Finite Group Databasebeta

← #6811851

Subgroups — #6811851

Order 512.

2013 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C212yesmaximal, complemented
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#34934C211yesmaximal, characteristic
4#33196C211yesmaximal, characteristic, complemented
5#30799C214yesmaximal, complemented
6#30659C214yesmaximal, complemented
7#29689C212yesmaximal, complemented
8#28816C212yesmaximal, complemented
9#28508C211yesmaximal, characteristic, complemented
10#28279C211yesmaximal, characteristic, complemented
11#27778C212yesmaximal, complemented
12#26232C212yesmaximal
13#25016C212yesmaximal, complemented
14#24709C212yesmaximal, complemented
15#24298C212yesmaximal, complemented
16#22799C211yesmaximal, characteristic, complemented
17C4×#475C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#529C2212yes—
21#3239—216no—
22C2×#501C2214yes—
23C2×#489C2214yes—
24C2×#484C2214yes—
25C2×#481C2214yes—
26C2×#475C2212yes—
27C2×#475C2211yescharacteristic
28C2×#474C2212yes—
29C2×#474C2212yes—
30C2×#474C2212yes—
31C2×#432C2212yes—
32#2868C2211yescharacteristic
33#2857C2212yes—
34#2554—22no—
35#2552C2211yescharacteristic
36#2551—21no—
37#2551C2212yes—
38#2548C2211yescharacteristic
39#2546—21no—
40#2541C2211yescharacteristic
41#2538C2211yescharacteristic
42#2512—21no—
43#2490C2212yes—
44#2445—21no—
45#2440—22no—
46#2440C2214yes—
47#2434—22no—
48#2426C2212yes—
49#2392—22no—
50D4×C4⋊C4—21no—
51D4×C4⋊C4C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×#127—21no—
54C4×#127C2211yescharacteristic
55C4×#121C2212yes—
56#2293—21no—
57C42×Q8C2211yescharacteristic
58C42×D4C2211yescharacteristic
59#2093C2214yes—
60#2076C2214yes—
61#2067C2214yes—
62#2059C2214yes—
63#2057C2214yes—
64#2042C2214yes—
65#2036C2214yes—
66#2034C2214yes—
67#2034C2214yes—
68#2023C2214yes—
69#1988C2211yescharacteristic
70#1988C2212yes—
71#1982C2212yes—
72#1957C2212yes—
73#1954C2212yes—
74#1948C2212yes—
75#1948C2211yescharacteristic
76#1948C2212yes—
77#1947C2212yes—
78#1944C2214yes—
79#1936C2214yes—
80#1924C2214yes—
81#1919C2212yes—
82#1874C2214yes—
83#1865C2212yes—
84#1852C2214yes—
85#1845C2212yes—
86#1842C2211yescharacteristic
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×#106C2212yes—
93#1770C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96C23×Q8—22no—
97D4×Q8—48no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127—21no—
109C2×#127C2311yescharacteristic
110#529—28no—
111#529—44no—
112C2×#126—22no—
113C2×#125C2312yes—
114C2×#124C2312yes—
115C2×#123—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8C2312yes—
119C2×#121C2312yes—
120C2×#121C2312yes—
121C2×C4×Q8—21no—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—22no—
126C2×C4×D4—22no—
127C2×C4×D4—22no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4D411yescharacteristic
131C2×C4×D4D411yescharacteristic
132C2×C4×D4C2312yes—
133C2×C4×D4C2311yescharacteristic
134C2×#117—22no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2314yes—
137C22×C42C2312yes—
138#501—216no—
139#489—216no—
140#484—216no—
141#481—216no—
142#475—28no—
143#475D418yescomplemented
144#474—28no—
145#474—28no—
146#474—44no—
147#432—28no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×#106C2312yes—
153C2×#106C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2311yescharacteristic
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×C22⋊C8C2314yes—
167C2×C22⋊C8C2314yes—
168C2×#97C2314yes—
169C2×C4×C8C2312yes—
170#398—22no—
171#398—24no—
172#397—22no—
173#396—22no—
174#396—22no—
175#395—24no—
176#395—24no—
177#394—22no—
178#394—22no—
179#394—22no—
180#392—22no—
181#391—21no—
182#391—22no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—22no—
188#390—22no—
189#390—21no—
190#390—21no—
191#390C2312yes—
192#390C2312yes—
193#389—22no—
194#389—22no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—22no—
199#389C2312yes—
200#389C2312yes—
201#389C2314yes—
202#388—22no—
203#388—22no—
204#388—22no—
205#386—22no—
206#385—24no—
207#385—24no—
208#384—21no—
209#384—21no—
210#384—21no—
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#384C2312yes—
214#384C2312yes—
215#380—22no—
216#380C2312yes—
217#380C2314yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—22no—
230C4×C22⋊C4D412yes—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4C2312yes—
233C43—21no—
234C43C2311yescharacteristic
235#340C2312yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2314yes—
240#340C2312yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—48no—
248C22×Q8—48no—
249C22×Q8—84no—
250C22×Q8C2411yescharacteristic
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4C2×D411yescharacteristic
256C22×D4C2412yes—
257C23×C4—24no—
258C23×C4—22no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—28no—
266C4⋊Q8—24no—
267C4⋊Q8—22no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—48no—
271C4⋊Q8—48no—
272C4⋊Q8—48no—
273C4⋊Q8—44no—
274C4⋊Q8—82no—
275C4⋊Q8C2×D414yes—
276#127—22no—
277#127—22no—
