Finite Group Databasebeta

← #6458260

Subgroups — #6458260

Order 512.

1998 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34267C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#27690C214yesmaximal, complemented
4#27688C214yesmaximal, complemented
5#27685C212yesmaximal, complemented
6#27683C212yesmaximal, complemented
7#27680C212yesmaximal, complemented
8#27677C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic, complemented
10C2×#2099C211yesmaximal, characteristic, complemented
11#23734C211yesmaximal, characteristic, complemented
12#23730C211yesmaximal, characteristic, complemented
13#21175C212yesmaximal, complemented
14#20915C212yesmaximal
15#20849C212yesmaximal, complemented
16#20652C212yesmaximal, complemented
17#20630C211yesmaximal, characteristic, complemented
18C2×#537C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×#520C2212yes—
21C2×C4×#41C2211yescharacteristic
22C2×#428C2212yes—
23C2×#427C2212yes—
24C2×#424C2212yes—
25C2×#417C2212yes—
26C2×#416C2212yes—
27C2×#408C2212yes—
28#2364—22no—
29#2363—21no—
30#2362—21no—
31#2361—22no—
32#2361—21no—
33#2360—21no—
34Q8×C4⋊C4—21no—
35Q8×C4⋊C4—21no—
36D4×C4⋊C4—22no—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4—22no—
40#2324C2212yes—
41#2323C2212yes—
42#2317C2214yes—
43#2317C2212yes—
44#2311C2211yescharacteristic
45C4×C22⋊Q8C2212yes—
46C4×#121C2212yes—
47#2297C2211yescharacteristic
48C2×#389C2211yescharacteristic
49#2285C2212yes—
50#2285C2211yescharacteristic
51C2×#384C2212yes—
52C2×#384C2211yescharacteristic
53#2274C2211yescharacteristic
54#2101C2214yes—
55#2100C2212yes—
56#2099C2212yes—
57#2099C2212yes—
58#2099C2212yes—
59#2099C2214yes—
60#2097C2212yes—
61#2096C2211yescharacteristic
62#2096C2212yes—
63#2095C2211yescharacteristic
64#2095C2212yes—
65#2094C2211yescharacteristic
66#2094C2212yes—
67#2093C2211yescharacteristic
68#2093C2212yes—
69#2092C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1948C2214yes—
73#1947C2214yes—
74#1875C2214yes—
75#1874C2214yes—
76#1873C2214yes—
77#1872C2214yes—
78#1854C2214yes—
79#1854C2214yes—
80#1853C2214yes—
81#1852C2214yes—
82#1805C2214yes—
83#1804C2214yes—
84#1803C2214yes—
85#1802C2214yes—
86#1798C2214yes—
87#1797C2212yes—
88#1797C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91C2×#340C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#537—24no—
100#537Q818yes—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C2312yes—
103C2×#121—21no—
104C2×#121—22no—
105C2×#121C2312yes—
106#520—28no—
107#520—44no—
108#520—44no—
109C4×#41—28no—
110C2×C4×Q8—22no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×D4—24no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D412yes—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D412yes—
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C2×#117C2311yescharacteristic
125C2×#117C2312yes—
126C2×#117C2314yes—
127C2×#117C2312yes—
128C2×#117C2311yescharacteristic
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2314yes—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141#428—44no—
142#427—44no—
143C2×#107C2312yes—
144C2×#107C2314yes—
145C2×#107C2312yes—
146C2×#106C2312yes—
147C2×#106C2314yes—
148C2×#106C2312yes—
149#424—28no—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152#417—44no—
153#416—44no—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160#408—28no—
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2314yes—
163#391—21no—
164#391C2311yescharacteristic
165#391C2311yescharacteristic
166#390—21no—
167#390—22no—
168#390C2311yescharacteristic
169#390C2311yescharacteristic
170#389—24no—
171#389—24no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#386—22no—
186#386—21no—
187#386—22no—
188#386—21no—
189#386—22no—
190#385—22no—
191#385—22no—
192#385—24no—
193#385—24no—
194#385—24no—
195#384—22no—
196#384—21no—
197#384—21no—
198#384—22no—
199#384C2314yes—
200#384C2312yes—
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2311yescharacteristic
204#382C2314yes—
205#382C2312yes—
206#380—21no—
207#380—22no—
208#380—21no—
209#380—22no—
210#380C2312yes—
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226C2×#95—21no—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#336C2311yescharacteristic
236#336C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2312yes—
240C2×#41—24no—
241C2×#41—22no—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41C2×Q814yes—
246C2×#41C2×Q814yes—
247C2×#41#4114yes—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2411yescharacteristic
250C22×D4—21no—
251C22×D4—22no—
252C22×D4C2×D412yes—
253C22×D4C2×D411yescharacteristic
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—24no—
258C23×C4—22no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C22⋊Q8—28no—
