Finite Group Databasebeta

← #6856366

Subgroups — #6856366

Order 512.

1859 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#36577C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic
3#32905C211yesmaximal, characteristic
4#32598C214yesmaximal, complemented
5#32546C214yesmaximal, complemented
6#29714C212yesmaximal
7#29329C212yesmaximal, complemented
8#29240C212yesmaximal, complemented
9#28186C212yesmaximal, complemented
10#27972C211yesmaximal, characteristic, complemented
11#27680C211yesmaximal, characteristic, complemented
12#25293C212yesmaximal, complemented
13#24992C212yesmaximal, complemented
14#24187C212yesmaximal, complemented
15Q8×C22⋊C8C211yesmaximal, characteristic, complemented
16#22043C212yesmaximal
17C4×#450C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic, complemented
19C22×C22⋊Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#461C2212yes—
23C2×#450C2211yescharacteristic
24C2×#450C2212yes—
25C2×C8×Q8C2212yescomplemented
26C2×C4×SD16C2212yes—
27#2857C2212yes—
28#2857C2211yescharacteristic
29#2582C2211yescharacteristic, complemented
30#2582C2212yescomplemented
31#2568C2212yescomplemented
32#2568C2211yescharacteristic, complemented
33#2538—21no—
34#2538—21no—
35#2538—22no—
36#2538C2212yes—
37#2538C2214yes—
38#2538C2212yes—
39#2453—21no—
40#2453—22no—
41#2442—22no—
42#2442—21no—
43C2×#396—22no—
44#2351—21no—
45Q8×C4⋊C4C2211yescharacteristic, complemented
46Q8×C22⋊C4—21no—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49C2×#390C2212yes—
50C2×#389C2212yes—
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53#2100C2214yescomplemented
54#2099C2212yescomplemented
55#2096C2212yescomplemented
56#2088C2214yescomplemented
57#2072C2214yescomplemented
58#2071C2214yes—
59#2066C2214yescomplemented
60#2055C2214yes—
61#2055C2214yes—
62#2043C2214yescomplemented
63#2013C2214yes—
64#1984C2214yescomplemented
65#1984C2212yescomplemented
66#1968C2214yes—
67#1956C2212yescomplemented
68#1948C2212yescomplemented
69#1934C2214yes—
70#1933C2214yescomplemented
71#1933C2214yescomplemented
72#1926C2214yes—
73#1919C2214yescomplemented
74#1880C2214yes—
75#1877C2214yes—
76#1877C2212yes—
77#1872C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1865C2214yescomplemented
83#1864C2212yes—
84#1853C2212yes—
85#1853C2211yescharacteristic
86#1834C2212yescomplemented
87#1803C2212yescomplemented
88#1802C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1786C2211yescharacteristic
92C4×#102C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C22×SD16C2314yes—
100#557—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#125—21no—
104C2×#125—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—48no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C22⋊Q8C2312yes—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—44no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D412yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—44no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4D412yes—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—22no—
136C22×C22⋊C4—22no—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#461—44no—
140#450—24no—
141#450—28no—
142C8×Q8—44no—
143C4×SD16—44no—
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C4×C8C2312yes—
161C2×C4×C8C2314yes—
162#400—22no—
163#400—22no—
164#400—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#396—24no—
169#396—24no—
170#396—24no—
171#396—24no—
172#391—21no—
173#391—22no—
174#391—21no—
175#391—22no—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#391C2311yescharacteristic
180#391C2312yes—
181#390—22no—
182#390—22no—
183#390C2314yes—
184#390C2314yes—
185#389—21no—
186#389—22no—
187#389—21no—
188#389—22no—
189#389—21no—
190#389—22no—
191#389—21no—
192#389—22no—
193#389—22no—
194#389—22no—
195#389C2314yes—
196#389C2312yes—
197#389C2311yescharacteristic
198#389C2314yes—
199#387—21no—
200#387—22no—
201#385—21no—
202#385—22no—
203#384—21no—
204#384—21no—
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#384C2312yes—
209#382—21no—
210#382—22no—
211#380—21no—
212#380—22no—
213#380—21no—
214#380—22no—
215#380—22no—
216#380—22no—
217#380—24no—
218#380C2311yescharacteristic
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C2×#95—22no—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8SD1618yescomplemented
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2×D411yescharacteristic
243C22×Q8C2×D412yes—
244C22×Q8C2411yescharacteristic
245C22×Q8C2412yes—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—42no—
249C22×D4—42no—
250C22×D4#4114yes—
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C2×SD16—24no—
265C2×SD16—28no—
266C2×SD16—42no—
267C2×SD16—42no—
268C2×SD16—44no—
269C2×SD16—44no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—44no—
