Finite Group Databasebeta

← #396506

Subgroups — #396506

Order 512.

2216 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×#2095C212yesmaximal, complemented
3C2×#1852C211yesmaximal, characteristic
4C2×#1808C211yesmaximal, characteristic
5C2×#1787C211yesmaximal, characteristic
6#13922C211yesmaximal, characteristic
7#13880C211yesmaximal, characteristic
8#11124C212yesmaximal, complemented
9#11124C212yesmaximal, complemented
10#11050C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2212yes—
17C22×#107C2211yescharacteristic
18C22×#103—28no—
19C22×#103—24no—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#385—22no—
28C2×#384C2212yes—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#2095—28no—
33#1852—24no—
34#1808—24no—
35#1787—24no—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1446—22no—
42#1444—22no—
43C4⋊#95C2212yes—
44C22⋊#95—22no—
45C23×Q8—21no—
46C23×Q8—22no—
47C23×Q8—24no—
48C23×Q8—24no—
49C23×Q8—22no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—42no—
54C23×Q8—42no—
55C23×Q8Q814yes—
56C23×Q8C2312yes—
57C24×C4—21no—
58C24×C4—21no—
59C24×C4C2311yescharacteristic
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—24no—
63C23×C8—42no—
64C23×C8D414yes—
65C23×C8C2312yes—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D412yes—
82C22×C42C2312yes—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—42no—
89C2×#107—42no—
90C2×#103—232no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—42no—
101C2×#103—42no—
102C2×#103D418yes—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8D414yes—
106#391—24no—
107#391—24no—
108#391—42no—
109#391—42no—
110#385—48no—
111#384—24no—
112#384—42no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C2×#95—24no—
120C2×#95—24no—
121C2×#95—24no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95D412yes—
125#340—22no—
126#340—22no—
127#340—24no—
128#336—24no—
129#336—42no—
130C25C2×D411yescharacteristic
131C22×Q8—24no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—28no—
136C22×Q8—216no—
137C22×Q8—28no—
138C22×Q8—216no—
139C22×Q8—28no—
140C22×Q8—28no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—44no—
150C22×Q8—44no—
151C22×Q8—48no—
152C22×Q8—48no—
153C22×Q8—48no—
154C22×Q8—48no—
155C22×Q8—48no—
156C22×Q8—48no—
157C22×Q8—48no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8—48no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—42no—
165C22×Q8C2×D414yes—
166C22×Q8C2×Q814yes—
167C22×Q8#4114yes—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—44no—
186C23×C4—82no—
187C23×C4C2×D414yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2×Q812yes—
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4#4112yes—
195C23×C4C2411yescharacteristic
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—28no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—28no—
211C22×C8—44no—
212C22×C8—48no—
213C22×C8—48no—
214C22×C8C2×D412yes—
215C22×C8C2×D414yes—
216C22×C8C2×D414yes—
217C22×C8C2×D412yes—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—84no—
247C2×C4⋊C4C2×D414yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4Q1618yes—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261#107—48no—
262#107—44no—
263#107—44no—
264#103—216no—
265#103—216no—
266#103—48no—
267#103—48no—
268C22⋊C8—28no—
269#95—24no—
270#95—48no—
271#95—48no—
272#95—48no—
273#95—42no—
274#95—42no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24C22⋊Q812yes—
280C24#12312yes—
281C24#12712yes—
282C24C2×Q1614yes—
283C24AGL(1,ℤ/8)14yes—
284C24C22×D411yescharacteristic
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—232no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—416no—
302C2×Q8—432no—
303C2×Q8—416no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—416no—
307C2×Q8—416no—
308C2×Q8—432no—
309C2×Q8—416no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—48no—
323C2×Q8C22⋊Q818yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—416no—
351C22×C4—416no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—816no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4C22⋊Q814yes—
384C22×C4C22⋊Q814yes—
385C22×C4C22⋊Q812yes—
386C22×C4#12314yes—
387C22×C4#12314yes—
388C22×C4#12312yes—
389C22×C4#12712yes—
390C22×C4C2×Q1618yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×Q811yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4C2×#4111yescharacteristic
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—416no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8#12014yes—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—48no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—816no—
437C22⋊C4—216no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C23—24no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—22no—
450C23—24no—
451C23—28no—
452C23—28no—
453C23—28no—
454C23—22no—
455C23—28no—
456C23#39914yes—
457C23#47718yes—
458C23#48418yes—
459C23#49414yes—
460C23#49614yes—
461C23C2×C22⋊Q811yescharacteristic
462C23C2×#12311yescharacteristic
463C23C2×#12711yescharacteristic
464C23D4212yes—
465C23#54612yes—
466C23C22×Q1612yes—
467C23C2×AGL(1,ℤ/8)12yes—
468C23#57712yes—
469C23#57812yes—
470Q8—28no—
471Q8—28no—
472Q8—432no—
473Q8—44no—
474Q8—44no—
475Q8—44no—
476Q8—44no—
477Q8—44no—
478Q8—44no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—432no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—816no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—84no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4#39914yes—
527C2×C4C2×#12011yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×C22⋊Q812yes—
530C2×C4C2×C22⋊Q811yescharacteristic
531C2×C4C2×#12311yescharacteristic
532C2×C4C2×#12311yescharacteristic
533C2×C4#52512yes—
534C2×C4#52511yescharacteristic
535C2×C4#53411yescharacteristic
536C2×C4#53511yescharacteristic
537C2×C4#53812yes—
538C2×C4#54312yes—
539C2×C4D4212yes—
540C2×C4#54612yes—
541C2×C4#54612yes—
542C2×C4#54612yes—
543C2×C4#55112yes—
544C2×C4#55112yes—
545C8—24no—
546C8—42no—
547C8—44no—
548C22—216no—
549C22—216no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22#209518yescomplemented
557C22C2×#39911yescharacteristic
558C22#241812yes—
559C22#243311yescharacteristic
560C22#243712yes—
561C22#253911yescharacteristic
562C22C2×#47712yes—
563C22#308212yes—
564C22C2×#48412yes—
565C22#309712yes—
566C22C2×#49411yescharacteristic
567C22C2×#49611yescharacteristic
568C22#315411yescharacteristic
569C22#315611yescharacteristic
570C22#326514yes—
571C22D4×Q1614yes—
572C22#329914yes—
573C22#330814yes—
574C4—24no—
575C4—48no—
576C4—48no—
577C4—84no—
578C4—161no—
579C4—161no—
580C4—161no—
581C4—161no—
582C4#240511yescharacteristic
583C4#241812yes—
584C4#243712yes—
585C4#248211yescharacteristic
586C4#262611yescharacteristic
587C4#284811yescharacteristic
588C2—28no—
589C2#1473411yescharacteristic
590C2C2×#209511yescharacteristic
591C2#2373611yescharacteristic
592C2#2486714yes—
593C2#2578011yescharacteristic
594C2#2578311yescharacteristic
595C2#2596314yes—
596C2#2833311yescharacteristic
597C2#2834611yescharacteristic
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×#2095C2semidirect
8. #11124C2semidirect
9. #11124C2semidirect
10. #11050C2semidirect
556. C22#2095semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×#20952
C2×#18522
C2×#18082
C2×#17872
#139222
#138802
#111242
#110502
#107142