277C22⋊Q8—24no—
278C22⋊Q8C2×Q818yes—
279#121—28no—
280#121—216no—
281#121—24no—
282#121C2×Q818yes—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—28no—
286C4×Q8—48no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—28no—
292C4×D4—416no—
293C4×D4—48no—
294C4×D4—48no—
295C4×D4—84no—
296#117—28no—
297#117—24no—
298#117—24no—
299#117—24no—
300#117—44no—
301#117—44no—
302#117—44no—
303#117#4118yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4—84no—
331C2×C4⋊C4—82no—
332C2×C4⋊C4—84no—
333C2×C4⋊C4—82no—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×Q814yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—24no—
362C2×C42—28no—
363C2×C42—22no—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42C2×D412yes—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2412yes—
376#107—42no—
377#107—42no—
378#107—44no—
379#106—42no—
380#106—42no—
381#106—44no—
382C4⋊C8—28no—
383C4⋊C8—28no—
384#103—42no—
385#103—42no—
386#103—44no—
387#102—42no—
388#102—42no—
389#102—44no—
390C22⋊C8—28no—
391C22⋊C8—28no—
392#95—22no—
393#95—22no—
394#95—21no—
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—24no—
404#95—24no—
405#95—22no—
406C24—21no—
407C24C22×D412yes—
408C24C22×D411yescharacteristic
409#41—216no—
410#41—216no—
411#41—216no—
412#41—48no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8C22⋊Q814yes—
419C2×Q8C22×Q812yes—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—28no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4C22⋊Q818yes—
434C2×D4C22⋊Q814yes—
435C2×D4C22×Q812yes—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—48no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—82no—
472C22×C4—84no—
473C22×C4—82no—
474C22×C4—84no—
475C22×C4—81no—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×Q814yes—
485C22×C4C22×Q812yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4114yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C4C2511yescharacteristic
491M16—48no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—48no—
502C4⋊C4—24no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—48no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4#12314yes—
533C4⋊C4#12314yes—
534C4⋊C4C22×Q814yes—
535C22⋊C4—28no—
536C22⋊C4—28no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—48no—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—81no—
546C22⋊C4—81no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4#12318yes—
550C42—24no—
551C42—24no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—48no—
557C42—44no—
558C42—42no—
559C42—44no—
560C42—82no—
561C42—82no—
562C42C2×#4118yes—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23C2×C22⋊Q814yes—
581C23C2×C22⋊Q812yes—
582C23C2×#12312yes—
583C23#53512yes—
584C23C23×D411yescharacteristic
585Q8—28no—
586Q8—24no—
587Q8—42no—
588D4—216no—
589D4—24no—
590D4—28no—
591D4—48no—
592D4—42no—
593D4—88no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—84no—
642C2×C4—164no—
643C2×C4—162no—
644C2×C4—164no—
645C2×C4—168no—
646C2×C4C2×C22⋊Q814yes—
647C2×C4C2×C22⋊Q812yes—
648C2×C4C2×#12314yes—
649C2×C4C2×#12314yes—
650C2×C4C2×#12312yes—
651C2×C4#53512yes—
652C2×C4#55318yes—
653C2×C4#55418yes—
654C2×C4#57612yes—
655C2×C4#57712yes—
656C2×C4C23×Q811yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4112yes—
659C2×C4C2×H(5,𝔽2)12yes—
660C8—42no—
661C8—42no—
662C8—44no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22#248714yes—
684C22C22×C22⋊Q811yescharacteristic
685C22C22×#12311yescharacteristic
686C22C2×#53511yescharacteristic
687C22C2×#57611yescharacteristic
688C22C2×#57711yescharacteristic
689C22#358611yescharacteristic
690C22#358611yescharacteristic
691C4—22no—
692C4—24no—
693C4—22no—
694C4—24no—
695C4—24no—
696C4—44no—
697C4—41no—
698C4—41no—
699C4—42no—
700C4—82no—
701C4—81no—
702C4—84no—
703C4—82no—
704C4—81no—
705C4—82no—
706C4#248714yes—
707C4C2×#55312yes—
708C4C2×#55412yes—
709C2—24no—
710C2—44no—
711C2—82no—
712C2C2×#248711yescharacteristic
713C2#6049411yescharacteristic
714C2#6049711yescharacteristic
715C2#6052111yescharacteristic
716C2#6052211yescharacteristic
717C2#6082211yescharacteristic
718C2#6083111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34183C2semidirect
3. #27690C2semidirect
4. #27688C2semidirect
5. #27685C2semidirect
6. #27683C2semidirect
7. #27680C2semidirect
8. #27677C2semidirect
9. #23747C2semidirect
10. C2×#2099C2semidirect
11. #23734C2semidirect
12. #23730C2semidirect
13. #21175C2semidirect
15. #20849C2semidirect
16. #20652C2semidirect
17. #20630C2semidirect

Maximal subgroups

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

subgroupindex
#342672
#341832
#276902
#276882
#276852
#276832
#276802
#276772
#237472
C2×#20992
#237342
#237302
#211752
#209152
#208492
#206522
#206302