Finite Group Databasebeta

← #6804723

Subgroups — #6804723

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36608C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic
3#33287C211yesmaximal, characteristic, complemented
4#32556C214yesmaximal, complemented
5#31827C214yesmaximal, complemented
6#30229C211yesmaximal, characteristic, complemented
7#30205C211yesmaximal, characteristic, complemented
8#29219C211yesmaximal, characteristic, complemented
9#29193C211yesmaximal, characteristic, complemented
10#29179C211yesmaximal, characteristic, complemented
11#28718C211yesmaximal, characteristic
12#28613C211yesmaximal, characteristic
13#28613C211yesmaximal, characteristic
14#27566C214yesmaximal, complemented
15#27446C214yesmaximal, complemented
16#22268C211yesmaximal, characteristic
17#22220C211yesmaximal, characteristic
18#21646C211yesmaximal, characteristic
19#21340C211yesmaximal, characteristic, complemented
20C2×#554—21no—
21C2×#492C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2862C2211yescharacteristic
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2581—21no—
31#2581—21no—
32#2581C2212yes—
33#2566—21no—
34#2566—21no—
35#2563—21no—
36#2538—21no—
37#2538C2212yes—
38#2514—21no—
39#2514—21no—
40#2514—21no—
41#2514—21no—
42#2514C2212yes—
43#2514C2212yes—
44#2452—21no—
45#2452—21no—
46#2452C2212yes—
47#2452C2212yes—
48#2356—21no—
49#2356—21no—
50#2356C2212yes—
51D4×C4⋊C4—21no—
52D4×C4⋊C4C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C4×#127C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2214yes—
61#2101C2214yes—
62#2099C2214yes—
63#2087C2214yes—
64#2073C2214yes—
65#2072C2214yes—
66#2068C2214yes—
67#2055C2214yes—
68#2055C2214yes—
69#2024C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1988C2214yes—
77#1985C2212yes—
78#1957C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1940C2212yes—
83#1936C2214yes—
84#1935C2214yes—
85#1935C2214yes—
86#1925C2214yes—
87#1921C2214yes—
88#1917C2212yes—
89#1917C2212yes—
90#1880C2214yes—
91#1877C2214yes—
92#1877C2214yes—
93#1877C2214yes—
94#1863C2214yes—
95#1862C2214yes—
96#1845C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1804C2214yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103#1762C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×SD16C2314yes—
108#554—48no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#125—21no—
117C2×#125—21no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C22×C4⋊C4—22no—
137#492—24no—
138#486—24no—
139#486—24no—
140C2×#107C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#97C2312yes—
151C2×#97C2312yes—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156#401—22no—
157#401—22no—
158#400—22no—
159#400—22no—
160#400—22no—
161#396—22no—
162#396—22no—
163#396—22no—
164#396—22no—
165#395—22no—
166#393—22no—
167#393—22no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391C2312yes—
179#391C2312yes—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#385—22no—
213#385—22no—
214#385—22no—
215#385—22no—
216#384—21no—
217#384—21no—
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2312yes—
223#383—22no—
224#383—22no—
225#383—22no—
226#382—21no—
227#382—21no—
228#382—21no—
229#382—21no—
230#382—21no—
231#382—21no—
232#382C2312yes—
233#382C2312yes—
234#380—22no—
235#380—22no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2311yescharacteristic
249C4×C4⋊C4C2311yescharacteristic
250C4×C4⋊C4C2311yescharacteristic
251#376—22no—
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#336C2312yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—21no—
271C22×Q8—21no—
272C22×Q8—22no—
273C22×Q8C2×D411yescharacteristic
274C22×Q8C2412yes—
275C22×D4—21no—
276C22×D4—21no—
277C22×D4C2412yes—
278C23×C4—22no—
279C23×C4—22no—
280C23×C4—22no—
281C2×SD16—28no—
282C2×SD16—44no—
283C2×SD16—44no—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296C4⋊Q8—82no—
297#127—22no—
298#127#4114yes—
299#125—44no—
300#125—44no—
301C22⋊Q8—48no—
302C22⋊Q8—48no—
303C22⋊Q8—48no—
304C4×Q8—24no—
305C4×Q8—42no—
306C4×Q8—42no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×D4—44no—
311C4×D4—44no—
312C4×D4—44no—
313C4×D4—44no—
314C4×D4—44no—
315C4×D4—44no—
316C4×D4—44no—
317C4×D4—82no—
318C4×D4—82no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—48no—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388#107—44no—
389C4⋊C8—42no—
390C4⋊C8—42no—
391C4⋊C8—42no—
392#103—44no—
393#103—44no—
394#103—44no—
395#102—28no—
396#102—28no—
397#102—44no—
398#97—42no—
399#97—42no—
400C4×C8—22no—
401C4×C8—22no—
402C4×C8—22no—
403C4×C8—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432C24—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—41no—
439C2×Q8—41no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—82no—
448C2×Q8—82no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4C2×#4114yes—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2511yescharacteristic
490SD16—88no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—44no—
569C42—44no—
570C42—44no—
571C42AGL(1,ℤ/8)14yes—
572C42C2×#4112yes—
573C422−1+412yes—
574C23—24no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23C23×D411yescharacteristic
582Q8—44no—
583Q8—44no—
584Q8—44no—
585Q8—84no—
586Q8—84no—
587Q8—164no—
588D4—28no—
589D4—48no—
590D4—48no—
591D4—44no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—88no—
648C2×C4—164no—
649C2×C4—164no—
650C2×C4#54812yes—
651C2×C4#54812yes—
652C2×C4C2×AGL(1,ℤ/8)12yes—
653C2×C4C2×AGL(1,ℤ/8)12yes—
654C2×C4C2×AGL(1,ℤ/8)12yes—
655C2×C4#57512yes—
656C2×C4#57712yes—
657C2×C4#57812yes—
658C2×C4C22×#4111yescharacteristic
659C2×C4C2×2−1+411yescharacteristic
660C2×C4#58512yes—
661C2×C4#58512yes—
662C8—42no—
663C8—42no—
664C8—42no—
665C8—42no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22#344911yescharacteristic
674C22C2×#54811yescharacteristic
675C22#350011yescharacteristic
676C22C22×AGL(1,ℤ/8)11yescharacteristic
677C22C2×#57511yescharacteristic
678C22C2×#57711yescharacteristic
679C22C2×#57811yescharacteristic
680C4—22no—
681C4—22no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4#336714yes—
700C4#337614yes—
701C4#352112yes—
702C4#355712yes—
703C2—84no—
704C2#3916311yescharacteristic
705C2#6096511yescharacteristic
706C2#6100811yescharacteristic
707C2C2×#336711yescharacteristic
708C2C2×#337611yescharacteristic
709C2#6253911yescharacteristic
710C2#6254511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36608C2semidirect
3. #33287C2semidirect
4. #32556C2semidirect
5. #31827C2semidirect
6. #30229C2semidirect
7. #30205C2semidirect
8. #29219C2semidirect
9. #29193C2semidirect
10. #29179C2semidirect
14. #27566C2semidirect
15. #27446C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#366082
#365592
#332872
#325562
#318272
#302292
#302052
#292192
#291932
#291792
#287182
#286132
#275662
#274462
#222682
#222202
#216462
#213402