Finite Group Databasebeta

← #6802697

Subgroups — #6802697

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36579C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic, complemented
3#32906C211yesmaximal, characteristic, complemented
4#32660C214yesmaximal, complemented
5#32618C214yesmaximal, complemented
6#29761C212yesmaximal
7#28315C212yesmaximal
8#28272C212yesmaximal, complemented
9#28121C212yesmaximal, complemented
10#27910C211yesmaximal, characteristic, complemented
11#27645C211yesmaximal, characteristic
12#25302C212yesmaximal, complemented
13#25063C212yesmaximal, complemented
14#24189C212yesmaximal, complemented
15#22510C211yesmaximal, characteristic, complemented
16#22085C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#484C2212yes—
22C2×#484C2212yes—
23C2×#481C2212yes—
24C2×#462C2212yes—
25C2×#451C2212yes—
26C2×#448C2211yescharacteristic
27C2×#446C2212yes—
28C2×C4×SD16C2212yes—
29#2857C2211yescharacteristic
30#2850C2212yes—
31#2592C2212yes—
32#2582—21no—
33#2582—21no—
34#2582—22no—
35#2582C2214yes—
36#2582C2212yes—
37#2582C2212yes—
38#2581C2212yes—
39#2568C2211yescharacteristic
40#2538C2211yescharacteristic
41#2481—22no—
42#2469—22no—
43#2421—21no—
44#2409—21no—
45C2×#398—22no—
46#2362—21no—
47#2355C2211yescharacteristic
48#2352—21no—
49C4×C22⋊Q8C2212yes—
50C4×#121C2211yescharacteristic
51C2×#391—21no—
52C2×#391C2212yes—
53C2×#389—21no—
54C2×#389C2212yes—
55C2×#384C2211yescharacteristic
56#2097C2214yes—
57#2095C2214yes—
58#2085C2214yes—
59#2084C2212yes—
60#2071C2212yes—
61#2068C2214yes—
62#2067C2214yes—
63#2066C2214yes—
64#2056C2214yes—
65#2042C2214yes—
66#2015C2214yes—
67#1985C2214yes—
68#1984C2212yes—
69#1973C2214yes—
70#1971C2214yes—
71#1956C2212yes—
72#1948C2212yes—
73#1943C2214yes—
74#1935C2214yes—
75#1928C2214yes—
76#1918C2214yes—
77#1880C2212yes—
78#1878C2214yes—
79#1878C2214yes—
80#1874C2211yescharacteristic
81#1872C2212yes—
82#1865C2214yes—
83#1865C2212yes—
84#1853C2211yescharacteristic
85#1845C2212yes—
86#1824C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1787C2211yescharacteristic
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×Q16C2314yes—
98C22×SD16C2312yes—
99#556—44no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#126—22no—
102C2×#124C2312yes—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×#121C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—44no—
110C2×C4×Q8—44no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D412yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D412yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#484—28no—
135#484—44no—
136#481—44no—
137#462—44no—
138#451—44no—
139#448—24no—
140#446—44no—
141C4×SD16—44no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2311yescharacteristic
160C2×#97C2314yes—
161C2×C4×C8C2312yes—
162#400—22no—
163#400—22no—
164#398—24no—
165#398—24no—
166#398—24no—
167#398—24no—
168#397—22no—
169#396—22no—
170#395—22no—
171#393—22no—
172#391—21no—
173#391—22no—
174#391—22no—
175#391—24no—
176#391—21no—
177#391—22no—
178#391—22no—
179#391C2312yes—
180#391C2314yes—
181#391C2312yes—
182#391C2312yes—
183#391C2314yes—
184#390C2312yes—
185#390C2311yescharacteristic
186#389—22no—
187#389—22no—
188#389—24no—
189#389—22no—
190#389—22no—
191#389—21no—
192#389—21no—
193#389C2314yes—
194#389C2312yes—
195#389C2311yescharacteristic
196#389C2314yes—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—21no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#386—22no—
207#386—22no—
208#386—21no—
209#385—22no—
210#384—21no—
211#384—21no—
212#384C2312yes—
213#384C2311yescharacteristic
214#384C2312yes—
215#382—21no—
216#382C2312yes—
217#380—21no—
218#380—21no—
219#380—21no—
220#380C2311yescharacteristic
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—21no—
227C2×#95—22no—
228C2×#95—22no—
229C2×#95—22no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2312yes—
238#336C2314yes—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—42no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2×D412yes—
