Finite Group Databasebeta

← #431044

Subgroups — #431044

Order 512.

1322 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic
2C2×#2097C211yesmaximal, characteristic
3C2×#2087C211yesmaximal, characteristic
4C2×#2058C211yesmaximal, characteristic
5C2×#1863C211yesmaximal, characteristic
6C2×#1847C211yesmaximal, characteristic
7#15230C211yesmaximal, characteristic
8#13976C211yesmaximal, characteristic
9#11378C211yesmaximal, characteristic
10#11325C211yesmaximal, characteristic
11#11323C211yesmaximal, characteristic
12#11122C211yesmaximal, characteristic
13#11052C211yesmaximal, characteristic
14#10928C211yesmaximal, characteristic
15#10877C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#475—24no—
19C2×#475—24no—
20C2×#475—24no—
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#97C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#382C2211yescharacteristic
40#2097—24no—
41#2087—24no—
42#2058—24no—
43#1863—24no—
44#1847—24no—
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#340C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55#1446—22no—
56#1444—22no—
57#1441—22no—
58#1436—22no—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4×#95C2211yescharacteristic
63C23×Q8—21no—
64C23×Q8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—41no—
74C2×C4⋊Q8—41no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—41no—
80C2×C4×Q8—41no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#475—232no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#107—22no—
102C2×#107—22no—
103C2×#107—41no—
104C2×#106—22no—
105C2×#106—22no—
106C2×#106—22no—
107C2×#106—22no—
108C2×#106—22no—
109C2×#106—22no—
110C2×#106—22no—
111C2×#106—22no—
112C2×#106—22no—
113C2×#106—41no—
114C2×#106—41no—
115C2×#106D414yes—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8D414yes—
119C2×#103—22no—
120C2×#103—22no—
121C2×#103—22no—
122C2×#103—22no—
123C2×#103—22no—
124C2×#103—22no—
125C2×#103—22no—
126C2×#103—22no—
127C2×#103—22no—
128C2×#103—41no—
129C2×#103D414yes—
130C2×#103Q814yescomplemented
131C2×#97—22no—
132C2×#97—22no—
133C2×#97—22no—
134#391—22no—
135#391—22no—
136#391—41no—
137#391Q814yescomplemented
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#384—22no—
143#384—41no—
144#382—22no—
145#382—22no—
146#382—41no—
147#382—41no—
148C2×#95—22no—
149C2×#95—21no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95—21no—
154C2×#95—22no—
155C2×#95—22no—
156C2×#95—22no—
157C2×#95D412yes—
158C2×#95D412yes—
159#340—22no—
160#340—22no—
161#340—22no—
162#340—22no—
163#340—22no—
164#340—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168#336D414yes—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—44no—
184C22×Q8—82no—
185C22×Q8C2×D412yes—
186C22×Q8C2×Q812yes—
187C22×Q8C2×Q812yes—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2411yescharacteristic
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
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×C8C2×D412yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C22×C8C2×Q812yes—
228C22×C8#4112yes—
229C4⋊Q8—28no—
230C4⋊Q8—44no—
231C4⋊Q8—44no—
232C4⋊Q8—44no—
233C4×Q8—28no—
234C4×Q8—44no—
235C4×Q8—44no—
236C4×Q8—44no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
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—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×Q812yes—
291C2×C4⋊C4C2×Q812yes—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×Q812yes—
312C2×C42#4112yes—
313#107—44no—
314#107—44no—
315#106—28no—
316#106—28no—
317#106—44no—
318#106—44no—
319C4⋊C8—28no—
320#103—28no—
321#103—28no—
322#103—28no—
323#103—44no—
324#97—28no—
325#95—28no—
326#95—24no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—42no—
333#95—82no—
334#95—82no—
335#95—82no—
336C24C22×D411yescharacteristic
