Finite Group Databasebeta

← #6591977

Subgroups — #6591977

Order 512.

1935 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2C4×#547C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#32958C212yesmaximal, complemented
5#29141C211yesmaximal, characteristic, complemented
6#28284C211yesmaximal, characteristic, complemented
7#27680C212yesmaximal, complemented
8#26528C212yesmaximal, complemented
9#26431C214yesmaximal, complemented
10#26106C212yesmaximal, complemented
11#25978C214yesmaximal, complemented
12#24846C212yesmaximal, complemented
13#22715C212yesmaximal, complemented
14#22650C211yesmaximal, characteristic, complemented
15#22431C212yesmaximal
16C4×#476C211yesmaximal, characteristic, complemented
17#20648C212yesmaximal, complemented
18C2×#554—21no—
19C2×#547C2211yescharacteristic
20C2×C4×#41C2212yes—
21C2×#477C2212yes—
22C2×#476C2211yescharacteristic
23C2×#476C2212yes—
24C2×#436C2212yes—
25C2×#423C2212yes—
26C2×#417C2214yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2567—21no—
30#2552—21no—
31#2552C2212yes—
32#2502—22no—
33#2440—22no—
34#2440C2214yes—
35#2365C2211yescharacteristic
36#2362C2212yes—
37#2361—21no—
38#2359—22no—
39#2354C2211yescharacteristic
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2211yescharacteristic
42D4×C4⋊C4C2212yes—
43#2344—21no—
44#2339C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C4×#123—22no—
49C4×#121—21no—
50C2×#384C2212yes—
51C42×D4C2211yescharacteristic
52C42×D4C2211yescharacteristic
53C2×C4×C4⋊C4—21no—
54#2099C2214yes—
55#2096C2214yes—
56#2086C2214yes—
57#2068C2214yes—
58#2066C2214yes—
59#2024C2214yes—
60#1987C2211yescharacteristic
61#1987C2212yes—
62#1981C2212yes—
63#1956C2212yes—
64#1948C2212yes—
65#1948C2211yescharacteristic
66#1948C2212yes—
67#1947C2212yes—
68#1939C2212yes—
69#1920C2212yes—
70#1894C2214yes—
71#1872C2214yes—
72#1865C2214yes—
73#1864C2212yes—
74#1862C2214yes—
75#1854C2214yes—
76#1853C2214yes—
77#1851C2214yes—
78#1842C2211yescharacteristic
79#1834C2212yes—
80#1809C2214yes—
81#1803C2212yes—
82#1803C2212yes—
83#1803C2212yes—
84#1803C2214yes—
85#1802C2211yescharacteristic
86#1802C2214yes—
87#1802C2212yes—
88#1802C2212yes—
89#1797C2212yes—
90#1788C2214yes—
91C4×#106C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1736C2214yes—
95C22×#41C2312yes—
96C22×M16C2314yes—
97#554—48no—
98#547—24no—
99#547Q818yescomplemented
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125—22no—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8C2312yes—
110C2×#121—21no—
111C4×#41—28no—
112C4×#41—44no—
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C2×#117C2312yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2314yes—
133C22×C42—22no—
134C22×C42—21no—
135C22×C42C2312yes—
136#477—44no—
137#476—24no—
138#476—44no—
139#436—28no—
140C2×#107C2314yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144#423—44no—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2312yes—
151#417—48no—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×C22⋊C8C2314yes—
160C2×C4×C8C2312yes—
161#398—24no—
162#398—24no—
163#395—24no—
164#391—21no—
165#391—22no—
166#391C2312yes—
167#391C2312yes—
168#390—21no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389C2312yes—
177#389C2314yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—24no—
189#385—24no—
190#385—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384—22no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2314yes—
199#384C2312yes—
200#381—22no—
201#380—22no—
202#380—21no—
203#380C2312yes—
204#380C2314yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226C2×#95—22no—
227C43C2311yescharacteristic
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—42no—
243C2×#41—42no—
244C2×#41C2×Q818yes—
245C22×Q8—21no—
246C22×Q8—22no—
247C22×Q8C2412yes—
248C22×D4—22no—
249C22×D4C2×D411yescharacteristic
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—24no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2412yes—
260C2×M16—28no—
261C2×M16—44no—
262C2×M16—44no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—42no—
270C4⋊Q8—42no—
271C4⋊Q8—44no—
272C4⋊Q8—82no—
273C4⋊Q8C2×Q814yes—
274#125—48no—
275#123—216no—
276C22⋊Q8—24no—
277C22⋊Q8—416no—
278C22⋊Q8—48no—
279C22⋊Q8C2×Q818yes—
280#121—28no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—42no—
285C4×Q8—44no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—41no—
293C4×D4—42no—