250C22×Q8C2412yes—
251C22×Q8C2412yes—
252C22×D4—21no—
253C22×D4C2411yescharacteristic
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C2×Q16—28no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×SD16—24no—
267C2×SD16—24no—
268C2×SD16—42no—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273#126—48no—
274#124—44no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279#121—22no—
280#121—22no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—42no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×Q8—42no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—24no—
295C4×D4—44no—
296#117—44no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4—84no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—44no—
354C2×C42—44no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363#107—28no—
364#107—44no—
365#106—42no—
366#106—44no—
367C4⋊C8—24no—
368C4⋊C8—44no—
369#103—24no—
370#103—24no—
371#103—28no—
372#103—42no—
373#103—44no—
374#103—44no—
375#103—44no—
376#102—42no—
377#102—44no—
378C22⋊C8—24no—
379C22⋊C8—22no—
380C22⋊C8#4114yes—
381#97—44no—
382C4×C8—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402C24—21no—
403C24C22×D411yescharacteristic
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—42no—
426C2×Q8—416no—
427C2×Q8—42no—
428C2×Q82−1+414yes—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4#13714yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4C2511yescharacteristic
482Q16—88no—
483SD16—48no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8C2×#4114yes—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—81no—
524C4⋊C4—82no—
525C4⋊C4—84no—
526C4⋊C4—82no—
527C4⋊C4—84no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—81no—
537C22⋊C4—82no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—22no—
542C42—24no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—44no—
547C42—44no—
548C42—42no—
549C42—82no—
550C42H(5,𝔽2)14yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23#55012yes—
563C23C2×#13712yes—
564C23#57812yes—
565C23C23×D411yescharacteristic
566Q8—24no—
567Q8—24no—
568Q8—48no—
569Q8—48no—
570Q8—44no—
571Q8—44no—
572Q8—42no—
573Q8—42no—
574Q8—44no—
575Q8—44no—
576D4—42no—
577D4—42no—
578D4—84no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4—84no—
634C2×C4#53412yes—
635C2×C4#53412yes—
636C2×C4#55012yes—
637C2×C4C2×#13712yes—
638C2×C4C2×#13712yes—
639C2×C4#57512yes—
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×H(5,𝔽2)12yes—
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×2−1+412yes—
644C8—24no—
645C8—42no—
646C8—44no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22#339814yes—
661C22C2×#53411yescharacteristic
662C22C2×#55011yescharacteristic
663C22#348611yescharacteristic
664C22C22×#13711yescharacteristic
665C22C2×#57511yescharacteristic
666C22C2×#57811yescharacteristic
667C22#358511yescharacteristic
668C4—22no—
669C4—22no—
670C4—22no—
671C4—42no—
672C4—42no—
673C4—41no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—81no—
678C4—82no—
679C4—81no—
680C4—81no—
681C4—82no—
682C4—82no—
683C4—84no—
684C4#319614yes—
685C4#352012yes—
686C4#353612yes—
687C2—24no—
688C2—161no—
689C2#3913311yescharacteristic
690C2C2×#319611yescharacteristic
691C2#6076411yescharacteristic
692C2C2×#339811yescharacteristic
693C2#6140511yescharacteristic
694C2#6183011yescharacteristic
695C2#6184211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2582C2semidirect
3. #32906C2semidirect
4. #32660C2semidirect
5. #32618C2semidirect
8. #28272C2semidirect
9. #28121C2semidirect
10. #27910C2semidirect
12. #25302C2semidirect
13. #25063C2semidirect
14. #24189C2semidirect
15. #22510C2semidirect
16. #22085C2semidirect

Maximal subgroups

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

subgroupindex
#365792
C2×#25822
#329062
#326602
#326182
#297612
#283152
#282722
#281212
#279102
#276452
#253022
#250632
#241892
#225102
#220852
C4×#4482