Finite Group Databasebeta

← #416714

Subgroups — #416714

Order 512.

1448 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#1983C211yesmaximal, characteristic
3C2×#1959C211yesmaximal, characteristic
4#14153C211yesmaximal, characteristic
5#11067C212yesmaximal, complemented
6#11066C212yesmaximal
7#11050C212yesmaximal, complemented
8#11049C212yesmaximal
9#10849C211yesmaximal, characteristic, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×#125—21no—
13C22×#124C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×#102C2211yescharacteristic
21C22×#97C2211yescharacteristic
22#2552—24no—
23#2552—24no—
24C2×#398—22no—
25C2×#391C2212yes—
26C2×#390C2212yes—
27C2×#389—21no—
28C2×#389C2212yes—
29C2×#384C2212yes—
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1983—24no—
35#1959—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1439—21no—
43#1439—21no—
44#1437—22no—
45C23×Q8C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×#125—28no—
51C2×#125—48no—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124Q814yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—42no—
92C2×#102—42no—
93C2×#102—42no—
94C2×#97—22no—
95C2×#97—22no—
96C2×#97—22no—
97#398—216no—
98#391—24no—
99#391—42no—
100#390—24no—
101#390—42no—
102#389—28no—
103#389—24no—
104#389—24no—
105#384—24no—
106#384—24no—
107#384—22no—
108#384D414yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4D414yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95D412yes—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#336—42no—
126#336—42no—
127C25—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8C2×Q812yes—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4C2×Q812yes—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8#4114yes—
175#125—432no—
176#124—28no—
177#124—28no—
178#124—44no—
179#124—44no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
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—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4#4114yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—44no—
238C2×C42—42no—
239C2×C42—81no—
240C2×C42—81no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×Q812yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C4⋊C8—44no—
248#103—48no—
249#103—44no—
250#103—44no—
251#102—48no—
252#102—44no—
253#102—44no—
254#97—44no—
255#95—28no—
256#95—24no—
257#95—24no—
258#95—44no—
259#95—44no—
260#95—42no—
261C24—22no—
262C24—41no—
263C24—41no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24C22×D411yescharacteristic
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4#12012yes—
357C22×C4#12012yes—
358C22×C4C22⋊Q812yes—
359C22×C4C22⋊Q812yes—
360C22×C4#12412yes—
361C22×C4#12412yes—
362C22×C4AGL(1,ℤ/8)14yes—
363C22×C4#13714yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×Q811yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—28no—
392C4⋊C4—28no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—48no—
402C4⋊C4—416no—
403C4⋊C4—416no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—84no—
411C4⋊C4—88no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—88no—
415C22⋊C4—88no—
416C42—48no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—42no—
421C42—84no—
422C42C22⋊Q814yes—
423C42C22⋊Q814yes—
424C42#12414yes—
425C42#12414yes—
426C23—44no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—44no—
431C23—81no—
432C23—81no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—81no—
439C23—81no—
440C23—82no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23C2×#12011yescharacteristic
450C23C2×C22⋊Q811yescharacteristic
451C23C2×#12411yescharacteristic
452C23#54612yes—
453C23#55012yes—
454C23C2×AGL(1,ℤ/8)12yes—
455C23C2×#13712yes—
456C23#57611yescharacteristic
457C23#57711yescharacteristic
458C23#57711yescharacteristic
459C23#57811yescharacteristic
460Q8—42no—
461Q8—42no—
462Q8—44no—
463Q8—81no—
464Q8—81no—
465Q8—82no—
466D4—42no—
467D4—42no—
468D4—44no—
469D4—81no—
470D4—81no—
471D4—82no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#39314yes—
526C2×C4#39314yes—
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×#12411yescharacteristic
530C2×C4C2×#12411yescharacteristic
531C2×C4#52912yes—
532C2×C4#52912yes—
533C2×C4#53211yescharacteristic
534C2×C4#53211yescharacteristic
535C2×C4#53511yescharacteristic
536C2×C4#53511yescharacteristic
537C2×C4#54012yes—
538C2×C4#54012yes—
539C2×C4#54612yes—
540C2×C4#55012yes—
541C2×C4#55312yes—
542C2×C4#55312yes—
543C2×C4#55412yes—
544C2×C4#55412yes—
545C8—44no—
546C8—81no—
547C8—81no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—84no—
552C22—84no—
553C22—161no—
554C22—161no—
555C22—162no—
556C22—162no—
557C22—161no—
558C22—161no—
559C22C2×#39311yescharacteristic
560C22#243111yescharacteristic
561C22#243111yescharacteristic
562C22#245412yes—
563C22#249012yes—
564C22#300311yescharacteristic
565C22#300411yescharacteristic
566C22#301211yescharacteristic
567C22#301311yescharacteristic
568C22#307912yes—
569C22#308311yescharacteristic
570C22#308411yescharacteristic
571C22#313611yescharacteristic
572C22#313711yescharacteristic
573C22#314112yes—
574C22#330612yes—
575C22#330812yes—
576C22#331412yes—
577C22#331612yes—
578C22#339712yes—
579C22#339812yes—
580C22#339912yes—
581C22#340012yes—
582C4—24no—
583C4—24no—
584C4—44no—
585C4—44no—
586C4—82no—
587C4—84no—
588C4—84no—
589C4—82no—
590C4—82no—
591C4#245412yes—
592C4#249012yes—
593C4#256611yescharacteristic
594C4#256611yescharacteristic
595C4#264111yescharacteristic
596C4#286111yescharacteristic
597C2—162no—
598C2#1522911yescharacteristic
599C2#2345211yescharacteristic
600C2#2345811yescharacteristic
601C2#2557711yescharacteristic
602C2#2557911yescharacteristic
603C2#2563711yescharacteristic
604C2#2564011yescharacteristic
605C2#2654212yes—
606C2#2654412yes—
607C2#2776312yes—
608C2#2776512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2552C2semidirect
5. #11067C2semidirect
7. #11050C2semidirect
9. #10849C2semidirect
10. #10835C2semidirect
11. #10703C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#19832
C2×#19592
#141532
#110672
#110662
#110502
#110492
#108492
#108352
#107032