Finite Group Databasebeta

← #519628

Subgroups — #519628

Order 512.

3102 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×#2357C211yesmaximal, characteristic
3C2×C42×Q8C211yesmaximal, characteristic, complemented
4#14707C211yesmaximal, characteristic, complemented
5#14700C212yesmaximal
6#14194C214yesmaximal, complemented
7#14188C214yesmaximal, complemented
8#14041C212yesmaximal, complemented
9#14022C212yesmaximal
10#13983C214yesmaximal, complemented
11#13925C211yesmaximal, characteristic, complemented
12#13823C214yesmaximal, complemented
13#13804C211yesmaximal, characteristic, complemented
14C4×#389C212yesmaximal, complemented
15C4×#390C211yesmaximal, characteristic
16C22×C4×Q8C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#2580C4116yescomplemented
20C2×#399C2212yes—
21C2×#393C2212yes—
22#2357—28no—
23C2×#390C2212yes—
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#385C2214yes—
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#381C2212yes—
37C42×Q8—28no—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2214yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2214yes—
49C2×C43C2212yes—
50C2×C43C2211yescharacteristic
51#1446C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1442C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2211yescharacteristic
75#1434C2212yes—
76C4×#95C2214yes—
77C4×#95C2212yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C2×C4×Q8—28no—
82C2×C4×Q8—28no—
83C2×C4×Q8—24no—
84C2×C4×Q8—28no—
85C2×C4×Q8—24no—
86C2×C4×Q8C2×C418yes—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4C2×C418yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123#399C2×C4116yes—
124#393C2×C4116yes—
125#390—24no—
126#390—28no—
127#390—24no—
128#390C2×C4116yes—
129#389—28no—
130#389C2×C4116yes—
131#388—216no—
132#388—216no—
133#388—216no—
134#388—48no—
135#385C2×C4132yes—
136#382—24no—
137#382C2×C418yes—
138#381—28no—
139C4×C4⋊C4—28no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—48no—
147C4×C4⋊C4D418yes—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—48no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C43—28no—
174C43—24no—
175C25C2412yes—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8C22×C414yes—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4C22×C414yes—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4C22×C418yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C4×Q8—264no—
208C4×Q8—232no—
209C4×D4—232no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C22×C414yes—
239C2×C4⋊C4C22×C418yes—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C22×C418yes—
242C2×C4⋊C4C22×C4116yes—
243C2×C4⋊C4C2×D418yes—
244C2×C4⋊C4#4118yes—
245C2×C4⋊C4#4114yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C4116yes—
266C2×C22⋊C4C22×C4116yes—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—28no—
286C2×C42—28no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—44no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C22×C418yes—
302C2×C42C22×C414yes—
303C2×C42C2×D418yes—
304C2×C42C2×D414yes—
305C2×C42#4118yes—
306C2×C42#4114yes—
307C2×C42#4114yes—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95C22×C418yes—
327#95C22×C418yes—
328#95C22×C4116yes—
329#95C2×D414yes—
330#95C2×D414yes—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24C23×C414yes—
342C24C2511yescharacteristic
343C2×Q8—216no—
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—28no—
347C2×Q8—216no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
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—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—416no—
417C22×C4—88no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4C23×C418yes—
421C22×C4C23×C414yes—
422C22×C4C23×C414yes—
423C22×C4C23×C412yes—
424C22×C4C23×C412yes—
425C22×C4C23×C414yes—
426C22×C4C23×C412yes—
427C22×C4C22×D414yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C42−1+412yes—
447C22×C42−1+412yes—
448C4⋊C4—216no—
449C4⋊C4—28no—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—416no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—416no—
460C22⋊C4—216no—
461C22⋊C4—232no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—88no—
467C42—216no—
468C42—216no—
469C42—216no—
470C42—216no—
471C42—216no—
472C42—28no—
473C42—416no—
474C42—48no—
475C42—48no—
476C42—48no—
477C42—44no—
478C42—88no—
479C42—84no—
480C42C4×D4116yes—
481C23—24no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23C24×C411yescharacteristic
505C23C23×D411yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4112yes—
508C23C22×#4112yes—
509C23C22×#4111yescharacteristic
510C23C22×#4111yescharacteristic
511C23C2×H(5,𝔽2)12yes—
512C23C2×2−1+411yescharacteristic
513C23C2×2−1+411yescharacteristic
514C23#58512yes—
515Q8—232no—
516D4—416no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—816no—
594C2×C4C2×C4×D414yes—
595C2×C4C2×C4×D414yes—
596C2×C4C2×C4×D414yes—
597C2×C4C4×#4118yes—
598C2×C4C4×#4114yes—
599C2×C4#51914yes—
600C2×C4#52518yes—
601C2×C4#54714yes—
602C2×C4#54814yes—
603C2×C4#55214yes—
604C2×C4#55614yes—
605C2×C4#55918yes—
606C22—48no—
607C22—44no—
608C22—48no—
609C22—44no—
610C22—44no—
611C22—82no—
612C22—84no—
613C22—82no—
614C22—82no—
615C22—84no—
616C22—82no—
617C22C22×C4×D411yescharacteristic
618C22C2×C4×#4112yes—
619C22C2×C4×#4111yescharacteristic
620C22C2×#51911yescharacteristic
621C22C4×H(5,𝔽2)12yes—
622C22C2×#52512yes—
623C22#344411yescharacteristic
624C22#344411yescharacteristic
625C22C2×#54711yescharacteristic
626C22C2×#54811yescharacteristic
627C22C2×#55211yescharacteristic
628C22C2×#55611yescharacteristic
629C22C2×#55912yes—
630C22#348012yes—
631C22#348012yes—
632C22#348112yes—
633C22#348812yes—
634C22#350712yes—
635C22#350811yescharacteristic
636C22#351011yescharacteristic
637C22#351112yes—
638C22#351512yes—
639C22#352212yes—
640C4—28no—
641C4—24no—
642C4—28no—
643C4—24no—
644C4—28no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—48no—
649C4—44no—
650C4—44no—
651C4—82no—
652C4—84no—
653C4#236218yes—
654C2—84no—
655C2C4×#52511yescharacteristic
656C2C2×#236211yescharacteristic
657C2C4×#55912yes—
658C2#3426911yescharacteristic
659C2#3431712yes—
660C2#3636112yes—
661C2#3637011yescharacteristic
662C2#3640311yescharacteristic
663C2#3658012yes—
664C2#3660111yescharacteristic
665C2#3660911yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C42×Q8C2semidirect
4. #14707C2semidirect
6. #14194C2semidirect
7. #14188C2semidirect
8. #14041C2semidirect
10. #13983C2semidirect
11. #13925C2semidirect
12. #13823C2semidirect
13. #13804C2semidirect
14. C4×#389C2semidirect
19. #2580C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#23572
C2×C42×Q82
#147072
#147002
#141942
#141882
#140412
#140222
#139832
#139252
#138232
#138042
C4×#3892
C4×#3902