Finite Group Databasebeta

← #950066

Subgroups — #950066

Order 512.

2102 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17272C212yesmaximal
2#17136C212yesmaximal
3#16876C214yesmaximal, complemented
4#16794C214yesmaximal, complemented
5#16186C212yesmaximal, complemented
6#16167C212yesmaximal, complemented
7#15135C212yesmaximal, complemented
8#15133C212yesmaximal, complemented
9#15131C212yesmaximal, complemented
10#14266C212yesmaximal, complemented
11#14129C212yesmaximal, complemented
12#13951C212yesmaximal, complemented
13#13871C211yesmaximal, characteristic
14#13871C211yesmaximal, characteristic
15#13847C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#399C2214yes—
20C2×#398C2214yes—
21C2×#396C2214yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#384C2212yes—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2212yes—
34C2×#376C2211yescharacteristic
35#1445C2211yescharacteristic
36#1445C2211yescharacteristic
37#1445C2214yes—
38#1445C2214yes—
39#1444C2214yes—
40#1443C2214yes—
41#1442C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1440C2212yes—
49#1440C2212yes—
50#1439C2212yes—
51#1439C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2211yescharacteristic
59#1438C2211yescharacteristic
60#1438C2212yes—
61#1438C2212yes—
62#1437C2212yes—
63#1437C2212yes—
64#1436C2212yes—
65#1436C2212yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2214yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C2×C4×D4—42no—
79C2×C4×D4—42no—
80C2×C4×D4Q818yes—
81C2×#117—28no—
82C2×#117—42no—
83C2×#117—44no—
84C2×#117—42no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114#399—48no—
115#398—48no—
116#396—48no—
117#389—28no—
118#389—44no—
119#389—44no—
120#389—44no—
121#384—44no—
122#380—28no—
123#380—44no—
124#380—42no—
125#380—42no—
126C4×C4⋊C4—42no—
127C4×C22⋊C4—44no—
128C4×C22⋊C4—44no—
129#376—42no—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2312yes—
157C25C2411yescharacteristic
158C25C2411yescharacteristic
159C22×D4—41no—
160C22×D4—41no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4C2×Q814yes—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4C2×Q814yes—
173C23×C4C2×Q814yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C4×D4—416no—
188#117—416no—
189#117—416no—
190C2×C4⋊C4—24no—
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—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C2×Q814yes—
225C2×C4⋊C4#4118yes—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4C2×Q814yes—
257C2×C22⋊C4C2×Q814yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—28no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—42no—
268C2×C42—44no—
269C2×C42—42no—
270C2×C42C2×Q814yes—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—42no—
281#95—41no—
282#95—41no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95—41no—
288#95—41no—
289#95—44no—
290#95—42no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24C22×Q812yes—
315C24C22×Q812yes—
316C24C2511yescharacteristic
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—48no—
321C2×D4—48no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
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—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4C22×Q812yes—
402C22×C4C22×Q812yes—
403C22×C4C22×Q812yes—
404C22×C4C22×Q812yes—
405C22×C4C22×Q812yes—
406C22×C4C2×#4118yes—
407C22×C4C2×#4114yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C4H(5,𝔽2)14yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C42−1+414yes—
413C4⋊C4—216no—
414C4⋊C4—44no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—88no—
420C4⋊C4—88no—
421C4⋊C4—88no—
422C4⋊C4—88no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C42—28no—
438C42—48no—
439C42—48no—
440C42—48no—
441C42—48no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23C23×Q811yescharacteristic
485C23C22×#4112yes—
486C23C2×H(5,𝔽2)11yescharacteristic
487C23C2×H(5,𝔽2)12yes—
488C23C2×H(5,𝔽2)11yescharacteristic
489C23C2×H(5,𝔽2)12yes—
490C23C2×2−1+412yes—
491C23#58512yes—
492C23#58512yes—
493D4—416no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—416no—
536C2×C4—416no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4#54018yes—
573C2×C4#54314yes—
574C2×C4#55114yes—
575C2×C4#55418yes—
576C2×C4#56014yes—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—42no—
585C22—44no—
586C22—44no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22C2×#54012yes—
596C22C2×#54311yescharacteristic
597C22#345811yescharacteristic
598C22#345811yescharacteristic
599C22C2×#55111yescharacteristic
600C22C2×#55412yes—
601C22#350312yes—
602C22#351412yes—
603C22#351512yes—
604C22#351712yes—
605C22#351812yes—
606C22C2×#56011yescharacteristic
607C22#353612yes—
608C22#353912yes—
609C22#353912yes—
610C22#354112yes—
611C22#354112yes—
612C22#354212yes—
613C22#354212yes—
614C22#354911yescharacteristic
615C22#355011yescharacteristic
616C4—48no—
617C4—42no—
618C4—44no—
619C4—44no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—84no—
624C4—82no—
625C4—82no—
626C4—84no—
627C4—84no—
628C4#274818yes—
629C2—82no—
630C2—82no—
631C2#3730611yescharacteristic
632C2#3748111yescharacteristic
633C2#3810412yes—
634C2#3810512yes—
635C2#4271412yes—
636C2#4273612yes—
637C2#4540612yes—
638C2C2×#274811yescharacteristic
639C2#5309812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16876C2semidirect
4. #16794C2semidirect
5. #16186C2semidirect
6. #16167C2semidirect
7. #15135C2semidirect
8. #15133C2semidirect
9. #15131C2semidirect
10. #14266C2semidirect
11. #14129C2semidirect
12. #13951C2semidirect

Maximal subgroups

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

subgroupindex
#172722
#171362
#168762
#167942
#161862
#161672
#151352
#151332
#151312
#142662
#141292
#139512
#138712
#138472