Finite Group Databasebeta

← #4458110

Subgroups — #4458110

Order 512.

2090 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17003C212yesmaximal, complemented
2#16883C214yesmaximal, complemented
3#16877C212yesmaximal, complemented
4#16358C212yesmaximal
5#16343C214yesmaximal
6#16339C212yesmaximal
7#15232C212yesmaximal, complemented
8#15152C212yesmaximal, complemented
9#15145C212yesmaximal, complemented
10#15133C212yesmaximal, complemented
11#14340C211yesmaximal, characteristic
12#14129C212yesmaximal
13#13880C212yesmaximal
14#13852C211yesmaximal, characteristic
15#13795C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#400C2214yes—
20C2×#398C2214yes—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#385C2214yes—
26C2×#385C2214yes—
27C2×#384C2211yescharacteristic
28C2×#384C2212yes—
29C2×#384C2212yes—
30C2×#382C2211yescharacteristic
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35#1445C2214yes—
36#1445C2214yes—
37#1444C2214yes—
38#1444C2214yes—
39#1444C2214yes—
40#1444C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2212yes—
46#1441C2212yes—
47#1441C2211yescharacteristic
48#1441C2211yescharacteristic
49#1441C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2212yes—
55#1439C2212yes—
56#1439C2211yescharacteristic
57#1439C2212yes—
58#1439C2211yescharacteristic
59#1439C2214yes—
60#1439C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C4×#95C2212yes—
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C2×#117—24no—
78C2×#117—42no—
79C2×#117—42no—
80C2×#117—42no—
81C2×#117—42no—
82C2×#117Q818yes—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115#400—48no—
116#398—48no—
117#398—48no—
118#398—48no—
119#389—44no—
120#389—44no—
121#385—48no—
122#385—48no—
123#384—42no—
124#384—44no—
125#384—44no—
126#382—42no—
127#380—28no—
128C4×C4⋊C4—42no—
129C4×C4⋊C4—42no—
130C4×C22⋊C4—28no—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C25C2411yescharacteristic
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C2×Q814yes—
170C23×C4#4114yes—
171C23×C4#4114yes—
172C23×C4C2414yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186#117—416no—
187#117—416no—
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—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—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—416no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C2×Q814yes—
238C2×C4⋊C4C2×Q814yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4C2×Q814yes—
256C2×C22⋊C4C2×Q814yes—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42C2×Q814yes—
271C2×C42C2×Q814yes—
272#95—24no—
273#95—24no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—24no—
278#95—24no—
279#95—28no—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—41no—
284#95—41no—
285#95—42no—
286#95—42no—
287#95—42no—
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—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300C24—22no—
301C24—22no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—22no—
307C24—21no—
308C24C22×Q812yes—
309C24C2×#4112yes—
310C24C2×#4112yes—
311C24C2511yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
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—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
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—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—48no—
391C22×C4—88no—
392C22×C4—84no—
393C22×C4—88no—
394C22×C4—84no—
395C22×C4#125116yes—
396C22×C4C22×Q812yes—
397C22×C4C22×Q812yes—
398C22×C4C22×Q812yes—
399C22×C4C22×Q812yes—
400C22×C4C22×Q812yes—
401C22×C4C22×Q812yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4H(5,𝔽2)14yes—
407C22×C4H(5,𝔽2)14yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C42−1+414yes—
412C22×C42−1+414yes—
413C22×C42−1+412yes—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—88no—
425C4⋊C4—88no—
426C4⋊C4—88no—
427C4⋊C4—88no—
428C4⋊C4—84no—
429C4⋊C4—88no—
430C4⋊C4—84no—
431C4⋊C4—88no—
432C22⋊C4—216no—
433C22⋊C4—216no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—88no—
438C42—28no—
439C42—28no—
440C42—48no—
441C42—48no—
442C42—48no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23C2×#12514yes—
466C23C23×Q811yescharacteristic
467C23C22×#4111yescharacteristic
468C23C22×#4111yescharacteristic
469C23C2×H(5,𝔽2)11yescharacteristic
470C23C2×H(5,𝔽2)11yescharacteristic
471C23C2×H(5,𝔽2)11yescharacteristic
472C23C2×H(5,𝔽2)12yes—
473C23C2×H(5,𝔽2)12yes—
474C23C2×2−1+411yescharacteristic
475C23C2×2−1+412yes—
476C23C2×2−1+412yes—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—232no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—816no—
561C2×C4C2×#12514yes—
562C2×C4C2×#12514yes—
563C2×C4#53814yes—
564C2×C4#53814yes—
565C2×C4#53914yes—
566C2×C4#53914yes—
567C2×C4#54314yes—
568C2×C4#54414yes—
569C22—24no—
570C22—28no—
571C22—42no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22C22×#12511yescharacteristic
581C22C2×#53811yescharacteristic
582C22C2×#53811yescharacteristic
583C22C2×#53911yescharacteristic
584C22C2×#53911yescharacteristic
585C22C2×#54311yescharacteristic
586C22C2×#54411yescharacteristic
587C22#349212yes—
588C22#349212yes—
589C22#350312yes—
590C22#350312yes—
591C22#350812yes—
592C22#350912yes—
593C22#353612yes—
594C22#353711yescharacteristic
595C22#353811yescharacteristic
596C22#354312yes—
597C22#354412yes—
598C22#354812yes—
599C22#354911yescharacteristic
600C22#355011yescharacteristic
601C22#355212yes—
602C22#355212yes—
603C4—24no—
604C4—44no—
605C4—44no—
606C4—44no—
607C4—44no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—84no—
613C4—84no—
614C4—84no—
615C4—84no—
616C4#264618yes—
617C2—44no—
618C2C2×#264611yescharacteristic
619C2#3555512yes—
620C2#3730611yescharacteristic
621C2#3730711yescharacteristic
622C2#3731212yes—
623C2#4250211yescharacteristic
624C2#4250412yes—
625C2#4250611yescharacteristic
626C2#4293912yes—
627C2#4294111yescharacteristic
628C2#4297711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17003C2semidirect
2. #16883C2semidirect
3. #16877C2semidirect
7. #15232C2semidirect
8. #15152C2semidirect
9. #15145C2semidirect
10. #15133C2semidirect

Maximal subgroups

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

subgroupindex
#170032
#168832
#168772
#163582
#163432
#163392
#152322
#151522
#151452
#151332
#143402
#141292
#138802
#138522
#137952