Finite Group Databasebeta

← #399154

Subgroups — #399154

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#382C211yesmaximal, characteristic
2C2×#2097C212yesmaximal, complemented
3C2×#1872C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×#1853C211yesmaximal, characteristic
6C2×#1852C211yesmaximal, characteristic
7#14702C211yesmaximal, characteristic
8#11372C212yesmaximal, complemented
9#11325C212yesmaximal, complemented
10#11052C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#382—22no—
25C2×#382—28no—
26C2×#382C2212yes—
27C2×#382C2212yes—
28C2×#380C2211yescharacteristic
29C22×#95—22no—
30#2097—28no—
31#1872—24no—
32#1872—24no—
33#1853—24no—
34#1852—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1436—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C22⋊#95—22no—
45C23×Q8C2311yescharacteristic
46C24×C4—21no—
47C24×C4—21no—
48C24×C4D412yes—
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8Q814yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4—42no—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C42—22no—
77C22×C42—22no—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—42no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8D414yes—
100C2×C22⋊C8D414yes—
101#391—24no—
102#391—24no—
103#382—216no—
104#382—24no—
105#382—24no—
106#382—24no—
107#382D418yes—
108#380—22no—
109#380—22no—
110C2×#95—216no—
111C2×#95—24no—
112C2×#95—24no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#336—24no—
123#336—24no—
124C25C2×D411yescharacteristic
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8C2×Q812yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8C2×D414yes—
185C22×C8C2×D414yes—
186C22⋊Q8—28no—
187C22⋊Q8—28no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
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—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—48no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×Q812yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C42—216no—
235C2×C42—216no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42C2×D414yes—
246C2×C42#4114yes—
247#107—48no—
248#106—48no—
249#103—48no—
250#103—48no—
251C22⋊C8—28no—
252C22⋊C8—28no—
253#95—216no—
254#95—216no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—42no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24C22⋊Q812yes—
267C24#12312yes—
268C24#12712yes—
269C24#13514yes—
270C24#13714yes—
271C24C22×D411yescharacteristic
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—216no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—416no—
330C22×C4—416no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4#12014yes—
366C22×C4C22⋊Q814yes—
367C22×C4C22⋊Q814yes—
368C22×C4C22⋊Q812yes—
369C22×C4#12314yes—
370C22×C4#12314yes—
371C22×C4#12314yes—
372C22×C4#12312yes—
373C22×C4#12712yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C22×Q811yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4112yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C42−1+412yes—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
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—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—48no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C22⋊C4—48no—
418C22⋊C4—84no—
419C42—216no—
420C42—24no—
421C42—24no—
422C42—24no—
423C42—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—24no—
440C23—24no—
441C23—22no—
442C23#39914yes—
443C23#47918yes—
444C23#48518yes—
445C23#49514yes—
446C23#49714yes—
447C23C2×#12011yescharacteristic
448C23C2×C22⋊Q811yescharacteristic
449C23C2×#12311yescharacteristic
450C23C2×#12311yescharacteristic
451C23C2×#12711yescharacteristic
452C23#53511yescharacteristic
453C23#53611yescharacteristic
454C23C2×#13512yes—
455C23C2×#13712yes—
456C23#57512yes—
457C23#57512yes—
458Q8—44no—
459Q8—44no—
460Q8—82no—
461Q8—82no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—432no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—48no—
495C2×C4—48no—
496C2×C4—816no—
497C2×C4—816no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—81no—
521C2×C4#39914yes—
522C2×C4C2×C22⋊Q811yescharacteristic
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×#12311yescharacteristic
525C2×C4C2×#12311yescharacteristic
526C2×C4#52512yes—
527C2×C4#52512yes—
528C2×C4#53912yes—
529C2×C4#53912yes—
530C2×C4D4212yes—
531C2×C4D4212yes—
532C2×C4#54712yes—
533C2×C4#54712yes—
534C2×C4#54812yes—
535C2×C4D4×Q812yes—
536C2×C4#55312yes—
537C2×C4#55412yes—
538C8—44no—
539C8—44no—
540C22—24no—
541C22—24no—
542C22—28no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—28no—
548C22—28no—
549C22—28no—
550C22#209718yescomplemented
551C22C2×#39911yescharacteristic
552C22#240911yescharacteristic
553C22#243311yescharacteristic
554C22#248711yescharacteristic
555C22#250111yescharacteristic
556C22#262311yescharacteristic
557C22#284911yescharacteristic
558C22#300212yes—
559C22#300712yes—
560C22C2×#47912yes—
561C22#308012yes—
562C22C2×#48512yes—
563C22#309112yes—
564C22#309312yes—
565C22#309512yes—
566C22C2×#49511yescharacteristic
567C22C2×#49711yescharacteristic
568C22#315311yescharacteristic
569C22#315311yescharacteristic
570C22#321112yes—
571C22#321512yes—
572C22#323212yes—
573C22#323612yes—
574C4—24no—
575C4—44no—
576C4—44no—
577C4—84no—
578C4—84no—
579C4—82no—
580C4—82no—
581C4—161no—
582C4—161no—
583C4#244312yes—
584C4#244312yes—
585C4#244912yes—
586C4#244912yes—
587C2—28no—
588C2#1477711yescharacteristic
589C2C2×#209711yescharacteristic
590C2#2373311yescharacteristic
591C2#2430712yes—
592C2#2572411yescharacteristic
593C2#2573011yescharacteristic
594C2#2768612yes—
595C2#2791211yescharacteristic
596C2#2791911yescharacteristic
597C2#2991312yes—
598C2#3298512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2097C2semidirect
8. #11372C2semidirect
9. #11325C2semidirect
10. #11052C2semidirect
550. C22#2097semidirect

Maximal subgroups

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

subgroupindex
C22×#3822
C2×#20972
C2×#18722
C2×#18532
C2×#18522
#147022
#113722
#113252
#110522
#107122