Finite Group Databasebeta

← #575495

Subgroups — #575495

Order 512.

3550 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×#2380C211yesmaximal, characteristic
2C2×#2366C211yesmaximal, characteristic
3C2×#2328C211yesmaximal, characteristic
4#14513C212yesmaximal, complemented
5#14513C214yesmaximal, complemented
6#14407C214yesmaximal, complemented
7#14401C212yesmaximal, complemented
8#14298C212yesmaximal, complemented
9#14164C212yesmaximal, complemented
10#14016C212yesmaximal, complemented
11#14015C212yesmaximal, complemented
12#14015C214yesmaximal, complemented
13#13975C214yesmaximal, complemented
14C22×#127C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2214yes—
18C23×C4⋊C4C2212yes—
19#2380C4116yescomplemented
20#2366—28no—
21#2328—28no—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#390C2214yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#386C2214yes—
31C2×#386C2212yes—
32C2×#386C2214yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#383C2212yes—
38C2×#382C2211yescharacteristic
39C2×#380C2214yes—
40C2×C4×C4⋊C4C2212yes—
41C2×#376C2211yescharacteristic
42C22×#95C2212yes—
43#1444C2214yes—
44#1444C2212yes—
45#1444C2212yes—
46#1444C2214yes—
47#1442C2214yes—
48#1442C2212yes—
49#1442C2212yes—
50#1442C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C23×D4C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2312yes—
79C2×#127C2×C418yes—
80C2×#127D418yes—
81C2×#127Q818yes—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4—216no—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C2×C4×D4—48no—
90C2×C4×D4—48no—
91C22×C4⋊C4—48no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#390—24no—
118#390—24no—
119#390—216no—
120#386—216no—
121#386—216no—
122#386—28no—
123#386—28no—
124#386C2×C4116yescomplemented
125#386C2×C4116yescomplemented
126#386C2×C4132yescomplemented
127#386C2×C4132yescomplemented
128#385—44no—
129#385—48no—
130#385—44no—
131#385—48no—
132#383C2×C4116yescomplemented
133#382—24no—
134#380—48no—
135C4×C4⋊C4—28no—
136#376—24no—
137C2×#95—44no—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C25C2412yes—
159C25C2412yes—
160C22×D4—41no—
161C22×D4—41no—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4C22×C414yes—
173C22×D4C22×C414yes—
174C22×D4C22×C4116yes—
175C22×D4C2×D414yes—
176C22×D4C2×D414yes—
177C22×D4C2×D4116yes—
178C22×D4C2×Q814yes—
179C22×D4C2×Q814yes—
180C22×D4C2×Q8116yes—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—28no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4C22×C418yes—
210C23×C4C22×C418yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222#127C4⋊C4164yescomplemented
223C4×D4—416no—
224C4×D4—416no—
225C4×D4—464no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—432no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4C22×C418yes—
282C2×C22⋊C4C22×C4116yes—
283C2×C22⋊C4C22×C4116yes—
284C2×C22⋊C4C22×C418yes—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—28no—
289C2×C42—22no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—41no—
293C2×C42—41no—
294C2×C42C22×C414yes—
295C2×C42C2×D414yes—
296C2×C42C2×Q814yes—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95C22×C4116yes—
314#95C22×C4116yes—
315#95C22×C418yes—
316#95C22×C418yes—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—24no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—24no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24C23×C414yes—
334C24C23×C414yes—
335C24C22×D414yes—
336C24C22×D414yes—
337C24C22×Q814yes—
338C24C22×Q814yes—
339C24C2511yescharacteristic
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—416no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—416no—
347C2×D4—416no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—416no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—44no—
358C2×D4C2×C4⋊C4116yes—
359C2×D4C2×C4⋊C4116yescomplemented
360C2×D4C2×C4⋊C4164yescomplemented
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—48no—
414C22×C4—416no—
415C22×C4—48no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—416no—
427C22×C4—48no—
428C22×C4—88no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—88no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4C23×C418yes—
444C22×C4C23×C418yes—
445C22×C4C23×C412yes—
446C22×C4C23×C414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q814yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)18yes—
453C22×C4H(5,𝔽2)12yes—
454C4⋊C4—216no—
455C4⋊C4—416no—
456C4⋊C4—48no—
457C4⋊C4—416no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C22⋊C4—48no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—88no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C22⋊C4—88no—
474C42—28no—
475C42—44no—
476C42—416no—
477C42C2×C4⋊C4116yescomplemented
478C23—24no—
479C23—28no—
480C23—28no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—44no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23C22×C4⋊C418yes—
523C23C22×C4⋊C418yes—
524C23C24×C411yescharacteristic
525C23C23×D411yescharacteristic
526C23C23×Q811yescharacteristic
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)14yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23#58514yes—
531C23#58511yescharacteristic
532C23#58511yescharacteristic
533D4—416no—
534D4—416no—
535D4—464no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—432no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—816no—
599C2×C4—816no—
600C2×C4—816no—
601C2×C4—816no—
602C2×C4—816no—
603C2×C4C22×C4⋊C414yes—
604C2×C4C22×C4⋊C418yes—
605C22—28no—
606C22—216no—
607C22—216no—
608C22—28no—
609C22—44no—
610C22—44no—
611C22—48no—
612C22—48no—
613C22—44no—
614C22—44no—
615C22—48no—
616C22—44no—
617C22—44no—
618C22—48no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—48no—
624C22—44no—
625C22—44no—
626C22—48no—
627C22C23×C4⋊C411yescharacteristic
628C22C4×H(5,𝔽2)11yescharacteristic
629C22C4×H(5,𝔽2)14yes—
630C22C4×H(5,𝔽2)11yescharacteristic
631C22#348412yes—
632C22#348411yescharacteristic
633C22#348511yescharacteristic
634C22#348512yes—
635C22#350111yescharacteristic
636C22#350114yes—
637C22#350111yescharacteristic
638C22#353212yes—
639C22#353312yes—
640C22#353712yes—
641C22#353914yes—
642C22#354012yes—
643C22#356214yes—
644C4—24no—
645C4—28no—
646C4—42no—
647C4—48no—
648C4—42no—
649C4—48no—
650C4—48no—
651C4—82no—
652C4—84no—
653C4—82no—
654C4—82no—
655C2—48no—
656C2—48no—
657C2#3424811yescharacteristic
658C2#3424812yes—
659C2#3451112yes—
660C2#3451812yes—
661C2#5200411yescharacteristic
662C2#5200912yes—
663C2#5203512yes—
664C2#5203912yes—
665C2#5205411yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14513C2semidirect
5. #14513C2semidirect
6. #14407C2semidirect
7. #14401C2semidirect
8. #14298C2semidirect
9. #14164C2semidirect
10. #14016C2semidirect
11. #14015C2semidirect
12. #14015C2semidirect
13. #13975C2semidirect
19. #2380C4semidirect
124. #386C2×C4semidirect
125. #386C2×C4semidirect
126. #386C2×C4semidirect
127. #386C2×C4semidirect
132. #383C2×C4semidirect
222. #127C4⋊C4semidirect
359. C2×D4C2×C4⋊C4semidirect
360. C2×D4C2×C4⋊C4semidirect
477. C42C2×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23802
C2×#23662
C2×#23282
#145132
#144072
#144012
#142982
#141642
#140162
#140152
#139752