275#125—48no—
276#125—44no—
277C22⋊Q8—22no—
278C22⋊Q8—24no—
279C22⋊Q8—22no—
280C22⋊Q8—24no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283C22⋊Q8—416no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—42no—
290C4×Q8—42no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—44no—
299C4×D4—44no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—48no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—48no—
353C2×C22⋊C4—84no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—44no—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2412yes—
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368#106—44no—
369#106—42no—
370C4⋊C8—24no—
371C4⋊C8—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375#103—42no—
376#103—44no—
377#102—24no—
378#102—28no—
379#102—42no—
380#102—42no—
381C22⋊C8—22no—
382C22⋊C8—24no—
383C22⋊C8—22no—
384C4×C8—24no—
385C4×C8—44no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403C24—24no—
404C24—22no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24C22×D411yescharacteristic
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—416no—
427C2×Q8C2×SD1614yes—
428C2×Q8C2×SD1614yes—
429C2×Q8C2×SD1618yes—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—48no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—48no—
440C2×D4C2×#4114yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—816no—
474C22×C4C2×SD1614yes—
475C22×C4C2×SD1618yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4112yes—
486C22×C42−1+412yes—
487C22×C4C2511yescharacteristic
488SD16—48no—
489SD16—88no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—42no—
532C4⋊C4—81no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—81no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—44no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C42—22no—
548C42—22no—
549C42—24no—
550C42—22no—
551C42—24no—
552C42—42no—
553C42—42no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—42no—
558C42H(5,𝔽2)14yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—24no—
564C23—48no—
565C23—48no—
566C23—42no—
567C23—48no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23#54812yes—
578C23C22×SD1612yes—
579C23#57812yes—
580C23C23×D411yescharacteristic
581Q8—28no—
582Q8—28no—
583Q8—48no—
584Q8—42no—
585Q8—44no—
586Q8—42no—
587Q8—44no—
588D4—24no—
589D4—24no—
590D4—48no—
591D4—42no—
592D4—42no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—88no—
642C2×C4—84no—
643C2×C4—88no—
644C2×C4—84no—
645C2×C4—88no—
646C2×C4#53412yes—
647C2×C4#53412yes—
648C2×C4#54812yes—
649C2×C4C22×SD1614yes—
650C2×C4C22×SD1614yes—
651C2×C4C22×SD1612yes—
652C2×C4C22×SD1612yes—
653C2×C4#57512yes—
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×H(5,𝔽2)12yes—
656C2×C4C2×2−1+411yescharacteristic
657C2×C4#58512yes—
658C8—42no—
659C8—44no—
660C8—42no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—816no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22#335914yes—
676C22C2×#53411yescharacteristic
677C22C2×#54811yescharacteristic
678C22#348711yescharacteristic
679C22C23×SD1611yescharacteristic
680C22C2×#57511yescharacteristic
681C22C2×#57811yescharacteristic
682C22#358511yescharacteristic
683C4—22no—
684C4—22no—
685C4—24no—
686C4—24no—
687C4—22no—
688C4—41no—
689C4—42no—
690C4—42no—
691C4—81no—
692C4—82no—
693C4—82no—
694C4—81no—
695C4—82no—
696C4—81no—
697C4—84no—
698C4—81no—
699C4#318714yes—
700C4#350512yes—
701C4#354112yes—
702C2—24no—
703C2—84no—
704C2#3972311yescharacteristic
705C2C2×#318711yescharacteristic
706C2#6076411yescharacteristic
707C2C2×#335911yescharacteristic
708C2#6131811yescharacteristic
709C2#6188111yescharacteristic
710C2#6190511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
1. #36577C2semidirect
4. #32598C2semidirect
5. #32546C2semidirect
7. #29329C2semidirect
8. #29240C2semidirect
9. #28186C2semidirect
10. #27972C2semidirect
11. #27680C2semidirect
12. #25293C2semidirect
13. #24992C2semidirect
14. #24187C2semidirect
15. Q8×C22⋊C8C2semidirect
18. C2×#557C22semidirect
25. C2×C8×Q8C22semidirect
29. #2582C22semidirect
30. #2582C22semidirect
31. #2568C22semidirect
32. #2568C22semidirect
45. Q8×C4⋊C4C22semidirect
53. #2100C22semidirect
54. #2099C22semidirect
55. #2096C22semidirect
56. #2088C22semidirect
57. #2072C22semidirect
59. #2066C22semidirect
62. #2043C22semidirect
64. #1984C22semidirect
65. #1984C22semidirect
67. #1956C22semidirect
68. #1948C22semidirect
70. #1933C22semidirect
71. #1933C22semidirect
73. #1919C22semidirect
82. #1865C22semidirect
86. #1834C22semidirect
87. #1803C22semidirect
240. C22×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#365772
C2×#25382
#329052
#325982
#325462
#297142
#293292
#292402
#281862
#279722
#276802
#252932
#249922
#241872
Q8×C22⋊C82
#220432
C4×#4502