Finite Group Databasebeta

← #6463620

Subgroups — #6463620

Order 512.

2008 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34275C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#29267C214yesmaximal, complemented
4#29263C214yesmaximal, complemented
5#29262C212yesmaximal, complemented
6#29260C212yesmaximal, complemented
7#29258C212yesmaximal, complemented
8#29254C212yesmaximal, complemented
9#23731C211yesmaximal, characteristic, complemented
10#23730C211yesmaximal, characteristic, complemented
11#23452C211yesmaximal, characteristic, complemented
12C2×#2023C211yesmaximal, characteristic, complemented
13#21263C212yesmaximal
14#21175C212yesmaximal, complemented
15#20879C212yesmaximal, complemented
16#20878C212yesmaximal, complemented
17#20754C211yesmaximal, characteristic, complemented
18C2×#535C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×#428C2212yes—
23C2×#427C2212yes—
24C2×#424C2212yes—
25C2×#417C2212yes—
26C2×#418C2212yes—
27C2×#416C2212yes—
28C22×#102C2212yes—
29#2369—21no—
30#2369—21no—
31#2365—21no—
32#2365—21no—
33#2363—21no—
34#2363—21no—
35#2361—21no—
36#2361—21no—
37#2358—22no—
38#2358—22no—
39#2345—22no—
40Q8×C22⋊C4—22no—
41#2331C2214yescomplemented
42#2328C2212yes—
43#2326C2212yes—
44#2317C2211yescharacteristic
45#2316C2212yes—
46C4×C4⋊Q8C2212yescomplemented
47C4×C4⋊Q8C2212yescomplemented
48C4×#127C2212yes—
49C4×#124C2212yes—
50#2297C2211yescharacteristic
51C2×#390C2211yescharacteristic
52#2285C2211yescharacteristic
53#2274C2212yes—
54C4×#117C2211yescharacteristic
55#2097C2212yescomplemented
56#2096C2211yescharacteristic, complemented
57#2095C2211yescharacteristic, complemented
58#2094C2212yes—
59#2094C2211yescharacteristic, complemented
60#2093C2212yes—
61#2093C2211yescharacteristic, complemented
62#2092C2214yes—
63#2092C2212yescomplemented
64#2024C2212yescomplemented
65#2024C2212yescomplemented
66#2023C2212yes—
67#2023C2212yes—
68#2023C2214yes—
69#2023C2212yescomplemented
70#2023C2212yescomplemented
71#1983C2214yes—
72#1983C2214yes—
73#1982C2214yes—
74#1982C2214yes—
75#1949C2214yescomplemented
76#1949C2214yescomplemented
77#1948C2214yescomplemented
78#1947C2214yescomplemented
79#1865C2214yescomplemented
80#1864C2214yes—
81#1864C2214yes—
82#1864C2214yescomplemented
83#1863C2214yescomplemented
84#1862C2214yes—
85#1862C2214yes—
86#1862C2214yescomplemented
87#1832C2214yescomplemented
88#1831C2212yescomplemented
89#1831C2212yescomplemented
90#1736C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C23×D4C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#535—24no—
101#535Q818yescomplemented
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#127C2312yes—
108C2×#124C2312yes—
109C2×#121—21no—
110C2×#121—21no—
111C2×#120—22no—
112#520—44no—
113#518—44no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—22no—
137C22×C42C2311yescharacteristic
138#428—44no—
139#427—44no—
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#106C2314yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146#424—28no—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149#418—44no—
150#417—28no—
151#416—44no—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#102—28no—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163#391—22no—
164#391—22no—
165#391—24no—
166#391—24no—
167#391C2311yescharacteristic
168#391C2311yescharacteristic
169#390—22no—
170#390—22no—
171#390—21no—
172#390—21no—
173#390C2311yescharacteristic
174#390C2312yes—
175#390C2312yes—
176#390C2311yescharacteristic
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—24no—
190#387—24no—
191#386—21no—
192#386—21no—
193#386—22no—
194#386—22no—
195#386—22no—
196#386—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384C2312yes—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382C2312yes—
204#382C2312yes—
205#380—21no—
206#380—21no—
207#380—24no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—24no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227#376C2312yes—
228C43C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2312yes—
234#336C2311yescharacteristic
235#336C2311yescharacteristic
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239#336C2312yes—
240C25—21no—
241C2×#41—22no—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41C2×Q814yes—
245C22×Q8—28no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—82no—
249C22×Q8—82no—
250C22×Q8C2411yescharacteristic
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—24no—
256C22×D4—22no—
257C22×D4—24no—
258C22×D4C2×Q814yes—
259C22×D4C2412yes—
260C22×D4C2411yescharacteristic
261C23×C4—22no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—24no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8—24no—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2414yes—
279C4⋊Q8—28no—
280C4⋊Q8—28no—
281C4⋊Q8—28no—
282C4⋊Q8—28no—
283#127—24no—
284#127C2×Q818yescomplemented
285#124—24no—
286#124C2×Q818yescomplemented
287#121—28no—
288#121—28no—
289#120—216no—
290C4×Q8—44no—
291C4×Q8—84no—
292C4×Q8—84no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—48no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—84no—
300C4×D4—84no—
301#117—24no—
302#117—24no—
303#117—24no—
304#117—24no—
305#117—44no—
306#117C2×Q814yes—
307#117#4114yes—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4#4118yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—48no—
