Finite Group Databasebeta

← #6592137

Subgroups — #6592137

Order 512.

2068 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36153C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic, complemented
3#33146C211yesmaximal, characteristic, complemented
4#32980C212yesmaximal, complemented
5#27961C211yesmaximal, characteristic, complemented
6#27645C211yesmaximal, characteristic, complemented
7#27603C212yesmaximal, complemented
8#26303C212yesmaximal, complemented
9#26225C214yesmaximal, complemented
10#24968C212yesmaximal, complemented
11#24958C214yesmaximal, complemented
12#24557C212yesmaximal, complemented
13#22713C212yesmaximal, complemented
14#22500C212yesmaximal, complemented
15#22401C211yesmaximal, characteristic, complemented
16C4×#480C211yesmaximal, characteristic, complemented
17#20702C212yesmaximal, complemented
18C2×#547C2211yescharacteristic
19C2×#546—21no—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22C2×#481C2212yes—
23C2×#480C2211yescharacteristic
24C2×#436C2212yes—
25C2×#407C2212yes—
26#2857C2211yescharacteristic
27#2847C2212yes—
28#2547C2211yescharacteristic
29#2537C2211yescharacteristic
30#2486C2212yes—
31#2427—21no—
32#2425—22no—
33#2417—21no—
34#2416C2212yes—
35#2412—22no—
36#2354—21no—
37#2354—22no—
38#2354—21no—
39#2354C2214yes—
40#2354C2212yes—
41#2354C2212yes—
42D4×C4⋊C4C2211yescharacteristic
43#2339—21no—
44C4×#121C2211yescharacteristic
45C2×#390—21no—
46C2×#388—22no—
47C2×#384C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yescomplemented
53#2084C2212yescomplemented
54#2077C2214yescomplemented
55#2076C2214yescomplemented
56#2076C2214yescomplemented
57#2071C2212yescomplemented
58#2033C2214yescomplemented
59#1990C2212yescomplemented
60#1988C2212yescomplemented
61#1954C2212yes—
62#1948C2212yescomplemented
63#1947C2212yescomplemented
64#1925C2212yescomplemented
65#1918C2212yescomplemented
66#1918C2214yescomplemented
67#1895C2214yescomplemented
68#1875C2214yescomplemented
69#1875C2214yescomplemented
70#1874C2211yescharacteristic
71#1870C2214yes—
72#1854C2214yescomplemented
73#1854C2214yescomplemented
74#1852C2211yescharacteristic
75#1852C2212yes—
76#1852C2214yescomplemented
77#1834C2212yescomplemented
78#1825C2214yescomplemented
79#1825C2214yescomplemented
80#1824C2212yes—
81#1824C2212yes—
82#1824C2212yes—
83#1824C2214yescomplemented
84#1823C2211yescharacteristic
85#1823C2212yes—
86#1809C2214yescomplemented
87#1802C2212yescomplemented
88#1797C2211yescharacteristic
89#1779C2212yes—
90C4×#107C2212yescomplemented
91C4×#102C2212yescomplemented
92C4×C22⋊C8C2211yescharacteristic
93#1736C2214yescomplemented
94#1736C2214yescomplemented
95C22×#41—21no—
96C23×D4D411yescharacteristic
97C24×C4C2312yes—
98C22×M16C2314yes—
99#547—28no—
100#546—48no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#124—21no—
103C2×#123—22no—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×#121—22no—
108C2×#121—21no—
109C2×#121C2311yescharacteristic
110C2×#121C2312yes—
111C2×#121C2312yes—
112C2×#120—22no—
113C2×C4×D4—28no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D412yes—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—22no—
125C2×#117—28no—
126C2×#117—44no—
127C2×#117C2314yes—
128C2×#117C2314yes—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—22no—
138C22×C22⋊C4—22no—
139C22×C22⋊C4—22no—
140C22×C22⋊C4D412yes—
141C22×C42—22no—
142C22×C42—21no—
143C22×C42C2311yescharacteristic
144#481—44no—
145#480—24no—
146#436—28no—
147C2×#107C2312yes—
148C2×#107C2314yes—
149C2×#107C2312yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155#407—44no—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161C2×C22⋊C8C2314yes—
162C2×C4×C8C2312yes—
163#398—22no—
164#397—22no—
165#397—22no—
166#396—22no—
167#394—22no—
168#394—22no—
169#391—21no—
170#391—21no—
171#390—22no—
172#390—24no—
173#390—22no—
174#390C2311yescharacteristic
175#389—21no—
176#389—21no—
177#389C2311yescharacteristic
178#389C2312yes—
179#388—24no—
180#388—24no—
181#388—24no—
182#388—24no—
183#388—21no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—21no—
188#386—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384C2312yes—
193#384C2312yes—
194#384C2311yescharacteristic
195#381—22no—
196#380—21no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—21no—
202#380C2312yes—
203#380C2314yes—
204#380C2314yes—
205#380C2312yes—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209#380C2311yescharacteristic
210C4×C4⋊C4—22no—
211C4×C4⋊C4—24no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95—21no—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#336C2312yes—
236#336C2312yes—
237C25C2×D412yes—
238C2×#41—44no—
239C2×#41—44no—
240C2×#41—44no—
241C22×Q8—21no—
242C22×D4—24no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—24no—
247C22×D4—42no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—24no—
254C23×C4—28no—
255C23×C4—24no—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—44no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2×D411yescharacteristic
267C23×C4C2412yes—
268C23×C4C2412yes—
269C23×C4C2411yescharacteristic
270C2×M16—28no—
271C2×M16—44no—
272C2×M16—44no—
273C22×C8C2412yes—
274C22×C8C2414yes—
275C22×C8C2412yes—
276C4⋊Q8—24no—
277#124—44no—
278#123—48no—
279#123—48no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282#121—28no—
283#121—22no—
284#121—22no—
285#121—44no—
286#121—48no—
287#121—44no—
288#120—48no—
289C4×D4—24no—
