Finite Group Databasebeta

← #392927

Subgroups — #392927

Order 512.

1586 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#520C211yesmaximal, characteristic
2#21171C211yesmaximal, characteristic, complemented
3C2×#1860C212yesmaximal
4C2×#1806C211yesmaximal, characteristic
5#20648C211yesmaximal, characteristic
6#13911C211yesmaximal, characteristic, complemented
7#10899C212yesmaximal, complemented
8#10892C212yesmaximal, complemented
9#10852C211yesmaximal, characteristic, complemented
10#10847C211yesmaximal, characteristic, complemented
11#10781C212yesmaximal
12C2×#520C2211yescharacteristic
13C2×#520C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#428C2212yes—
16C2×#423C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C22×#104C2212yes—
19C2×#417C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382C2212yes—
22#2276—21no—
23#2275—22no—
24#2275—21no—
25#2274—21no—
26#2274—21no—
27#2274C2211yescharacteristic
28#2274C2211yescharacteristic
29C42×Q8—21no—
30C42×D4—22no—
31C42×D4—21no—
32C4×#117—21no—
33C4×#117C2211yescharacteristic
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1948—21no—
38#1948—21no—
39#1947—21no—
40#1947—21no—
41#1860—28no—
42#1806C418yescomplemented
43#1803—21no—
44#1803—21no—
45#1802—21no—
46#1802—21no—
47C2×#343C2212yes—
48C2×#343C2212yes—
49#1736C2212yes—
50#1736C2212yes—
51C2×#339C2212yes—
52C2×#339C2212yes—
53C2×#339C2212yes—
54#1442—22no—
55C4⋊#95—22no—
56C4⋊#95C2212yes—
57C22×#41C2311yescharacteristic
58C22×M16C2312yes—
59C22×M16C2312yes—
60#520—22no—
61#520—22no—
62#520—22no—
63#520—22no—
64#520—22no—
65#520—22no—
66#520D414yes—
67#520Q814yes—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41C2×C414yes—
71C4×#41Q814yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×#117—21no—
85C2×#117—21no—
86C2×#117—21no—
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C42—21no—
99C22×C42—22no—
100C22×C42D412yes—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103#428—24no—
104#428—24no—
105C2×#107—22no—
106C2×#106—22no—
107#423—22no—
108#423—22no—
109#423—22no—
110#423C2×C414yescomplemented
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×#104—24no—
116C2×#104—24no—
117C2×#104C2×C418yes—
118#417—24no—
119#417—24no—
120C2×#103—22no—
121C2×#102—22no—
122#384—21no—
123#384—21no—
124#384—21no—
125#384—21no—
126#382—24no—
127#382—24no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4C2×C414yescomplemented
148C4×C4⋊C4D411yescharacteristic
149C4×C4⋊C4D411yescharacteristic
150C4×C4⋊C4D411yescharacteristic
151C4×C4⋊C4D411yescharacteristic
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—22no—
155C4×C22⋊C4—22no—
156C4×C22⋊C4—22no—
157C4×C22⋊C4—22no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—22no—
161C4×C22⋊C4—22no—
162#376—22no—
163#376—21no—
164C2×#95—22no—
165C2×#95—21no—
166C2×#95—24no—
167C2×#95—24no—
168C2×#95D412yes—
169C43—22no—
170C43—21no—
171#343—24no—
172#343—24no—
173#340—22no—
174#340—22no—
175#340—22no—
176#340—22no—
177#339—24no—
178#339—24no—
179#339C2×C418yescomplemented
180C2×#41—21no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41C22×C412yes—
187C2×#41C2×D412yes—
188C2×#41C2×Q812yes—
189C2×#41C2×Q812yes—
190C2×#41#4112yes—
191C2×#41#4112yes—
192C22×Q8C2×D411yescharacteristic
193C22×D4—21no—
194C22×D4C2×D411yescharacteristic
195C23×C4—21no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16—22no—
218C2×M16C22×C414yes—
219C22×C8—22no—
220C22×C8—22no—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8C4⋊C414yes—
228C4×Q8C4⋊C414yes—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—28no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—28no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4C4⋊C414yes—
244C4×D4C4⋊C414yes—
245#117—24no—
246#117—24no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—24no—
254#117—24no—
255#117—24no—
256#117—24no—
257#117C22×C412yes—
258#117C2×D412yes—
259#117C2×Q812yes—
260#117C2×Q812yes—
261#117#4112yes—
262#117#4112yes—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—41no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×Q812yes—
302C2×C4⋊C4#4112yes—
303C2×C4⋊C4#4112yes—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—44no—
336C2×C42—84no—
337C2×C42C22×C414yes—
338C2×C42C22×C412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×Q812yes—
344C2×C42#4112yes—
345C2×C42#4112yes—
346#107—44no—
347#106—44no—
348C4⋊C8—24no—
349C4⋊C8—24no—
350C4⋊C8—42no—
351C4⋊C8—42no—
352#104—216no—
353#103—44no—
354#102—44no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—48no—
367#95—84no—
368C24—21no—
369C24C22×D411yescharacteristic
370#41—22no—
371#41—22no—
372#41—22no—
373#41—22no—
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—22no—
382#41C4×D414yes—
383#41C4×Q814yes—
384#41#12514yes—
385#41C4⋊Q814yes—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8C2×C4⋊C412yes—
390C2×Q8#12112yes—
391C2×Q8C22⋊Q812yes—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4C2×C4⋊C412yes—
402C2×D4#12112yes—
