Finite Group Databasebeta

← #392794

Subgroups — #392794

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2274C211yesmaximal, characteristic
2C2×#1927C212yesmaximal, complemented
3C2×#1873C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×#1853C211yesmaximal, characteristic
6#14258C211yesmaximal, characteristic
7#10932C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10807C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#106C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#2274—28no—
25C2×C4×C4⋊C4—22no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4—22no—
29#1927—28no—
30#1873C418yescomplemented
31#1872—24no—
32#1853—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1441—22no—
41#1440—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8C2×C414yes—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—24no—
59C2×#117—24no—
60C2×#117—24no—
61C2×#117—24no—
62C2×#117C2×C414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D414yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103C2×C418yescomplemented
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8C2×C414yes—
99#391—24no—
100#391—24no—
101#380—22no—
102#380—22no—
103C4×C4⋊C4—216no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4D418yes—
108C4×C22⋊C4—216no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340C2×C418yescomplemented
121#336—24no—
122#336—24no—
123C25C2×D411yescharacteristic
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C22×C412yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—82no—
148C23×C4—82no—
149C23×C4C22×C412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
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×C8C22×C414yes—
173C22⋊Q8—28no—
174C22⋊Q8—28no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177#117—216no—
178#117—216no—
179#117—216no—
180#117—216no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—216no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—216no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4#4114yes—
217C2×C4⋊C4#4114yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—82no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—28no—
239C2×C42—28no—
240C2×C42—82no—
241C2×C42—82no—
242C2×C42—82no—
243C2×C42—82no—
244C2×C42C2×D414yes—
245C2×C42#4114yes—
246#106—48no—
247#103—216no—
248#103—48no—
249C22⋊C8—28no—
250C22⋊C8—44no—
251#95—28no—
252#95—28no—
253#95—44no—
254#95—42no—
255#95—44no—
256#95—82no—
257#95—82no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24C4×D412yes—
267C24#12112yes—
268C24#12312yes—
269C24C22×D411yescharacteristic
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—24no—
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—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—216no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—88no—
354C22×C4—88no—
355C22×C4#11714yes—
356C22×C4C4×D414yes—
357C22×C4C4×D412yes—
358C22×C4#12014yes—
359C22×C4#12112yes—
360C22×C4#12314yes—
361C22×C4#12312yes—
362C22×C4#12414yes—
363C22×C4#12614yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4C2×#4112yes—
373C22×C4C2×#4111yescharacteristic
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C4⋊C4—216no—
389C4⋊C4—216no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—48no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4C4×D418yes—
406C4⋊C4#12118yes—
407C22⋊C4—216no—
408C22⋊C4—216no—
409C22⋊C4—48no—
410C22⋊C4—84no—
411C42—216no—
412C42—24no—
413C42—24no—
414C42—24no—
415C42—24no—
416C42—88no—
417C42—88no—
418C23—24no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23#38814yes—
441C23C2×C4×D411yescharacteristic
442C23C2×#12011yescharacteristic
443C23C2×#12111yescharacteristic
444C23C2×#12311yescharacteristic
445C23C2×#12311yescharacteristic
446C23#52511yescharacteristic
447C23#52511yescharacteristic
448C23#57712yes—
449C23#57712yes—
450C23#57712yes—
451C23#57812yes—
452Q8—44no—
453Q8—44no—
454Q8—82no—
455Q8—82no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—816no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4#38814yes—
518C2×C4C2×#11711yescharacteristic
519C2×C4C2×C4×D412yes—
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C4×#4112yes—
522C2×C4C2×#12112yes—
523C2×C4#52512yes—
524C2×C4#52512yes—
525C2×C4#52512yes—
526C2×C4#52512yes—
527C2×C4C2×#12411yescharacteristic
528C2×C4C2×#12611yescharacteristic
529C2×C4#52912yes—
530C2×C4#52912yes—
531C2×C4#52912yes—
532C2×C4#52912yes—
533C2×C4#53212yes—
534C8—44no—
535C8—82no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—44no—
546C22—44no—
547C22C2×#38811yescharacteristic
548C22C4×#12011yescharacteristic
549C22C4×#12311yescharacteristic
550C22#284711yescharacteristic
551C22#284911yescharacteristic
552C22#285311yescharacteristic
553C22#285411yescharacteristic
554C22#294712yes—
555C22#295212yes—
556C22#300912yes—
557C22#301312yes—
558C22#304211yescharacteristic
559C22#304211yescharacteristic
560C22#304311yescharacteristic
561C22#304411yescharacteristic
562C22#309712yes—
563C22#309712yes—
564C22#309912yes—
565C22#309912yes—
566C22#312012yes—
567C22#312112yes—
568C22#312312yes—
569C22#312512yes—
570C4—24no—
571C4—48no—
572C4—48no—
573C4—44no—
574C4—44no—
575C4—82no—
576C4—82no—
577C4—161no—
578C4—161no—
579C4#229012yes—
580C4C4×#12112yes—
581C4#285012yes—
582C4#285212yes—
583C2—44no—
584C2#1469611yescharacteristic
585C2#2106411yescharacteristic
586C2#2106811yescharacteristic
587C2#2146112yes—
588C2#2159112yes—
589C2#3291411yescharacteristic
590C2#3291811yescharacteristic
591C2#3298812yes—
592C2#3310512yes—
593C2#3311911yescharacteristic
594C2#3312111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1927C2semidirect
7. #10932C2semidirect
8. #10908C2semidirect
9. #10807C2semidirect
30. #1873C4semidirect
92. C2×#103C2×C4semidirect
120. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22742
C2×#19272
C2×#18732
C2×#18722
C2×#18532
#142582
#109322
#109082
#108072
#107312
#107132