Finite Group Databasebeta

← #6471246

Subgroups — #6471246

Order 512.

1982 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#35006C211yesmaximal, characteristic, complemented
2C2×#2367C211yesmaximal, characteristic, complemented
3#33096C212yesmaximal, complemented
4#32954C211yesmaximal, characteristic, complemented
5#32903C211yesmaximal, characteristic, complemented
6#29950C211yesmaximal, characteristic, complemented
7#29712C211yesmaximal, characteristic
8#29623C212yesmaximal, complemented
9#27086C214yesmaximal, complemented
10#26907C214yesmaximal, complemented
11#26776C214yesmaximal, complemented
12#26608C214yesmaximal, complemented
13#22047C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21452C212yesmaximal, complemented
16C2×#539—21no—
17C22×#124C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#475C2212yescomplemented
21C2×#460C2212yescomplemented
22C2×#448C2212yes—
23C2×#447C2212yes—
24C2×#442C2214yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2639—21no—
29#2618—21no—
30#2560—22no—
31#2558—22no—
32#2512C2212yescomplemented
33#2506C2212yescomplemented
34#2452C2212yescomplemented
35#2445C2212yescomplemented
36#2367—22no—
37#2367—22no—
38#2367C2214yes—
39#2367C2214yes—
40#2325—21no—
41C4×#121C2212yes—
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#388—21no—
45C2×#388—21no—
46C2×#385—21no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4—21no—
50#2093C2212yes—
51#2071C2212yes—
52#2067C2214yes—
53#2054C2214yes—
54#2042C2214yes—
55#2042C2214yes—
56#2023C2212yes—
57#2012C2214yes—
58#2012C2214yes—
59#2011C2212yes—
60#1973C2214yescomplemented
61#1935C2214yescomplemented
62#1928C2214yescomplemented
63#1926C2214yescomplemented
64#1925C2214yescomplemented
65#1921C2214yescomplemented
66#1920C2214yescomplemented
67#1917C2214yescomplemented
68#1917C2214yescomplemented
69#1880C2214yescomplemented
70#1879C2214yes—
71#1877C2214yescomplemented
72#1874C2212yes—
73#1874C2212yes—
74#1865C2214yescomplemented
75#1864C2214yes—
76#1853C2212yes—
77#1852C2212yes—
78#1817C2212yes—
79#1803C2212yescomplemented
80#1803C2212yescomplemented
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1787C2212yes—
84#1786C2212yes—
85#1762C2214yes—
86#1749C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91C22×D8C2314yes—
92#539—48no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#126—22no—
96C2×#125—22no—
97C2×#124—24no—
98C2×#124—24no—
99C2×#124—22no—
100C2×#124C2314yes—
101C2×#123—22no—
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—44no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—44no—
116C2×C4×D4C2314yes—
117C2×C4×D4C2312yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#475—44no—
134#460—44no—
135#448—28no—
136#447—28no—
137#442—48no—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×#97C2314yes—
153#400—22no—
154#398—22no—
155#397—22no—
156#396—22no—
157#396—22no—
158#394—22no—
159#394—22no—
160#393—22no—
161#391—21no—
162#391—22no—
163#391C2314yes—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#389—21no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#388—24no—
180#388—24no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—24no—
186#388—24no—
187#387—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#385—24no—
192#385—24no—
193#385—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#380—21no—
197#380C2312yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—24no—
207C4×C22⋊C4—24no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8#4114yes—
229C22×Q8C2412yes—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—42no—
236C22×D4—42no—
237C22×D4#4114yes—
238C22×D4C2412yes—
239C22×D4C2412yes—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—24no—
244C23×C4—22no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—28no—
256C2×SD16—44no—
257C2×SD16—44no—
258C2×D8—28no—
259C2×D8—44no—
260C2×D8—44no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263C4⋊Q8—42no—
264#127—42no—
265#126—48no—
266#125—48no—
267#124—216no—
268#124—28no—
269#124—28no—
270#123—48no—
271#123—48no—
272C22⋊Q8—48no—
273C22⋊Q8—44no—
274#121—24no—
275#121—24no—
276#121—44no—
