Finite Group Databasebeta

← #411951

Subgroups — #411951

Order 512.

1354 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2026C212yesmaximal
3C2×#1803C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6#11385C212yesmaximal
7#11292C212yesmaximal
8#11067C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C22×#125C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2552—24no—
20#2552—24no—
21C2×#398—22no—
22C2×#391C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2026—28no—
31#1803—24no—
32C4×#103—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1441—22no—
43C4⋊#95—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#125—24no—
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—24no—
54C2×#125—42no—
55C2×#125—42no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8Q814yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
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⋊C4D411yescharacteristic
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×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103D414yes—
92C2×#103Q814yes—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96#398—216no—
97#391—24no—
98#391—42no—
99#389—28no—
100#389—24no—
101#389—24no—
102#384—28no—
103#384—22no—
104#384—22no—
105#384—22no—
106#384D414yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4D414yes—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D412yes—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—24no—
124#340—42no—
125#336—24no—
126#336—42no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8C2×D412yes—
139C22×Q8C2×Q812yes—
140C22×Q8C2×Q812yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C2×D412yes—
172C22×C8C2×Q812yes—
173#125—48no—
174#125—48no—
175#125—48no—
176#125—48no—
177C4×Q8—28no—
178C4×Q8—28no—
179C4×Q8—44no—
180C4×Q8—44no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
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—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4—81no—
229C2×C4⋊C4—81no—
230C2×C4⋊C4—81no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×Q812yes—
238C2×C4⋊C4C2×Q812yes—
239C2×C4⋊C4#4112yes—
240C2×C4⋊C4#4112yes—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×Q812yes—
256C2×C42#4112yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C4⋊C8—44no—
260#103—28no—
261#103—28no—
262#103—48no—
263C4×C8—44no—
264#95—28no—
265#95—24no—
266#95—24no—
267#95—44no—
268#95—44no—
269#95—82no—
270#95—82no—
271C24C22×D411yescharacteristic
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8C4⋊Q814yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
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—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4C22⋊Q812yes—
363C22×C4C22⋊Q812yes—
364C22×C4#12412yes—
365C22×C4#12412yes—
366C22×C4#13512yes—
367C22×C4#13512yes—
368C22×C4#13712yes—
369C22×C4#13712yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×Q811yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C42−1+411yescharacteristic
383C22×C42−1+411yescharacteristic
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—416no—
416C4⋊C4—416no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—48no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4—88no—
432C4⋊C4—84no—
433C4⋊C4#12114yes—
434C4⋊C4C22⋊Q814yes—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—48no—
440C42—48no—
441C42C22⋊Q814yes—
442C42C22⋊Q814yes—
443C42#12414yes—
444C42#12414yes—
445C23C2×#12011yescharacteristic
446C23C2×C22⋊Q811yescharacteristic
447C23C2×#12411yescharacteristic
448C23D4211yescharacteristic
449C23#54611yescharacteristic
450C23#54811yescharacteristic
451C23D4×Q811yescharacteristic
452C23C2×#13511yescharacteristic
453C23C2×#13511yescharacteristic
454C23C2×#13711yescharacteristic
455C23C2×#13711yescharacteristic
456C23#57611yescharacteristic
457C23#57711yescharacteristic
458C23#57711yescharacteristic
459C23#57811yescharacteristic
460Q8—24no—
461Q8—24no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
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—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4#39314yes—
531C2×C4#39314yes—
532C2×C4#45214yes—
533C2×C4#45214yes—
534C2×C4#47914yes—
535C2×C4#47914yes—
536C2×C4#48814yes—
537C2×C4#49114yes—
538C2×C4C2×#12111yescharacteristic
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×C22⋊Q811yescharacteristic
541C2×C4C2×C22⋊Q811yescharacteristic
542C2×C4C2×#12411yescharacteristic
543C2×C4C2×#12411yescharacteristic
544C2×C4C2×C4⋊Q811yescharacteristic
545C2×C4#53211yescharacteristic
546C2×C4#53311yescharacteristic
547C2×C4#53311yescharacteristic
548C2×C4#53511yescharacteristic
549C2×C4#53611yescharacteristic
550C2×C4#54211yescharacteristic
551C2×C4#54211yescharacteristic
552C2×C4#54411yescharacteristic
553C2×C4#54411yescharacteristic
554C2×C4D4211yescharacteristic
555C2×C4#54611yescharacteristic
556C2×C4#54711yescharacteristic
557C2×C4#54711yescharacteristic
558C2×C4#54811yescharacteristic
559C2×C4D4×Q811yescharacteristic
560C2×C4#55411yescharacteristic
561C2×C4#55411yescharacteristic
562C2×C4#55511yescharacteristic
563C2×C4#55511yescharacteristic
564C2×C4#55611yescharacteristic
565C2×C4#55611yescharacteristic
566C8—42no—
567C8—42no—
568C8—82no—
569C22C2×#39311yescharacteristic
570C22#240811yescharacteristic
571C22#240911yescharacteristic
572C22#243911yescharacteristic
573C22#244011yescharacteristic
574C22#249111yescharacteristic
575C22#249211yescharacteristic
576C22C2×#45211yescharacteristic
577C22C2×#45211yescharacteristic
578C22#301211yescharacteristic
579C22#301311yescharacteristic
580C22C2×#47911yescharacteristic
581C22C2×#47911yescharacteristic
582C22#308311yescharacteristic
583C22#308411yescharacteristic
584C22C2×#48811yescharacteristic
585C22C2×#49111yescharacteristic
586C22#313611yescharacteristic
587C22#313711yescharacteristic
588C22#327611yescharacteristic
589C22#327811yescharacteristic
590C22#328411yescharacteristic
591C22#328711yescharacteristic
592C22#330211yescharacteristic
593C22#330411yescharacteristic
594C22#333111yescharacteristic
595C22#333311yescharacteristic
596C22#336111yescharacteristic
597C22#336511yescharacteristic
598C22#337211yescharacteristic
599C22#337411yescharacteristic
600C22#338111yescharacteristic
601C22#338311yescharacteristic
602C22#338511yescharacteristic
603C22#338711yescharacteristic
604C4—24no—
605C4—44no—
606C4—44no—
607C4—44no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—84no—
612C4—162no—
613C4#202618yescomplemented
614C4#243911yescharacteristic
615C4#244011yescharacteristic
616C4#249111yescharacteristic
617C4#249211yescharacteristic
618C4#254711yescharacteristic
619C4#256311yescharacteristic
620C4#262811yescharacteristic
621C4#285711yescharacteristic
622C2#1512511yescharacteristic
623C2C2×#202611yescharacteristic
624C2#2345811yescharacteristic
625C2#2424111yescharacteristic
626C2#2424211yescharacteristic
627C2#2433111yescharacteristic
628C2#2433211yescharacteristic
629C2#2602311yescharacteristic
630C2#2602711yescharacteristic
631C2#2604311yescharacteristic
632C2#2604711yescharacteristic
633C2#2777011yescharacteristic
634C2#2777711yescharacteristic
635C2#2781611yescharacteristic
636C2#2782411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
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
613. C4#2026semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#20262
C2×#18032
C2×C4×#1032
#139262
#113852
#112922
#110672
#108492
#108412
#108342