Finite Group Databasebeta

← #852751

Subgroups — #852751

Order 512.

2974 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16729C214yesmaximal, complemented
2#15369C214yesmaximal, complemented
3#15369C214yesmaximal, complemented
4#15229C212yesmaximal, complemented
5#15216C211yesmaximal, characteristic, complemented
6#15198C212yesmaximal, complemented
7#15124C211yesmaximal, characteristic
8#14698C212yesmaximal, complemented
9#14375C214yesmaximal, complemented
10#14025C212yesmaximal, complemented
11#13970C212yesmaximal, complemented
12#13916C212yesmaximal, complemented
13C4×#381C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×#121C2212yes—
17C22×C4×D4C2212yes—
18C23×C42C2211yescharacteristic
19C2×#392C2214yes—
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#386C2214yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#381C2212yes—
30C2×#381C2214yes—
31C2×#380C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1443C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1443C2214yes—
38#1442C2214yes—
39#1440C2214yes—
40#1440C2214yes—
41#1440C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1438C2214yes—
48#1438C2214yes—
49#1438C2214yes—
50#1438C2214yes—
51#1437C2214yes—
52#1437C2214yes—
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2212yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60#1434C2212yes—
61#1434C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C4×#95C2212yes—
66C4×#95C2212yes—
67C23×D4C2312yes—
68C23×D4C2314yes—
69C23×D4C2314yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2314yes—
74C2×#127—28no—
75C2×#127—42no—
76C2×#125—42no—
77C2×#125—44no—
78C2×#121—28no—
79C2×#121—28no—
80C2×#121—44no—
81C2×C4×D4—28no—
82C2×C4×D4—44no—
83C2×C4×D4—44no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C42—24no—
102C22×C42—42no—
103C22×C42—42no—
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109#392—48no—
110#390—44no—
111#390—44no—
112#390—44no—
113#390—44no—
114#386—216no—
115#386—48no—
116#386—48no—
117#386—48no—
118#386—48no—
119#381—28no—
120#381—48no—
121#380—44no—
122C4×C22⋊C4—42no—
123C4×C22⋊C4—42no—
124C2×#95C2312yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2311yescharacteristic
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C25C2414yes—
149C25C2411yescharacteristic
150C22×D4—28no—
151C22×D4—28no—
152C22×D4—28no—
153C22×D4—28no—
154C22×D4—28no—
155C22×D4—28no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—28no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4#4118yes—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—28no—
180C23×C4—24no—
181C23×C4—28no—
182C23×C4—42no—
183C23×C4—41no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4#4114yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204#127—416no—
205#125—416no—
206#121—432no—
207C4×D4—432no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
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×C22⋊C4—44no—
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—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—416no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—28no—
277#95—24no—
278#95—28no—
279#95—24no—
280#95—42no—
281#95—42no—
282#95—44no—
283#95—41no—
284#95—42no—
285#95—41no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95#4118yes—
297C24—22no—
298C24—22no—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24C2×#4118yes—
319C24C2×#4112yes—
320C24H(5,𝔽2)14yes—
321C24C2511yescharacteristic
322C2×D4—216no—
323C2×D4—216no—
324C2×D4—216no—
325C2×D4—416no—
326C2×D4—416no—
327C2×D4—416no—
328C2×D4—416no—
329C2×D4—416no—
330C2×D4—416no—
331C2×D4—416no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—88no—
337C2×D4—88no—
338C2×D4—88no—
339C2×D4—88no—
340C2×D4—88no—
341C2×D4—88no—
342C2×D4—88no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
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—28no—
373C22×C4—216no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—416no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—416no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4C2×#4114yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)14yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C4H(5,𝔽2)14yes—
438C4⋊C4—416no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—416no—
445C22⋊C4—416no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C42—48no—
456C42—48no—
457C42—48no—
458C42—44no—
459C42—44no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—42no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23#54014yes—
516C23C22×#4111yescharacteristic
517C23C22×#4112yes—
518C23C22×#4112yes—
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×H(5,𝔽2)12yes—
523C23C2×H(5,𝔽2)12yes—
524C23#58512yes—
525D4—432no—
526D4—832no—
527D4—832no—
528C2×C4—24no—
529C2×C4—22no—
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—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—416no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—832no—
600C2×C4#53814yes—
601C2×C4#53818yes—
602C2×C4#55114yes—
603C2×C4#55118yes—
604C2×C4#56014yes—
605C22—24no—
606C22—28no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—48no—
615C22—48no—
616C22—48no—
617C22—48no—
618C22—48no—
619C22—48no—
620C22—48no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22#344411yescharacteristic
630C22C2×#53811yescharacteristic
631C22C2×#53812yes—
632C22C2×#54011yescharacteristic
633C22C2×#55112yes—
634C22C2×#55111yescharacteristic
635C22#347512yes—
636C22#348112yes—
637C22#349412yes—
638C22#350812yes—
639C22#351312yes—
640C22#351712yes—
641C22#351912yes—
642C22#351912yes—
643C22#351912yes—
644C22C2×#56011yescharacteristic
645C22#353711yescharacteristic
646C22#353911yescharacteristic
647C22#354212yes—
648C22#354312yes—
649C22#355812yes—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—42no—
654C4—44no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4—84no—
660C4#276018yes—
661C2—44no—
662C2—88no—
663C2#3508111yescharacteristic
664C2#3636612yes—
665C2#3728311yescharacteristic
666C2#3805612yes—
667C2#3899712yes—
668C2#4005312yes—
669C2#4560412yes—
670C2C2×#276011yescharacteristic
671C2#5938212yes—
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. #16729C2semidirect
2. #15369C2semidirect
3. #15369C2semidirect
4. #15229C2semidirect
5. #15216C2semidirect
6. #15198C2semidirect
8. #14698C2semidirect
9. #14375C2semidirect
10. #14025C2semidirect
11. #13970C2semidirect
12. #13916C2semidirect
13. C4×#381C2semidirect

Maximal subgroups

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

subgroupindex
#167292
#153692
#152292
#152162
#151982
#151242
#146982
#143752
#140252
#139702
#139162
C4×#3812