Finite Group Databasebeta

← #366964

Subgroups — #366964

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2565C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1771C211yesmaximal, characteristic
6C2×#1763C211yesmaximal, characteristic
7#13910C211yesmaximal, characteristic
8#11373C211yesmaximal, characteristic
9#11372C211yesmaximal, characteristic
10#11050C211yesmaximal, characteristic
11#11050C211yesmaximal, characteristic
12#10928C211yesmaximal, characteristic
13#10926C211yesmaximal, characteristic
14#10843C211yesmaximal, characteristic
15#10841C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2565—24no—
27#2565C418yescomplemented
28C2×#398—22no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#389—22no—
32C2×#387—22no—
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#1992—24no—
41#1989—24no—
42#1803—24no—
43#1771—24no—
44#1763—24no—
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#340C2211yescharacteristic
51C2×#340C2211yescharacteristic
52C2×#340C2211yescharacteristic
53C2×#340C2211yescharacteristic
54C2×#340C2211yescharacteristic
55C2×#340C2211yescharacteristic
56C2×#340C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1444—22no—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C23×Q8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×#125—22no—
68C2×#125—22no—
69C2×#125—22no—
70C2×#125—22no—
71C2×#125—22no—
72C2×#125C2×C414yes—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8C2×C414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4D412yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C42—21no—
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8D414yes—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—22no—
112C2×#103—41no—
113C2×#103—41no—
114C2×#103Q814yes—
115C2×#97—22no—
116C2×#97—22no—
117C2×#97—22no—
118C2×#97—22no—
119C2×#97—41no—
120C2×#97D414yes—
121C2×C4×C8—22no—
122C2×C4×C8—22no—
123C2×C4×C8—41no—
124#398—216no—
125#391—22no—
126#391—22no—
127#391C2×C414yes—
128#391C2×C414yes—
129#389—48no—
130#387—216no—
131#384—22no—
132#384—22no—
133#384—22no—
134#384—22no—
135#384—22no—
136#384—41no—
137#384—41no—
138#384C2×C414yes—
139#382—22no—
140#382C2×C414yes—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4C2×C414yes—
144C4×C4⋊C4D414yes—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—21no—
151C2×#95D412yes—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#340—22no—
160#340—22no—
161#340—22no—
162#340—22no—
163#340—22no—
164#336—22no—
165#336—22no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
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×Q8C22×C412yes—
178C22×Q8C2×Q812yes—
179C22×Q8#4112yes—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2411yescharacteristic
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
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×C8C2×D412yes—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8C2×D412yes—
219C22×C8C2×Q812yes—
220C22×C8#4112yes—
221#125—28no—
222#125—28no—
223#125—44no—
224#125—44no—
225C4×Q8—28no—
226C4×Q8—28no—
227C4×Q8—44no—
228C4×Q8—44no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
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—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
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—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4C22⋊C418yes—
291C2×C4⋊C4C22×C412yes—
292C2×C4⋊C4C22×C412yes—
293C2×C4⋊C4C22×C412yes—
294C2×C4⋊C4C2×Q812yes—
295C2×C4⋊C4#4112yes—
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—24no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42—42no—
309C2×C42C22×C412yes—
310C2×C42C22×C412yes—
311C2×C42C22×C412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42#4112yes—
318C2×C42#4112yes—
319C4⋊C8—28no—
320#103—28no—
321#103—28no—
322#103—44no—
323#103—44no—
324#97—28no—
325#97—44no—
326C4×C8—44no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95C22⋊C418yes—
334C24C22×D411yescharacteristic
335C2×Q8—22no—
336C2×Q8—22no—
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—41no—
351C2×Q8—41no—
352C2×Q8—41no—