278#127—44no—
279#126—48no—
280#125—24no—
281#125—42no—
282#124—24no—
283#124—42no—
284#123—48no—
285C22⋊Q8—24no—
286C22⋊Q8—416no—
287C22⋊Q8—42no—
288C22⋊Q8—48no—
289#121—24no—
290#121—24no—
291#121—42no—
292#121—42no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×Q8—41no—
298C4×Q8—44no—
299C4×Q8C2×D414yes—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—48no—
307C4×D4—48no—
308C4×D4—44no—
309C4×D4—82no—
310C4×D4—84no—
311#117—48no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—48no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—21no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42C2×D412yes—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D412yes—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—28no—
381#107—28no—
382#107—44no—
383#106—28no—
384#106—24no—
385#106C2×D418yes—
386C4⋊C8—24no—
387C4⋊C8—22no—
388C4⋊C8—24no—
389C4⋊C8—42no—
390C4⋊C8C2×D414yes—
391#103—28no—
392#103—42no—
393#103C2×D418yes—
394#102—28no—
395#102—28no—
396#102—44no—
397#102—44no—
398C22⋊C8—28no—
399C22⋊C8—44no—
400#97—28no—
401C4×C8—24no—
402#95—22no—
403#95—22no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—24no—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414C24—22no—
415C24C22×D412yes—
416C2×Q8—24no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—48no—
423C2×Q8—48no—
424C2×Q8—44no—
425C2×Q8—42no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—44no—
429C2×Q8—84no—
430C2×Q8—832no—
431C2×Q8C22×D412yes—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—81no—
449C2×D4—81no—
450C2×D4—82no—
451C2×D4—82no—
452C2×D4#13714yes—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—42no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—84no—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2511yescharacteristic
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8C22×D414yes—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—48no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—81no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4C22×D414yes—
534C4⋊C4C22×D412yes—
535C4⋊C42−1+412yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—48no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C42—24no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—21no—
559C42—21no—
560C42—41no—
561C42—42no—
562C42—42no—
563C42—44no—
564C42—42no—
565C42—42no—
566C42—81no—
567C42AGL(1,ℤ/8)14yes—
568C42C22×D412yes—
569C42H(5,𝔽2)12yes—
570C422−1+412yes—
571C422−1+412yes—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23C2×#13714yes—
586C23C23×D411yescharacteristic
587Q8—22no—
588Q8—22no—
589Q8—48no—
590Q8—41no—
591Q8—41no—
592Q8—816no—
593Q8—88no—
594D4—28no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—42no—
599D4—84no—
600D4—88no—
601D4—88no—
602D4—84no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—48no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—48no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—816no—
652C2×C4—88no—
653C2×C4#53612yes—
654C2×C4#53612yes—
655C2×C4D4212yes—
656C2×C4D4212yes—
657C2×C4C2×AGL(1,ℤ/8)12yes—
658C2×C4C2×AGL(1,ℤ/8)12yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C2×#13712yes—
661C2×C4#57612yes—
662C2×C4C23×D411yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×2−1+411yescharacteristic
665C2×C4C2×2−1+411yescharacteristic
666C2×C4C2×2−1+411yescharacteristic
667C2×C4#58511yescharacteristic
668C8—24no—
669C8—44no—
670C8—42no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22C2×#53611yescharacteristic
686C22C2×D4211yescharacteristic
687C22#348911yescharacteristic
688C22C22×AGL(1,ℤ/8)11yescharacteristic
689C22C22×#13711yescharacteristic
690C22C2×#57611yescharacteristic
691C22#358611yescharacteristic
692C4—22no—
693C4—22no—
694C4—22no—
695C4—42no—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—41no—
702C4—42no—
703C4—82no—
704C4—84no—
705C4—82no—
706C4—81no—
707C4#323914yescomplemented
708C4#327414yes—
709C4#349711yescharacteristic
710C4#349911yescharacteristic
711C4#354711yescharacteristic
712C4#355011yescharacteristic
713C2—44no—
714C2—162no—
715C2#3934311yescharacteristic
716C2C2×#323911yescharacteristic
717C2#6088811yescharacteristic
718C2C2×#327411yescharacteristic
719C2D4×#13711yescharacteristic
720C2#6211111yescharacteristic
721C2#6212711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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. C2×#3239C2semidirect
2. D4×C4⋊Q8C2semidirect
4. #33196C2semidirect
5. #30799C2semidirect
6. #30659C2semidirect
7. #29689C2semidirect
8. #28816C2semidirect
9. #28508C2semidirect
10. #28279C2semidirect
11. #27778C2semidirect
13. #25016C2semidirect
14. #24709C2semidirect
15. #24298C2semidirect
16. #22799C2semidirect
17. C4×#475C2semidirect
143. #475D4semidirect
707. C4#3239semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
D4×C4⋊Q82
#349342
#331962
#307992
#306592
#296892
#288162
#285082
#282792
#277782
#262322
#250162
#247092
#242982
#227992
C4×#4752