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—48no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—88no—
366C2×Q8—88no—
367C2×Q8—84no—
368C2×Q8—84no—
369C2×Q8C4⋊Q814yes—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4C22⋊Q814yes—
433C22×C4#12712yes—
434C22×C4#12712yes—
435C22×C4#13714yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×Q811yescharacteristic
441C22×C4C22×Q811yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C42−1+411yescharacteristic
447C22×C42−1+411yescharacteristic
448C22×C42−1+411yescharacteristic
449C22×C42−1+411yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8#12014yes—
477C2×C8C22⋊Q814yes—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—88no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—48no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C23C2×C22⋊Q811yescharacteristic
525C23C2×#12711yescharacteristic
526C23#53611yescharacteristic
527C23D4211yescharacteristic
528C23D4211yescharacteristic
529C23#54711yescharacteristic
530C23D4×Q811yescharacteristic
531C23C2×#13712yes—
532C23#57512yes—
533C23#57712yes—
534C23#57712yes—
535Q8—24no—
536Q8—24no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—42no—
543Q8—816no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—82no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4#49714yes—
610C2×C4C2×#12011yescharacteristic
611C2×C4C2×C22⋊Q811yescharacteristic
612C2×C4#52511yescharacteristic
613C2×C4C2×C4⋊Q811yescharacteristic
614C2×C4#53511yescharacteristic
615C2×C4#53611yescharacteristic
616C2×C4#53611yescharacteristic
617C2×C4#53611yescharacteristic
618C2×C4#53911yescharacteristic
619C2×C4#53911yescharacteristic
620C2×C4#54411yescharacteristic
621C2×C4#54411yescharacteristic
622C2×C4D4211yescharacteristic
623C2×C4D4211yescharacteristic
624C2×C4#54711yescharacteristic
625C2×C4#54711yescharacteristic
626C2×C4#54711yescharacteristic
627C2×C4#54711yescharacteristic
628C2×C4#54711yescharacteristic
629C2×C4D4×Q811yescharacteristic
630C2×C4D4×Q811yescharacteristic
631C2×C4D4×Q811yescharacteristic
632C2×C4D4×Q811yescharacteristic
633C2×C4D4×Q811yescharacteristic
634C2×C4#55411yescharacteristic
635C2×C4#55411yescharacteristic
636C2×C4#56311yescharacteristic
637C2×C4#56311yescharacteristic
638C8—24no—
639C8—24no—
640C8—42no—
641C8—42no—
642C22#243311yescharacteristic
643C22#244011yescharacteristic
644C22#244911yescharacteristic
645C22#254211yescharacteristic
646C22#254211yescharacteristic
647C22#267711yescharacteristic
648C22#268011yescharacteristic
649C22#307812yes—
650C22#308412yes—
651C22C2×#49711yescharacteristic
652C22#315311yescharacteristic
653C22#315411yescharacteristic
654C22#315411yescharacteristic
655C22#323712yes—
656C22#324412yes—
657C22#326612yes—
658C22#326612yes—
659C22#327112yes—
660C22#327512yes—
661C22#334812yes—
662C22#335612yes—
663C22#338712yes—
664C22#338812yes—
665C4—24no—
666C4—44no—
667C4—42no—
668C4—44no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—161no—
676C4—161no—
677C4#240611yescharacteristic
678C4#243411yescharacteristic
679C4#244011yescharacteristic
680C4#244911yescharacteristic
681C4#267711yescharacteristic
682C4#268011yescharacteristic
683C4#272011yescharacteristic
684C4#272011yescharacteristic
685C2#1543011yescharacteristic
686C2#2578411yescharacteristic
687C2#2579711yescharacteristic
688C2#2605412yes—
689C2#2635612yes—
690C2#2852211yescharacteristic
691C2#2852411yescharacteristic
692C2#2854211yescharacteristic
693C2#2854911yescharacteristic
694C2#3096912yes—
695C2#3110712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
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
130. C2×#103Q8semidirect
137. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
C2×#20972
C2×#20872
C2×#20582
C2×#18632
C2×#18472
#152302
#139762
#113782
#113252
#113232
#111222
#110522
#109282
#108772