Finite Group Databasebeta

← #198030

Subgroups — #198030

Order 512.

1133 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2#34145C211yesmaximal, characteristic
3#21329C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#17486C211yesmaximal, characteristic
6#9236C214yesmaximal, complemented
7#9232C214yesmaximal, complemented
8C2×#1303C211yesmaximal, characteristic
9#7866C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3249C414yescomplemented
13#3230—21no—
14#3230C412yescomplemented
15#3209—21no—
16#3209C412yescomplemented
17#3189—22no—
18#3170—22no—
19#3104—22no—
20#3104C2214yes—
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24#2370—22no—
25#2369—21no—
26#2368—22no—
27#2365—21no—
28#2360—22no—
29#2296—21no—
30#2295—22no—
31#2295C2212yes—
32C42×Q8C2212yes—
33C4×#117—21no—
34C4×#117C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1995—22no—
37#1994C2214yes—
38#1733—21no—
39#1733—21no—
40#1503—22no—
41#1503C2212yes—
42#1503C2212yes—
43C2×#328—22no—
44#1493—22no—
45#1487C2214yes—
46#1481—22no—
47#1303C2212yes—
48#1303C2212yes—
49#1276C2214yes—
50C22×M16—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533C2×C412yes—
62#533C2×C412yes—
63#533C2×C412yes—
64#533D412yes—
65#533D412yes—
66#533C2312yes—
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8D411yescharacteristic
70C2×C4⋊Q8D411yescharacteristic
71C2×#125—21no—
72C2×#125—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8D412yes—
77C2×C4×Q8—22no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8C2312yes—
82C2×#117—21no—
83C2×#117—21no—
84C2×#117D411yescharacteristic
85C2×#117D411yescharacteristic
86C22×C4⋊C4—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#500—22no—
90#494—22no—
91#487—22no—
92#487—22no—
93#484—24no—
94#484—24no—
95#477—24no—
96#477C2×C418yescomplemented
97#467—28no—
98#462—24no—
99#462C2×C418yescomplemented
100#451—24no—
101#451—24no—
102C4×Q16—28no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431C2×C412yes—
109#431C2312yes—
110#431C2312yes—
111#423—21no—
112#423—21no—
113#423—21no—
114#423C2312yes—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8C2×C412yes—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123#391—24no—
124#391—22no—
125#389—24no—
126#389—22no—
127#389—22no—
128#389—22no—
129#387—24no—
130#387—24no—
131#387—24no—
132#385—24no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4D412yes—
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C4×C22⋊C4—24no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149#376—22no—
150#376D412yes—
151C2×#95D412yes—
152C43—22no—
153C43C2312yes—
154#339—24no—
155#339—24no—
156#328—24no—
157#328—24no—
158Q8⋊C8—28no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8C2×D412yes—
163C23×C4—22no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C2×Q16—28no—
168C2×M16—22no—
169C2×M16—41no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—22no—
173C22×C8—41no—
174C22×C8—41no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8C22⋊C412yes—
186C4⋊Q8C22⋊C412yes—
187C4⋊Q8C22⋊C412yes—
188C4⋊Q8C22⋊C412yes—
189C4⋊Q8C22×C412yes—
190C4⋊Q8C22×C412yes—
191#125—22no—
192#125—22no—
193#125—22no—
194#125—22no—
195#125—22no—
196#125—22no—
197#125—41no—
198#125—41no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—41no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C22⋊Q8C22⋊C414yes—
214C22⋊Q8C22⋊C414yes—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—48no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8C22×C414yes—
231C4×Q8C2×D414yes—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117—22no—
239#117—22no—
240#117—22no—
241#117—41no—
242#117—41no—
243#117Q1614yescomplemented
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4C2×D412yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—41no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—41no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42#4111yescharacteristic
289C2×C42#4111yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42C2411yescharacteristic
293#107—28no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—22no—
