Finite Group Databasebeta

← #6432374

Subgroups — #6432374

Order 512.

2274 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2947C212yesmaximal
2#34253C211yesmaximal, characteristic
3#34209C211yesmaximal, characteristic, complemented
4#23691C214yesmaximal, complemented
5#23689C214yesmaximal, complemented
6#23685C212yesmaximal, complemented
7#23683C212yesmaximal, complemented
8#23679C212yesmaximal, complemented
9#23677C212yesmaximal, complemented
10#23657C211yesmaximal, characteristic, complemented
11#23627C211yesmaximal, characteristic
12#23615C212yesmaximal, complemented
13#23545C212yesmaximal
14C2×#2043C211yesmaximal, characteristic, complemented
15#23432C211yesmaximal, characteristic
16#20430C211yesmaximal, characteristic, complemented
17#20417C212yesmaximal, complemented
18C22×#137C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×#519C2212yes—
22#2947—216no—
23C2×#441C2214yes—
24C2×#440C2214yes—
25C2×C4×Q16C2214yes—
26C2×C4×SD16C2214yes—
27C2×#428C2212yes—
28C2×#424C2212yes—
29C2×#419C2212yes—
30C2×#416C2212yes—
31C22×#103C2212yes—
32C22×#103C2212yes—
33C2×#405C2212yes—
34#2365—21no—
35#2365—21no—
36#2365—21no—
37#2365—21no—
38#2353—22no—
39#2352—22no—
40#2347—21no—
41#2346—21no—
42#2345—21no—
43#2344—21no—
44#2344—22no—
45#2344—22no—
46#2327C2214yes—
47#2320C2212yes—
48#2319C2212yes—
49#2316C2211yescharacteristic
50#2311C2211yescharacteristic
51#2310C2212yes—
52C4×C4⋊Q8C2212yes—
53C4×C4⋊Q8C2212yes—
54#2297C2211yescharacteristic
55#2295C2212yes—
56#2293C2212yes—
57C2×#389C2211yescharacteristic
58#2285C2211yescharacteristic
59#2274C2211yescharacteristic
60C4×#117C2211yescharacteristic
61#2085C2214yes—
62#2084C2212yes—
63#2083C2212yes—
64#2074C2212yes—
65#2073C2212yes—
66#2072C2211yescharacteristic
67#2072C2211yescharacteristic
68#2071C2211yescharacteristic
69#2071C2211yescharacteristic
70#2068C2214yes—
71#2068C2214yes—
72#2066C2214yes—
73#2066C2214yes—
74#2056C2214yes—
75#2056C2214yes—
76#2055C2214yes—
77#2055C2214yes—
78#2043C2212yes—
79#2043C2212yes—
80#2043C2214yes—
81#2016C2212yes—
82#2016C2212yes—
83#2013C2212yes—
84#2013C2212yes—
85#1768C2214yes—
86#1767C2212yes—
87C4×C4⋊C8C2212yes—
88#1763C2214yes—
89#1763C2214yes—
90C4×#103C2214yes—
91C4×#103C2214yes—
92#1736C2212yes—
93#1735C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C22×#41C2311yescharacteristic
97C23×Q8C2312yes—
98C2×#137—28no—
99C2×#137—28no—
100C2×#137—28no—
101C22×Q16C2314yes—
102C22×SD16C2314yes—
103C22×M16C2312yes—
104C23×C8C2312yes—
105C2×C4⋊Q8—28no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110#520—24no—
111#520D418yes—
112#519—28no—
113#519—28no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4C2312yes—
121C2×#117—21no—
122C2×#117—24no—
123C2×#117—22no—
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#441—216no—
136#440—216no—
137C4×Q16—216no—
138C4×SD16—216no—
139#428—28no—
140C2×#107C2314yes—
141C2×#106C2314yes—
142#424—28no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145#419—28no—
146#416—28no—
147C2×#103—28no—
148C2×#103—28no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159#405—28no—
160C2×#97C2314yes—
161C2×C4×C8C2314yes—
162#391—22no—
163#391—22no—
164#391C2314yes—
165#391C2314yes—
166#391C2312yes—
167#391C2311yescharacteristic
168#391C2311yescharacteristic
169#390C2312yes—
170#390C2311yescharacteristic
171#390C2311yescharacteristic
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—24no—
193#387—24no—
194#387—24no—
195#387—24no—
196#386—21no—
197#386—21no—
198#386—22no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#383—22no—
202#382—22no—
203#382C2312yes—
204#381—24no—
205#381—22no—
206#380—21no—
207#380—21no—
208#380—21no—
209#380—21no—
210#380—22no—
211#380—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4D412yes—
233#376—22no—
234#376C2312yes—
235C2×#95—21no—
236C43C2312yes—
237#340C2314yes—
238#340C2314yes—
239#340C2312yes—
240#340C2312yes—
241#340C2312yes—
242#340C2312yes—
243#336C2311yescharacteristic
244#336C2311yescharacteristic
245#336C2312yes—
246#336C2312yes—
247#336C2312yes—
248C2×#41—22no—
249C2×#41—24no—
250C2×#41—24no—
251C2×#41C2×D414yes—
252C22×Q8—22no—
253C22×Q8—24no—
254C22×Q8—24no—
255C22×Q8—24no—
256C22×Q8—24no—
257C22×Q8—24no—
258C22×Q8#4118yes—
259C22×Q8C2411yescharacteristic
260C22×Q8C2414yes—
261C22×D4—21no—
262C22×D4C2411yescharacteristic
263C23×C4—22no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—22no—
267C23×C4—21no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2411yescharacteristic
