Finite Group Databasebeta

← #6800010

Subgroups — #6800010

Order 512.

1897 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#36561C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3#32954C211yesmaximal, characteristic, complemented
4#32082C214yesmaximal, complemented
5#30878C214yesmaximal, complemented
6#29951C212yesmaximal, complemented
7#28400C212yesmaximal, complemented
8#27769C212yesmaximal, complemented
9#25856C211yesmaximal, characteristic, complemented
10#25856C212yesmaximal, complemented
11#25276C212yesmaximal, complemented
12#24730C212yesmaximal, complemented
13#24209C211yesmaximal, characteristic
14#24209C212yesmaximal
15#22625C211yesmaximal, characteristic, complemented
16#22041C212yesmaximal
17C4×#448C211yesmaximal, characteristic
18C2×#554—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C2×#487C2214yes—
23C2×#487C2212yes—
24C2×#475C2212yes—
25C2×#461C2214yes—
26C2×#461C2212yes—
27C2×#448C2212yes—
28C2×#448C2211yescharacteristic
29C2×C4×SD16C2212yes—
30#2857C2211yescharacteristic
31#2857C2212yes—
32#2564—22no—
33#2556—22no—
34#2551C2211yescharacteristic
35#2546—21no—
36#2546—21no—
37#2546—22no—
38#2546C2214yes—
39#2546C2212yes—
40#2546C2212yes—
41#2542C2211yescharacteristic
42#2506C2212yes—
43#2492—21no—
44#2452C2212yes—
45#2417—21no—
46C2×#398—22no—
47#2365C2211yescharacteristic
48#2360—21no—
49D4×C4⋊C4C2211yescharacteristic
50C4×#121C2212yes—
51C4×#121C2211yescharacteristic
52C2×#391C2212yes—
53C2×#389—21no—
54C2×#389—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56#2096C2214yes—
57#2066C2214yes—
58#2066C2214yes—
59#2043C2214yes—
60#2043C2214yes—
61#2043C2214yes—
62#2043C2212yes—
63#2043C2214yes—
64#2024C2214yes—
65#2013C2212yes—
66#2013C2214yes—
67#1981C2214yes—
68#1981C2212yes—
69#1968C2214yes—
70#1956C2214yes—
71#1956C2212yes—
72#1939C2212yes—
73#1933C2214yes—
74#1927C2214yes—
75#1916C2214yes—
76#1880C2212yes—
77#1877C2214yes—
78#1874C2211yescharacteristic
79#1865C2212yes—
80#1865C2214yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1803C2212yes—
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1787C2212yes—
87#1787C2211yescharacteristic
88#1787C2212yes—
89#1787C2212yes—
90#1787C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×Q8C2312yes—
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99#554—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8—48no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2314yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2312yes—
113C2×#125—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—21no—
119C2×#121C2311yescharacteristic
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—44no—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4D411yescharacteristic
137C22×C4⋊C4D411yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#487—44no—
143#487—48no—
144#475—44no—
145#461—48no—
146#461—44no—
147#448—24no—
148#448—28no—
149C4×SD16—44no—
150C2×#106C2314yes—
151C2×#106C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2314yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#102C2312yes—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2311yescharacteristic
169C2×C22⋊C8C2311yescharacteristic
170C2×C4×C8C2312yes—
171C2×C4×C8C2314yes—
172#400—22no—
173#399—22no—
174#398—22no—
175#398—24no—
176#398—24no—
177#398—24no—
178#398—24no—
179#396—22no—
180#395—22no—
181#393—22no—
182#391—21no—
183#391—21no—
184#391—22no—
185#391—22no—
186#391—22no—
187#391C2314yes—
188#391C2314yes—
189#390C2311yescharacteristic
190#390C2312yes—
191#389—21no—
192#389—22no—
193#389—24no—
194#389—22no—
195#389—24no—
196#389—22no—
197#389—22no—
198#389—22no—
199#389—21no—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2311yescharacteristic
204#389C2312yes—
205#388—22no—
206#388—21no—
207#387—22no—
208#387—22no—
209#386—21no—
210#386—22no—
211#385—22no—
212#385—22no—
213#384C2311yescharacteristic
214#384C2312yes—
215#380—21no—
216#380—22no—
217#380—21no—
218#380C2312yes—
219#380C2311yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D412yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—21no—
229C4×C22⋊C4—22no—
230C2×#95—22no—
231C2×#95—22no—
232C2×#95D412yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238#336C2312yes—
239#336C2314yes—
240C22×Q8—24no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—22no—
248C22×Q8—24no—
249C22×Q8—42no—
250C22×Q8—44no—
251C22×Q8—44no—
252C22×Q8C2414yes—
253C22×Q8C2412yes—
254C22×D4—21no—
255C22×D4C2411yescharacteristic
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C2×SD16—24no—
267C2×SD16—24no—
268C2×SD16—28no—
269C2×SD16—42no—
270C2×SD16—44no—