298C4⋊C8—41no—
299C4⋊C8—41no—
300C4⋊C8—41no—
301C4⋊C8C22×C414yes—
302#103—28no—
303#103—28no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—41no—
308C4×C8—22no—
309C4×C8—22no—
310C4×C8—22no—
311C4×C8—41no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95C2×D412yes—
320#95C2×D412yes—
321C24#12011yescharacteristic
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—44no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—82no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×Q8C2×C22⋊C414yes—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—44no—
345C22×C4—81no—
346C22×C4—82no—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12011yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12311yescharacteristic
361C22×C4#12311yescharacteristic
362C22×C4C2×Q1612yes—
363C22×C4#13512yes—
364C22×C4#13712yes—
365C22×C4#13712yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369Q16—88no—
370M16—42no—
371M16—42no—
372M16—82no—
373C2×C8—24no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—82no—
377C2×C8—82no—
378C2×C8—82no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—84no—
409C4⋊C4C2×C22⋊C414yes—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C42—24no—
417C42—24no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—44no—
422C42—48no—
423C42—42no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—42no—
429C42—44no—
430C42—82no—
431C42—84no—
432C42C2×C22⋊C411yescharacteristic
433C42C2×C22⋊C411yescharacteristic
434C42C4×D411yescharacteristic
435C42C4×D411yescharacteristic
436C42C4×D411yescharacteristic
437C42C4×D411yescharacteristic
438C42#12111yescharacteristic
439C42#12111yescharacteristic
440C42#12111yescharacteristic
441C42#12111yescharacteristic
442C42#12311yescharacteristic
443C42#12311yescharacteristic
444C42C2×Q1614yes—
445C42C23×C411yescharacteristic
446C42C22×D411yescharacteristic
447C42C2×#4111yescharacteristic
448C42C2×#4111yescharacteristic
449C23—41no—
450C23—41no—
451C23#38611yescharacteristic
452C23#38611yescharacteristic
453C23#45112yes—
454C23#45212yes—
455C23C2×#12011yescharacteristic
456Q8—28no—
457Q8—44no—
458Q8—44no—
459Q8—82no—
460Q8—82no—
461Q8—82no—
462Q8—82no—
463Q8—82no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—88no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4#38611yescharacteristic
498C2×C4#38611yescharacteristic
499C2×C4#38611yescharacteristic
500C2×C4#38611yescharacteristic
501C2×C4#38611yescharacteristic
502C2×C4#38611yescharacteristic
503C2×C4C4×Q1612yes—
504C2×C4C4×Q1612yes—
505C2×C4#44112yes—
506C2×C4#44112yes—
507C2×C4#45112yes—
508C2×C4#45212yes—
509C2×C4#46212yes—
510C2×C4#46212yes—
511C2×C4#46412yes—
512C2×C4#46412yes—
513C2×C4C22×C22⋊C411yescharacteristic
514C2×C4C2×C4×D411yescharacteristic
515C2×C4C2×C4×D411yescharacteristic
516C2×C4C2×#12111yescharacteristic
517C2×C4C2×#12111yescharacteristic
518C2×C4C2×#12311yescharacteristic
519C2×C4C22×Q1611yescharacteristic
520C2×C4C2×#13511yescharacteristic
521C2×C4C2×#13711yescharacteristic
522C2×C4C2×#13711yescharacteristic
523C8—44no—
524C8—82no—
525C8—82no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—42no—
533C22—42no—
534C22#187812yes—
535C22#187812yes—
536C22C2×#38611yescharacteristic
537C22C2×#45111yescharacteristic
538C22C2×#45211yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—44no—
547C4—44no—
548C4—44no—
549C4—44no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4#187812yes—
555C4#187812yes—
556C4#290112yes—
557C4C2×C4×Q1611yescharacteristic
558C4C2×#44111yescharacteristic
559C4C2×#46211yescharacteristic
560C4C2×#46411yescharacteristic
561C2—21no—
562C2—21no—
563C2—22no—
564C2—22no—
565C2C2×#187811yescharacteristic
566C2#2098011yescharacteristic
567C2#2100011yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9236C2semidirect
7. #9232C2semidirect
12. #3249C4semidirect
14. #3230C4semidirect
16. #3209C4semidirect
96. #477C2×C4semidirect
99. #462C2×C4semidirect
243. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#609432
#341452
#213292
C2×#15032
#174862
#92362
#92322
C2×#13032
#78662