294C4×D4—48no—
295C4×D4—48no—
296C4×D4—48no—
297C4×D4C2×Q814yes—
298C4×D4C2×Q814yes—
299C4×D4#4114yes—
300C4×D4#4114yes—
301#117—24no—
302#117—48no—
303#117—42no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—48no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—84no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—48no—
362C2×C42—44no—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—44no—
374#106—24no—
375#106—42no—
376#106—44no—
377C4⋊C8—24no—
378C4⋊C8—22no—
379C4⋊C8—22no—
380C4⋊C8—24no—
381C4⋊C8—44no—
382C4⋊C8—42no—
383#103—44no—
384#103—44no—
385#103—44no—
386#103—44no—
387#102—24no—
388#102—24no—
389#102—28no—
390C22⋊C8—28no—
391C4×C8—24no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—21no—
409C24C22×D412yes—
410#41—216no—
411#41—48no—
412#41—48no—
413#41—48no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—82no—
424C2×Q8—82no—
425C2×Q8C22×Q814yes—
426C2×D4—28no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4C22⋊Q814yes—
435C2×D4C22×Q812yes—
436C2×D4C2×#4112yes—
437C2×D4H(5,𝔽2)12yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—81no—
471C22×C4—82no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×Q814yes—
484C22×C4C2×#4114yes—
485C22×C4C2511yescharacteristic
486M16—88no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—22no—
496C4⋊C4—24no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—81no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4C22⋊Q814yes—
531C4⋊C4AGL(1,ℤ/8)14yes—
532C4⋊C4C22×Q812yes—
533C4⋊C4C2×#4112yes—
534C22⋊C4—28no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—48no—
543C22⋊C4—82no—
544C22⋊C4C22⋊Q818yes—
545C42—24no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—21no—
552C42—21no—
553C42—41no—
554C42—44no—
555C42—44no—
556C42—44no—
557C42—44no—
558C42—42no—
559C42—42no—
560C42—42no—
561C42#13714yes—
562C42C22×Q812yes—
563C42C2×#4112yes—
564C42C2×#4112yes—
565C42H(5,𝔽2)12yes—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23C2×C22⋊Q814yes—
580C23C23×D411yescharacteristic
581Q8—24no—
582Q8—24no—
583Q8—42no—
584Q8—42no—
585Q8—84no—
586Q8—164no—
587D4—28no—
588D4—22no—
589D4—22no—
590D4—22no—
591D4—416no—
592D4#55414yescomplemented
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—48no—
617C2×C4—416no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—84no—
643C2×C4—88no—
644C2×C4—168no—
645C2×C4C2×C22⋊Q812yes—
646C2×C4C2×C22⋊Q814yes—
647C2×C4C2×C22⋊Q812yes—
648C2×C4#54612yes—
649C2×C4#54612yes—
650C2×C4C2×AGL(1,ℤ/8)14yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4C2×#13712yes—
653C2×C4C2×#13712yes—
654C2×C4C2×#13712yes—
655C2×C4#57712yes—
656C2×C4C23×Q811yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×H(5,𝔽2)11yescharacteristic
661C2×C4#58511yescharacteristic
662C8—42no—
663C8—44no—
664C8—42no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22C22×C22⋊Q811yescharacteristic
680C22C2×#54611yescharacteristic
681C22#348811yescharacteristic
682C22C22×AGL(1,ℤ/8)11yescharacteristic
683C22C22×#13711yescharacteristic
684C22C2×#57711yescharacteristic
685C22#358611yescharacteristic
686C4—22no—
687C4—24no—
688C4—22no—
689C4—22no—
690C4—24no—
691C4—41no—
692C4—44no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—81no—
697C4—82no—
698C4—84no—
699C4—82no—
700C4—82no—
701C4#307914yes—
702C4#333114yes—
703C4C2×#55411yescharacteristic
704C4#347511yescharacteristic
705C4#350111yescharacteristic
706C4#351911yescharacteristic
707C2—44no—
708C2—44no—
709C2#3615411yescharacteristic
710C2C2×#307911yescharacteristic
711C2#6049711yescharacteristic
712C2C2×#333111yescharacteristic
713C2#6116111yescharacteristic
714C2#6196011yescharacteristic
715C2#6199911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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
1. #36180C2semidirect
3. #33197C2semidirect
4. #32958C2semidirect
5. #29141C2semidirect
6. #28284C2semidirect
7. #27680C2semidirect
8. #26528C2semidirect
9. #26431C2semidirect
10. #26106C2semidirect
11. #25978C2semidirect
12. #24846C2semidirect
13. #22715C2semidirect
14. #22650C2semidirect
16. C4×#476C2semidirect
17. #20648C2semidirect
99. #547Q8semidirect
592. D4#554semidirect

Maximal subgroups

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

subgroupindex
#361802
C4×#5472
#331972
#329582
#291412
#282842
#276802
#265282
#264312
#261062
#259782
#248462
#227152
#226502
#224312
C4×#4762
#206482