Finite Group Databasebeta

← #6795539

Subgroups — #6795539

Order 512.

1813 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36179C211yesmaximal, characteristic
2#34945C211yesmaximal, characteristic, complemented
3#33196C211yesmaximal, characteristic, complemented
4#32304C214yesmaximal, complemented
5#31960C214yesmaximal, complemented
6#29925C212yesmaximal, complemented
7#28689C211yesmaximal, characteristic, complemented
8#28540C212yesmaximal, complemented
9#28279C211yesmaximal, characteristic, complemented
10#27736C212yesmaximal, complemented
11#27681C212yesmaximal, complemented
12#25703C212yesmaximal, complemented
13#24661C212yesmaximal, complemented
14#24120C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic, complemented
16#21987C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#547—21no—
20C2×#547C2212yes—
21C2×#500C2212yes—
22C2×#475C2212yes—
23C2×#474C2212yes—
24C2×#460C2211yescharacteristic
25C2×C4×SD16C2212yes—
26C2×#432C2212yes—
27#2868C2211yescharacteristic
28#2856C2212yes—
29#2573—22no—
30#2571C2211yescharacteristic
31#2568C2211yescharacteristic
32#2547C2211yescharacteristic
33#2542—21no—
34#2542—21no—
35#2542C2212yes—
36#2538C2211yescharacteristic
37#2514—21no—
38#2501—22no—
39#2501—22no—
40#2501C2214yes—
41#2494C2212yes—
42#2447—21no—
43#2421C2212yes—
44#2401—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#124—21no—
48C4×C22⋊Q8C2212yes—
49#2295—21no—
50C2×#384C2212yes—
51#2283C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2100C2214yes—
55#2095C2214yes—
56#2092C2214yes—
57#2083C2214yes—
58#2073C2214yes—
59#2067C2214yes—
60#2034C2214yes—
61#2033C2214yes—
62#2033C2214yes—
63#2000C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1982C2212yes—
67#1982C2211yescharacteristic
68#1973C2214yes—
69#1954C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1936C2214yes—
77#1926C2212yes—
78#1924C2214yes—
79#1880C2212yes—
80#1872C2214yes—
81#1853C2214yes—
82#1852C2214yes—
83#1852C2214yes—
84#1847C2214yes—
85#1842C2211yescharacteristic
86#1836C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×#41—22no—
96C22×SD16C2312yes—
97#547—44no—
98#547—44no—
99#547—48no—
100#547—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#124—21no—
107C2×#123—24no—
108C2×#123—24no—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4C2314yes—
130C22×C42C2312yes—
131#500—44no—
132#475—44no—
133#474—44no—
134#460—24no—
135C4×SD16—44no—
136#432—28no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158#401—22no—
159#400—22no—
160#400—22no—
161#400—22no—
162#399—24no—
163#399—24no—
164#399—24no—
165#397—22no—
166#397—22no—
167#397—22no—
168#396—22no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391C2312yes—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—22no—
178#390—22no—
179#390C2312yes—
180#390C2314yes—
181#389—21no—
182#389—21no—
183#389—21no—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384—22no—
195#384—22no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#384C2314yes—
201#383—22no—
202#382—21no—
203#382C2312yes—
204#382C2312yes—
205#381—22no—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226#376—22no—
227C43—21no—
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C2×#41—48no—
239C2×#41—48no—
240C2×#41—48no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8C2411yescharacteristic
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2412yes—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266#127—22no—
267#127—22no—
268#124—44no—
269#123—416no—
270#123—416no—
271#123—416no—
272C22⋊Q8—24no—
273C22⋊Q8—42no—
274C22⋊Q8—48no—
275C22⋊Q8—48no—
276C22⋊Q8—48no—
277#121—44no—
278#121—44no—
279#121—48no—
280#121—48no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—41no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—44no—
295C4×D4—44no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—44no—
362#107—44no—
363#107—44no—
364#106—42no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—42no—
370C4⋊C8#4114yescomplemented
371#103—42no—
372#103—44no—
373#102—24no—
374#102—42no—
375#102—44no—
376#102—42no—
377#102—44no—
378C22⋊C8—28no—
379C22⋊C8—44no—
380#97—28no—
381C4×C8—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—22no—
401C24C22×D412yes—
402#41—832no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q82−1+412yes—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4AGL(1,ℤ/8)14yes—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—42no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2511yescharacteristic
469SD16—48no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8C2×#4114yes—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—82no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—21no—
535C42—21no—
536C42—41no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—44no—
541C42—42no—
542C42—42no—
543C42—81no—
544C42AGL(1,ℤ/8)14yes—
545C42C2×#4112yes—
546C42H(5,𝔽2)12yes—
547C42H(5,𝔽2)12yes—
548C422−1+412yes—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23C2×AGL(1,ℤ/8)14yes—
563C23C23×D411yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—84no—
569Q8—88no—
570D4—28no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—84no—
576D4—816no—
577D4—84no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—48no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—816no—
627C2×C4—88no—
628C2×C4#53512yes—
629C2×C4#53512yes—
630C2×C4#54712yes—
631C2×C4#54712yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4#57812yes—
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×2−1+411yescharacteristic
641C2×C4C2×2−1+411yescharacteristic
642C2×C4#58511yescharacteristic
643C8—24no—
644C8—44no—
645C8—42no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22C2×#53511yescharacteristic
661C22C2×#54711yescharacteristic
662C22#348911yescharacteristic
663C22C22×AGL(1,ℤ/8)11yescharacteristic
664C22C22×AGL(1,ℤ/8)11yescharacteristic
665C22C2×#57811yescharacteristic
666C22#358611yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—41no—
675C4—81no—
676C4—82no—
677C4—84no—
678C4—82no—
679C4—82no—
680C4—81no—
681C4—81no—
682C4#321714yes—
683C4#335314yes—
684C4#351711yescharacteristic
685C4#352011yescharacteristic
686C4#354711yescharacteristic
687C4#355211yescharacteristic
688C2—44no—
689C2—162no—
690C2#3892011yescharacteristic
691C2C2×#321711yescharacteristic
692C2#6083711yescharacteristic
693C2C2×#335311yescharacteristic
694C2#6125211yescharacteristic
695C2#6211111yescharacteristic
696C2#6213211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34945C2semidirect
3. #33196C2semidirect
4. #32304C2semidirect
5. #31960C2semidirect
6. #29925C2semidirect
7. #28689C2semidirect
8. #28540C2semidirect
9. #28279C2semidirect
10. #27736C2semidirect
11. #27681C2semidirect
12. #25703C2semidirect
13. #24661C2semidirect
14. #24120C2semidirect
15. #22799C2semidirect
16. #21987C2semidirect
17. C4×#460C2semidirect
370. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#361792
#349452
#331962
#323042
#319602
#299252
#286892
#285402
#282792
#277362
#276812
#257032
#246612
#241202
#227992
#219872
C4×#4602