Finite Group Databasebeta

← #6794846

Subgroups — #6794846

Order 512.

1947 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36161C211yesmaximal, characteristic
2C2×#2538C211yesmaximal, characteristic
3#32954C211yesmaximal, characteristic, complemented
4#32009C214yesmaximal, complemented
5#30572C214yesmaximal, complemented
6#29894C212yesmaximal, complemented
7#28228C212yesmaximal, complemented
8#27950C211yesmaximal, characteristic, complemented
9#27680C211yesmaximal, characteristic, complemented
10#25341C212yesmaximal, complemented
11#25017C212yesmaximal, complemented
12#24380C212yesmaximal, complemented
13#24123C212yesmaximal, complemented
14#24010C212yesmaximal, complemented
15D4×C22⋊C8C211yesmaximal, characteristic
16#21941C212yesmaximal, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#485C2214yescomplemented
23C2×#475C2212yescomplemented
24C2×#467C2214yescomplemented
25C2×#451C2211yescharacteristic
26C2×C4×Q16C2212yescomplemented
27C2×C8×D4C2212yes—
28C22×C22⋊C8C2212yes—
29#2857C2211yescharacteristic
30#2848C2212yes—
31#2547C2211yescharacteristic
32#2538—21no—
33#2538—21no—
34#2538—22no—
35#2538C2214yes—
36#2538C2212yes—
37#2538C2211yescharacteristic
38#2538C2212yes—
39#2485C2212yes—
40#2480—22no—
41#2466—22no—
42#2447—21no—
43#2435—21no—
44#2415C2212yes—
45C2×#396—22no—
46#2367—21no—
47#2360—21no—
48#2354C2211yescharacteristic
49C4×C22⋊Q8C2211yescharacteristic
50C2×#390—21no—
51C2×#389—21no—
52C2×#384C2211yescharacteristic
53#2280C2212yes—
54C2×#380C2212yes—
55#2099C2212yescomplemented
56#2096C2212yescomplemented
57#2083C2214yescomplemented
58#2077C2214yescomplemented
59#2075C2214yescomplemented
60#2036C2214yescomplemented
61#2036C2214yescomplemented
62#2036C2214yescomplemented
63#2035C2214yescomplemented
64#2035C2214yescomplemented
65#2001C2214yescomplemented
66#1990C2212yescomplemented
67#1954C2212yes—
68#1948C2212yescomplemented
69#1892C2214yescomplemented
70#1878C2212yescomplemented
71#1873C2214yescomplemented
72#1872C2212yes—
73#1872C2211yescharacteristic
74#1869C2214yes—
75#1865C2214yescomplemented
76#1863C2212yescomplemented
77#1854C2214yescomplemented
78#1854C2214yescomplemented
79#1853C2212yes—
80#1853C2212yes—
81#1852C2212yes—
82#1852C2212yes—
83#1852C2211yescharacteristic
84#1845C2212yescomplemented
85#1808C2214yescomplemented
86#1807C2214yescomplemented
87#1803C2212yescomplemented
88#1803C2212yescomplemented
89#1797C2211yescharacteristic
90#1787C2212yes—
91#1787C2211yescharacteristic
92#1779C2212yes—
93C4×#103C2212yescomplemented
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×D4D411yescharacteristic
97C24×C4C2312yes—
98C22×Q16C2312yes—
99C23×C8C2314yes—
100#556—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#126—22no—
103C2×#124—21no—
104C2×#124—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—48no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C22⋊Q8C2314yes—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—44no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×D4—44no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D412yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×#117C2312yes—
129C2×#117C2312yes—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4D411yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—22no—
138C22×C22⋊C4—22no—
139C22×C22⋊C4D412yes—
140C22×C42—21no—
141C22×C42C2311yescharacteristic
142#485—48no—
143#475—44no—
144#467—48no—
145#451—24no—
146C4×Q16—44no—
147C8×D4—44no—
