Finite Group Databasebeta

← #6513868

Subgroups — #6513868

Order 512.

1846 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34976C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#27677C212yesmaximal, complemented
4#27648C212yesmaximal, complemented
5#27630C212yesmaximal, complemented
6#27626C212yesmaximal, complemented
7#25056C212yesmaximal, complemented
8#25016C212yesmaximal, complemented
9#24994C212yesmaximal, complemented
10#24967C212yesmaximal, complemented
11#24953C214yesmaximal, complemented
12#24883C214yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21964C211yesmaximal, characteristic, complemented
15#21849C211yesmaximal, characteristic, complemented
16#20737C211yesmaximal, characteristic, complemented
17#20648C211yesmaximal, characteristic, complemented
18C2×#520—21no—
19C2×C4×#41C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#484C2214yes—
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#417C2212yes—
25#2609—22no—
26#2582C2211yescharacteristic
27#2538C2211yescharacteristic
28#2538C2211yescharacteristic
29#2537C2211yescharacteristic
30#2485—21no—
31#2485C2212yes—
32#2407—21no—
33#2407C2212yes—
34#2392—22no—
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4C2212yes—
37D4×C4⋊C4—22no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2214yes—
40D4×C4⋊C4C2212yes—
41#2317—22no—
42#2317—21no—
43C4×C22⋊Q8—21no—
44C4×C22⋊Q8C2212yes—
45C4×#121—21no—
46C4×#121C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49#2285C2211yescharacteristic
50C2×#384C2212yes—
51C2×#384C2211yescharacteristic
52#2099C2214yes—
53#2093C2214yes—
54#2083C2214yes—
55#2077C2214yes—
56#2076C2214yes—
57#2072C2214yes—
58#2036C2214yes—
59#2034C2214yes—
60#1984C2212yes—
61#1954C2212yes—
62#1948C2212yes—
63#1948C2212yes—
64#1947C2212yes—
65#1947C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1927C2214yes—
69#1925C2214yes—
70#1920C2214yes—
71#1918C2214yes—
72#1874C2214yes—
73#1870C2214yes—
74#1852C2214yes—
75#1852C2214yes—
76#1845C2212yes—
77#1834C2212yes—
78#1825C2212yes—
79#1824C2214yes—
80#1809C2214yes—
81#1803C2211yescharacteristic
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2212yes—
85#1802C2211yescharacteristic
86#1802C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89#1797C2212yes—
90#1797C2212yes—
91#1788C2212yes—
92C4×#103C2212yes—
93C4×#102C2212yes—
94#1736C2212yes—
95C22×#41D411yescharacteristic
96C22×#41C2311yescharacteristic
97C22×M16C2312yes—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2312yes—
100C2×#121—21no—
101C2×#121C2312yes—
102#520—48no—
103C4×#41—24no—
104C4×#41—24no—
105C4×#41—24no—
106C4×#41—24no—
107C2×C4×Q8—22no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—24no—
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4D412yes—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117—21no—
127C2×#117D412yes—
128C2×#117D411yescharacteristic
129C2×#117C2311yescharacteristic
130C22×C4⋊C4—22no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2314yes—
135C22×C4⋊C4C2312yes—
136C22×C42—21no—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#484—48no—
140#431—24no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145#423—24no—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2312yes—
150#417—44no—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C4×C8C2312yes—
161#398—22no—
162#397—22no—
163#396—22no—
164#396—22no—
165#394—22no—
166#394—22no—
167#394—22no—
168#392—22no—
169#391—21no—
170#391C2312yes—
171#390—21no—
172#390C2312yes—
173#389—22no—
174#389—24no—
175#389—22no—
176#389—24no—
177#389—21no—
178#389—21no—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—22no—
189#385—24no—
190#385—24no—
191#385—24no—
192#384—22no—
193#384—21no—
194#384—22no—
195#384—21no—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2314yes—
202#384C2311yescharacteristic
203#382—22no—
204#382—21no—
205#382—21no—
206#380—22no—
207#380—21no—
208#380—21no—
209#380—21no—
210#380—21no—
211#380C2312yes—
212#380C2312yes—
213#380C2312yes—
214#380C2314yes—
215#380C2312yes—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2314yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#336C2314yes—
236#336C2314yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—41no—
245C2×#41—41no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2411yescharacteristic
248C22×D4—21no—
249C22×D4—22no—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2411yescharacteristic
253C23×C4—22no—
254C23×C4—24no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C2×M16—24no—
265C2×M16—42no—
266C2×M16—42no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22⋊Q8—24no—
271C22⋊Q8—42no—
272C22⋊Q8—44no—
273#121—24no—
274#121—42no—
275#121—44no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—48no—
284C4×D4—28no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—416no—
295C4×D4—48no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—84no—
299#117—24no—
300#117—28no—
