Finite Group Databasebeta

← #406290

Subgroups — #406290

Order 512.

1572 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2C2×#1894C211yesmaximal, characteristic
3C2×#1873C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6#13951C211yesmaximal, characteristic
7#11267C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#11190C212yesmaximal, complemented
10#11067C212yesmaximal, complemented
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2502—24no—
21#2502—24no—
22C2×#398—21no—
23C2×#398—21no—
24C2×#391C2212yes—
25C2×#389C2212yes—
26C2×#385—22no—
27C2×#384C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1894—24no—
31#1873—24no—
32#1823—24no—
33#1736—24no—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1439—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46#1434—21no—
47C23×Q8C2311yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—24no—
54C22×M16—42no—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8D414yes—
64C2×#117—22no—
65C2×#117—24no—
66C2×#117—42no—
67C2×#117D414yes—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4D414yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4D412yes—
88C22×C42—21no—
89C22×C42C2312yes—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—24no—
95C2×#103—42no—
96C2×#103—42no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100#398—28no—
101#398—28no—
102#391—24no—
103#391—24no—
104#389—24no—
105#389—42no—
106#385—216no—
107#384—24no—
108#384—24no—
109#380—22no—
110#380—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—24no—
122#340—24no—
123#340—22no—
124#340—22no—
125#340—24no—
126#340—24no—
127#340—24no—
128#336—24no—
129#336—42no—
130C25C2×D411yescharacteristic
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8C2×D412yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—81no—
155C23×C4—81no—
156C23×C4—81no—
157C23×C4—81no—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4C2411yescharacteristic
168C2×M16—28no—
169C2×M16—48no—
170C2×M16—48no—
171C2×M16—44no—
172C2×M16—44no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8#4114yes—
190C22⋊Q8—28no—
191C22⋊Q8—28no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194#117—28no—
195#117—44no—
196#117—48no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4#4114yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—81no—
252C2×C22⋊C4—81no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—44no—
261#103—44no—
262#103—48no—
263#103—44no—
264C22⋊C8—44no—
265#95—28no—
266#95—28no—
267#95—22no—
268#95—22no—
269#95—44no—
270#95—42no—
271#95—44no—
272#95—82no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24#12012yes—
282C24#12112yes—
283C24#12412yes—
284C24C22×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
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—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—48no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4#12012yes—
375C22×C4#12114yes—
376C22×C4#12114yes—
377C22×C4#12112yes—
378C22×C4#12314yes—
379C22×C4#12314yes—
380C22×C4#12412yes—
381C22×C4#13714yes—
382C22×C4#13714yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4C2×#4112yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C42−1+412yes—
393M16—416no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—88no—
431C4⋊C4—88no—
432C4⋊C4—88no—
433C4⋊C4#12314yes—
434C22⋊C4—216no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—84no—
438C42—48no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23#39414yes—
466C23C2×#12011yescharacteristic
467C23C2×#12111yescharacteristic
468C23C2×#12411yescharacteristic
469C23#54612yes—
470C23#54712yes—
471C23C2×#13712yes—
472C23C2×#13712yes—
473C23#57611yescharacteristic
474C23#57611yescharacteristic
475C23#57711yescharacteristic
476C23#57711yescharacteristic
477Q8—44no—
478Q8—42no—
479Q8—42no—
480Q8—81no—
481Q8—81no—
482Q8—82no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—28no—
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—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—816no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#39414yes—
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×#12311yescharacteristic
548C2×C4C2×#12311yescharacteristic
549C2×C4C2×#12311yescharacteristic
550C2×C4#52511yescharacteristic
551C2×C4#52912yes—
552C2×C4#53212yes—
553C2×C4#53411yescharacteristic
554C2×C4#53611yescharacteristic
555C2×C4#53712yes—
556C2×C4#53912yes—
557C2×C4#54612yes—
558C2×C4#54712yes—
559C2×C4#55112yes—
560C2×C4#55112yes—
561C2×C4#55212yes—
562C2×C4#55212yes—
563C8—44no—
564C8—81no—
565C8—81no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22C2×#39411yescharacteristic
578C22#241211yescharacteristic
579C22#241311yescharacteristic
580C22#243012yes—
581C22#248812yes—
582C22#300412yes—
583C22#300811yescharacteristic
584C22#300911yescharacteristic
585C22#303911yescharacteristic
586C22#303911yescharacteristic
587C22#304511yescharacteristic
588C22#304511yescharacteristic
589C22#313611yescharacteristic
590C22#313611yescharacteristic
591C22#313912yes—
592C22#331412yes—
593C22#331612yes—
594C22#333112yes—
595C22#333312yes—
596C22#334412yes—
597C22#334612yes—
598C22#335612yes—
599C22#335712yes—
600C4—24no—
601C4—44no—
602C4—44no—
603C4—44no—
604C4—82no—
605C4—82no—
606C4—84no—
607C4—82no—
608C4—162no—
609C4#241912yes—
610C4#248611yescharacteristic
611C4#249812yes—
612C4#253411yescharacteristic
613C4#261811yescharacteristic
614C4#285411yescharacteristic
615C2—44no—
616C2#1492711yescharacteristic
617C2#2350711yescharacteristic
618C2#2350911yescharacteristic
619C2#2454611yescharacteristic
620C2#2454811yescharacteristic
621C2#2459611yescharacteristic
622C2#2459811yescharacteristic
623C2#2555912yes—
624C2#2556112yes—
625C2#2770712yes—
626C2#2772112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11267C2semidirect
8. #11207C2semidirect
9. #11190C2semidirect
10. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#18942
C2×#18732
C2×#18232
C2×#17362
#139512
#112672
#112072
#111902
#110672
#107152