Finite Group Databasebeta

← #411949

Subgroups — #411949

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2025C212yesmaximal
3C2×#1802C211yesmaximal, characteristic
4C2×C4×#102C211yesmaximal, characteristic
5#13906C211yesmaximal, characteristic
6#11385C212yesmaximal, complemented
7#11293C212yesmaximal, complemented
8#11066C212yesmaximal
9#10849C211yesmaximal, characteristic, complemented
10#10841C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C22×#125C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2552—24no—
20#2552—24no—
21C2×#398—22no—
22C2×#390C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29C2×#380—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2025—28no—
33#1802—24no—
34C4×#102—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1439—22no—
45C4⋊#95C2212yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—24no—
54C2×#125—24no—
55C2×#125—42no—
56C2×#125—42no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4Q814yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×#102D414yes—
94C2×#102Q814yes—
95C2×C4×C8—22no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98#398—216no—
99#390—24no—
100#390—42no—
101#389—28no—
102#389—24no—
103#389—24no—
104#384—28no—
105#384—22no—
106#384—22no—
107#384—22no—
108#384D414yes—
109#380—44no—
110#380—44no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4D414yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D412yes—
120#340—24no—
121#340—24no—
122#340—22no—
123#340—22no—
124#340—24no—
125#340—24no—
126#340—42no—
127#336—24no—
128#336—42no—
129C25—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4C2×D412yes—
142C22×D4C2×Q812yes—
143C22×D4C2×Q812yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8C2×D412yes—
181C22×C8C2×Q812yes—
182#125—48no—
183#125—48no—
184#125—48no—
185#125—48no—
186C4×D4—28no—
187C4×D4—28no—
188C4×D4—44no—
189C4×D4—44no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—81no—
234C2×C4⋊C4—81no—
235C2×C4⋊C4—81no—
236C2×C4⋊C4—81no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×Q812yes—
244C2×C4⋊C4C2×Q812yes—
245C2×C4⋊C4#4112yes—
246C2×C4⋊C4#4112yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—82no—
255C2×C22⋊C4—82no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×Q812yes—
268C2×C42#4112yes—
269C2×C42#4112yes—
270C2×C42#4112yes—
271C4⋊C8—44no—
272#102—28no—
273#102—28no—
274#102—48no—
275C4×C8—44no—
276#95—28no—
277#95—24no—
278#95—24no—
279#95—44no—
280#95—44no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24C22×D411yescharacteristic
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4C4⋊Q814yes—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4#12012yes—
386C22×C4#12012yes—
387C22×C4C22⋊Q812yes—
388C22×C4C22⋊Q812yes—
389C22×C4#12412yes—
390C22×C4#12412yes—
391C22×C4#13512yes—
392C22×C4#13512yes—
393C22×C4AGL(1,ℤ/8)12yes—
394C22×C4AGL(1,ℤ/8)12yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C22×Q811yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C4H(5,𝔽2)11yescharacteristic
407C22×C42−1+411yescharacteristic
408C22×C42−1+411yescharacteristic
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—416no—
441C4⋊C4—416no—
442C4⋊C4—42no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—88no—
456C4⋊C4—84no—
457C4⋊C4#12114yes—
458C4⋊C4C22⋊Q814yes—
459C22⋊C4—48no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—48no—
467C42C22⋊Q814yes—
468C42C22⋊Q814yes—
469C42#12414yes—
470C42#12414yes—
471C23—24no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23—81no—
500C23—81no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23C2×#12011yescharacteristic
507C23C2×C22⋊Q811yescharacteristic
508C23C2×#12411yescharacteristic
509C23D4211yescharacteristic
510C23#54611yescharacteristic
511C23#54811yescharacteristic
512C23D4×Q811yescharacteristic
513C23C2×#13511yescharacteristic
514C23C2×#13511yescharacteristic