148C2×#107C2312yes—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8C2312yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×C22⋊C8—28no—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2311yescharacteristic
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×C22⋊C8C2314yes—
167C2×C22⋊C8C2311yescharacteristic
168C2×C4×C8C2312yes—
169#397—22no—
170#397—22no—
171#397—22no—
172#396—24no—
173#396—24no—
174#396—24no—
175#396—24no—
176#396—22no—
177#394—22no—
178#394—22no—
179#391—21no—
180#391—21no—
181#391C2311yescharacteristic
182#390—21no—
183#390—21no—
184#390—22no—
185#390—24no—
186#390—22no—
187#389—22no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389—24no—
192#389—21no—
193#389C2312yes—
194#389C2311yescharacteristic
195#388—22no—
196#388—22no—
197#388—22no—
198#387—22no—
199#387—21no—
200#385—22no—
201#385—22no—
202#385—22no—
203#385—21no—
204#384—21no—
205#384—21no—
206#384C2311yescharacteristic
207#384C2312yes—
208#384C2312yes—
209#381—22no—
210#380—21no—
211#380—22no—
212#380—22no—
213#380C2312yes—
214#380C2314yes—
215#380C2312yes—
216#380C2312yes—
217#380C2314yes—
218#380C2312yes—
219#380C2312yes—
220#380C2311yescharacteristic
221#379—22no—
222#379—22no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C2×#95—22no—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#336C2312yes—
238#336C2312yes—
239C25C2×D412yes—
240C22×Q8—24no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—42no—
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C22×D4—24no—
249C22×D4—24no—
250C22×D4—42no—
251C22×D4C2×D411yescharacteristic
252C22×D4C2×D412yes—
253C22×D4C2×D411yescharacteristic
254C23×C4—21no—
255C23×C4—24no—
256C23×C4—24no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2412yes—
267C23×C4C2412yes—
268C23×C4C2411yescharacteristic
269C2×Q16—24no—
270C2×Q16—24no—
271C2×Q16—42no—
272C22×C8—28no—
273C22×C8—44no—
274C22×C8—44no—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—24no—
279#126—48no—
280#124—44no—
281#124—44no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—48no—
290C22⋊Q8—48no—
291C22⋊Q8—44no—
292C22⋊Q8—44no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—44no—
304C4×D4—42no—
305C4×D4—42no—
306C4×D4—44no—
307#117—44no—
308#117—44no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
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—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—48no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—82no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—44no—
361C2×C22⋊C4—48no—
362C2×C22⋊C4—84no—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—24no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—22no—
374C2×C42—44no—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—42no—
380#107—44no—
381#107—44no—
382#106—44no—
383#106—44no—
384C4⋊C8—24no—
385#103—28no—
386#103—24no—
387#103—24no—
388#103—42no—
389#103—42no—
390#103—44no—
391#103—44no—
392C22⋊C8—24no—
393C22⋊C8—22no—
394C22⋊C8—28no—
395C22⋊C8—44no—
396C22⋊C8—44no—
397C22⋊C8#4114yes—
398C4×C8—24no—
399#95—24no—
400#95—24no—
401#95—21no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—21no—
407#95—22no—
408#95—21no—
409#95—21no—
410#95—22no—
411#95—22no—
412C24—24no—
413C24—24no—
414C24—24no—
415C24—24no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24C22×D412yes—
420C24C22×D412yes—
421C24C22×D411yescharacteristic
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—48no—
433C2×Q8—48no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8H(5,𝔽2)14yes—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—416no—
448C2×D4—42no—
