Finite Group Databasebeta

← #790829

Subgroups — #790829

Order 512.

2226 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16932C212yesmaximal
2#16185C212yesmaximal, complemented
3#15931C214yesmaximal, complemented
4#15569C212yesmaximal
5#15152C211yesmaximal, characteristic, complemented
6#15125C212yesmaximal, complemented
7#15023C214yesmaximal, complemented
8#14254C212yesmaximal, complemented
9#14161C211yesmaximal, characteristic, complemented
10#13939C212yesmaximal, complemented
11#13921C212yesmaximal, 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×#388C2214yes—
23C2×#384C2211yescharacteristic
24C2×#383C2214yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33#1446C2214yes—
34#1446C2212yes—
35#1446C2212yes—
36#1445C2212yes—
37#1443C2214yes—
38#1443C2214yes—
39#1442C2214yes—
40#1442C2211yescharacteristic
41#1442C2212yes—
42#1442C2214yes—
43#1441C2211yescharacteristic
44#1441C2212yes—
45#1440C2212yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2212yes—
51#1439C2214yes—
52#1438C2211yescharacteristic
53#1438C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
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⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2314yes—
109C22×C42C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112#400—28no—
113#394—24no—
114#393—48no—
115#388—48no—
116#384—42no—
117#383—48no—
118#380—28no—
119#380—44no—
120#380—44no—
121C4×C4⋊C4—28no—
122C4×C4⋊C4—42no—
123C4×C4⋊C4—42no—
124C4×C22⋊C4—28no—
125C4×C22⋊C4—28no—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2311yescharacteristic
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2311yescharacteristic
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C25C2412yes—
157C22×Q8—24no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8—44no—
161C22×Q8—44no—
162C22×D4—24no—
163C22×D4—22no—
164C22×D4—41no—
165C22×D4—42no—
166C22×D4—41no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4#4118yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2411yescharacteristic
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180C23×C4C2414yes—
181C23×C4C2411yescharacteristic
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C4⋊Q8—416no—
189C4×D4—416no—
190#117—432no—
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—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4#4118yes—
218C2×C4⋊C4#4114yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
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×C42—22no—
247C2×C42—22no—
248C2×C42—28no—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—28no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—28no—
259C2×C42—24no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42#4114yes—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—22no—
272#95—24no—
273#95—28no—
274#95—28no—
275#95—24no—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—41no—
283#95—44no—
284#95—41no—
285#95—41no—
286#95—42no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24C2×#4114yes—
310C24C2511yescharacteristic
311C2×Q8—48no—
312C2×Q8—88no—
313C2×Q8—84no—
314C2×Q8—88no—
315C2×Q8—84no—
316C2×D4—48no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—48no—
320C2×D4—44no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—216no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4C2×#4112yes—
400C22×C4C2×#4118yes—
401C22×C4C2×#4114yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4114yes—
408C22×C4C2×#4114yes—
409C22×C4H(5,𝔽2)14yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C42−1+412yes—
413C22×C42−1+412yes—
414C4⋊C4—216no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C22⋊C4—216no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—416no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C42—28no—
434C42—416no—
435C42—416no—
436C42—48no—
437C42—44no—
438C42—44no—
439C42—84no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
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—
475C23C22×#4111yescharacteristic
476C23C22×#4112yes—
477C23C22×#4112yes—
478C23C22×#4111yescharacteristic
479C23C2×H(5,𝔽2)11yescharacteristic
480C23C2×H(5,𝔽2)11yescharacteristic
481C23C2×H(5,𝔽2)12yes—
482C23C2×2−1+411yescharacteristic
483C23C2×2−1+411yescharacteristic
484C23#58512yes—
485C23#58511yescharacteristic
486Q8—816no—
487D4—416no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—432no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4#52918yes—
570C2×C4#53814yes—
571C2×C4#55114yes—
572C2×C4#55118yes—
573C2×C4#55514yes—
574C2×C4#55614yes—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22C2×#52912yes—
591C22C2×#53811yescharacteristic
592C22C2×#55111yescharacteristic
593C22C2×#55112yes—
594C22C2×#55511yescharacteristic
595C22C2×#55611yescharacteristic
596C22#347512yes—
597C22#348312yes—
598C22#349212yes—
599C22#349312yes—
600C22#349412yes—
601C22#350511yescharacteristic
602C22#350711yescharacteristic
603C22#351111yescharacteristic
604C22#351112yes—
605C22#351312yes—
606C22#351512yes—
607C22#351812yes—
608C22#353811yescharacteristic
609C22#354011yescharacteristic
610C22#354612yes—
611C22#354911yescharacteristic
612C22#356111yescharacteristic
613C4—24no—
614C4—44no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—44no—
619C4—48no—
620C4—42no—
621C4—84no—
622C4—82no—
623C4—81no—
624C4—84no—
625C4—84no—
626C4#257218yes—
627C2—84no—
628C2C2×#257211yescharacteristic
629C2#3577412yes—
630C2#3645912yes—
631C2#3976011yescharacteristic
632C2#4297511yescharacteristic
633C2#4491411yescharacteristic
634C2#4500312yes—
635C2#4541011yescharacteristic
636C2#5920111yescharacteristic
637C2#5927612yes—
638C2#5995211yescharacteristic
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. #16185C2semidirect
3. #15931C2semidirect
5. #15152C2semidirect
6. #15125C2semidirect
7. #15023C2semidirect
8. #14254C2semidirect
9. #14161C2semidirect
10. #13939C2semidirect
11. #13921C2semidirect
12. #13876C2semidirect
15. #13801C2semidirect

Maximal subgroups

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

subgroupindex
#169322
#161852
#159312
#155692
#151522
#151252
#150232
#142542
#141612
#139392
#139212
#138762
#138712
C4×#3942
#138012