Finite Group Databasebeta

← #460168

Subgroups — #460168

Order 512.

1306 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×#2083C211yesmaximal, characteristic, complemented
3C2×#1803C211yesmaximal, characteristic, complemented
4#15150C211yesmaximal, characteristic
5#11268C212yesmaximal, complemented
6#11227C211yesmaximal, characteristic, complemented
7#11124C212yesmaximal, complemented
8#11096C212yesmaximal
9#11042C211yesmaximal, characteristic
10#10970C212yesmaximal
11#10789C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19#2421—24no—
20#2421—24no—
21C2×#400—21no—
22C2×#397—21no—
23C2×#397—21no—
24C2×#397—21no—
25C2×#391C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#388—21no—
28C2×#387—21no—
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#2083—24no—
35#1803—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
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—21no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—42no—
79C2×#107—42no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—42no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91#400—28no—
92#397—28no—
93#397—44no—
94#397—44no—
95#391—22no—
96#391—22no—
97#389—22no—
98#389D414yes—
99#388—28no—
100#387—28no—
101#384—22no—
102#384—22no—
103#382—22no—
104#382—22no—
105#380—22no—
106#380—22no—
107#380—42no—
108#380—42no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#340—42no—
129#336—24no—
130#336—22no—
131C25C2×D411yescharacteristic
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8C2×D412yes—
141C22×Q8#4114yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
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×C8#4114yes—
178C22⋊Q8—48no—
179C22⋊Q8—44no—
180C22⋊Q8—44no—
181C4×Q8—28no—
182C4×Q8—44no—
183C4×Q8—48no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
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—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—44no—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247#107—48no—
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—42no—
262#95—44no—
263#95—44no—
264#95—42no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—22no—
274C24—22no—
275C24AGL(1,ℤ/8)12yes—
276C24C22×D411yescharacteristic
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8#12314yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
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—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
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—42no—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4#13512yes—
373C22×C4AGL(1,ℤ/8)12yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4H(5,𝔽2)12yes—
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C42−1+412yes—
383C22×C42−1+411yescharacteristic
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
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—
397C2×C8—42no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—48no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—84no—
420C4⋊C4—84no—
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—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23C2×#12011yescharacteristic
457C23#53511yescharacteristic
458C23#53711yescharacteristic
459C23#54612yes—
460C23#54712yes—
461C23C2×#13511yescharacteristic
462C23C2×AGL(1,ℤ/8)11yescharacteristic
463C23C2×AGL(1,ℤ/8)11yescharacteristic
464C23#57511yescharacteristic
465C23#57812yes—
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—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
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—82no—
538C2×C4—82no—
539C2×C4#44914yes—
540C2×C4C2×#12311yescharacteristic
541C2×C4#53211yescharacteristic
542C2×C4#53411yescharacteristic
543C2×C4#53511yescharacteristic
544C2×C4#53511yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53711yescharacteristic
547C2×C4#53711yescharacteristic
548C2×C4#53812yes—
549C2×C4#53912yes—
550C2×C4#53912yes—
551C2×C4#54012yes—
552C2×C4#54612yes—
553C2×C4#54812yes—
554C2×C4#55112yes—
555C2×C4#55212yes—
556C2×C4#56212yes—
557C2×C4#56312yes—
558C8—44no—
559C8—82no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—82no—
567C22—82no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22#240511yescharacteristic
572C22#243211yescharacteristic
573C22#261511yescharacteristic
574C22#269512yes—
575C22#270612yes—
576C22C2×#44911yescharacteristic
577C22#300611yescharacteristic
578C22#301111yescharacteristic
579C22#301111yescharacteristic
580C22#321211yescharacteristic
581C22#321711yescharacteristic
582C22#322312yes—
583C22#324711yescharacteristic
584C22#325111yescharacteristic
585C22#326112yes—
586C22#330012yes—
587C22#330812yes—
588C22#330812yes—
589C22#332712yes—
590C22#334212yes—
591C22#334312yes—
592C22#334912yes—
593C22#335312yes—
594C4—24no—
595C4—44no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—82no—
600C4—82no—
601C4—81no—
602C4—82no—
603C4—162no—
604C4—162no—
605C4#249811yescharacteristic
606C4#262311yescharacteristic
607C4#266111yescharacteristic
608C4#269612yes—
609C4#270712yes—
610C4#281411yescharacteristic
611C2—82no—
612C2#1635911yescharacteristic
613C2#2403511yescharacteristic
614C2#2405111yescharacteristic
615C2#2570511yescharacteristic
616C2#2571611yescharacteristic
617C2#2966211yescharacteristic
618C2#2967411yescharacteristic
619C2#3161212yes—
620C2#3162412yes—
621C2#3194012yes—
622C2#3196012yes—
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×#2083C2semidirect
3. C2×#1803C2semidirect
5. #11268C2semidirect
6. #11227C2semidirect
7. #11124C2semidirect
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×#20832
C2×#18032
#151502
#112682
#112272
#111242
#110962
#110422
#109702
#107892