Finite Group Databasebeta

← #6528509

Subgroups — #6528509

Order 512.

1848 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35239C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3#29109C212yesmaximal
4#29104C212yesmaximal
5#27965C212yesmaximal, complemented
6#27961C212yesmaximal, complemented
7#27830C214yesmaximal, complemented
8#27822C212yesmaximal, complemented
9#27813C212yesmaximal, complemented
10#27744C214yesmaximal, complemented
11#27736C212yesmaximal, complemented
12#27727C212yesmaximal, complemented
13#21841C211yesmaximal, characteristic, complemented
14#21834C211yesmaximal, characteristic, complemented
15#20738C211yesmaximal, characteristic
16#20733C211yesmaximal, characteristic
17#20535C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8—21no—
21C22×C4×D4C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24#2857C2212yes—
25#2851C2212yes—
26#2666—22no—
27#2571C2211yescharacteristic, complemented
28#2570C2212yescomplemented
29#2552C2211yescharacteristic, complemented
30#2552C2211yescharacteristic, complemented
31#2551C2211yescharacteristic, complemented
32#2550C2212yescomplemented
33#2542—21no—
34#2542—21no—
35#2542—22no—
36#2542C2212yes—
37#2542C2212yes—
38#2542C2214yes—
39#2442—21no—
40#2440—21no—
41#2438—22no—
42C2×#399—22no—
43#2331—21no—
44#2330—21no—
45C2×#390C2212yes—
46#2285C2211yescharacteristic
47C2×#384C2212yes—
48#2283C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2100C2214yescomplemented
51#2100C2214yescomplemented
52#2099C2214yescomplemented
53#2099C2214yescomplemented
54#2073C2214yes—
55#2071C2214yes—
56#2025C2214yes—
57#2023C2214yes—
58#1993C2212yescomplemented
59#1993C2212yescomplemented
60#1992C2214yescomplemented
61#1990C2214yescomplemented
62#1990C2212yescomplemented
63#1990C2212yescomplemented
64#1988C2212yescomplemented
65#1988C2212yescomplemented
66#1987C2214yescomplemented
67#1936C2214yes—
68#1936C2214yes—
69#1934C2214yes—
70#1934C2214yes—
71#1836C2214yescomplemented
72#1834C2214yescomplemented
73#1824C2212yes—
74#1824C2211yescharacteristic
75#1824C2212yes—
76#1824C2212yes—
77#1824C2211yescharacteristic
78#1824C2212yes—
79#1824C2212yes—
80#1804C2212yes—
81#1804C2212yes—
82#1802C2212yes—
83#1802C2212yes—
84#1786C2212yes—
85#1786C2211yescharacteristic
86#1786C2211yescharacteristic
87#1771C2214yescomplemented
88#1770C2212yescomplemented
89C8×C4⋊C4C2212yescomplemented
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2212yes—
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—21no—
107C2×#125C2312yes—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—48no—
116C2×C22⋊Q8—48no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—24no—
120C2×C4×D4—24no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#432—24no—
137#431—24no—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2312yes—
149C2×#97C2312yes—
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#400—22no—
154#400—22no—
155#399—24no—
156#399—24no—
157#399—24no—
158#399—24no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#395—22no—
164#395—22no—
165#391—22no—
166#391—22no—
167#390—22no—
168#390—22no—
169#390C2314yes—
170#390C2314yes—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—22no—
189#384—22no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2314yes—
194#384C2314yes—
195#383—21no—
196#382—21no—
197#382—21no—
198#382C2311yescharacteristic
199#382C2312yes—
200#382C2311yescharacteristic
201#381—21no—
202#380—22no—
203#380—21no—
204#380C2311yescharacteristic
205#380C2312yes—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218#376—21no—
219C2×#95D412yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236C25—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×D4—24no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—42no—
248C22×D4C2412yes—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272#125—44no—
273#125—44no—
274C22⋊Q8—48no—
275C22⋊Q8—48no—