449C2×D4#13714yes—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—88no—
493C22×C4—88no—
494C22×C4#13514yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C2×#4112yes—
503C22×C42−1+412yes—
504C22×C4C2511yescharacteristic
505Q16—48no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—22no—
514C2×C8—88no—
515C2×C8C2×#4114yes—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4—81no—
544C4⋊C4—81no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—42no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—42no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—84no—
564C22⋊C4—84no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—44no—
571C42—44no—
572C42—44no—
573C42—42no—
574C42—42no—
575C42—82no—
576C42H(5,𝔽2)14yes—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—48no—
595C23—48no—
596C23—48no—
597C23—48no—
598C23—84no—
599C23—84no—
600C23—84no—
601C23#54712yes—
602C23C2×#13512yes—
603C23C2×#13714yes—
604C23#57812yes—
605C23C23×D411yescharacteristic
606Q8—24no—
607Q8—24no—
608Q8—48no—
609Q8—42no—
610Q8—42no—
611Q8—42no—
612Q8—42no—
613Q8—84no—
614D4—28no—
615D4—48no—
616D4—44no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—44no—
637C2×C4—48no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4—88no—
673C2×C4#53412yes—
674C2×C4#53412yes—
675C2×C4#54712yes—
676C2×C4C2×#13512yes—
677C2×C4C2×#13512yes—
678C2×C4C2×#13712yes—
679C2×C4C22×#4111yescharacteristic
680C2×C4C2×H(5,𝔽2)12yes—
681C2×C4C2×H(5,𝔽2)12yes—
682C2×C4C2×2−1+411yescharacteristic
683C8—24no—
684C8—44no—
685C8—42no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—816no—
707C22#334214yes—
708C22C2×#53411yescharacteristic
709C22C2×#54711yescharacteristic
710C22#348411yescharacteristic
711C22C22×#13511yescharacteristic
712C22C22×#13711yescharacteristic
713C22C2×#57811yescharacteristic
714C22#358511yescharacteristic
715C4—22no—
716C4—22no—
717C4—22no—
718C4—42no—
719C4—42no—
720C4—44no—
721C4—41no—
722C4—81no—
723C4—82no—
724C4—81no—
725C4—81no—
726C4—82no—
727C4—82no—
728C4—84no—
729C4—81no—
730C4#319014yes—
731C4#351912yes—
732C4#353612yes—
733C2—24no—
734C2—44no—
735C2—82no—
736C2#3891611yescharacteristic
737C2C2×#319011yescharacteristic
738C2#6076411yescharacteristic
739C2C2×#334211yescharacteristic
740C2#6125411yescharacteristic
741C2#6156611yescharacteristic
742C2#6160211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32954C2semidirect
4. #32009C2semidirect
5. #30572C2semidirect
6. #29894C2semidirect
7. #28228C2semidirect
8. #27950C2semidirect
9. #27680C2semidirect
10. #25341C2semidirect
11. #25017C2semidirect
12. #24380C2semidirect
13. #24123C2semidirect
14. #24010C2semidirect
16. #21941C2semidirect
17. C4×#451C2semidirect
22. C2×#485C22semidirect
23. C2×#475C22semidirect
24. C2×#467C22semidirect
26. C2×C4×Q16C22semidirect
55. #2099C22semidirect
56. #2096C22semidirect
57. #2083C22semidirect
58. #2077C22semidirect
59. #2075C22semidirect
60. #2036C22semidirect
61. #2036C22semidirect
62. #2036C22semidirect
63. #2035C22semidirect
64. #2035C22semidirect
65. #2001C22semidirect
66. #1990C22semidirect
68. #1948C22semidirect
69. #1892C22semidirect
70. #1878C22semidirect
71. #1873C22semidirect
75. #1865C22semidirect
76. #1863C22semidirect
77. #1854C22semidirect
78. #1854C22semidirect
84. #1845C22semidirect
85. #1808C22semidirect
86. #1807C22semidirect
87. #1803C22semidirect
88. #1803C22semidirect
93. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#361612
C2×#25382
#329542
#320092
#305722
#298942
#282282
#279502
#276802
#253412
#250172
#243802
#241232
#240102
D4×C22⋊C82
#219412
C4×#4512