Finite Group Databasebeta

← #204885

Subgroups — #204885

Order 512.

1110 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
1#35298C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic, complemented
3#20431C211yesmaximal, characteristic, complemented
4C2×#1561C211yesmaximal, characteristic, complemented
5C2×#1557C211yesmaximal, characteristic
6C2×#1552C211yesmaximal, characteristic, complemented
7C2×#1547C211yesmaximal, characteristic
8#9710C211yesmaximal, characteristic, complemented
9#9708C211yesmaximal, characteristic, complemented
10#9704C211yesmaximal, characteristic, complemented
11#9703C211yesmaximal, characteristic, complemented
12#9700C211yesmaximal, characteristic, complemented
13#9698C211yesmaximal, characteristic, complemented
14#9696C211yesmaximal, characteristic, complemented
15#9692C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#405—21no—
23#2868C2211yescharacteristic
24#2862C2211yescharacteristic
25#2858C2211yescharacteristic
26#2858C2211yescharacteristic
27#2620—21no—
28#2620—21no—
29#2566C414yes—
30#2545—22no—
31#2500C414yes—
32#2432—22no—
33#2326—21no—
34#2326—21no—
35C4×#117C2211yescharacteristic
36#1940—22no—
37#1851—22no—
38#1842C2211yescharacteristic
39#1842C2211yescharacteristic
40#1831C2211yescharacteristic
41#1831C2211yescharacteristic
42#1768—21no—
43#1768—21no—
44#1767C2211yescharacteristic
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48#1729—21no—
49#1561—24no—
50#1557—24no—
51#1552—24no—
52#1547—24no—
53C2×#332C2211yescharacteristic
54C2×#332C2211yescharacteristic
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#331C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C2×#331C2211yescharacteristic
69C22×#41—21no—
70C22×M16—21no—
71C22×M16—21no—
72C23×C8—21no—
73#533—22no—
74#533—22no—
75#533—22no—
76#533—22no—
77#532—22no—
78#532—22no—
79#532—22no—
80#532—22no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#125C2×C412yes—
87C2×#124—21no—
88C2×#124—21no—
89C2×#124—21no—
90C2×#124—22no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×#121—21no—
94C2×#121C2×C412yes—
95C2×C4×Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×D4—21no—
98C2×C4×D4C2×C412yes—
99C2×#117C2×C412yes—
100C2×#117D411yescharacteristic
101C2×#117D411yescharacteristic
102C22×C42C2311yescharacteristic
103#432—22no—
104#432—22no—
105#423—22no—
106#423—22no—
107#423—22no—
108#423D414yes—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8D414yes—
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C22⋊C8—21no—
125#405—44no—
126C2×#97—21no—
127C2×#97—21no—
128C2×#97—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131#396—24no—
132#396—24no—
133#394—24no—
134#394—24no—
135#391—21no—
136#391—21no—
137#391—22no—
138#390—21no—
139#390—21no—
140#390—22no—
141#390C2×C414yes—
142#390C2×C414yes—
143#388—24no—
144#388—24no—
145#385—24no—
146#385—24no—
147#384—21no—
148#384—21no—
149#384C2×C414yes—
150#382C2×C414yes—
151C4×C4⋊C4C2×C412yes—
152C4×C4⋊C4D412yes—
153C4×C22⋊C4C2×C412yes—
154C4×C22⋊C4D412yes—
155#376D411yescharacteristic
156#376D411yescharacteristic
157C43C2311yescharacteristic
158C43C2311yescharacteristic
159#336—22no—
160#336—22no—
161#336—21no—
162#336—21no—
163#336—21no—
164#336—21no—
165#332—22no—
166#332—22no—
167#332—22no—
168#332—22no—
169#332—22no—
170#332—22no—
171#332—22no—
172#332—22no—
173#332—22no—
174#332—22no—
175#332—22no—
176#332—22no—
177#331—22no—
178#331—22no—
179#331—22no—
180#331—22no—
181C2×#41—42no—
182C2×#41—42no—
183C2×#41—82no—
184C2×#41—82no—
185C22×Q8—21no—
186C22×Q8—21no—
187C22×Q8—21no—
188C22×D4—22no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4C22×C412yes—
192C23×C4—22no—
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C2×M16—24no—
197C2×M16—42no—
198C2×M16—42no—
199C2×M16—42no—
200C2×M16—82no—
201C22×C8—24no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—42no—
211C22×C8—42no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—82no—
225C4⋊Q8—82no—
226#127—21no—
227#127—21no—
228#127—21no—
229#127—21no—
230#125—24no—
231#125—24no—
232#124—42no—
233#124—42no—
234#124—42no—
235#124—42no—
236#124—42no—
237#124—42no—
238#124—82no—
239#124—82no—
240C22⋊Q8—42no—
241C22⋊Q8—42no—
242C22⋊Q8—42no—
243C22⋊Q8—42no—
244#121—24no—
245#121—24no—
246#121—42no—
247#121—42no—
248C4×Q8—42no—