301#117—22no—
302#117—22no—
303#117—48no—
304#117—44no—
305#117—44no—
306#117—82no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—48no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—82no—
335C2×C4⋊C4—84no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—21no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—44no—
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42#4114yes—
376C2×C42#4114yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—44no—
381#107—44no—
382#107—44no—
383#106—44no—
384C4⋊C8—24no—
385C4⋊C8—24no—
386C4⋊C8—22no—
387C4⋊C8—22no—
388#103—28no—
389#103—24no—
390#103—42no—
391#102—24no—
392#102—42no—
393#102—44no—
394C22⋊C8—24no—
395C22⋊C8—44no—
396C22⋊C8—44no—
397C4×C8—24no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—22no—
409#95—22no—
410C24—21no—
411C24C22×D411yescharacteristic
412C24C22×D412yes—
413#41—232no—
414#41—28no—
415#41—44no—
416#41—44no—
417#41—44no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8AGL(1,ℤ/8)14yes—
426C2×Q8H(5,𝔽2)12yes—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—82no—
443C2×D4#13714yes—
444C2×D42−1+412yes—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—84no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—84no—
488C22×C4#12314yes—
489C22×C4#12314yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2511yescharacteristic
501M16—84no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—24no—
510C2×C8—22no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—48no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—82no—
544C4⋊C4—84no—
545C4⋊C4—82no—
546C4⋊C4—84no—
547C4⋊C4—81no—
548C4⋊C4—81no—
549C4⋊C4—82no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—48no—
557C22⋊C4—42no—
558C22⋊C4—44no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C42—24no—
566C42—24no—
567C42—22no—
568C42—22no—
569C42—42no—
570C42—42no—
571C42—42no—
572C42—48no—
573C42—44no—
574C42—44no—
575C42#12314yes—
576C42#12314yes—
577C42C2×#4112yes—
578C42C2×#4112yes—
579C42C2×#4112yes—
580C42C2×#4112yes—
581C23—22no—
582C23—22no—
583C23—21no—
584C23—22no—
585C23—21no—
586C23—21no—
587C23—22no—
588C23—21no—
589C23—21no—
590C23—22no—
591C23—22no—
592C23—21no—
593C23—22no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23C2×#12312yes—
601C23C2×#13714yes—
602C23C23×D411yescharacteristic
603Q8—28no—
604Q8—22no—
605Q8—22no—
606Q8—41no—
607Q8—41no—
608D4—216no—
609D4—28no—
610D4—22no—
611D4—22no—
612D4—48no—
613D4—41no—
614D4—41no—
615D4—88no—
616C2×C4—216no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—48no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—48no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—84no—
668C2×C4—84no—
669C2×C4—168no—
670C2×C4—164no—
671C2×C4C2×#12312yes—
672C2×C4C2×#12312yes—
673C2×C4C2×#12312yes—
674C2×C4C2×#12312yes—
675C2×C4C2×#12312yes—
676C2×C4C2×AGL(1,ℤ/8)14yes—
677C2×C4C2×AGL(1,ℤ/8)12yes—
678C2×C4C2×#13712yes—
679C2×C4#57612yes—
680C2×C4#57812yes—
681C2×C4C22×#4111yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4C2×H(5,𝔽2)11yescharacteristic
684C2×C4C2×2−1+411yescharacteristic
685C2×C4#58511yescharacteristic
686C2×C4#58511yescharacteristic
687C8—42no—
688C8—42no—
689C8—44no—
690C22—24no—
691C22—22no—
692C22—24no—
693C22—22no—
694C22—24no—
695C22—22no—
696C22—41no—
697C22—42no—
698C22—41no—
699C22—42no—
700C22—41no—
701C22—42no—
702C22—42no—
703C22—41no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22—81no—
711C22C22×#12311yescharacteristic
712C22#348811yescharacteristic
713C22#348911yescharacteristic
714C22C22×AGL(1,ℤ/8)11yescharacteristic
715C22C22×#13711yescharacteristic
716C22C2×#57611yescharacteristic
717C22C2×#57811yescharacteristic
718C4—22no—
719C4—24no—
720C4—22no—
721C4—22no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—82no—
726C4—84no—
727C4—82no—
728C4—81no—
729C4—81no—
730C4—82no—
731C4—82no—
732C4#309214yes—
733C4#309814yes—
734C4#349211yescharacteristic
735C4#349311yescharacteristic
736C4#349411yescharacteristic
737C4#349511yescharacteristic
738C2—44no—
739C2—42no—
740C2—82no—
741C2#3551611yescharacteristic
742C2C2×#309211yescharacteristic
743C2C2×#309811yescharacteristic
744C2#6198411yescharacteristic
745C2#6200111yescharacteristic
746C2#6212711yescharacteristic
747C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34976C2
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. #27677C2semidirect
4. #27648C2semidirect
5. #27630C2semidirect
6. #27626C2semidirect
7. #25056C2semidirect
8. #25016C2semidirect
9. #24994C2semidirect
10. #24967C2semidirect
11. #24953C2semidirect
12. #24883C2semidirect
13. #21982C2semidirect
14. #21964C2semidirect
15. #21849C2semidirect
16. #20737C2semidirect
17. #20648C2semidirect

Maximal subgroups

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

subgroupindex
#349762
#341832
#276772
#276482
#276302
#276262
#250562
#250162
#249942
#249672
#249532
#248832
#219822
#219642
#218492
#207372
#206482