Finite Group Databasebeta

← #464344

Subgroups — #464344

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic, complemented
2C2×#2035C212yesmaximal
3C2×#1923C211yesmaximal, characteristic
4#15178C211yesmaximal, characteristic
5#15125C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11122C212yesmaximal, complemented
8#11077C212yesmaximal, complemented
9#11010C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11#10861C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C2×#487—28no—
16C2×#487—24no—
17C22×#106C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#393—21no—
27C2×#391C2212yes—
28C2×#380C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30#2035—28no—
31#1923—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1446—21no—
38#1444—21no—
39#1444—21no—
40#1442—21no—
41#1439—21no—
42#1439—21no—
43#1439—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×Q8C2312yes—
49C24×C4C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8—42no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—42no—
65C2×C22⋊Q8—42no—
66C2×C22⋊Q8—42no—
67C2×C22⋊Q8—42no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#487—232no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—42no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103D418yes—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—42no—
103C2×C4×C8—24no—
104C2×C4×C8D414yes—
105#393—44no—
106#391—24no—
107#391—42no—
108#380—24no—
109#380—42no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123#340—24no—
124#340—42no—
125#340—42no—
126#336—22no—
127#336—22no—
128C25C2×D411yescharacteristic
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—44no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—82no—
147C22×Q8C2×D414yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D414yes—
188C4⋊Q8—28no—
189C4⋊Q8—28no—
190C4⋊Q8—48no—
191C4⋊Q8—48no—
192C4⋊Q8—44no—
193C4⋊Q8—44no—
194C22⋊Q8—48no—
195C22⋊Q8—48no—
196C22⋊Q8—48no—
197C22⋊Q8—48no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4#4114yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4SD1614yes—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42C2×D412yes—
244C2×C42#4112yes—
245#106—48no—
246C4⋊C8—44no—
247#103—216no—
248#103—216no—
249#103—48no—
250#103—48no—
251C22⋊C8—44no—
252C22⋊C8—44no—
253C4×C8—28no—
254#95—22no—
255#95—22no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—82no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—22no—
274C24—22no—
275C24C2×SD1612yes—
276C24C22×D411yescharacteristic
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—48no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—88no—
307C2×Q8—88no—
308C2×Q8—84no—
309C2×Q8—84no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4C2×SD1612yes—
377C22×C4C2×SD1612yes—
378C22×C4#13512yes—
379C22×C4#13712yes—
380C22×C4#13712yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4111yescharacteristic
385C22×C4H(5,𝔽2)11yescharacteristic
386C22×C4H(5,𝔽2)12yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C42−1+411yescharacteristic
390C22×C42−1+411yescharacteristic
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—42no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8#12014yes—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—416no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C22⋊C4—28no—
428C22⋊C4—416no—
429C22⋊C4—44no—
430C22⋊C4—88no—
431C42—22no—
432C42—22no—
433C42—24no—
434C42—41no—
435C42—41no—
436C42—42no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—24no—
441C23—22no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—41no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23C2×#12111yescharacteristic
461C23#53511yescharacteristic
462C23#53611yescharacteristic
463C23D4212yes—
464C23#54612yes—
465C23C22×SD1611yescharacteristic
466C23C2×#13511yescharacteristic
467C23C2×#13711yescharacteristic
468C23C2×#13711yescharacteristic
469C23#57712yes—
470C23#57812yes—
471Q8—44no—
472Q8—44no—
473Q8—44no—
474Q8—44no—
475Q8—816no—
476Q8—82no—
477Q8—82no—
478Q8—82no—
479Q8—82no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4#46514yes—
538C2×C4#47014yes—
539C2×C4C2×#12011yescharacteristic
540C2×C4#53211yescharacteristic
541C2×C4#53511yescharacteristic
542C2×C4#53511yescharacteristic
543C2×C4#53512yes—
544C2×C4#53711yescharacteristic
545C2×C4#53711yescharacteristic
546C2×C4#53712yes—
547C2×C4#53812yes—
548C2×C4#54212yes—
549C2×C4#54211yescharacteristic
550C2×C4#54211yescharacteristic
551C2×C4#54612yes—
552C2×C4#54612yes—
553C2×C4#54712yes—
554C2×C4#55912yes—
555C2×C4H(3,𝔽4)12yes—
556C2×C4#56212yes—
557C8—24no—
558C8—42no—
559C8—81no—
560C8—81no—
561C22—44no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22—82no—
567C22—81no—
568C22—81no—
569C22—82no—
570C22—81no—
571C22—81no—
572C22#240811yescharacteristic
573C22#254411yescharacteristic
574C22#262011yescharacteristic
575C22#267012yes—
576C22#267212yes—
577C22C2×#46511yescharacteristic
578C22C2×#47011yescharacteristic
579C22#305711yescharacteristic
580C22#305911yescharacteristic
581C22#320611yescharacteristic
582C22#321011yescharacteristic
583C22#322512yes—
584C22#322911yescharacteristic
585C22#323111yescharacteristic
586C22#324212yes—
587C22#327012yes—
588C22#327012yes—
589C22D4×SD1612yes—
590C22#328212yes—
591C22#329812yes—
592C22#330212yes—
593C22#332912yes—
594C22#333112yes—
595C4—24no—
596C4—24no—
597C4—42no—
598C4—84no—
599C4—81no—
600C4—84no—
601C4—81no—
602C4—84no—
603C4—162no—
604C4#243011yescharacteristic
605C4#261511yescharacteristic
606C4#266211yescharacteristic
607C4#271012yes—
608C4#271212yes—
609C4#281911yescharacteristic
610C2—82no—
611C2#1650411yescharacteristic
612C2#2426411yescharacteristic
613C2#2427311yescharacteristic
614C2#2860011yescharacteristic
615C2#2860811yescharacteristic
616C2#2984511yescharacteristic
617C2#2984811yescharacteristic
618C2#3052012yes—
619C2#3052212yes—
620C2#3063312yes—
621C2#3063612yes—
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
1. C22×#487C2semidirect
6. #11384C2semidirect
7. #11122C2semidirect
8. #11077C2semidirect
9. #11010C2semidirect
11. #10861C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4872
C2×#20352
C2×#19232
#151782
#151252
#113842
#111222
#110772
#110102
#109702
#108612