Finite Group Databasebeta

← #790826

Subgroups — #790826

Order 512.

2226 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16652C212yesmaximal
2#16177C212yesmaximal, complemented
3#16146C214yesmaximal, complemented
4#16047C212yesmaximal
5#15152C211yesmaximal, characteristic, complemented
6#15125C212yesmaximal, complemented
7#15004C214yesmaximal, complemented
8#14162C212yesmaximal, complemented
9#14155C211yesmaximal, characteristic, complemented
10#13939C212yesmaximal, complemented
11#13936C212yesmaximal, complemented
12#13876C212yesmaximal, complemented
13#13871C212yesmaximal
14C4×#394C211yesmaximal, characteristic
15#13801C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2212yes—
19C2×#400C2212yes—
20C2×#394C2211yescharacteristic
21C2×#393C2214yes—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#388C2214yes—
25C2×#384C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2212yes—
34#1446C2214yes—
35#1445C2212yes—
36#1445C2212yes—
37#1445C2212yes—
38#1443C2214yes—
39#1443C2214yes—
40#1442C2212yes—
41#1442C2214yes—
42#1442C2211yescharacteristic
43#1442C2214yes—
44#1441C2212yes—
45#1441C2211yescharacteristic
46#1440C2214yes—
47#1440C2212yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2212yes—
53#1438C2214yes—
54#1438C2211yescharacteristic
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73C4×#95C2211yescharacteristic
74C4×#95C2214yes—
75C23×Q8C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—42no—
81C2×C4×D4—28no—
82C2×C4×D4—42no—
83C2×#117—28no—
84C2×#117—28no—
85C2×#117—44no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2314yes—
112#400—44no—
113#394—24no—
114#393—48no—
115#391—28no—
116#391—44no—
117#388—48no—
118#384—42no—
119#380—28no—
120#380—44no—
121#380—44no—
122C4×C4⋊C4—28no—
123C4×C4⋊C4—42no—
124C4×C4⋊C4—42no—
125C4×C22⋊C4—28no—
126C4×C22⋊C4—28no—
127C2×#95C2314yes—
128C2×#95C2311yescharacteristic
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2311yescharacteristic
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C25C2412yes—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×D4—24no—
166C22×D4—22no—
167C22×D4—41no—
168C22×D4—42no—
169C22×D4—41no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4#4118yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2411yescharacteristic
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C4⋊Q8—416no—
192C4×D4—416no—
193#117—432no—
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—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4#4118yes—
221C2×C4⋊C4#4114yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
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—22no—
250C2×C42—22no—
251C2×C42—28no—
252C2×C42—28no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—28no—
261C2×C42—24no—
262C2×C42—44no—
263C2×C42—42no—
264C2×C42—44no—
265C2×C42—42no—
266C2×C42—44no—
267C2×C42#4114yes—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—22no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—24no—
281#95—41no—
282#95—44no—
283#95—44no—
284#95—42no—
285#95—44no—
286#95—44no—
287#95—41no—
288#95—41no—
289#95—42no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—42no—
297#95—44no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24C2×#4114yes—
313C24C2511yescharacteristic
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—84no—
318C2×Q8—84no—
319C2×Q8—84no—
320C2×Q8—84no—
321C2×D4—48no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—48no—
325C2×D4—44no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—21no—
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—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4118yes—
407C22×C4C2×#4114yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4114yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C42−1+414yes—
419C4⋊C4—216no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C22⋊C4—216no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—416no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C42—28no—
439C42—416no—
440C42—48no—
441C42—44no—
442C42—44no—
443C42—84no—
444C42—88no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
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—
480C23C22×#4111yescharacteristic
481C23C22×#4112yes—
482C23C22×#4112yes—
483C23C22×#4111yescharacteristic
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×H(5,𝔽2)11yescharacteristic
486C23C2×H(5,𝔽2)11yescharacteristic
487C23C2×H(5,𝔽2)11yescharacteristic
488C23C2×2−1+412yes—
489C23#58512yes—
490C23#58511yescharacteristic
491Q8—816no—
492D4—416no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—432no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4#52918yes—
575C2×C4#53814yes—
576C2×C4#55114yes—
577C2×C4#55118yes—
578C2×C4#55514yes—
579C2×C4#55514yes—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22C2×#52912yes—
596C22C2×#53811yescharacteristic
597C22C2×#55111yescharacteristic
598C22C2×#55112yes—
599C22C2×#55511yescharacteristic
600C22C2×#55511yescharacteristic
601C22#347512yes—
602C22#348212yes—
603C22#349212yes—
604C22#349312yes—
605C22#349512yes—
606C22#350511yescharacteristic
607C22#350611yescharacteristic
608C22#350811yescharacteristic
609C22#351112yes—
610C22#351412yes—
611C22#351612yes—
612C22#351712yes—
613C22#353711yescharacteristic
614C22#354011yescharacteristic
615C22#355011yescharacteristic
616C22#355412yes—
617C22#355811yescharacteristic
618C4—24no—
619C4—42no—
620C4—44no—
621C4—42no—
622C4—42no—
623C4—48no—
624C4—42no—
625C4—44no—
626C4—81no—
627C4—84no—
628C4—84no—
629C4—84no—
630C4—82no—
631C4#257218yes—
632C2—84no—
633C2C2×#257211yescharacteristic
634C2#3577012yes—
635C2#3631512yes—
636C2#3950311yescharacteristic
637C2#4293711yescharacteristic
638C2#4525912yes—
639C2#4533011yescharacteristic
640C2#4540811yescharacteristic
641C2#5920211yescharacteristic
642C2#5927512yes—
643C2#5995111yescharacteristic
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
2. #16177C2semidirect
3. #16146C2semidirect
5. #15152C2semidirect
6. #15125C2semidirect
7. #15004C2semidirect
8. #14162C2semidirect
9. #14155C2semidirect
10. #13939C2semidirect
11. #13936C2semidirect
12. #13876C2semidirect
15. #13801C2semidirect

Maximal subgroups

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

subgroupindex
#166522
#161772
#161462
#160472
#151522
#151252
#150042
#141622
#141552
#139392
#139362
#138762
#138712
C4×#3942
#138012