Finite Group Databasebeta

← #6615079

Subgroups — #6615079

Order 512.

1681 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2C4×#554C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#33149C212yesmaximal, complemented
5#29218C211yesmaximal, characteristic, complemented
6#28284C211yesmaximal, characteristic, complemented
7#27164C214yesmaximal, complemented
8#26662C214yesmaximal, complemented
9#24971C212yesmaximal
10#24702C212yesmaximal, complemented
11#24155C212yesmaximal, complemented
12#24122C212yesmaximal
13#22965C212yesmaximal, complemented
14#22795C211yesmaximal, characteristic
15#22771C212yesmaximal, complemented
16C4×#462C211yesmaximal, characteristic, complemented
17#21385C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×#533C2212yes—
20C2×#477C2212yes—
21C2×#477C2212yes—
22C2×#462C2211yescharacteristic
23C2×C4×Q16C2212yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2552—21no—
28#2542—21no—
29#2538C2212yes—
30#2441—22no—
31#2429—22no—
32#2423C2214yes—
33#2370—21no—
34#2369C2211yescharacteristic
35#2368—22no—
36#2365—21no—
37#2360C2212yes—
38#2355C2211yescharacteristic
39#2354—21no—
40Q8×C4⋊C4C2212yes—
41#2349—21no—
42D4×C4⋊C4C2211yescharacteristic
43#2345C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#125—21no—
47C4×C22⋊Q8—22no—
48C4×C22⋊Q8C2212yes—
49C42×Q8C2211yescharacteristic
50C42×D4—21no—
51C42×D4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53#2100C2214yes—
54#2076C2214yes—
55#2068C2214yes—
56#2036C2214yes—
57#2036C2214yes—
58#2001C2214yes—
59#1990C2212yes—
60#1987C2212yes—
61#1987C2212yes—
62#1987C2212yes—
63#1981C2212yes—
64#1981C2211yescharacteristic
65#1954C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1947C2211yescharacteristic
69#1947C2212yes—
70#1942C2214yes—
71#1933C2214yes—
72#1928C2212yes—
73#1923C2214yes—
74#1921C2214yes—
75#1921C2214yes—
76#1878C2212yes—
77#1878C2214yes—
78#1872C2214yes—
79#1872C2214yes—
80#1863C2214yes—
81#1852C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1824C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1797C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2214yes—
93C4×#103C2214yes—
94C4×#103C2212yes—
95C22×Q16C2312yes—
96#554—24no—
97#554—24no—
98#533—28no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125—21no—
106C2×#125—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2314yes—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42—22no—
128C22×C42C2312yes—
129#477—28no—
130#477—44no—
131#462—24no—
132C4×Q16—28no—
133#431—44no—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#399—24no—
154#398—24no—
155#396—24no—
156#391—21no—
157#391—22no—
158#391—22no—
159#391C2312yes—
160#391C2312yes—
161#390—22no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—24no—
170#389—22no—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#388—24no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#385—22no—
182#385—22no—
183#385—22no—
184#384—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—21no—
192#380—22no—
193#380—22no—
194#380C2312yes—
195#380C2314yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C2×#95—22no—
224C43—21no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×Q8C2411yescharacteristic
241C22×D4C2×D411yescharacteristic
242C23×C4—22no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C2×Q16—24no—
248C2×Q16—24no—
249C2×Q16—42no—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259C4⋊Q8#4114yes—
260C4⋊Q8#4114yes—
261#125—28no—
262#125—48no—
263C22⋊Q8—216no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266C22⋊Q8—28no—
267C22⋊Q8—44no—
268C22⋊Q8—48no—
269C22⋊Q8—84no—
270C4×Q8—22no—
271C4×Q8—28no—
272C4×Q8—22no—
273C4×Q8—24no—
274C4×Q8—22no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—41no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×Q8—82no—
281C4×Q8#4114yes—
282C4×D4—28no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—42no—
286C4×D4—44no—
287C4×D4Q1618yescomplemented
288#117—48no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—84no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—24no—
359#107—28no—
360#107—44no—
361C4⋊C8—22no—
362C4⋊C8—24no—
363C4⋊C8—22no—
364C4⋊C8—24no—
365C4⋊C8—42no—
366#103—24no—
367#103—24no—
368#103—24no—
369#103—28no—
370#103—24no—
371#103—44no—
372#103—44no—
373C22⋊C8—44no—
374C22⋊C8—44no—
375C4×C8—24no—
376C4×C8—44no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24C22×D412yes—
396C2×Q8—24no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—81no—
411C2×Q8—81no—
412C2×Q8C2×#4112yes—
413C2×Q8C2×#4114yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4C2×Q1614yes—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4C2×Q1618yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454Q16—48no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—82no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—84no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4C2×Q1614yes—
502C4⋊C4C2×#4112yes—
503C4⋊C4C2×#4114yes—
504C4⋊C42−1+412yes—
505C22⋊C4—28no—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—84no—
511C22⋊C4C2×Q1618yes—
512C42—24no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—41no—
522C42—48no—
523C42—42no—
524C42—44no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—44no—
529C42—81no—
530C42—82no—
531C42—82no—
532C42C2×Q1614yes—
533C42C2×#4112yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C42H(5,𝔽2)12yes—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23C22×Q1614yes—
544C23C23×D411yescharacteristic
545Q8—22no—
546Q8—24no—
547Q8—24no—
548Q8—22no—
549Q8—41no—
550Q8—42no—
551Q8—41no—
552Q8—42no—
553Q8—84no—
554Q8—88no—
555D4—28no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—216no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—84no—
606C2×C4—88no—
607C2×C4—88no—
608C2×C4#52512yes—
609C2×C4#52512yes—
610C2×C4#54612yes—
611C2×C4#54612yes—
612C2×C4C22×Q1612yes—
613C2×C4C22×Q1614yes—
614C2×C4C22×Q1612yes—
615C2×C4C22×Q1612yes—
616C2×C4#57512yes—
617C2×C4#57712yes—
618C2×C4C22×#4111yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4#58511yescharacteristic
624C8—42no—
625C8—42no—
626C8—44no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22C2×#52511yescharacteristic
635C22C2×#54611yescharacteristic
636C22#348911yescharacteristic
637C22C23×Q1611yescharacteristic
638C22C2×#57511yescharacteristic
639C22C2×#57711yescharacteristic
640C22#358511yescharacteristic
641C4—22no—
642C4—22no—
643C4—24no—
644C4—22no—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—42no—
649C4—44no—
650C4—41no—
651C4—42no—
652C4—42no—
653C4—82no—
654C4—82no—
655C4—81no—
656C4—84no—
657C4#310414yes—
658C4#329914yes—
659C4#347511yescharacteristic
660C4#348211yescharacteristic
661C4#351111yescharacteristic
662C4#351711yescharacteristic
663C2—44no—
664C2#3640411yescharacteristic
665C2C2×#310411yescharacteristic
666C2#6059711yescharacteristic
667C2C2×#329911yescharacteristic
668C2#6116011yescharacteristic
669C2#6210011yescharacteristic
670C2#6214411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #33198C2semidirect
4. #33149C2semidirect
5. #29218C2semidirect
6. #28284C2semidirect
7. #27164C2semidirect
8. #26662C2semidirect
10. #24702C2semidirect
11. #24155C2semidirect
13. #22965C2semidirect
15. #22771C2semidirect
16. C4×#462C2semidirect
17. #21385C2semidirect
287. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#361342
C4×#5542
#331982
#331492
#292182
#282842
#271642
#266622
#249712
#247022
#241552
#241222
#229652
#227952
#227712
C4×#4622
#213852