515C23C2×AGL(1,ℤ/8)11yescharacteristic
516C23C2×AGL(1,ℤ/8)11yescharacteristic
517C23#57611yescharacteristic
518C23#57711yescharacteristic
519C23#57711yescharacteristic
520C23#57811yescharacteristic
521D4—24no—
522D4—24no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—42no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—816no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4#39314yes—
586C2×C4#39314yes—
587C2×C4#44914yes—
588C2×C4#44914yes—
589C2×C4#47814yes—
590C2×C4#47814yes—
591C2×C4#48814yes—
592C2×C4#49014yes—
593C2×C4C2×#12111yescharacteristic
594C2×C4C2×C22⋊Q811yescharacteristic
595C2×C4C2×C22⋊Q811yescharacteristic
596C2×C4C2×C22⋊Q811yescharacteristic
597C2×C4C2×#12411yescharacteristic
598C2×C4C2×#12411yescharacteristic
599C2×C4C2×C4⋊Q811yescharacteristic
600C2×C4#53211yescharacteristic
601C2×C4#53311yescharacteristic
602C2×C4#53311yescharacteristic
603C2×C4#53511yescharacteristic
604C2×C4#53611yescharacteristic
605C2×C4#54211yescharacteristic
606C2×C4#54211yescharacteristic
607C2×C4#54411yescharacteristic
608C2×C4#54411yescharacteristic
609C2×C4D4211yescharacteristic
610C2×C4#54611yescharacteristic
611C2×C4#54711yescharacteristic
612C2×C4#54711yescharacteristic
613C2×C4#54811yescharacteristic
614C2×C4D4×Q811yescharacteristic
615C2×C4#55411yescharacteristic
616C2×C4#55411yescharacteristic
617C2×C4#55511yescharacteristic
618C2×C4#55511yescharacteristic
619C2×C4#55611yescharacteristic
620C2×C4#55611yescharacteristic
621C8—42no—
622C8—42no—
623C8—82no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22C2×#39311yescharacteristic
640C22#240811yescharacteristic
641C22#240911yescharacteristic
642C22#243911yescharacteristic
643C22#244011yescharacteristic
644C22#249111yescharacteristic
645C22#249211yescharacteristic
646C22C2×#44911yescharacteristic
647C22C2×#44911yescharacteristic
648C22#301211yescharacteristic
649C22#301311yescharacteristic
650C22C2×#47811yescharacteristic
651C22C2×#47811yescharacteristic
652C22#308311yescharacteristic
653C22#308411yescharacteristic
654C22C2×#48811yescharacteristic
655C22C2×#49011yescharacteristic
656C22#313611yescharacteristic
657C22#313711yescharacteristic
658C22#327211yescharacteristic
659C22#327411yescharacteristic
660C22#328311yescharacteristic
661C22#328411yescharacteristic
662C22#330311yescharacteristic
663C22#330511yescharacteristic
664C22#333011yescharacteristic
665C22#333211yescharacteristic
666C22#336011yescharacteristic
667C22#336411yescharacteristic
668C22#336611yescharacteristic
669C22#336811yescharacteristic
670C22#338011yescharacteristic
671C22#338111yescharacteristic
672C22#338411yescharacteristic
673C22#338611yescharacteristic
674C4—24no—
675C4—44no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—84no—
681C4—162no—
682C4#202518yescomplemented
683C4#243911yescharacteristic
684C4#244011yescharacteristic
685C4#249111yescharacteristic
686C4#249211yescharacteristic
687C4#254711yescharacteristic
688C4#256311yescharacteristic
689C4#262811yescharacteristic
690C4#285711yescharacteristic
691C2—84no—
692C2#1512511yescharacteristic
693C2C2×#202511yescharacteristic
694C2#2345811yescharacteristic
695C2#2423711yescharacteristic
696C2#2423811yescharacteristic
697C2#2432711yescharacteristic
698C2#2432811yescharacteristic
699C2#2602111yescharacteristic
700C2#2602511yescharacteristic
701C2#2604111yescharacteristic
702C2#2604511yescharacteristic
703C2#2776811yescharacteristic
704C2#2777511yescharacteristic
705C2#2781411yescharacteristic
706C2#2782211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
1. C2×#2552C2semidirect
6. #11385C2semidirect
7. #11293C2semidirect
9. #10849C2semidirect
10. #10841C2semidirect
11. #10834C2semidirect
682. C4#2025semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#20252
C2×#18022
C2×C4×#1022
#139062
#113852
#112932
#110662
#108492
#108412
#108342