249C4×Q8—42no—
250C4×Q8—42no—
251C4×Q8—42no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—42no—
255C4×D4—42no—
256#117—22no—
257#117—22no—
258#117—24no—
259#117—24no—
260#117—41no—
261#117—41no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4C22⋊C412yes—
270C2×C4⋊C4C22⋊C412yes—
271C2×C4⋊C4C22×C412yes—
272C2×C4⋊C4C22×C412yes—
273C2×C4⋊C4C22×C412yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4C22⋊C412yes—
282C2×C22⋊C4C22⋊C412yes—
283C2×C42—22no—
284C2×C42C22×C412yes—
285C2×C42C22×C412yes—
286C2×C42C22×C411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42#4112yes—
298C2×C42#4112yes—
299C2×C42C2411yescharacteristic
300C4⋊C8—24no—
301C4⋊C8—24no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—42no—
315C4⋊C8—82no—
316C4⋊C8—82no—
317C4⋊C8C2×D412yes—
318C4⋊C8C2×D412yes—
319C4⋊C8C2×D412yes—
320C4⋊C8C2×D412yes—
321C22⋊C8—42no—
322#97—42no—
323#97—42no—
324#97—42no—
325C4×C8—42no—
326C4×C8—42no—
327#95—24no—
328#95—24no—
329#95C22⋊C412yes—
330#95C22⋊C412yes—
331#95C22⋊C412yes—
332#95C22⋊C412yes—
333C24—22no—
334C24C2×C22⋊C411yescharacteristic
335#41—88no—
336#41—84no—
337#41—84no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—84no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—84no—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C412yes—
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4#12112yes—
389C22×C4#12112yes—
390C22×C4#12412yes—
391C22×C4#12712yes—
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395M16—48no—
396M16—84no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—48no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C22⋊C4—24no—
437C22⋊C4—42no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—42no—
456C42—42no—
457C42—44no—
458C42C4×D412yes—
459C42C4×D412yes—
460C42#12112yes—
461C42#12112yes—
462C42#12412yes—
463C42#12712yes—
464C42AGL(1,ℤ/8)12yes—
465C42#13712yes—
466C42C22×D411yescharacteristic
467C42C22×D411yescharacteristic
468C42C2×#4111yescharacteristic
469C42C2×#4111yescharacteristic
470C23—22no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—42no—
479C23#39012yes—
480C23C22×C22⋊C411yescharacteristic
481Q8—82no—
482Q8—82no—
483Q8—81no—
484Q8—81no—
485Q8—81no—
486Q8—81no—
487Q8—81no—
488Q8—81no—
489Q8—81no—
490Q8—81no—
491D4—82no—
492D4—82no—
493D4—88no—
494D4—82no—
495D4—81no—
496D4—81no—
497D4—81no—
498D4—81no—
499D4—81no—
500D4—81no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—88no—
534C2×C4#39012yes—
535C2×C4#39012yes—
536C2×C4#39012yes—
537C2×C4#41714yes—
538C2×C4C2×C4×D411yescharacteristic
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C2×#12111yescharacteristic
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×#12411yescharacteristic
543C2×C4C2×#12711yescharacteristic
544C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
545C2×C4C2×#13711yescharacteristic
546C2×C4#57511yescharacteristic
547C2×C4#57511yescharacteristic
548C2×C4#57612yes—
549C2×C4#57712yes—
550C8—42no—
551C8—42no—
552C8—42no—
553C8—42no—
554C8—81no—
555C8—81no—
556C8—81no—
557C8—81no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—42no—
562C22—42no—
563C22—82no—
564C22—84no—
565C22—84no—
566C22—82no—
567C22—82no—
568C22C2×#39011yescharacteristic
569C22#288912yes—
570C22C2×#41711yescharacteristic
571C22#289611yescharacteristic
572C22#289712yes—
573C4—22no—
574C4—24no—
575C4—22no—
576C4—21no—
577C4—21no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4#303611yescharacteristic
586C4#303611yescharacteristic
587C4#303711yescharacteristic
588C4#303711yescharacteristic
589C4#313612yes—
590C4#315411yescharacteristic
591C4#315511yescharacteristic
592C2—42no—
593C2—162no—
594C2#1776512yes—
595C2#1776912yes—
596C2#2120611yescharacteristic
597C2#2122011yescharacteristic
598C2#2125811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21978C2semidirect
3. #20431C2semidirect
4. C2×#1561C2semidirect
6. C2×#1552C2semidirect
8. #9710C2semidirect
9. #9708C2semidirect
10. #9704C2semidirect
11. #9703C2semidirect
12. #9700C2semidirect
13. #9698C2semidirect
14. #9696C2semidirect
15. #9692C2semidirect

Maximal subgroups

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

subgroupindex
#352982
#219782
#204312
C2×#15612
C2×#15572
C2×#15522
C2×#15472
#97102
#97082
#97042
#97032
#97002
#96982
#96962
#96922