Finite Group Databasebeta

← #431047

Subgroups — #431047

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×#2092C211yesmaximal, characteristic
3C2×#2088C211yesmaximal, characteristic, complemented
4C2×#2057C211yesmaximal, characteristic, complemented
5C2×#1864C211yesmaximal, characteristic
6C2×#1847C211yesmaximal, characteristic, complemented
7#15230C211yesmaximal, characteristic, complemented
8#13977C211yesmaximal, characteristic
9#11377C211yesmaximal, characteristic, complemented
10#11325C211yesmaximal, characteristic, complemented
11#11324C211yesmaximal, characteristic, complemented
12#11123C211yesmaximal, characteristic
13#11051C211yesmaximal, characteristic
14#10927C211yesmaximal, characteristic
15#10877C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#474—24no—
19C2×#474—24no—
20C2×#474—24no—
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#97C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#385—22no—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2211yescharacteristic
41#2092—24no—
42#2088—24no—
43#2057—24no—
44#1864—24no—
45#1847—24no—
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#340C2211yescharacteristic
51C2×#340C2211yescharacteristic
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55C2×#336C2211yescharacteristic
56#1446—22no—
57#1441—22no—
58#1439—22no—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4×#95C2211yescharacteristic
63C23×Q8—21no—
64C23×D4C2311yescharacteristic
65C24×C4—21no—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—41no—
75C2×C4⋊Q8—41no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—41no—
81C2×C4×D4—41no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—22no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—22no—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#474—232no—
100C2×#107—22no—
101C2×#107—22no—
102C2×#107—22no—
103C2×#107—22no—
104C2×#107—22no—
105C2×#107—22no—
106C2×#107—22no—
107C2×#107—22no—
108C2×#107—22no—
109C2×#107—41no—
110C2×#107—41no—
111C2×#107D414yes—
112C2×#106—22no—
113C2×#106—22no—
114C2×#106—22no—
115C2×#106—22no—
116C2×#106—22no—
117C2×#106—41no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—22no—
120C2×C4⋊C8D414yes—
121C2×#102—22no—
122C2×#102—22no—
123C2×#102—22no—
124C2×#102—22no—
125C2×#102—22no—
126C2×#102—22no—
127C2×#102—22no—
128C2×#102—22no—
129C2×#102—22no—
130C2×#102—41no—
131C2×#102D414yes—
132C2×#102Q814yescomplemented
133C2×#97—22no—
134C2×#97—22no—
135C2×#97—22no—
136#390—22no—
137#390—22no—
138#390—41no—
139#390Q814yescomplemented
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#385—48no—
145#384—22no—
146#384—41no—
147#382—22no—
148#382—22no—
149#382—41no—
150#382—41no—
151C2×#95—22no—
152C2×#95—21no—
153C2×#95—22no—
154C2×#95—22no—
155C2×#95—21no—
156C2×#95—22no—
157C2×#95—22no—
158C2×#95D412yes—
159C2×#95D412yes—
160#340—22no—
161#340—22no—
162#340—22no—
163#340—22no—
164#340—22no—
165#340—22no—
166#336—22no—
167#336—22no—
168#336—22no—
169#336D414yes—
170C25—21no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8—44no—
175C22×Q8—82no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C2×D412yes—
188C22×D4C2×Q812yes—
189C22×D4C2×Q812yes—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—82no—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8C2×D412yes—
228C22×C8C2×D412yes—
229C22×C8C2×D412yes—
230C22×C8C2×D412yes—
231C22×C8C2×D412yes—
232C22×C8C2×D412yes—
233C22×C8C2×Q812yes—
234C22×C8#4112yes—
235C4⋊Q8—28no—
236C4⋊Q8—44no—
237C4⋊Q8—44no—
238C4⋊Q8—44no—
239C4×D4—28no—
240C4×D4—44no—
241C4×D4—44no—
242C4×D4—44no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
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—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×Q812yes—
294C2×C4⋊C4C2×Q812yes—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—82no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×Q812yes—
321C2×C42#4112yes—
322#107—28no—
323#107—28no—
324#107—44no—
325#107—44no—
326#106—44no—
327#106—44no—
328C4⋊C8—28no—
329#102—28no—
330#102—28no—
331#102—28no—
332#102—44no—
333#97—28no—
334#95—24no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—42no—
340#95—42no—
341#95—82no—
342#95—82no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—41no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24C22×D411yescharacteristic
355C2×Q8—48no—
356C2×Q8—88no—
357C2×Q8—88no—
358C2×Q8—84no—