352C2×C22⋊C4—82no—
353C2×C22⋊C4—82no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372#107—42no—
373#107—42no—
374#107—44no—
375#106—42no—
376#106—42no—
377#106—44no—
378C4⋊C8—28no—
379C4⋊C8—28no—
380#103—28no—
381#103—42no—
382#103—42no—
383#102—28no—
384#102—28no—
385#102—28no—
386#102—42no—
387#102—42no—
388#102—44no—
389#102—44no—
390#95—21no—
391#95—21no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—22no—
407C24—21no—
408C24—24no—
409C24—42no—
410C24—42no—
411C24—81no—
412C24—81no—
413C24—81no—
414C24—81no—
415C24C22×D411yescharacteristic
416#41—216no—
417#41—48no—
418C2×Q8—28no—
419C2×Q8—28no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×Q8—816no—
428C2×Q8C22×Q812yes—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—28no—
432C2×D4—28no—
433C2×D4—216no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—48no—
445C2×D4—82no—
446C2×D4—82no—
447C2×D4—82no—
448C2×D4—82no—
449C2×D4C22×Q814yes—
450C2×D4C22×Q812yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—81no—
481C22×C4—81no—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×Q812yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4114yes—
494C22×C4H(5,𝔽2)14yes—
495C22×C4C2511yescharacteristic
496M16—48no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—48no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4C22⋊Q814yes—
531C4⋊C4C22⋊Q814yes—
532C4⋊C4C2×#4118yes—
533C22⋊C4—24no—
534C22⋊C4—28no—
535C22⋊C4—28no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—44no—
544C22⋊C4—42no—
545C22⋊C4—81no—
546C22⋊C4—81no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4C22⋊Q818yes—
550C42—22no—
551C42—22no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—24no—
557C42—42no—
558C42—42no—
559C42—44no—
560C42—44no—
561C42—42no—
562C42—84no—
563C42—84no—
564C42C22×Q814yes—
565C42C2×#4114yes—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—24no—
571C23—22no—
572C23—41no—
573C23—42no—
574C23—41no—
575C23—42no—
576C23—41no—
577C23—42no—
578C23—41no—
579C23—42no—
580C23—42no—
581C23—41no—
582C23—42no—
583C23—41no—
584C23—48no—
585C23—84no—
586C23—84no—
587C23—162no—
588C23—162no—
589C23—162no—
590C23—162no—
591C23C2×C22⋊Q812yes—
592C23#52512yes—
593C23#53512yes—
594C23C23×D411yescharacteristic
595Q8—24no—
596Q8—416no—
597Q8—42no—
598Q8—88no—
599D4—28no—
600D4—24no—
601D4—48no—
602D4—48no—
603D4—44no—
604D4—42no—
605D4—88no—
606D4—88no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—28no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—48no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—88no—
652C2×C4—164no—
653C2×C4—164no—
654C2×C4—164no—
655C2×C4—162no—
656C2×C4C2×C22⋊Q814yes—
657C2×C4C2×C22⋊Q814yes—
658C2×C4C2×C22⋊Q812yes—
659C2×C4#52512yes—
660C2×C4#53512yes—
661C2×C4#55418yes—
662C2×C4#55918yes—
663C2×C4#57612yes—
664C2×C4#57712yes—
665C2×C4C23×Q811yescharacteristic
666C2×C4C22×#4112yes—
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)12yes—
669C8—42no—
670C8—42no—
671C8—44no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—42no—
679C22—44no—
680C22—42no—
681C22—44no—
682C22—42no—
683C22—44no—
684C22—81no—
685C22—82no—
686C22—82no—
687C22—81no—
688C22—81no—
689C22—82no—
690C22—81no—
691C22—82no—
692C22—168no—
693C22#258014yes—
694C22C22×C22⋊Q811yescharacteristic
695C22C2×#52511yescharacteristic
696C22C2×#53511yescharacteristic
697C22C2×#57611yescharacteristic
698C22C2×#57711yescharacteristic
699C22#358511yescharacteristic
700C22#358611yescharacteristic
701C4—24no—
702C4—24no—
703C4—22no—
704C4—22no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—41no—
709C4—48no—
710C4—41no—
711C4—82no—
712C4—81no—
713C4—82no—
714C4—82no—
715C4—81no—
716C4#258014yes—
717C4C2×#55412yes—
718C4C2×#55912yes—
719C2—24no—
720C2—82no—
721C2—84no—
722C2C2×#258011yescharacteristic
723C2#6049411yescharacteristic
724C2#6049511yescharacteristic
725C2#6057811yescharacteristic
726C2#6058211yescharacteristic
727C2#6082211yescharacteristic
728C2#6082911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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
2. #34216C2semidirect
3. #29267C2semidirect
4. #29263C2semidirect
5. #29262C2semidirect
6. #29260C2semidirect
7. #29258C2semidirect
8. #29254C2semidirect
9. #23731C2semidirect
10. #23730C2semidirect
11. #23452C2semidirect
12. C2×#2023C2semidirect
14. #21175C2semidirect
15. #20879C2semidirect
16. #20878C2semidirect
17. #20754C2semidirect
41. #2331C22semidirect
46. C4×C4⋊Q8C22semidirect
47. C4×C4⋊Q8C22semidirect
55. #2097C22semidirect
56. #2096C22semidirect
57. #2095C22semidirect
59. #2094C22semidirect
61. #2093C22semidirect
63. #2092C22semidirect
64. #2024C22semidirect
65. #2024C22semidirect
69. #2023C22semidirect
70. #2023C22semidirect
75. #1949C22semidirect
76. #1949C22semidirect
77. #1948C22semidirect
78. #1947C22semidirect
79. #1865C22semidirect
82. #1864C22semidirect
83. #1863C22semidirect
86. #1862C22semidirect
87. #1832C22semidirect
88. #1831C22semidirect
89. #1831C22semidirect
101. #535Q8semidirect
284. #127C2×Q8semidirect
286. #124C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#342752
#342162
#292672
#292632
#292622
#292602
#292582
#292542
#237312
#237302
#234522
C2×#20232
#212632
#211752
#208792
#208782
#207542