290C4×D4—28no—
291C4×D4—24no—
292C4×D4—28no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4#4118yes—
298#117—28no—
299#117—28no—
300#117—44no—
301#117—44no—
302#117—48no—
303#117—48no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—82no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4#4114yescomplemented
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—48no—
357C2×C22⋊C4—48no—
358C2×C22⋊C4—84no—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—48no—
370C2×C42—44no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—24no—
376#107—42no—
377#107—44no—
378#106—44no—
379C4⋊C8—24no—
380#102—24no—
381#102—42no—
382#102—44no—
383C22⋊C8—28no—
384C22⋊C8—22no—
385C22⋊C8—22no—
386C22⋊C8—24no—
387C22⋊C8—44no—
388C22⋊C8—44no—
389C4×C8—24no—
390#95—21no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—22no—
403C24—24no—
404C24—24no—
405C24—24no—
406C24—24no—
407C24—21no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24C22×D412yes—
412C24C22×D412yes—
413C24C22×D411yescharacteristic
414#41—816no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—416no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—44no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4#12314yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4AGL(1,ℤ/8)14yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C2×#4118yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4C2511yescharacteristic
504M16—88no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—22no—
512C2×C8—24no—
513C2×C8—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—81no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4#12314yes—
542C4⋊C4C2×#4114yes—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—44no—
554C22⋊C4—42no—
555C22⋊C4—44no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—81no—
559C22⋊C4—82no—
560C22⋊C4—84no—
561C22⋊C4—84no—
562C22⋊C4#12318yes—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—48no—
567C42—48no—
568C42—44no—
569C42—44no—
570C42—42no—
571C42C2×#4114yes—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—48no—
590C23—48no—
591C23—48no—
592C23—48no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—84no—
601C23—84no—
602C23—84no—
603C23C2×#12314yes—
604C23#54612yes—
605C23C2×AGL(1,ℤ/8)12yes—
606C23#57812yes—
607C23C23×D411yescharacteristic
608Q8—84no—
609D4—28no—
610D4—48no—
611D4—44no—
612D4—42no—
613D4—42no—
614D4—88no—
615D4—84no—
616D4—164no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—28no—
626C2×C4—24no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—24no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—48no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4—48no—
641C2×C4—48no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—84no—
668C2×C4—88no—
669C2×C4—88no—
670C2×C4—84no—
671C2×C4—164no—
672C2×C4C2×#12312yes—
673C2×C4C2×#12314yes—
674C2×C4C2×#12312yes—
675C2×C4#54612yes—
676C2×C4C2×AGL(1,ℤ/8)12yes—
677C2×C4C2×AGL(1,ℤ/8)12yes—
678C2×C4#57512yes—
679C2×C4C22×#4111yescharacteristic
680C2×C4C22×#4112yes—
681C2×C4C2×H(5,𝔽2)11yescharacteristic
682C2×C4#58512yes—
683C8—42no—
684C8—44no—
685C8—42no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—816no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22—81no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22#333014yes—
715C22C22×#12311yescharacteristic
716C22C2×#54611yescharacteristic
717C22#348511yescharacteristic
718C22C22×AGL(1,ℤ/8)11yescharacteristic
719C22C2×#57511yescharacteristic
720C22C2×#57811yescharacteristic
721C22#358511yescharacteristic
722C4—22no—
723C4—24no—
724C4—22no—
725C4—22no—
726C4—42no—
727C4—41no—
728C4—44no—
729C4—82no—
730C4—81no—
731C4—82no—
732C4—81no—
733C4—82no—
734C4—81no—
735C4—84no—
736C4#309414yes—
737C4#347512yes—
738C4#351312yes—
739C2—24no—
740C2—44no—
741C2—82no—
742C2—161no—
743C2#3615511yescharacteristic
744C2C2×#309411yescharacteristic
745C2#6052611yescharacteristic
746C2C2×#333011yescharacteristic
747C2#6116011yescharacteristic
748C2#6172011yescharacteristic
749C2#6173011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36153C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2354C2semidirect
3. #33146C2semidirect
4. #32980C2semidirect
5. #27961C2semidirect
6. #27645C2semidirect
7. #27603C2semidirect
8. #26303C2semidirect
9. #26225C2semidirect
10. #24968C2semidirect
11. #24958C2semidirect
12. #24557C2semidirect
13. #22713C2semidirect
14. #22500C2semidirect
15. #22401C2semidirect
16. C4×#480C2semidirect
17. #20702C2semidirect
52. #2100C22semidirect
53. #2084C22semidirect
54. #2077C22semidirect
55. #2076C22semidirect
56. #2076C22semidirect
57. #2071C22semidirect
58. #2033C22semidirect
59. #1990C22semidirect
60. #1988C22semidirect
62. #1948C22semidirect
63. #1947C22semidirect
64. #1925C22semidirect
65. #1918C22semidirect
66. #1918C22semidirect
67. #1895C22semidirect
68. #1875C22semidirect
69. #1875C22semidirect
72. #1854C22semidirect
73. #1854C22semidirect
76. #1852C22semidirect
77. #1834C22semidirect
78. #1825C22semidirect
79. #1825C22semidirect
83. #1824C22semidirect
86. #1809C22semidirect
87. #1802C22semidirect
90. C4×#107C22semidirect
91. C4×#102C22semidirect
93. #1736C22semidirect
94. #1736C22semidirect
330. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
#361532
C2×#23542
#331462
#329802
#279612
#276452
#276032
#263032
#262252
#249682
#249582
#245572
#227132
#225002
#224012
C4×#4802
#207022