Finite Group Databasebeta

← #460221

Subgroups — #460221

Order 512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2421C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#1803C211yesmaximal, characteristic, complemented
4#15135C211yesmaximal, characteristic
5#11227C211yesmaximal, characteristic, complemented
6#11208C212yesmaximal, complemented
7#11199C212yesmaximal
8#11067C212yesmaximal, complemented
9#11042C211yesmaximal, characteristic
10#10970C212yesmaximal
11#10789C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2212yes—
18#2421—24no—
19#2421—24no—
20C2×#400—21no—
21C2×#399—21no—
22C2×#397—21no—
23C2×#391C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#388—21no—
27C2×#387—21no—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#2100—24no—
34#1803—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1444—21no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C23×Q8C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—42no—
55C2×C4×Q8—22no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4D411yescharacteristic
74C22×C22⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—22no—
79C2×C4⋊C8—42no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89#400—28no—
90#399—44no—
91#397—28no—
92#391—22no—
93#391—22no—
94#389—22no—
95#389—24no—
96#389—42no—
97#389D414yes—
98#388—28no—
99#387—28no—
100#384—22no—
101#384—22no—
102#382—22no—
103#382—22no—
104#380—22no—
105#380—22no—
106#380—24no—
107#380—42no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#340—42no—
127#340—42no—
128#336—22no—
129#336—42no—
130C25C2×D411yescharacteristic
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8C2×D412yes—
140C22×Q8#4114yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22⋊Q8—48no—
178C22⋊Q8—44no—
179C22⋊Q8—44no—
180C4×Q8—28no—
181C4×Q8—44no—
182C4×Q8—48no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
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—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4#4114yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—82no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—44no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C4⋊C8—44no—
249#103—48no—
250#103—48no—
251C22⋊C8—48no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—22no—
259#95—22no—
260#95—44no—
261#95—44no—
262#95—42no—
263#95—42no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24AGL(1,ℤ/8)12yes—
275C24C22×D411yescharacteristic
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8#12314yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
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#12012yes—
370C22×C4#12012yes—
371C22×C4#13512yes—
372C22×C4#13712yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C4H(5,𝔽2)12yes—
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C42−1+411yescharacteristic
381C22×C42−1+411yescharacteristic
382C22×C42−1+411yescharacteristic
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—48no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—88no—
421C22⋊C4—44no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—44no—
425C22⋊C4—88no—
426C42—22no—
427C42—22no—
428C42—24no—
429C42—48no—
430C42—41no—
431C42—41no—
432C42—42no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23C2×#12011yescharacteristic
457C23#53611yescharacteristic
458C23#53711yescharacteristic
459C23#54612yes—
460C23#54612yes—
461C23C2×#13511yescharacteristic
462C23C2×AGL(1,ℤ/8)11yescharacteristic
463C23C2×#13711yescharacteristic
464C23#57511yescharacteristic
465C23#57712yes—
466C23#57812yes—
467Q8—24no—
468Q8—24no—
469Q8—42no—
470Q8—44no—
471Q8—44no—
472Q8—42no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4#45214yes—
540C2×C4C2×#12311yescharacteristic
541C2×C4#53211yescharacteristic
542C2×C4#53411yescharacteristic
543C2×C4#53511yescharacteristic
544C2×C4#53611yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53711yescharacteristic
547C2×C4#53711yescharacteristic
548C2×C4#53912yes—
549C2×C4#53912yes—
550C2×C4#53912yes—
551C2×C4#54212yes—
552C2×C4#54612yes—
553C2×C4#55012yes—
554C2×C4#55112yes—
555C2×C4#55212yes—
556C2×C4#56012yes—
557C2×C4#56212yes—
558C8—82no—
559C8—82no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—82no—
567C22—82no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22#240511yescharacteristic
572C22#243111yescharacteristic
573C22#261611yescharacteristic
574C22#268712yes—
575C22#269312yes—
576C22C2×#45211yescharacteristic
577C22#300611yescharacteristic
578C22#301011yescharacteristic
579C22#301111yescharacteristic
580C22#323311yescharacteristic
581C22#323811yescharacteristic
582C22#324212yes—
583C22#324711yescharacteristic
584C22#325211yescharacteristic
585C22#325812yes—
586C22#330112yes—
587C22#330212yes—
588C22#330812yes—
589C22#331012yes—
590C22#331612yes—
591C22#331712yes—
592C22#332712yes—
593C22#332812yes—
594C4—24no—
595C4—44no—
596C4—42no—
597C4—84no—
598C4—81no—
599C4—82no—
600C4—81no—
601C4—81no—
602C4—82no—
603C4—82no—
604C4—162no—
605C4#249811yescharacteristic
606C4#262311yescharacteristic
607C4#266311yescharacteristic
608C4#269412yes—
609C4#270512yes—
610C4#281311yescharacteristic
611C2—82no—
612C2#1636011yescharacteristic
613C2#2403611yescharacteristic
614C2#2405111yescharacteristic
615C2#2557011yescharacteristic
616C2#2558311yescharacteristic
617C2#2969211yescharacteristic
618C2#2969911yescharacteristic
619C2#3136512yes—
620C2#3137812yes—
621C2#3155912yes—
622C2#3156212yes—
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
2. C2×#2100C2semidirect
3. C2×#1803C2semidirect
5. #11227C2semidirect
6. #11208C2semidirect
8. #11067C2semidirect
11. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24212
C2×#21002
C2×#18032
#151352
#112272
#112082
#111992
#110672
#110422
#109702
#107892