Finite Group Databasebeta

← #924624

Subgroups — #924624

Order 512.

2748 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17149C214yesmaximal, complemented
2#16690C214yesmaximal, complemented
3#16382C214yesmaximal, complemented
4#15369C214yesmaximal, complemented
5#15229C212yesmaximal, complemented
6#15216C211yesmaximal, characteristic, complemented
7C4⋊#401C212yesmaximal, complemented
8#15124C211yesmaximal, characteristic
9#14257C212yesmaximal, complemented
10#14025C212yesmaximal, complemented
11#13970C212yesmaximal, complemented
12#13916C212yesmaximal, complemented
13#13853C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×#121C2212yes—
17C22×C4×D4C2212yes—
18C22×#117C2211yescharacteristic
19C2×#401C2214yes—
20C2×#392C2214yes—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2214yes—
28C2×#381C2214yes—
29C2×#380C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#1445C2212yes—
33#1445C2212yes—
34#1443C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1442C2214yes—
38#1440C2212yes—
39#1440C2212yes—
40#1440C2214yes—
41#1440C2214yes—
42#1440C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1438C2214yes—
51#1438C2214yes—
52#1438C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C4×#95C2212yes—
66C4×#95C2212yes—
67C23×D4C2312yes—
68C23×D4C2314yes—
69C23×D4C2314yes—
70C24×C4C2311yescharacteristic
71C24×C4C2314yes—
72C2×#127—28no—
73C2×#127—42no—
74C2×#125—42no—
75C2×#125—44no—
76C2×#121—28no—
77C2×#121—28no—
78C2×#121—44no—
79C2×C4×D4—28no—
80C2×C4×D4—44no—
81C2×C4×D4—44no—
82C2×#117—24no—
83C2×#117—42no—
84C2×#117—42no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#401—48no—
111#392—48no—
112#390—44no—
113#390—44no—
114#390—44no—
115#390—44no—
116#386—48no—
117#386—48no—
118#386—48no—
119#381—48no—
120#380—44no—
121C4×C22⋊C4—42no—
122C4×C22⋊C4—42no—
123C2×#95C2314yes—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C25C2414yes—
147C25C2411yescharacteristic
148C22×D4—28no—
149C22×D4—28no—
150C22×D4—28no—
151C22×D4—28no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—28no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4#4118yes—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—28no—
174C23×C4—28no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4#4114yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2411yescharacteristic
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192#127—416no—
193#125—416no—
194#121—432no—
195C4×D4—432no—
196#117—416no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
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×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—22no—
268#95—24no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—28no—
273#95—28no—
274#95—24no—
275#95—42no—
276#95—42no—
277#95—44no—
278#95—42no—
279#95—42no—
280#95—42no—
281#95—42no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291C24—24no—
292C24—22no—
293C24—24no—
294C24—22no—
295C24—24no—
296C24—22no—
297C24—21no—
298C24—21no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—22no—
312C24—24no—
313C24C2×#4118yes—
314C24C2×#4112yes—
315C24C2511yescharacteristic
316C2×D4—216no—
317C2×D4—216no—
318C2×D4—216no—
319C2×D4—416no—
320C2×D4—416no—
321C2×D4—416no—
322C2×D4—416no—
323C2×D4—416no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—88no—
329C2×D4—88no—
330C2×D4—88no—
331C2×D4—88no—
332C2×D4—88no—
333C2×D4—88no—
334C2×D4—88no—
335C2×D4—88no—
336C2×D4—88no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—28no—
354C22×C4—216no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
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—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—416no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—416no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4C2×#4112yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4112yes—
417C22×C4H(5,𝔽2)14yes—
418C22×C4H(5,𝔽2)14yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)14yes—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—416no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C22⋊C4—44no—
429C22⋊C4—416no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—416no—
433C22⋊C4—44no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C42—48no—
444C42—48no—
445C42—44no—
446C42—44no—
447C23—22no—
448C23—28no—
449C23—24no—
450C23—28no—
451C23—24no—
452C23—28no—
453C23—22no—
454C23—22no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—41no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—44no—
471C23—42no—
472C23—44no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—41no—
477C23—44no—
478C23—41no—
479C23—44no—
480C23—41no—
481C23—44no—
482C23—42no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23C22×#4111yescharacteristic
504C23C22×#4112yes—
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×H(5,𝔽2)12yes—
508C23C2×H(5,𝔽2)12yes—
509C23C2×H(5,𝔽2)12yes—
510C23#58512yes—
511C23#58512yes—
512D4—432no—
513D4—832no—
514D4—832no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—416no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—832no—
585C2×C4#53814yes—
586C2×C4#53818yes—
587C2×C4#55114yes—
588C2×C4#55118yes—
589C2×C4#56014yes—
590C22—48no—
591C22—48no—
592C22—48no—
593C22—44no—
594C22—48no—
595C22—48no—
596C22—42no—
597C22—44no—
598C22—48no—
599C22—42no—
600C22—42no—
601C22—48no—
602C22—82no—
603C22—82no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22—84no—
608C22—84no—
609C22—82no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22C2×#53812yes—
615C22C2×#53811yescharacteristic
616C22#346012yes—
617C22C2×#55111yescharacteristic
618C22C2×#55112yes—
619C22#350811yescharacteristic
620C22#351211yescharacteristic
621C22#351312yes—
622C22#351312yes—
623C22#351912yes—
624C22#352112yes—
625C22C2×#56011yescharacteristic
626C22#353711yescharacteristic
627C22#353911yescharacteristic
628C22#354212yes—
629C22#354312yes—
630C22#354612yes—
631C22#354612yes—
632C22#354612yes—
633C22#354612yes—
634C22#356212yes—
635C4—44no—
636C4—44no—
637C4—42no—
638C4—82no—
639C4—84no—
640C4—82no—
641C4—84no—
642C4—82no—
643C4—84no—
644C4—82no—
645C4#276018yes—
646C2—88no—
647C2—82no—
648C2#3731711yescharacteristic
649C2#3751511yescharacteristic
650C2#3825212yes—
651C2#3906612yes—
652C2#4260212yes—
653C2#4261612yes—
654C2#4506112yes—
655C2C2×#276011yescharacteristic
656C2#5305912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15124C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17149C2semidirect
2. #16690C2semidirect
3. #16382C2semidirect
4. #15369C2semidirect
5. #15229C2semidirect
6. #15216C2semidirect
7. C4⋊#401C2semidirect
9. #14257C2semidirect
10. #14025C2semidirect
11. #13970C2semidirect
12. #13916C2semidirect
13. #13853C2semidirect

Maximal subgroups

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

subgroupindex
#171492
#166902
#163822
#153692
#152292
#152162
C4⋊#4012
#151242
#142572
#140252
#139702
#139162
#138532