276C22⋊Q8—416no—
277C22⋊Q8—416no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—84no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—44no—
286C4×D4—44no—
287#117—82no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—48no—
342C2×C22⋊C4—48no—
343C2×C22⋊C4—44no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42#4114yes—
353C2×C42#4114yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C4⋊C8—24no—
359C4⋊C8—24no—
360#102—28no—
361#102—24no—
362#102—24no—
363#102—42no—
364#102—44no—
365#102—42no—
366C22⋊C8—22no—
367C22⋊C8—22no—
368C22⋊C8—24no—
369#97—24no—
370#97—44no—
371C4×C8—24no—
372C4×C8—44no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24C22×D411yescharacteristic
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—816no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D42−1+414yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—816no—
456C22×C4#12414yes—
457C22×C4#12414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2511yescharacteristic
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4AGL(1,ℤ/8)14yes—
515C4⋊C4AGL(1,ℤ/8)14yes—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C42—24no—
526C42—24no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—82no—
532C42#12414yes—
533C42#12414yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C42C2×#4112yes—
537C42C2×#4112yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—24no—
543C23—48no—
544C23—48no—
545C23—48no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23C2×#12412yes—
557C23#57712yes—
558C23#57812yes—
559C23C23×D411yescharacteristic
560Q8—88no—
561Q8—88no—
562Q8—164no—
563D4—24no—
564D4—24no—
565D4—48no—
566D4—42no—
567D4—42no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4—88no—
619C2×C4—88no—
620C2×C4C2×#12412yes—
621C2×C4C2×#12412yes—
622C2×C4C2×#12412yes—
623C2×C4C2×#12412yes—
624C2×C4C2×#12412yes—
625C2×C4C2×AGL(1,ℤ/8)14yes—
626C2×C4C2×AGL(1,ℤ/8)14yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×2−1+412yes—
632C2×C4#58512yes—
633C8—42no—
634C8—42no—
635C8—44no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—816no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22#313714yes—
651C22C22×#12411yescharacteristic
652C22#348511yescharacteristic
653C22#348611yescharacteristic
654C22C22×AGL(1,ℤ/8)11yescharacteristic
655C22C22×AGL(1,ℤ/8)11yescharacteristic
656C22C2×#57711yescharacteristic
657C22C2×#57811yescharacteristic
658C4—24no—
659C4—24no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—81no—
668C4—81no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—84no—
674C4#313814yes—
675C4#349512yes—
676C4#349512yes—
677C2—24no—
678C2—84no—
679C2#3564411yescharacteristic
680C2C2×#313711yescharacteristic
681C2C2×#313811yescharacteristic
682C2#6172811yescharacteristic
683C2#6173711yescharacteristic
684C2#6183711yescharacteristic
685C2#6184311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35239C2semidirect
5. #27965C2semidirect
6. #27961C2semidirect
7. #27830C2semidirect
8. #27822C2semidirect
9. #27813C2semidirect
10. #27744C2semidirect
11. #27736C2semidirect
12. #27727C2semidirect
13. #21841C2semidirect
14. #21834C2semidirect
17. #20535C2semidirect
27. #2571C22semidirect
28. #2570C22semidirect
29. #2552C22semidirect
30. #2552C22semidirect
31. #2551C22semidirect
32. #2550C22semidirect
50. #2100C22semidirect
51. #2100C22semidirect
52. #2099C22semidirect
53. #2099C22semidirect
58. #1993C22semidirect
59. #1993C22semidirect
60. #1992C22semidirect
61. #1990C22semidirect
62. #1990C22semidirect
63. #1990C22semidirect
64. #1988C22semidirect
65. #1988C22semidirect
66. #1987C22semidirect
71. #1836C22semidirect
72. #1834C22semidirect
87. #1771C22semidirect
88. #1770C22semidirect
89. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#352392
C2×#25422
#291092
#291042
#279652
#279612
#278302
#278222
#278132
#277442
#277362
#277272
#218412
#218342
#207382
#207332
#205352