Finite Group Databasebeta

← #923339

Subgroups — #923339

Order 512.

2544 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17255C214yesmaximal, complemented
2#17049C214yesmaximal, complemented
3#16750C214yesmaximal, complemented
4#15978C214yesmaximal, complemented
5#15216C211yesmaximal, characteristic, complemented
6#15216C212yesmaximal, complemented
7#15195C211yesmaximal, characteristic, complemented
8#15183C212yesmaximal, complemented
9#14337C212yesmaximal
10#13994C212yesmaximal, complemented
11#13974C212yesmaximal, complemented
12#13876C212yesmaximal, complemented
13#13851C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C2×#389C2212yes—
17C2×#388C2214yes—
18C2×#388C2214yes—
19C2×#388C2214yes—
20C2×#384C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#1446C2214yes—
33#1446C2212yes—
34#1445C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1440C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2212yes—
56#1438C2212yes—
57#1438C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66#1434C2214yes—
67C22×#41—28no—
68C22×#41—24no—
69C22×#41—42no—
70C22×#41—44no—
71C22×#41—42no—
72C23×Q8C2311yescharacteristic
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C2×#125—44no—
78C2×#125—42no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2311yescharacteristic
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2314yes—
103#389—44no—
104#388—48no—
105#388—48no—
106#388—48no—
107#384—42no—
108#380—28no—
109#380—28no—
110#380—44no—
111#380—44no—
112#380—44no—
113#380—44no—
114C4×C4⋊C4—44no—
115C4×C22⋊C4—28no—
116C4×C22⋊C4—42no—
117C4×C22⋊C4—44no—
118C4×C22⋊C4—42no—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2312yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C25C2412yes—
144C25C2411yescharacteristic
145C2×#41—416no—
146C2×#41—432no—
147C2×#41—432no—
148C2×#41—416no—
149C2×#41—416no—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×Q8—41no—
154C22×Q8—41no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—22no—
159C22×D4—24no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4—42no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4#4118yes—
178C23×C4#4114yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190#125—416no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—28no—
257C2×C42—44no—
258C2×C42—44no—
259#95—24no—
260#95—24no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—21no—
290C24—22no—
291C24—22no—
292C24—21no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24C2×#4112yes—
306C24C2×#4114yes—
307C24C2511yescharacteristic
308#41—4128no—
309C2×Q8—48no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—44no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—44no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4—44no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—44no—
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—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
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—216no—
349C22×C4—28no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
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—24no—
394C22×C4—24no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—44no—
401C22×C4—48no—
402C22×C4C2×#4112yes—
403C22×C4C2×#4114yes—
404C22×C4C2×#4114yes—
405C22×C4C2×#4112yes—
406C22×C4H(5,𝔽2)14yes—
407C22×C4H(5,𝔽2)12yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C42−1+414yes—
411C4⋊C4—416no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C22⋊C4—216no—
419C22⋊C4—44no—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—416no—
424C22⋊C4—416no—
425C22⋊C4—44no—
426C22⋊C4—48no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C42—48no—
432C42—44no—
433C42—44no—
434C42—88no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
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—42no—
491C23C22×#4111yescharacteristic
492C23C22×#4112yes—
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×H(5,𝔽2)12yes—
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×H(5,𝔽2)12yes—
497C23C2×2−1+412yes—
498C23#58512yes—
499C23#58512yes—
500Q8—416no—
501D4—416no—
502D4—432no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—432no—
530C2×C4—416no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4#53818yes—
574C2×C4#53814yes—
575C2×C4#55114yes—
576C2×C4#55118yes—
577C2×C4H(3,𝔽4)14yes—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22C2×#53811yescharacteristic
603C22C2×#53812yes—
604C22#346012yes—
605C22C2×#55112yes—
606C22C2×#55111yescharacteristic
607C22#349212yes—
608C22#349512yes—
609C22#350612yes—
610C22#350911yescharacteristic
611C22#351211yescharacteristic
612C22#351212yes—
613C22C2×H(3,𝔽4)11yescharacteristic
614C22#353711yescharacteristic
615C22#353911yescharacteristic
616C22#354012yes—
617C22#354112yes—
618C22#354212yes—
619C22#354612yes—
620C22#354712yes—
621C22#354712yes—
622C22#354812yes—
623C4—44no—
624C4—42no—
625C4—44no—
626C4—44no—
627C4—82no—
628C4—84no—
629C4—84no—
630C4—84no—
631C4—84no—
632C4—82no—
633C4#273618yes—
634C2—82no—
635C2—84no—
636C2#3739311yescharacteristic
637C2#3751511yescharacteristic
638C2#3797112yes—
639C2#3956512yes—
640C2#4244812yes—
641C2#4299312yes—
642C2#4525012yes—
643C2C2×#273611yescharacteristic
644C2#5304612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17255C2semidirect
2. #17049C2semidirect
3. #16750C2semidirect
4. #15978C2semidirect
5. #15216C2semidirect
6. #15216C2semidirect
7. #15195C2semidirect
8. #15183C2semidirect
10. #13994C2semidirect
11. #13974C2semidirect
12. #13876C2semidirect

Maximal subgroups

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

subgroupindex
#172552
#170492
#167502
#159782
#152162
#151952
#151832
#143372
#139942
#139742
#138762
#138512