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×Q8C4×Q814yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4C2×C22⋊C414yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C412yes—
422C22×C4C2×C22⋊C414yes—
423C22×C4#12412yes—
424C22×C4#12412yes—
425C22×C4#12712yes—
426C22×C4#12712yes—
427C22×C4AGL(1,ℤ/8)14yes—
428C22×C4#13714yes—
429C22×C4C23×C411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×Q811yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4H(5,𝔽2)11yescharacteristic
440C22×C42−1+411yescharacteristic
441C22×C42−1+411yescharacteristic
442C22×C42−1+411yescharacteristic
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
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—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8C22⋊Q814yes—
471C4⋊C4—216no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—416no—
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—88no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—88no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—48no—
517C42—84no—
518C42C4×D414yes—
519C42C4×D414yes—
520C42#12114yes—
521C42#12114yes—
522C23C22×C22⋊C411yescharacteristic
523C23C2×#12411yescharacteristic
524C23C2×#12711yescharacteristic
525C23#54611yescharacteristic
526C23#54711yescharacteristic
527C23#54811yescharacteristic
528C23D4×Q811yescharacteristic
529C23C2×AGL(1,ℤ/8)12yes—
530C23C2×#13712yes—
531C23#57712yes—
532C23#57812yes—
533Q8—24no—
534Q8—24no—
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
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—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—81no—
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—82no—
595C2×C4—82no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4#39014yes—
605C2×C4#39014yes—
606C2×C4#41718yes—
607C2×C4C2×C4×D411yescharacteristic
608C2×C4C2×C4×D411yescharacteristic
609C2×C4C2×C4×Q811yescharacteristic
610C2×C4C4×#4111yescharacteristic
611C2×C4#51911yescharacteristic
612C2×C4#52011yescharacteristic
613C2×C4C2×#12111yescharacteristic
614C2×C4C2×#12111yescharacteristic
615C2×C4C2×C22⋊Q811yescharacteristic
616C2×C4#52511yescharacteristic
617C2×C4#52911yescharacteristic
618C2×C4#53311yescharacteristic
619C2×C4#53611yescharacteristic
620C2×C4#53711yescharacteristic
621C2×C4#54111yescharacteristic
622C2×C4#54211yescharacteristic
623C2×C4#54611yescharacteristic
624C2×C4#54611yescharacteristic
625C2×C4#54711yescharacteristic
626C2×C4#54711yescharacteristic
627C2×C4#54811yescharacteristic
628C2×C4#54811yescharacteristic
629C2×C4D4×Q811yescharacteristic
630C2×C4D4×Q811yescharacteristic
631C2×C4#55211yescharacteristic
632C2×C4#55211yescharacteristic
633C2×C4#55411yescharacteristic
634C2×C4#55611yescharacteristic
635C8—24no—
636C8—42no—
637C8—42no—
638C8—42no—
639C22C2×#39011yescharacteristic
640C22#233911yescharacteristic
641C22Q8×C22⋊C411yescharacteristic
642C22#248911yescharacteristic
643C22#249211yescharacteristic
644C22#254011yescharacteristic
645C22#254211yescharacteristic
646C22C2×#41712yes—
647C22#289812yes—
648C22#313712yes—
649C22#314112yes—
650C22#315111yescharacteristic
651C22#315211yescharacteristic
652C22#315411yescharacteristic
653C22#315611yescharacteristic
654C22#331012yes—
655C22#331212yes—
656C22#334312yes—
657C22#334412yes—
658C22#336812yes—
659C22#337412yes—
660C22#338612yes—
661C22#338712yes—
662C4—24no—
663C4—24no—
664C4—28no—
665C4—44no—
666C4—44no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—161no—
673C4—161no—
674C4C4×C22⋊Q811yescharacteristic
675C4#232311yescharacteristic
676C4#236011yescharacteristic
677C4#236211yescharacteristic
678C4#241511yescharacteristic
679C4#241711yescharacteristic
680C4#242711yescharacteristic
681C4#242911yescharacteristic
682C2#1399711yescharacteristic
683C2#2120011yescharacteristic
684C2#2120511yescharacteristic
685C2#2249112yes—
686C2#2252412yes—
687C2#2774112yes—
688C2#2782712yes—
689C2#2842311yescharacteristic
690C2#2843511yescharacteristic
691C2#2854511yescharacteristic
692C2#2855011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #2565C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25652
C2×#19922
C2×#19892
C2×#18032
C2×#17712
C2×#17632
#139102
#113732
#113722
#110502
#109282
#109262
#108432
#108412