Finite Group Databasebeta

← #1040870

Subgroups — #1040870

Order 512.

2038 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17159C212yesmaximal
2#16932C212yesmaximal
3#16930C214yesmaximal, complemented
4#16243C212yesmaximal, complemented
5#15802C214yesmaximal, complemented
6#15239C212yesmaximal, complemented
7#15152C211yesmaximal, characteristic, complemented
8#15132C212yesmaximal, complemented
9#14604C212yesmaximal, complemented
10#13993C212yesmaximal, complemented
11#13911C211yesmaximal, characteristic, complemented
12#13879C212yesmaximal, complemented
13#13879C212yesmaximal, complemented
14#13871C212yesmaximal
15C4×#394C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C22×#117C2212yes—
18C2×#400C2212yes—
19C2×#397C2214yes—
20C2×#394C2211yescharacteristic
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2212yes—
25C2×#382C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33#1445C2212yes—
34#1443C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1443C2214yes—
38#1442C2214yes—
39#1442C2212yes—
40#1442C2212yes—
41#1442C2214yes—
42#1442C2211yescharacteristic
43#1442C2212yes—
44#1442C2214yes—
45#1441C2211yescharacteristic
46#1441C2212yes—
47#1440C2214yes—
48#1440C2212yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2211yescharacteristic
57#1438C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4×#95C2211yescharacteristic
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C2×C4×D4—28no—
78C2×C4×D4—42no—
79C2×#117—44no—
80C2×#117—44no—
81C2×#117—44no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2314yes—
109#400—28no—
110#397—48no—
111#394—24no—
112#389—44no—
113#384—42no—
114#384—42no—
115#382—44no—
116#382—44no—
117#380—28no—
118#380—44no—
119#380—44no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—42no—
122C4×C22⋊C4—28no—
123C4×C22⋊C4—28no—
124C2×#95C2311yescharacteristic
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2311yescharacteristic
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C25C2412yes—
155C22×D4—24no—
156C22×D4—22no—
157C22×D4—41no—
158C22×D4—42no—
159C22×D4—41no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4#4118yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2414yes—
172C23×C4C2411yescharacteristic
173C23×C4C2412yes—
174C23×C4C2411yescharacteristic
175C23×C4C2411yescharacteristic
176C23×C4C2412yes—
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C4×D4—416no—
182#117—432no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4#4114yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
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×C42—22no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—44no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42#4114yes—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—28no—
265#95—22no—
266#95—24no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—41no—
275#95—42no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—41no—
281#95—41no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24C2×#4114yes—
304C24C2511yescharacteristic
305C2×D4—48no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—48no—
309C2×D4—44no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—216no—
337C22×C4—216no—
338C22×C4—216no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4C2×#4112yes—
391C22×C4C2×#4112yes—
392C22×C4C2×#4114yes—
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4114yes—
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)14yes—
401C22×C42−1+412yes—
402C22×C42−1+412yes—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—88no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—416no—
420C22⋊C4—48no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C42—28no—
425C42—48no—
426C42—416no—
427C42—44no—
428C42—44no—
429C42—48no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
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—
465C23C22×#4112yes—
466C23C22×#4111yescharacteristic
467C23C22×#4111yescharacteristic
468C23C2×H(5,𝔽2)11yescharacteristic
469C23C2×H(5,𝔽2)12yes—
470C23C2×H(5,𝔽2)11yescharacteristic
471C23C2×2−1+411yescharacteristic
472C23C2×2−1+411yescharacteristic
473C23#58512yes—
474C23#58511yescharacteristic
475C23#58512yes—
476D4—416no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—432no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4#53914yes—
559C2×C4#53918yes—
560C2×C4#55114yes—
561C2×C4#55318yes—
562C2×C4#55614yes—
563C2×C4H(3,𝔽4)14yes—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22C2×#53912yes—
580C22C2×#53911yescharacteristic
581C22C2×#55111yescharacteristic
582C22C2×#55312yes—
583C22C2×#55611yescharacteristic
584C22#347512yes—
585C22#349312yes—
586C22#349312yes—
587C22#349412yes—
588C22#350511yescharacteristic
589C22#350611yescharacteristic
590C22#350712yes—
591C22#351111yescharacteristic
592C22#351712yes—
593C22C2×H(3,𝔽4)11yescharacteristic
594C22#353611yescharacteristic
595C22#354011yescharacteristic
596C22#354612yes—
597C22#355111yescharacteristic
598C22#355412yes—
599C22#356212yes—
600C22#356311yescharacteristic
601C22#356512yes—
602C4—24no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—44no—
608C4—84no—
609C4—84no—
610C4—81no—
611C4—84no—
612C4—82no—
613C4—84no—
614C4—82no—
615C4#278218yes—
616C2—84no—
617C2#3975911yescharacteristic
618C2#3993712yes—
619C2#4265612yes—
620C2#4268611yescharacteristic
621C2#4490712yes—
622C2#4511511yescharacteristic
623C2#4608411yescharacteristic
624C2C2×#278211yescharacteristic
625C2#5308011yescharacteristic
626C2#5920111yescharacteristic
627C2#5924312yes—
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. #16930C2semidirect
4. #16243C2semidirect
5. #15802C2semidirect
6. #15239C2semidirect
7. #15152C2semidirect
8. #15132C2semidirect
9. #14604C2semidirect
10. #13993C2semidirect
11. #13911C2semidirect
12. #13879C2semidirect
13. #13879C2semidirect

Maximal subgroups

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

subgroupindex
#171592
#169322
#169302
#162432
#158022
#152392
#151522
#151322
#146042
#139932
#139112
#138792
#138712
C4×#3942