271C2×SD16—44no—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C4⋊Q8—28no—
276C4⋊Q8—28no—
277C4⋊Q8—24no—
278C4⋊Q8—24no—
279C4⋊Q8—24no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286C4⋊Q8—48no—
287C4⋊Q8—48no—
288C4⋊Q8—44no—
289#125—48no—
290C22⋊Q8—48no—
291C22⋊Q8—44no—
292C22⋊Q8—48no—
293C22⋊Q8—44no—
294C22⋊Q8—44no—
295#121—22no—
296#121#4114yes—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—42no—
300C4×Q8—42no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×D4—24no—
305C4×D4—44no—
306C4×D4—48no—
307C4×D4—44no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—48no—
334C2×C4⋊C4—48no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4SD1618yescomplemented
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—44no—
366C2×C42—44no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375#106—42no—
376#106—44no—
377C4⋊C8—24no—
378C4⋊C8—44no—
379#103—28no—
380#103—24no—
381#103—24no—
382#103—28no—
383#103—42no—
384#103—44no—
385#103—44no—
386#103—44no—
387#103—44no—
388#103—44no—
389#103—44no—
390#103—44no—
391#102—42no—
392C22⋊C8—22no—
393C22⋊C8—24no—
394C22⋊C8—22no—
395C4×C8—24no—
396C4×C8—44no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—22no—
402#95—21no—
403#95—21no—
404#95—22no—
405#95—21no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95C2×D412yes—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414C24—21no—
415C24C22×D411yescharacteristic
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—48no—
429C2×Q8—48no—
430C2×Q8—42no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—82no—
440C2×Q8—82no—
441C2×Q8—88no—
442C2×Q8—88no—
443C2×Q8H(5,𝔽2)14yes—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4C2×#4112yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4C2×SD1614yes—
487C22×C4C2×SD1618yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C2×#4112yes—
498C22×C42−1+412yes—
499C22×C4C2511yescharacteristic
500SD16—48no—
501SD16—88no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
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—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—81no—
541C4⋊C4—82no—
542C4⋊C4C2×SD1614yes—
543C4⋊C4C2×SD1618yes—
544C4⋊C4C2×SD1614yes—
545C4⋊C4#13714yes—
546C4⋊C4C2×#4112yes—
547C22⋊C4—24no—
548C22⋊C4—44no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—44no—
554C22⋊C4—81no—
555C22⋊C4—82no—
556C42—22no—
557C42—22no—
558C42—24no—
559C42—24no—
560C42—22no—
561C42—42no—
562C42—42no—
563C42—44no—
564C42—44no—
565C42—44no—
566C42—42no—
567C42H(5,𝔽2)14yes—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—22no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23#54812yes—
580C23C22×SD1612yes—
581C23#57812yes—
582C23C23×D411yescharacteristic
583Q8—24no—
584Q8—24no—
585Q8—48no—
586Q8—42no—
587Q8—42no—
588Q8—44no—
589Q8—44no—
590Q8—44no—
591Q8—44no—
592Q8—88no—
593Q8—84no—
594Q8—164no—
595D4—24no—
596D4—48no—
597D4—42no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—84no—
648C2×C4—88no—
649C2×C4—84no—
650C2×C4—164no—
651C2×C4#53512yes—
652C2×C4#53512yes—
653C2×C4#54812yes—
654C2×C4C22×SD1614yes—
655C2×C4C22×SD1612yes—
656C2×C4C22×SD1614yes—
657C2×C4C22×SD1612yes—
658C2×C4C2×#13714yes—
659C2×C4C2×#13712yes—
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)12yes—
662C2×C4C2×H(5,𝔽2)12yes—
663C2×C4C2×2−1+411yescharacteristic
664C8—42no—
665C8—44no—
666C8—42no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22#335914yes—
681C22C2×#53511yescharacteristic
682C22C2×#54811yescharacteristic
683C22#348411yescharacteristic
684C22C23×SD1611yescharacteristic
685C22C22×#13711yescharacteristic
686C22C2×#57811yescharacteristic
687C22#358611yescharacteristic
688C4—22no—
689C4—24no—
690C4—24no—
691C4—22no—
692C4—24no—
693C4—22no—
694C4—42no—
695C4—41no—
696C4—42no—
697C4—44no—
698C4—81no—
699C4—82no—
700C4—82no—
701C4—81no—
702C4—82no—
703C4—84no—
704C4#320614yes—
705C4#351912yes—
706C4#353612yes—
707C2—24no—
708C2—82no—
709C2#3912011yescharacteristic
710C2C2×#320611yescharacteristic
711C2#6081411yescharacteristic
712C2C2×#335911yescharacteristic
713C2#6132211yescharacteristic
714C2#6156211yescharacteristic
715C2#6160211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2546C2semidirect
3. #32954C2semidirect
4. #32082C2semidirect
5. #30878C2semidirect
6. #29951C2semidirect
7. #28400C2semidirect
8. #27769C2semidirect
9. #25856C2semidirect
10. #25856C2semidirect
11. #25276C2semidirect
12. #24730C2semidirect
15. #22625C2semidirect
338. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#25462
#329542
#320822
#308782
#299512
#284002
#277692
#258562
#252762
#247302
#242092
#226252
#220412
C4×#4482