Finite Group Databasebeta

← #414809

Subgroups — #414809

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2C2×#1984C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×C4×#103C211yesmaximal, characteristic
6#13926C211yesmaximal, characteristic
7#11293C212yesmaximal
8#11219C212yesmaximal
9#11190C212yesmaximal
10#11124C212yesmaximal
11#10841C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2582—24no—
21#2582—24no—
22C2×#398—22no—
23C2×#391—21no—
24C2×#391—21no—
25C2×#391C2212yes—
26C2×#389—21no—
27C2×#389C2212yes—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1984—24no—
32#1947—24no—
33#1803—24no—
34C4×#103—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8D411yescharacteristic
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—42no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8D414yes—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103Q814yescomplemented
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99#398—216no—
100#391—28no—
101#391—24no—
102#391—24no—
103#391—44no—
104#389—28no—
105#389—24no—
106#389—24no—
107#384—22no—
108#384—22no—
109#384—22no—
110#384D414yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—22no—
125#340—22no—
126#340—42no—
127#336—24no—
128#336—24no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8Q1614yes—
147C22×Q8C2×D412yes—
148C22×Q8C2×Q812yes—
149C22×Q8#4112yes—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
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—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8C2×D414yes—
180C22×C8C2×Q812yes—
181C22×C8#4112yes—
182C4×Q8—28no—
183C4×Q8—28no—
184C4×Q8—44no—
185C4×Q8—44no—
186C4×Q8—44no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×Q8—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—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—42no—
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—42no—
234C2×C4⋊C4—81no—
235C2×C4⋊C4—81no—
236C2×C4⋊C4—81no—
237C2×C4⋊C4—81no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×Q812yes—
243C2×C4⋊C4#4112yes—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C2×C42#4112yes—
262#107—48no—
263C4⋊C8—28no—
264#103—28no—
265#103—28no—
266#103—48no—
267C4×C8—44no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—82no—
275#95—82no—
276C24C22×D411yescharacteristic
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—28no—
281C2×Q8—28no—
282C2×Q8—28no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8C22⋊Q814yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4#12012yes—
372C22×C4#12012yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4#12312yes—
376C22×C4#12312yes—
377C22×C4C2×Q1614yes—
378C22×C4C2×Q1612yes—
379C22×C4#13512yes—
380C22×C4AGL(1,ℤ/8)12yes—
381C22×C4#13712yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×Q811yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C42−1+411yescharacteristic
394C22×C42−1+411yescharacteristic
395C22×C42−1+411yescharacteristic
396C2×C8—24no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—42no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—41no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—88no—
444C4⋊C4—84no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—48no—
450C42—84no—
451C42#12114yes—
452C42#12114yes—
453C42#12314yes—
454C42#12414yes—
455C23C2×#12011yescharacteristic
456C23C2×#12111yescharacteristic
457C23C2×#12311yescharacteristic
458C23#54811yescharacteristic
459C23#54811yescharacteristic
460C23D4×Q811yescharacteristic
461C23#55011yescharacteristic
462C23C22×Q1611yescharacteristic
463C23C2×#13511yescharacteristic
464C23C2×AGL(1,ℤ/8)11yescharacteristic
465C23C2×#13711yescharacteristic
466C23#57611yescharacteristic
467C23#57711yescharacteristic
468C23#57711yescharacteristic
469C23#57811yescharacteristic
470Q8—24no—
471Q8—24no—
472Q8—42no—
473Q8—42no—
474Q8—42no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—416no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4#39414yes—
542C2×C4#39414yes—
543C2×C4#44914yes—
544C2×C4#45114yes—
545C2×C4#46714yes—
546C2×C4#46814yes—
547C2×C4#48414yes—
548C2×C4#48514yes—
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×#12111yescharacteristic
551C2×C4C2×C22⋊Q811yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4C2×#12411yescharacteristic
555C2×C4#52911yescharacteristic
556C2×C4#52911yescharacteristic
557C2×C4#53311yescharacteristic
558C2×C4#53311yescharacteristic
559C2×C4#53611yescharacteristic
560C2×C4#53711yescharacteristic
561C2×C4#54111yescharacteristic
562C2×C4#54111yescharacteristic
563C2×C4#54211yescharacteristic
564C2×C4#54211yescharacteristic
565C2×C4#54611yescharacteristic
566C2×C4#54611yescharacteristic
567C2×C4#54711yescharacteristic
568C2×C4#54711yescharacteristic
569C2×C4#54811yescharacteristic
570C2×C4#54811yescharacteristic
571C2×C4D4×Q811yescharacteristic
572C2×C4D4×Q811yescharacteristic
573C2×C4#55211yescharacteristic
574C2×C4#55211yescharacteristic
575C2×C4#55411yescharacteristic
576C2×C4#55611yescharacteristic
577C8—42no—
578C8—42no—
579C8—44no—
580C22C2×#39411yescharacteristic
581C22#240711yescharacteristic
582C22#240911yescharacteristic
583C22#242111yescharacteristic
584C22#242311yescharacteristic
585C22#242611yescharacteristic
586C22#242911yescharacteristic
587C22C2×#45111yescharacteristic
588C22C2×#44911yescharacteristic
589C22#301211yescharacteristic
590C22#301311yescharacteristic
591C22C2×#46711yescharacteristic
592C22C2×#46811yescharacteristic
593C22#306211yescharacteristic
594C22#306311yescharacteristic
595C22C2×#48511yescharacteristic
596C22C2×#48411yescharacteristic
597C22#309811yescharacteristic
598C22#309911yescharacteristic
599C22#336111yescharacteristic
600C22#336311yescharacteristic
601C22#336311yescharacteristic
602C22#336411yescharacteristic
603C22#336811yescharacteristic
604C22#337311yescharacteristic
605C22#337411yescharacteristic
606C22#337511yescharacteristic
607C22#338111yescharacteristic
608C22Q8×Q1611yescharacteristic
609C22#338611yescharacteristic
610C22#338711yescharacteristic
611C22#339411yescharacteristic
612C22#339511yescharacteristic
613C22#339911yescharacteristic
614C22#340011yescharacteristic
615C4—24no—
616C4—44no—
617C4—44no—
618C4—82no—
619C4—82no—
620C4—84no—
621C4—82no—
622C4—82no—
623C4—162no—
624C4#203618yes—
625C4#241511yescharacteristic
626C4#241711yescharacteristic
627C4#242711yescharacteristic
628C4#242911yescharacteristic
629C4#248911yescharacteristic
630C4#249211yescharacteristic
631C4#262511yescharacteristic
632C4#285511yescharacteristic
633C2#1517911yescharacteristic
634C2C2×#203611yescharacteristic
635C2#2348511yescharacteristic
636C2#2418611yescharacteristic
637C2#2419111yescharacteristic
638C2#2432811yescharacteristic
639C2#2433311yescharacteristic
640C2#2499511yescharacteristic
641C2#2499611yescharacteristic
642C2#2506111yescharacteristic
643C2#2506211yescharacteristic
644C2#2529411yescharacteristic
645C2#2530311yescharacteristic
646C2#2547111yescharacteristic
647C2#2547911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
95. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#19842
C2×#19472
C2×#18032
C2×C4×#1032
#139262
#112932
#112192
#111902
#111242
#108412