Finite Group Databasebeta

← #366780

Subgroups — #366780

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×#1874C211yesmaximal, characteristic
4C2×#1824C211yesmaximal, characteristic, complemented
5C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
6C22×#340C211yesmaximal, characteristic, complemented
7#13881C211yesmaximal, characteristic, complemented
8#11372C212yesmaximal, complemented
9#11051C212yesmaximal, complemented
10#10925C212yesmaximal, complemented
11#10842C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C4×C8C2212yes—
21#2504—24no—
22#2504C418yescomplemented
23C2×#397—22no—
24C2×#390C2212yes—
25C2×#386—22no—
26C2×#382C2212yes—
27C2×#382C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1893—24no—
31#1874—24no—
32#1824—24no—
33C8×C22⋊C4—24no—
34C2×#340—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1442—22no—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C23×D4—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—42no—
55C23×C8D414yes—
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121C2×C414yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4C2×C414yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D412yes—
86C22×C42C2312yes—
87C2×#102—24no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×#102—42no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8D414yes—
97C2×C4×C8—24no—
98C2×C4×C8—24no—
99C2×C4×C8—42no—
100#397—216no—
101#390—24no—
102#390C2×C418yescomplemented
103#386—216no—
104#382—24no—
105#382—24no—
106#382—42no—
107#382C2×C418yescomplemented
108#380—22no—
109#380—22no—
110C4×C22⋊C4C2×C414yes—
111C4×C22⋊C4D414yes—
112C2×#95—24no—
113C2×#95—24no—
114C2×#95—21no—
115C2×#95—24no—
116C2×#95D412yes—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—24no—
124#336—24no—
125C25—21no—
126C25C2×D411yescharacteristic
127C22×D4—24no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4C22×C412yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—44no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—44no—
157C23×C4—82no—
158C23×C4—82no—
159C23×C4—82no—
160C23×C4C22×C412yes—
161C23×C4C2×D414yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—28no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—28no—
184C22×C8—44no—
185C22×C8—48no—
186C22×C8—48no—
187C22×C8C2×D412yes—
188C22×C8C2×D414yes—
189C22×C8C2×D412yes—
190C22×C8#4114yes—
191#121—28no—
192#121—28no—
193#121—44no—
194#121—44no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4#4114yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—216no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—84no—
242C2×C22⋊C4—84no—
243C2×C22⋊C4C22⋊C418yes—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42C22×C414yes—
249C2×C42C2×D414yes—
250#102—48no—
251#102—44no—
252#102—44no—
253C22⋊C8—28no—
254C4×C8—48no—
255#95—216no—
256#95—48no—
257#95—42no—
258#95—84no—
259#95C22⋊C418yes—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—41no—
266C24—41no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24C2×C22⋊C412yes—
274C24#12412yes—
275C24#12712yes—
276C24#13514yes—
277C24#13514yes—
278C24C22×D411yescharacteristic
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—48no—
284C2×D4—416no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—88no—
300C2×D4—84no—
301C2×D4—84no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—48no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—48no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—88no—
350C22×C4—816no—
351C22×C4—88no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4C2×C22⋊C418yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C4×D414yes—
359C22×C4C4×D414yes—
360C22×C4#12114yes—
361C22×C4#12114yes—
362C22×C4#12412yes—
363C22×C4#12712yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C42−1+412yes—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—416no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8#12314yes—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—48no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—48no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C22⋊C4—216no—
413C22⋊C4—48no—
414C22⋊C4—816no—
415C22⋊C4—84no—
416C42—24no—
417C42—24no—
418C42—24no—
419C42—24no—
420C23—28no—
421C23—28no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—22no—
429C23—28no—
430C23—28no—
431C23—22no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23—84no—
437C23—84no—
438C23—82no—
439C23—82no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—82no—
445C23—81no—
446C23—84no—
447C23—84no—
448C23—81no—
449C23#39014yes—
450C23#41618yes—
451C23#48818yes—
452C23#49514yes—
453C23#49514yes—
454C23C22×C22⋊C411yescharacteristic
455C23C2×#12411yescharacteristic
456C23C2×#12711yescharacteristic
457C23#54612yes—
458C23#54712yes—
459C23C2×#13512yes—
460C23C2×#13512yes—
461C23#57612yes—
462C23#57712yes—
463D4—42no—
464D4—44no—
465D4—42no—
466D4—816no—
467D4—81no—
468D4—81no—
469D4—82no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—816no—
494C2×C4—816no—
495C2×C4—82no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—84no—
511C2×C4—81no—
512C2×C4#39014yes—
513C2×C4C2×C4×D411yescharacteristic
514C2×C4C2×C4×D411yescharacteristic
515C2×C4C4×#4112yes—
516C2×C4#52012yes—
517C2×C4C2×#12111yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×#12311yescharacteristic
520C2×C4#52511yescharacteristic
521C2×C4#52912yes—
522C2×C4#53411yescharacteristic
523C2×C4#53611yescharacteristic
524C2×C4#53912yes—
525C2×C4#54612yes—
526C2×C4#54612yes—
527C2×C4#54712yes—
528C2×C4#54712yes—
529C2×C4#55112yes—
530C2×C4#55212yes—
531C8—24no—
532C8—42no—
533C8—44no—
534C22—24no—
535C22—24no—
536C22—216no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—216no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—88no—
546C22—161no—
547C22—161no—
548C22—164no—
549C22—161no—
550C22—161no—
551C22#195818yes—
552C22C2×#39011yescharacteristic
553C22#233912yes—
554C22#248812yes—
555C22#250111yescharacteristic
556C22#253911yescharacteristic
557C22C2×#41612yes—
558C22#289712yes—
559C22C2×#48812yes—
560C22#313612yes—
561C22C2×#49511yescharacteristic
562C22C2×#49511yescharacteristic
563C22#315411yescharacteristic
564C22#315511yescharacteristic
565C22#329714yes—
566C22#330114yes—
567C22#333714yes—
568C22#333914yes—
569C4—24no—
570C4—48no—
571C4—84no—
572C4—82no—
573C4—84no—
574C4—161no—
575C4—161no—
576C4C4×#12311yescharacteristic
577C4#232211yescharacteristic
578C4#235912yes—
579C4#241211yescharacteristic
580C4#241311yescharacteristic
581C4#242512yes—
582C2—28no—
583C2—162no—
584C2#1399611yescharacteristic
585C2C2×#195811yescharacteristic
586C2#2120611yescharacteristic
587C2#2248814yes—
588C2#2770114yes—
589C2#2792611yescharacteristic
590C2#2793211yescharacteristic
591C2#2833111yescharacteristic
592C2#2833411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
4. C2×#1824C2semidirect
5. C2×C8×C22⋊C4C2semidirect
6. C22×#340C2semidirect
7. #13881C2semidirect
8. #11372C2semidirect
9. #11051C2semidirect
10. #10925C2semidirect
11. #10842C2semidirect
22. #2504C4semidirect
102. #390C2×C4semidirect
107. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#18932
C2×#18742
C2×#18242
C2×C8×C22⋊C42
C22×#3402
#138812
#113722
#110512
#109252
#108422