277#121—44no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×D4—28no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—48no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—82no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—42no—
346C2×C42—42no—
347C2×C42—44no—
348C2×C42—82no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357#107—44no—
358#106—28no—
359C4⋊C8—42no—
360C4⋊C8—42no—
361#103—28no—
362#103—44no—
363#103—44no—
364#102—28no—
365#102—28no—
366#102—44no—
367#102—44no—
368#102—44no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371#97—44no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—24no—
386C24—22no—
387C24—22no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C22×D411yescharacteristic
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—48no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—48no—
405C2×Q8—42no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×Q8C2×#4114yes—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—48no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—48no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4—81no—
430C2×D4C2×#4114yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—84no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—816no—
472C22×C4AGL(1,ℤ/8)14yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2511yescharacteristic
485SD16—88no—
486D8—416no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4H(5,𝔽2)14yes—
517C22⋊C4—216no—
518C22⋊C4—216no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—42no—
542C42—41no—
543C42—41no—
544C42—44no—
545C42—44no—
546C42—44no—
547C42—84no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—24no—
553C23—48no—
554C23—48no—
555C23—42no—
556C23—42no—
557C23—48no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23#52512yes—
574C23C2×AGL(1,ℤ/8)12yes—
575C23#57512yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—48no—
580Q8—42no—
581Q8—42no—
582Q8—162no—
583D4—28no—
584D4—48no—
585D4—44no—
586D4—44no—
587D4—44no—
588D4—88no—
589D4—162no—
590D4—162no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—44no—
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—82no—
642C2×C4—82no—
643C2×C4—816no—
644C2×C4—84no—
645C2×C4—164no—
646C2×C4#52512yes—
647C2×C4C2×AGL(1,ℤ/8)12yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)12yes—
652C2×C4#58512yes—
653C8—44no—
654C8—44no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—816no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22#311514yes—
677C22C2×#52511yescharacteristic
678C22#348411yescharacteristic
679C22#348711yescharacteristic
680C22C22×AGL(1,ℤ/8)11yescharacteristic
681C22C2×#57511yescharacteristic
682C22#358511yescharacteristic
683C22#358611yescharacteristic
684C4—22no—
685C4—22no—
686C4—22no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—41no—
691C4—41no—
692C4—44no—
693C4—42no—
694C4—44no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4#345912yes—
700C4#350512yes—
701C2—24no—
702C2—84no—
703C2—162no—
704C2#3498711yescharacteristic
705C2C2×#311511yescharacteristic
706C2#6056511yescharacteristic
707C2#6157211yescharacteristic
708C2#6158111yescharacteristic
709C2#6189111yescharacteristic
710C2#6189411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#29712C2
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. #35006C2semidirect
2. C2×#2367C2semidirect
3. #33096C2semidirect
4. #32954C2semidirect
5. #32903C2semidirect
6. #29950C2semidirect
8. #29623C2semidirect
9. #27086C2semidirect
10. #26907C2semidirect
11. #26776C2semidirect
12. #26608C2semidirect
13. #22047C2semidirect
14. #21916C2semidirect
15. #21452C2semidirect
20. C2×#475C22semidirect
21. C2×#460C22semidirect
32. #2512C22semidirect
33. #2506C22semidirect
34. #2452C22semidirect
35. #2445C22semidirect
60. #1973C22semidirect
61. #1935C22semidirect
62. #1928C22semidirect
63. #1926C22semidirect
64. #1925C22semidirect
65. #1921C22semidirect
66. #1920C22semidirect
67. #1917C22semidirect
68. #1917C22semidirect
69. #1880C22semidirect
71. #1877C22semidirect
74. #1865C22semidirect
79. #1803C22semidirect
80. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#23672
#330962
#329542
#329032
#299502
#297122
#296232
#270862
#269072
#267762
#266082
#220472
#219162
#214522