Finite Group Databasebeta

← #396507

Subgroups — #396507

Order 512.

2144 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×#2097C212yesmaximal, complemented
3C2×#1853C211yesmaximal, characteristic
4C2×#1808C211yesmaximal, characteristic
5C2×#1787C211yesmaximal, characteristic
6#13922C211yesmaximal, characteristic
7#13881C211yesmaximal, characteristic
8#11124C212yesmaximal, complemented
9#11122C212yesmaximal, complemented
10#11052C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#106C2212yes—
18C22×#103—28no—
19C22×#103—24no—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#382C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#2097—28no—
32#1853—24no—
33#1808—24no—
34#1787—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1446—22no—
41#1442—22no—
42C4⋊#95C2212yes—
43C22⋊#95—22no—
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—42no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D412yes—
81C22×C42C2312yes—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—42no—
89C2×#103—232no—
90C2×#103—24no—
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—42no—
100C2×#103—42no—
101C2×#103D418yes—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8D414yes—
105#391—24no—
106#391—24no—
107#391—42no—
108#391—42no—
109#382—24no—
110#382—42no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C2×#95—24no—
118C2×#95—24no—
119C2×#95—24no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—24no—
123C2×#95D412yes—
124#340—22no—
125#340—22no—
126#340—24no—
127#336—24no—
128#336—42no—
129C25C2×D411yescharacteristic
130C22×Q8—24no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—28no—
135C22×Q8—216no—
136C22×Q8—28no—
137C22×Q8—216no—
138C22×Q8—28no—
139C22×Q8—28no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—44no—
149C22×Q8—44no—
150C22×Q8—48no—
151C22×Q8—48no—
152C22×Q8—48no—
153C22×Q8—48no—
154C22×Q8—48no—
155C22×Q8—48no—
156C22×Q8—48no—
157C22×Q8—44no—
158C22×Q8—44no—
159C22×Q8—48no—
160C22×Q8—42no—
161C22×Q8—44no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8C2×D414yes—
165C22×Q8C2×Q814yes—
166C22×Q8#4114yes—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4—82no—
186C23×C4C2×D414yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×Q812yes—
191C23×C4#4112yes—
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4C2411yescharacteristic
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—28no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—28no—
210C22×C8—44no—
211C22×C8—48no—
212C22×C8—48no—
213C22×C8C2×D412yes—
214C22×C8C2×D414yes—
215C22×C8C2×D414yes—
216C22×C8C2×D412yes—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
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—22no—
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—48no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C2×D414yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257#107—44no—
258#107—44no—
259#106—48no—
260#103—216no—
261#103—216no—
262#103—48no—
263#103—48no—
264C22⋊C8—28no—
265#95—24no—
266#95—48no—
267#95—48no—
268#95—48no—
269#95—42no—
270#95—42no—
271#95—84no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24C22⋊Q812yes—
277C24#12312yes—
278C24#12712yes—
279C24#13514yes—
280C24#13714yes—
281C24C22×D411yescharacteristic
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—232no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
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—416no—
299C2×Q8—432no—
300C2×Q8—416no—
301C2×Q8—48no—
302C2×Q8—48no—
303C2×Q8—416no—
304C2×Q8—416no—
305C2×Q8—432no—
306C2×Q8—416no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
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—48no—
320C2×Q8C22⋊Q818yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
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—416no—
349C22×C4—416no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
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—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—816no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4C22⋊Q814yes—
382C22×C4C22⋊Q814yes—
383C22×C4C22⋊Q812yes—
384C22×C4#12314yes—
385C22×C4#12314yes—
386C22×C4#12312yes—
387C22×C4#12712yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×Q811yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4C2×#4111yescharacteristic
396C22×C4H(5,𝔽2)12yes—
397C22×C4H(5,𝔽2)12yes—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—22no—
403C2×C8—24no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—416no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8#12014yes—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—48no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C22⋊C4—48no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—22no—
446C23—24no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—22no—
451C23—28no—
452C23#39914yes—
453C23#47918yes—
454C23#48518yes—
455C23#49514yes—
456C23#49714yes—
457C23C2×C22⋊Q811yescharacteristic
458C23C2×#12311yescharacteristic
459C23C2×#12711yescharacteristic
460C23D4212yes—
461C23#54612yes—
462C23C2×#13512yes—
463C23C2×#13712yes—
464C23#57712yes—
465C23#57812yes—
466Q8—28no—
467Q8—28no—
468Q8—432no—
469Q8—44no—
470Q8—44no—
471Q8—44no—
472Q8—44no—
473Q8—44no—
474Q8—44no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—432no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—816no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4#39914yes—
523C2×C4C2×#12011yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4C2×C22⋊Q812yes—
526C2×C4C2×C22⋊Q811yescharacteristic
527C2×C4C2×#12311yescharacteristic
528C2×C4C2×#12311yescharacteristic
529C2×C4#52512yes—
530C2×C4#52511yescharacteristic
531C2×C4#53411yescharacteristic
532C2×C4#53511yescharacteristic
533C2×C4#53812yes—
534C2×C4#54312yes—
535C2×C4D4212yes—
536C2×C4#54612yes—
537C2×C4#54612yes—
538C2×C4#54612yes—
539C2×C4#55112yes—
540C2×C4#55112yes—
541C8—24no—
542C8—42no—
543C8—44no—
544C22—216no—
545C22—216no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22#209718yescomplemented
553C22C2×#39911yescharacteristic
554C22#241812yes—
555C22#243311yescharacteristic
556C22#243712yes—
557C22#253911yescharacteristic
558C22C2×#47912yes—
559C22#308212yes—
560C22C2×#48512yes—
561C22#309712yes—
562C22C2×#49511yescharacteristic
563C22C2×#49711yescharacteristic
564C22#315411yescharacteristic
565C22#315611yescharacteristic
566C22#326614yes—
567C22#328514yes—
568C22#330114yes—
569C22#331214yes—
570C4—24no—
571C4—48no—
572C4—84no—
573C4—84no—
574C4—161no—
575C4—161no—
576C4—161no—
577C4—161no—
578C4#240511yescharacteristic
579C4#241812yes—
580C4#243712yes—
581C4#248211yescharacteristic
582C4#262611yescharacteristic
583C4#284811yescharacteristic
584C2—28no—
585C2#1473411yescharacteristic
586C2C2×#209711yescharacteristic
587C2#2373611yescharacteristic
588C2#2487514yes—
589C2#2578211yescharacteristic
590C2#2578411yescharacteristic
591C2#2596514yes—
592C2#2833411yescharacteristic
593C2#2834511yescharacteristic
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×#2097C2semidirect
8. #11124C2semidirect
9. #11122C2semidirect
10. #11052C2semidirect
552. C22#2097semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×#20972
C2×#18532
C2×#18082
C2×#17872
#139222
#138812
#111242
#111222
#110522
#107142