271#137—264no—
272C2×Q16—28no—
273C2×Q16—28no—
274C2×Q16—28no—
275C2×SD16—28no—
276C2×SD16—28no—
277C2×SD16—28no—
278C2×M16—24no—
279C2×M16—24no—
280C2×M16#4118yes—
281C22×C8—24no—
282C22×C8—24no—
283C22×C8—24no—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2414yes—
287C4⋊Q8—28no—
288C4⋊Q8—28no—
289C4⋊Q8—28no—
290C4⋊Q8—28no—
291C4×Q8—24no—
292C4×Q8—28no—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—48no—
297C4×Q8—48no—
298C4×Q8C2×D418yes—
299C4×D4—216no—
300C4×D4—24no—
301C4×D4C2×D418yes—
302#117—24no—
303#117—28no—
304#117—24no—
305#117—24no—
306#117—24no—
307#117—416no—
308#117—84no—
309#117C2×D414yes—
310#117#4114yes—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—82no—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D414yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—22no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2412yes—
374#107—28no—
375#106—28no—
376C4⋊C8—28no—
377C4⋊C8—28no—
378#103—24no—
379#103—24no—
380#103—28no—
381#103—28no—
382#103—28no—
383#103—28no—
384#103—28no—
385#103—28no—
386#102—24no—
387#102—24no—
388#97—28no—
389C4×C8—28no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95C2×D412yes—
409#95C2×D412yes—
410#95C2×D412yes—
411#95C2×D412yes—
412C24—21no—
413C24C22×D411yescharacteristic
414#41—216no—
415#41—48no—
416C2×Q8—28no—
417C2×Q8—21no—
418C2×Q8—24no—
419C2×Q8—216no—
420C2×Q8—216no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—44no—
428C2×Q8—48no—
429C2×Q8—48no—
430C2×Q8—44no—
431C2×Q8C22×D412yes—
432C2×Q8C2×#4118yes—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C22×D412yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—28no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4C2511yescharacteristic
483Q16—232no—
484SD16—232no—
485M16—216no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—48no—
495C2×C8C2×#4118yes—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—24no—
501C4⋊C4—28no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4#12114yes—
524C4⋊C4#12114yes—
525C4⋊C4C22×D414yes—
526C22⋊C4—24no—
527C22⋊C4—28no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—82no—
533C22⋊C4—81no—
534C22⋊C4—81no—
535C22⋊C4—82no—
536C22⋊C4#12118yes—
537C42—28no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—24no—
542C42—24no—
543C42—22no—
544C42—22no—
545C42—42no—
546C42—42no—
547C42—44no—
548C42—48no—
549C42—48no—
550C42—42no—
551C42—82no—
552C42—82no—
553C42C22×D414yes—
554C42C2×#4114yes—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23C2×#12112yes—
567C23#52512yes—
568C23#53212yes—
569C23C23×D411yescharacteristic
570Q8—28no—
571Q8—24no—
572Q8—28no—
573Q8—416no—
574Q8—44no—
575Q8—44no—
576Q8—42no—
577D4—24no—
578D4—48no—
579D4—42no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—416no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—88no—
629C2×C4—84no—
630C2×C4—162no—
631C2×C4C2×#12114yes—
632C2×C4C2×#12114yes—
633C2×C4C2×#12112yes—
634C2×C4#52512yes—
635C2×C4#52518yes—
636C2×C4#52918yes—
637C2×C4#53212yes—
638C2×C4#57712yes—
639C2×C4#57812yes—
640C2×C4C23×D411yescharacteristic
641C2×C4C22×#4112yes—
642C2×C4C22×#4112yes—
643C2×C4C22×#4111yescharacteristic
644C8—28no—
645C8—42no—
646C8—42no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22#239214yes—
661C22C22×#12111yescharacteristic
662C22C2×#52511yescharacteristic
663C22C2×#53211yescharacteristic
664C22C2×#57711yescharacteristic
665C22C2×#57811yescharacteristic
666C22#358611yescharacteristic
667C22#358611yescharacteristic
668C4—24no—
669C4—24no—
670C4—22no—
671C4—22no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—41no—
676C4—41no—
677C4—42no—
678C4—48no—
679C4—82no—
680C4—82no—
681C4—81no—
682C4—81no—
683C4—82no—
684C4#239214yes—
685C4C2×#52512yes—
686C4C2×#52912yes—
687C2—24no—
688C2—82no—
689C2C2×#239211yescharacteristic
690C2#6042911yescharacteristic
691C2#6043211yescharacteristic
692C2#6058211yescharacteristic
693C2#6058311yescharacteristic
694C2#6072311yescharacteristic
695C2#6072611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #34209C2semidirect
4. #23691C2semidirect
5. #23689C2semidirect
6. #23685C2semidirect
7. #23683C2semidirect
8. #23679C2semidirect
9. #23677C2semidirect
10. #23657C2semidirect
12. #23615C2semidirect
14. C2×#2043C2semidirect
16. #20430C2semidirect
17. #20417C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29472
#342532
#342092
#236912
#236892
#236852
#236832
#236792
#236772
#236572
#236272
#236152
#235452
C2×#20432
#234322
#204302
#204172