Finite Group Databasebeta

← #6629568

Subgroups — #6629568

Order 512.

2016 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2C4×#518C211yesmaximal, characteristic
3#33076C214yesmaximal, complemented
4#33074C214yesmaximal, complemented
5#33072C212yesmaximal, complemented
6#33072C212yesmaximal, complemented
7#33068C212yesmaximal, complemented
8#33066C212yesmaximal, complemented
9#21171C211yesmaximal, characteristic, complemented
10#21169C211yesmaximal, characteristic, complemented
11#21039C212yesmaximal, complemented
12#21036C212yesmaximal, complemented
13#20879C212yesmaximal
14#20876C212yesmaximal
15#20755C211yesmaximal, characteristic, complemented
16#20646C211yesmaximal, characteristic
17C2×#1802C211yesmaximal, characteristic
18C2×#518C2211yescharacteristic
19C2×#518C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#418C2212yes—
26C22×#102C2212yes—
27#2859C2212yescomplemented
28#2859C2212yescomplemented
29#2647C2214yescomplemented
30#2571—21no—
31#2571—21no—
32#2552—21no—
33#2552—21no—
34#2442—22no—
35#2442—22no—
36#2317C2212yes—
37#2316C2212yes—
38#2285C2211yescharacteristic
39C2×#384C2211yescharacteristic
40#2275—21no—
41#2275—21no—
42#2275C2212yes—
43#2275C2212yes—
44#2274C2211yescharacteristic
45#2274C2211yescharacteristic
46#2273—22no—
47C42×D4—21no—
48C42×D4—21no—
49C42×D4C2212yes—
50C42×D4C2212yes—
51C4×#117C2211yescharacteristic
52C4×#117C2211yescharacteristic
53C2×C4×C22⋊C4—22no—
54#1949C2212yescomplemented
55#1949C2212yescomplemented
56#1948C2211yescharacteristic, complemented
57#1948C2212yescomplemented
58#1948C2211yescharacteristic, complemented
59#1948C2212yescomplemented
60#1947C2211yescharacteristic, complemented
61#1947C2212yescomplemented
62#1947C2211yescharacteristic, complemented
63#1947C2212yescomplemented
64#1921C2214yescomplemented
65#1921C2214yescomplemented
66#1920C2214yescomplemented
67#1920C2214yescomplemented
68#1919C2214yescomplemented
69#1919C2214yescomplemented
70#1918C2214yescomplemented
71#1918C2214yescomplemented
72#1864C2214yes—
73#1864C2214yes—
74#1864C2214yes—
75#1864C2214yes—
76#1862C2214yes—
77#1862C2214yes—
78#1862C2214yes—
79#1862C2214yes—
80#1832C2214yescomplemented
81#1831C2212yescomplemented
82#1831C2212yescomplemented
83#1804C2214yes—
84#1802C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2214yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1729C2212yes—
94#1729C2212yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×#125—22no—
100C2×#125—22no—
101#518—24no—
102#518—24no—
103#518—24no—
104#518—24no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—24no—
112C2×C4×D4Q818yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C22⋊C4—22no—
131C22×C42—22no—
132C22×C42—22no—
133C22×C42C2311yescharacteristic
134#428—28no—
135#427—28no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2314yes—
142#423—24no—
143C2×C4⋊C8—24no—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149#418—28no—
150C2×#102—28no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155#400—24no—
156#400—24no—
157#398—24no—
158#398—24no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2314yes—
166#389C2314yes—
167#389C2314yes—
168#389C2314yes—
169#385—24no—
170#384—22no—
171#384—21no—
172#384—21no—
173#384C2312yes—
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2312yes—
177#382—22no—
178#382—22no—
179#382—22no—
180#382C2312yes—
181#380—21no—
182#380—21no—
183#380—24no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4D411yescharacteristic
191C4×C4⋊C4D411yescharacteristic
192C4×C4⋊C4D411yescharacteristic
193C4×C4⋊C4D411yescharacteristic
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2311yescharacteristic
202C4×C22⋊C4—21no—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220#376—21no—
221#376C2312yes—
222C2×#95—22no—
223C2×#95—22no—
224C43—21no—
225C43C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239#336C2314yes—
240C25—21no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4C2×Q814yes—
246C22×D4#4114yes—
247C22×D4#4114yes—
248C22×D4C2412yes—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—28no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—82no—
259C23×C4—82no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2414yes—
272#125—84no—
273#125—84no—
274C4×D4—28no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4C2×Q818yes—
291C4×D4C2×Q818yes—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—24no—
299#117—24no—
300#117#4114yes—
301#117#4114yes—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×Q814yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C22⋊C4—216no—
335C2×C22⋊C4—216no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—84no—
