4#348512yes—
671C4#352112yes—
672C2—44no—
673C2—82no—
674C2#3600911yescharacteristic
675C2C2×#303911yescharacteristic
676C2#6043111yescharacteristic
677C2#6171911yescharacteristic
678C2#6173511yescharacteristic
679C2#6254211yescharacteristic
680C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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
2. #36180C2semidirect
4. #29868C2semidirect
5. #29717C2semidirect
6. #28717C2semidirect
7. #28581C2semidirect
8. #28161C2semidirect
9. #27647C2semidirect
10. #25324C2semidirect
11. #25219C2semidirect
12. #24680C2semidirect
13. #24546C2semidirect
14. #23677C2semidirect
15. #23630C2semidirect
16. #22700C2semidirect
17. #22502C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33312
#361802
#343162
#298682
#297172
#287172
#285812
#281612
#276472
#253242
#252192
#246802
#245462
#236772
#236302
#227002
#225022