403C2×D4C22⋊Q812yes—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4C4×D412yes—
454C22×C4C4×D412yes—
455C22×C4#12011yescharacteristic
456C22×C4#12011yescharacteristic
457C22×C4#12011yescharacteristic
458C22×C4#12011yescharacteristic
459C22×C4#12114yes—
460C22×C4C22⋊Q812yes—
461C22×C4C22⋊Q814yes—
462C22×C4C22⋊Q812yes—
463C22×C4C23×C411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×Q811yescharacteristic
468C22×C4C22×Q811yescharacteristic
469C22×C4C2×#4111yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4111yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4C2×#4111yescharacteristic
475C22×C4C2×#4111yescharacteristic
476C22×C4C2×#4111yescharacteristic
477M16—24no—
478M16—24no—
479M16—42no—
480M16—42no—
481M16—42no—
482M16—42no—
483M16—42no—
484M16—42no—
485C2×C8—24no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8—42no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—48no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4C2×C4⋊C412yes—
519C4⋊C4C2×C4⋊C412yes—
520C4⋊C4#12112yes—
521C4⋊C4#12112yes—
522C4⋊C4C22⋊Q812yes—
523C4⋊C4C22⋊Q812yes—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C42—28no—
535C42—28no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—24no—
543C42—24no—
544C42—24no—
545C42—24no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—44no—
550C42—42no—
551C42—81no—
552C42—81no—
553C42—82no—
554C42—82no—
555C42—82no—
556C42—82no—
557C42—82no—
558C42C2×C4⋊C412yes—
559C42C2×C4⋊C412yes—
560C42C4×D412yes—
561C42C4×D412yes—
562C42C22⋊Q812yes—
563C42C22⋊Q812yes—
564C42#12312yes—
565C42#12312yes—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23C2×C4×D411yescharacteristic
578C23C2×#12011yescharacteristic
579C23C2×#12111yescharacteristic
580C23C2×C22⋊Q811yescharacteristic
581C23C2×C22⋊Q811yescharacteristic
582C23#52511yescharacteristic
583C23#52511yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8#38914yes—
588D4—24no—
589D4—24no—
590D4—24no—
591D4—24no—
592D4—22no—
593D4—22no—
594D4—22no—
595D4#38914yes—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—48no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—84no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—88no—
647C2×C4—88no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4#38512yes—
658C2×C4#38512yes—
659C2×C4#38512yes—
660C2×C4#38512yes—
661C2×C4#45412yes—
662C2×C4#45512yes—
663C2×C4C22×C4⋊C411yescharacteristic
664C2×C4C2×C4×D411yescharacteristic
665C2×C4C2×C4×D411yescharacteristic
666C2×C4C2×C4×Q811yescharacteristic
667C2×C4C4×#4111yescharacteristic
668C2×C4C4×#4111yescharacteristic
669C2×C4C2×#12111yescharacteristic
670C2×C4C2×#12111yescharacteristic
671C2×C4C2×C22⋊Q811yescharacteristic
672C2×C4C2×C22⋊Q811yescharacteristic
673C2×C4C2×C22⋊Q811yescharacteristic
674C2×C4C2×#12311yescharacteristic
675C2×C4#52511yescharacteristic
676C2×C4#52511yescharacteristic
677C2×C4#52511yescharacteristic
678C2×C4#52511yescharacteristic
679C2×C4C2×#12511yescharacteristic
680C2×C4#52911yescharacteristic
681C2×C4#52911yescharacteristic
682C2×C4#52911yescharacteristic
683C2×C4#52911yescharacteristic
684C2×C4C2×C4⋊Q811yescharacteristic
685C2×C4#53211yescharacteristic
686C2×C4#53211yescharacteristic
687C2×C4#53311yescharacteristic
688C2×C4#53311yescharacteristic
689C2×C4#53311yescharacteristic
690C2×C4#53311yescharacteristic
691C8—44no—
692C8—82no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—24no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22C2×#38511yescharacteristic
713C22C4×#12111yescharacteristic
714C22C4×C22⋊Q811yescharacteristic
715C22#284711yescharacteristic
716C22#284811yescharacteristic
717C22#285111yescharacteristic
718C22#285711yescharacteristic
719C22C2×#45511yescharacteristic
720C22C2×#45411yescharacteristic
721C22#302111yescharacteristic
722C22#302211yescharacteristic
723C4—24no—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—44no—
731C4—42no—
732C4—42no—
733C4—44no—
734C4—42no—
735C4—84no—
736C4—162no—
737C4#186014yescomplemented
738C4C2×#38911yescharacteristic
739C4#229611yescharacteristic
740C4C4×#12111yescharacteristic
741C4C4×C22⋊Q811yescharacteristic
742C4#285111yescharacteristic
743C4#285411yescharacteristic
744C4#285511yescharacteristic
745C4#285711yescharacteristic
746C2—21no—
747C2—21no—
748C2—21no—
749C2—21no—
750C2—44no—
751C2#1469711yescharacteristic
752C2C2×#186011yescharacteristic
753C2#2086511yescharacteristic
754C2#3292111yescharacteristic
755C2#3292211yescharacteristic
756C2#3297011yescharacteristic
757C2#3297111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21171C2semidirect
6. #13911C2semidirect
7. #10899C2semidirect
8. #10892C2semidirect
9. #10852C2semidirect
10. #10847C2semidirect
42. #1806C4semidirect
110. #423C2×C4semidirect
147. C4×C4⋊C4C2×C4semidirect
179. #339C2×C4semidirect
737. C4#1860semidirect

Maximal subgroups

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

subgroupindex
C4×#5202
#211712
C2×#18602
C2×#18062
#206482
#139112
#108992
#108922
#108522
#108472
#107812