360C2×C42—84no—
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×Q814yes—
368C2×C42#4114yes—
369C2×C42C2412yes—
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—24no—
374#107—24no—
375#107—28no—
376#106—24no—
377#106—24no—
378#106—28no—
379C4⋊C8—22no—
380C4⋊C8—22no—
381C4⋊C8—24no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384#102—28no—
385#102—28no—
386#102—28no—
387#102—28no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—22no—
407C24—24no—
408C24—24no—
409C24—24no—
410C24—81no—
411C24—81no—
412C24—81no—
413C24—81no—
414C24C22×D411yescharacteristic
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4#125116yescomplemented
425C2×D4C22×Q814yes—
426C2×D4C2×#4114yes—
427C2×D4C2×#4114yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—816no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4C22⋊Q814yes—
466C22×C4C22⋊Q814yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×Q812yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2511yescharacteristic
483M16—48no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—48no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4C22×Q814yes—
524C4⋊C4C2×#4112yes—
525C4⋊C4C2×#4112yes—
526C4⋊C4C2×#4112yes—
527C4⋊C4C2×#4112yes—
528C22⋊C4—28no—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—28no—
537C22⋊C4—28no—
538C22⋊C4—44no—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—24no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—42no—
548C42—42no—
549C42—44no—
550C42—42no—
551C42—42no—
552C42—81no—
553C42—81no—
554C42—82no—
555C42—82no—
556C42—82no—
557C42—82no—
558C42C22⋊Q814yes—
559C42C22⋊Q814yes—
560C42C22×Q812yes—
561C42C22×Q812yes—
562C42C2×#4112yes—
563C42C2×#4112yes—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—216no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23C2×C22⋊Q812yes—
583C23#52512yes—
584C23#52512yes—
585C23C23×D411yescharacteristic
586D4—216no—
587D4—24no—
588D4—24no—
589D4—24no—
590D4C2×#12518yes—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—28no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—416no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4—88no—
639C2×C4—88no—
640C2×C4—88no—
641C2×C4—88no—
642C2×C4—164no—
643C2×C4C2×C22⋊Q812yes—
644C2×C4C2×C22⋊Q812yes—
645C2×C4C2×C22⋊Q812yes—
646C2×C4C2×C22⋊Q812yes—
647C2×C4C2×C22⋊Q812yes—
648C2×C4#52512yes—
649C2×C4#52512yes—
650C2×C4C2×#12514yes—
651C2×C4#52914yes—
652C2×C4#52914yes—
653C2×C4#53314yes—
654C2×C4#57612yes—
655C2×C4#57712yes—
656C2×C4C23×Q811yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C8—42no—
663C8—42no—
664C8—44no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—28no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—816no—
679C22#285114yes—
680C22C22×C22⋊Q811yescharacteristic
681C22C2×#52511yescharacteristic
682C22C2×#52511yescharacteristic
683C22C2×#57611yescharacteristic
684C22C2×#57711yescharacteristic
685C22#358511yescharacteristic
686C22#358611yescharacteristic
687C4—22no—
688C4—22no—
689C4—22no—
690C4—44no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—84no—
700C4—82no—
701C4—82no—
702C4#285114yes—
703C4#308114yes—
704C4C22×#12511yescharacteristic
705C4C2×#52911yescharacteristic
706C4C2×#52911yescharacteristic
707C4C2×#53311yescharacteristic
708C2—24no—
709C2—48no—
710C2C2×#285111yescharacteristic
711C2C2×#308111yescharacteristic
712C2#6049311yescharacteristic
713C2#6059011yescharacteristic
714C2#6059111yescharacteristic
715C2#6059611yescharacteristic
716C2#6059711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #33076C2semidirect
4. #33074C2semidirect
5. #33072C2semidirect
6. #33072C2semidirect
7. #33068C2semidirect
8. #33066C2semidirect
9. #21171C2semidirect
10. #21169C2semidirect
11. #21039C2semidirect
12. #21036C2semidirect
15. #20755C2semidirect
27. #2859C22semidirect
28. #2859C22semidirect
29. #2647C22semidirect
54. #1949C22semidirect
55. #1949C22semidirect
56. #1948C22semidirect
57. #1948C22semidirect
58. #1948C22semidirect
59. #1948C22semidirect
60. #1947C22semidirect
61. #1947C22semidirect
62. #1947C22semidirect
63. #1947C22semidirect
64. #1921C22semidirect
65. #1921C22semidirect
66. #1920C22semidirect
67. #1920C22semidirect
68. #1919C22semidirect
69. #1919C22semidirect
70. #1918C22semidirect
71. #1918C22semidirect
80. #1832C22semidirect
81. #1831C22semidirect
82. #1831C22semidirect
424. C2×D4#125semidirect

Maximal subgroups

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

subgroupindex
#352862
C4×#5182
#330762
#330742
#330722
#330682
#330662
#211712
#211692
#210392
#210362
#208792
#208762
#207552
#206462
C2×#18022