Finite Group Databasebeta

← #366820

Subgroups — #366820

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C2×#1894C211yesmaximal, characteristic
3C2×#1873C211yesmaximal, characteristic
4C2×#1851C211yesmaximal, characteristic
5C2×#1824C211yesmaximal, characteristic
6C2×#1736C211yesmaximal, characteristic
7#13801C211yesmaximal, characteristic
8#11358C212yesmaximal, complemented
9#11077C212yesmaximal, complemented
10#10932C212yesmaximal, complemented
11#10850C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21#2503—24no—
22#2503C418yescomplemented
23C2×#400—21no—
24C2×#400—21no—
25C2×#391C2212yes—
26C2×#382C2212yes—
27C2×#381—22no—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1894—24no—
32#1873—24no—
33#1851—24no—
34#1824—24no—
35#1736—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1437—22no—
45C4⋊#95C2212yes—
46#1434—21no—
47C4×#95—21no—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—24no—
55C22×M16—42no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2×C414yes—
65C2×#117—22no—
66C2×#117—24no—
67C2×#117—42no—
68C2×#117D414yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2×C414yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D412yes—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42C2312yes—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—42no—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102#400—28no—
103#400—28no—
104#391—24no—
105#391C2×C418yes—
106#382—24no—
107#382C2×C418yes—
108#381—216no—
109#380—22no—
110#380D414yes—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—42no—
113C4×C22⋊C4C2×C414yes—
114C4×C22⋊C4D414yes—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D411yescharacteristic
124C2×#95D411yescharacteristic
125#340—24no—
126#340—24no—
127#340—22no—
128#340—22no—
129#340—24no—
130#340—24no—
131#340—24no—
132#336—24no—
133C25C2×D411yescharacteristic
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8C22×C412yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4C22×C412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C2×M16—28no—
170C2×M16—48no—
171C2×M16—48no—
172C2×M16—44no—
173C2×M16—44no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8#4114yes—
191C22⋊Q8—28no—
192C22⋊Q8—28no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195#117—28no—
196#117—44no—
197#117—48no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4#4114yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—81no—
243C2×C22⋊C4—81no—
244C2×C22⋊C4C22⋊C418yes—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—44no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42—82no—
254C2×C42C22×C414yes—
255C2×C42C2×D414yes—
256C4⋊C8—48no—
257#103—48no—
258#103—44no—
259#103—44no—
260C22⋊C8—44no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—24no—
266#95—44no—
267#95—44no—
268#95—82no—
269#95C22⋊C414yes—
270#95C22⋊C414yes—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24C2×C22⋊C412yes—
280C24#12112yes—
281C24#12112yes—
282C24C22×D411yescharacteristic
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
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—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—416no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
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—44no—
355C22×C4—44no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—48no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—88no—
368C22×C4C2×C22⋊C414yes—
369C22×C4C2×C22⋊C414yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C4×D414yes—
372C22×C4C4×D414yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4#12414yes—
376C22×C4#12714yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C42−1+412yes—
387M16—416no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4#12314yes—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—84no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—24no—
433C42—48no—
434C42—88no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23#39014yes—
458C23C22×C22⋊C411yescharacteristic
459C23C2×#12111yescharacteristic
460C23C2×#12111yescharacteristic
461C23#54612yes—
462C23#54712yes—
463C23#57512yes—
464C23#57512yes—
465C23#57711yescharacteristic
466C23#57711yescharacteristic
467C23#57811yescharacteristic
468C23#57811yescharacteristic
469Q8—42no—
470Q8—44no—
471Q8—42no—
472Q8—82no—
473Q8—81no—
474Q8—81no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—816no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—84no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4#39014yes—
535C2×C4C2×C4×D411yescharacteristic
536C2×C4C2×C4×D411yescharacteristic
537C2×C4C4×#4112yes—
538C2×C4#52012yes—
539C2×C4C2×#12311yescharacteristic
540C2×C4#52511yescharacteristic
541C2×C4C2×#12411yescharacteristic
542C2×C4#52912yes—
543C2×C4C2×#12711yescharacteristic
544C2×C4#53411yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53912yes—
547C2×C4#54612yes—
548C2×C4#54612yes—
549C2×C4#54712yes—
550C2×C4#54712yes—
551C2×C4#55112yes—
552C2×C4#55212yes—
553C8—44no—
554C8—81no—
555C8—81no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22—44no—
565C22—42no—
566C22—44no—
567C22C2×#39011yescharacteristic
568C22#233912yes—
569C22#241211yescharacteristic
570C22#241311yescharacteristic
571C22#242512yes—
572C22#289612yes—
573C22#289811yescharacteristic
574C22#289811yescharacteristic
575C22#304011yescharacteristic
576C22#304011yescharacteristic
577C22#304012yes—
578C22#304211yescharacteristic
579C22#304411yescharacteristic
580C22#304611yescharacteristic
581C22#304611yescharacteristic
582C22#331512yes—
583C22#331712yes—
584C22#332312yes—
585C22#332712yes—
586C22#334812yes—
587C22#334912yes—
588C22#335112yes—
589C22#335212yes—
590C4—24no—
591C4—44no—
592C4—44no—
593C4—44no—
594C4—84no—
595C4—82no—
596C4—84no—
597C4—82no—
598C4—82no—
599C4C4×#12311yescharacteristic
600C4#232211yescharacteristic
601C4#235912yes—
602C4#248812yes—
603C4#250111yescharacteristic
604C4#253911yescharacteristic
605C2—44no—
606C2#1399611yescharacteristic
607C2#2126111yescharacteristic
608C2#2126211yescharacteristic
609C2#2249612yes—
610C2#2249912yes—
611C2#2454311yescharacteristic
612C2#2454511yescharacteristic
613C2#2459311yescharacteristic
614C2#2459511yescharacteristic
615C2#2522512yes—
616C2#2522912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
8. #11358C2semidirect
9. #11077C2semidirect
10. #10932C2semidirect
11. #10850C2semidirect
22. #2503C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#18942
C2×#18732
C2×#18512
C2×#18242
C2×#17362
#138012
#113582
#110772
#109322
#108502