Finite Group Databasebeta

← #770344

Subgroups — #770344

Order 512.

2594 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic
2C2×#2329C211yesmaximal, characteristic
3#15746C214yesmaximal, complemented
4#15682C214yesmaximal, complemented
5#15438C214yesmaximal, complemented
6#15333C214yesmaximal, complemented
7#15092C212yesmaximal
8#15092C212yesmaximal
9#14159C211yesmaximal, characteristic
10#14149C212yesmaximal
11#14139C211yesmaximal, characteristic
12#14130C212yesmaximal
13#13965C211yesmaximal, characteristic
14#13880C211yesmaximal, characteristic
15#13778C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#398—28no—
22C2×#398C2214yes—
23C2×#398C2214yes—
24#2329—28no—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#389C2214yes—
29C2×#387C2212yes—
30C2×#387C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#384C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1446C2212yes—
46#1445C2212yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2212yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8Q818yes—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120#398—216no—
121#398—216no—
122#391—28no—
123#391—44no—
124#389—28no—
125#389—48no—
126#387—28no—
127#387—28no—
128#385—28no—
129#385—44no—
130#385—44no—
131#385—44no—
132#385—44no—
133#384—44no—
134#380—24no—
135#380Q8116yescomplemented
136C4×C22⋊C4—28no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—24no—
139C2×#95—24no—
140C2×#95—24no—
141C2×#95D418yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C25C2411yescharacteristic
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C2×Q814yes—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×Q814yes—
202C23×C4C2×Q818yes—
203C23×C4#4114yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C22⋊Q8—232no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—416no—
245C2×C4⋊C4—416no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×Q814yes—
263C2×C4⋊C4C2×Q818yes—
264C2×C4⋊C4C2×Q8116yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4C2×Q818yes—
275C2×C22⋊C4C2×Q818yes—
276C2×C22⋊C4C2×Q818yes—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42—42no—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95C2×D418yes—
312#95C2×D418yes—
313#95C2×Q8116yes—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C22×D412yes—
319C24C22×Q812yes—
320C24C22×Q814yes—
321C24C2×#4112yes—
322C24H(5,𝔽2)14yes—
323C24C2511yescharacteristic
324C2×Q8—28no—
325C2×Q8—48no—
326C2×Q8—82no—
327C2×Q8—82no—
328C2×Q8—84no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—416no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—416no—
399C22×C4—44no—
400C22×C4—416no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4C22⋊Q8116yes—
415C22×C4C4⋊Q8116yes—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D414yes—
420C22×C4C22×Q818yes—
421C22×C4C22×Q818yes—
422C22×C4C22×Q814yes—
423C22×C4C22×Q814yes—
424C22×C4C22×Q812yes—
425C22×C4C22×Q812yes—
426C22×C4C22×Q814yes—
427C22×C4C22×Q814yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)14yes—
431C4⋊C4—28no—
432C4⋊C4—216no—
433C4⋊C4—416no—
434C4⋊C4—416no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—416no—
438C4⋊C4—416no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—88no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C4⋊C4—88no—
447C22⋊C4—216no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—48no—
451C42—48no—
452C42—84no—
453C42—84no—
454C42—84no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—28no—
461C23—24no—
462C23—28no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23C2×C22⋊Q814yes—
468C23C2×C4⋊Q814yes—
469C23#53514yes—
470C23#55418yes—
471C23Q8218yes—
472C23C23×D411yescharacteristic
473C23C23×Q811yescharacteristic
474C23C23×Q812yes—
475C23C22×#4111yescharacteristic
476C23C2×H(5,𝔽2)12yes—
477C23C2×H(5,𝔽2)12yes—
478C23#58512yes—
479C23#58512yes—
480C23#58512yes—
481Q8—88no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—416no—
510C2×C4—416no—
511C2×C4—416no—
512C2×C4—416no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
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—88no—
557C2×C4—88no—
558C2×C4—88no—
559C2×C4C2×C22⋊Q814yes—
560C2×C4C2×C22⋊Q814yes—
561C2×C4C2×C4⋊Q818yes—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—216no—
569C22—216no—
570C22#256718yes—
571C22C22×C22⋊Q811yescharacteristic
572C22C22×C4⋊Q811yescharacteristic
573C22C2×#53511yescharacteristic
574C22C2×#55412yes—
575C22C2×Q8212yes—
576C22#348411yescharacteristic
577C22#348511yescharacteristic
578C22#348511yescharacteristic
579C22#348711yescharacteristic
580C22#350112yes—
581C22#350114yes—
582C22#350412yes—
583C22#350414yes—
584C22#350512yes—
585C22#351212yes—
586C22#353314yes—
587C22#353414yes—
588C4—24no—
589C4—24no—
590C4—28no—
591C4—44no—
592C4—44no—
593C4—44no—
594C4—44no—
595C4—48no—
596C4—48no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—82no—
602C2—28no—
603C2C2×#256711yescharacteristic
604C2#3534911yescharacteristic
605C2#3539111yescharacteristic
606C2#3563911yescharacteristic
607C2#3564011yescharacteristic
608C2#3695611yescharacteristic
609C2#3718511yescharacteristic
610C2#4312414yes—
611C2#4312614yes—
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
3. #15746C2semidirect
4. #15682C2semidirect
5. #15438C2semidirect
6. #15333C2semidirect
135. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#3982
C2×#23292
#157462
#156822
#154382
#153332
#150922
#141592
#141492
#141392
#141302
#139652
#138802
#137782