359C2×Q8—84no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4C4⋊Q814yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
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—48no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4C22⋊Q814yes—
449C22×C4#12712yes—
450C22×C4#12712yes—
451C22×C4AGL(1,ℤ/8)14yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×Q811yescharacteristic
457C22×C4C22×Q811yescharacteristic
458C22×C4C2×#4111yescharacteristic
459C22×C4C2×#4111yescharacteristic
460C22×C4H(5,𝔽2)11yescharacteristic
461C22×C4H(5,𝔽2)11yescharacteristic
462C22×C42−1+411yescharacteristic
463C22×C42−1+411yescharacteristic
464C22×C42−1+411yescharacteristic
465C22×C42−1+411yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8#12014yes—
493C2×C8C22⋊Q814yes—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—48no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—84no—
524C4⋊C4—88no—
525C4⋊C4—84no—
526C22⋊C4—416no—
527C22⋊C4—48no—
528C22⋊C4—88no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C23—24no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—82no—
555C23—82no—
556C23—81no—
557C23—81no—
558C23—81no—
559C23—81no—
560C23—82no—
561C23—82no—
562C23—82no—
563C23—82no—
564C23C2×C22⋊Q811yescharacteristic
565C23C2×#12711yescharacteristic
566C23#53611yescharacteristic
567C23D4211yescharacteristic
568C23D4211yescharacteristic
569C23#54711yescharacteristic
570C23D4×Q811yescharacteristic
571C23C2×AGL(1,ℤ/8)12yes—
572C23#57512yes—
573C23#57612yes—
574C23#57812yes—
575Q8—816no—
576D4—24no—
577D4—24no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—816no—
618C2×C4—82no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4#49614yes—
644C2×C4C2×#12011yescharacteristic
645C2×C4C2×C22⋊Q811yescharacteristic
646C2×C4#52511yescharacteristic
647C2×C4C2×C4⋊Q811yescharacteristic
648C2×C4#53511yescharacteristic
649C2×C4#53611yescharacteristic
650C2×C4#53611yescharacteristic
651C2×C4#53611yescharacteristic
652C2×C4#53911yescharacteristic
653C2×C4#53911yescharacteristic
654C2×C4#54411yescharacteristic
655C2×C4#54411yescharacteristic
656C2×C4D4211yescharacteristic
657C2×C4D4211yescharacteristic
658C2×C4#54711yescharacteristic
659C2×C4#54711yescharacteristic
660C2×C4#54711yescharacteristic
661C2×C4#54711yescharacteristic
662C2×C4#54711yescharacteristic
663C2×C4D4×Q811yescharacteristic
664C2×C4D4×Q811yescharacteristic
665C2×C4D4×Q811yescharacteristic
666C2×C4D4×Q811yescharacteristic
667C2×C4D4×Q811yescharacteristic
668C2×C4#55411yescharacteristic
669C2×C4#55411yescharacteristic
670C2×C4#56311yescharacteristic
671C2×C4#56311yescharacteristic
672C8—24no—
673C8—24no—
674C8—42no—
675C8—42no—
676C22—44no—
677C22—48no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—84no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—84no—
686C22—84no—
687C22#243311yescharacteristic
688C22#244011yescharacteristic
689C22#244911yescharacteristic
690C22#254211yescharacteristic
691C22#254211yescharacteristic
692C22#267711yescharacteristic
693C22#268011yescharacteristic
694C22#307712yes—
695C22#308312yes—
696C22C2×#49611yescharacteristic
697C22#315311yescharacteristic
698C22#315511yescharacteristic
699C22#315611yescharacteristic
700C22#323512yes—
701C22#324312yes—
702C22#326412yes—
703C22#326412yes—
704C22#326912yes—
705C22#327712yes—
706C22#334712yes—
707C22#335312yes—
708C22#338612yes—
709C22#338912yes—
710C4—24no—
711C4—42no—
712C4—44no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—161no—
720C4—161no—
721C4#240611yescharacteristic
722C4#243411yescharacteristic
723C4#244011yescharacteristic
724C4#244911yescharacteristic
725C4#267711yescharacteristic
726C4#268011yescharacteristic
727C4#272011yescharacteristic
728C4#272011yescharacteristic
729C2—84no—
730C2#1543011yescharacteristic
731C2#2578811yescharacteristic
732C2#2579211yescharacteristic
733C2#2605212yes—
734C2#2635212yes—
735C2#2852111yescharacteristic
736C2#2852311yescharacteristic
737C2#2854011yescharacteristic
738C2#2854411yescharacteristic
739C2#3096612yes—
740C2#3110012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2088C2semidirect
4. C2×#2057C2semidirect
6. C2×#1847C2semidirect
7. #15230C2semidirect
9. #11377C2semidirect
10. #11325C2semidirect
11. #11324C2semidirect
15. #10877C2semidirect
132. C2×#102Q8semidirect
139. #390Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#20922
C2×#20882
C2×#20572
C2×#18642
C2×#18472
#152302
#139772
#113772
#113252
#113242
#111232
#110512
#109272
#108772