Finite Group Databasebeta

← #198035

Subgroups — #198035

Order 512.

1190 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic
3C2×#1994C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5#17486C211yesmaximal, characteristic
6#9236C214yesmaximal, complemented
7#9232C214yesmaximal, complemented
8#8095C211yesmaximal, characteristic
9C2×#1276C211yesmaximal, 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
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8—21no—
26C4×#125—22no—
27C4×C22⋊Q8—22no—
28C4×C22⋊Q8—22no—
29#2296—21no—
30#2295—21no—
31#2295C2212yes—
32#2285C2211yescharacteristic
33C42×Q8—21no—
34C42×Q8C2212yes—
35C4×#117—21no—
36C2×C43C2211yescharacteristic
37#1994—21no—
38#1994—21no—
39#1994C2212yes—
40#1994C2212yes—
41C2×#339—21no—
42C2×#339—21no—
43#1520—22no—
44#1518—22no—
45#1503C2214yes—
46#1493—22no—
47#1487C2214yes—
48#1481—22no—
49#1303C2214yes—
50#1276C2212yes—
51#1276C2212yes—
52C22×M16—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#533—21no—
62#533—21no—
63#533C2×C412yes—
64#533C2×C412yes—
65#533C2×C412yes—
66#533D412yes—
67#533D412yes—
68#533C2312yes—
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4⋊Q8C2×C411yescharacteristic
71C2×C4⋊Q8D411yescharacteristic
72C2×C4⋊Q8D411yescharacteristic
73C2×#125—21no—
74C2×#125—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8D412yes—
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8—21no—
84C2×C4×Q8C2312yes—
85C2×#117—21no—
86C2×#117—21no—
87C2×#117—21no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93#500—22no—
94#494—22no—
95#487—22no—
96#487—22no—
97#484—24no—
98#484—24no—
99#477—24no—
100#477C2×C418yescomplemented
101#467—28no—
102#462—24no—
103#462C2×C418yescomplemented
104#451—24no—
105#451—24no—
106C4×Q16—28no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431C2×C412yes—
113#431C2312yes—
114#431C2312yes—
115#423—21no—
116#423—21no—
117#423—21no—
118#423C2312yes—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8C2×C412yes—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127#391—22no—
128#391—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#387—24no—
134#387—24no—
135#385—24no—
136#384—21no—
137#382D412yes—
138#380—21no—
139#380D412yes—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4C2312yes—
150C4×C22⋊C4—24no—
151#376—22no—
152C43—21no—
153C43—21no—
154C43—22no—
155C43D411yescharacteristic
156C43D411yescharacteristic
157C43C2311yescharacteristic
158C43C2311yescharacteristic
159#339—22no—
160#339—22no—
161#339—22no—
162#339—22no—
163#328—28no—
164Q8⋊C8—28no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8C2×D412yes—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C2×Q16—28no—
175C2×M16—22no—
176C2×M16—41no—
177C2×M16—42no—
178C2×M16—42no—
179C22×C8—22no—
180C22×C8—41no—
181C22×C8—41no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8C22⋊C412yes—
193C4⋊Q8C22⋊C412yes—
194C4⋊Q8C22⋊C412yes—
195C4⋊Q8C22⋊C412yes—
196C4⋊Q8C22×C412yes—
197C4⋊Q8C22×C412yes—
198#125—22no—
199#125—22no—
200#125—22no—
201#125—22no—
202#125—22no—
203#125—22no—
204#125—41no—
205#125—41no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—41no—
217C22⋊Q8—41no—
218C22⋊Q8—41no—
219C22⋊Q8—41no—
220C22⋊Q8C22⋊C414yes—
221C22⋊Q8C22⋊C414yes—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—48no—
233C4×Q8—48no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—41no—
238C4×Q8C22×C414yes—
239C4×Q8C2×D414yes—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—41no—
247#117—41no—
248#117—42no—
249#117—42no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4C2×D412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—41no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—41no—
290C2×C42Q1614yescomplemented
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42#4111yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42#4111yescharacteristic
305C2×C42#4111yescharacteristic
306C2×C42C2411yescharacteristic
307#107—28no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—41no—
313C4⋊C8—41no—
314C4⋊C8—41no—
315C4⋊C8C22×C414yes—
316#103—28no—
317#103—28no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321C22⋊C8—41no—
322C4×C8—22no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—41no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95C2×D414yes—
332C24#12011yescharacteristic
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—82no—
343C2×Q8—82no—
344C2×Q8C2×C22⋊C414yes—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12011yescharacteristic
370C22×C4#12011yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12311yescharacteristic
376C22×C4#12311yescharacteristic
377C22×C4C2×Q1612yes—
378C22×C4#13512yes—
379C22×C4#13712yes—
380C22×C4#13712yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384Q16—88no—
385M16—42no—
386M16—42no—
387M16—82no—
388C2×C8—24no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—82no—
392C2×C8—82no—
393C2×C8—82no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4C2×C22⋊C414yes—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—44no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—21no—
433C42—21no—
434C42—24no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—48no—
446C42—48no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42C2×C22⋊C411yescharacteristic
453C42C2×C22⋊C411yescharacteristic
454C42C4×D411yescharacteristic
455C42C4×D411yescharacteristic
456C42C4×D411yescharacteristic
457C42C4×D411yescharacteristic
458C42#12111yescharacteristic
459C42#12111yescharacteristic
460C42#12111yescharacteristic
461C42#12111yescharacteristic
462C42#12311yescharacteristic
463C42#12311yescharacteristic
464C42C2×Q1612yes—
465C42C2×Q1612yes—
466C42C23×C411yescharacteristic
467C42C22×D411yescharacteristic
468C42C2×#4111yescharacteristic
469C42C2×#4111yescharacteristic
470C23—41no—
471C23—41no—
472C23#38611yescharacteristic
473C23#38611yescharacteristic
474C23#45112yes—
475C23#45212yes—
476C23C2×#12011yescharacteristic
477Q8—28no—
478Q8—44no—
479Q8—44no—
480Q8—44no—
481Q8—44no—
482Q8—82no—
483Q8—82no—
484Q8—82no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—48no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4#38611yescharacteristic
523C2×C4#38611yescharacteristic
524C2×C4#38611yescharacteristic
525C2×C4#38611yescharacteristic
526C2×C4#38611yescharacteristic
527C2×C4#38611yescharacteristic
528C2×C4C4×Q1612yes—
529C2×C4C4×Q1612yes—
530C2×C4#44112yes—
531C2×C4#44112yes—
532C2×C4#45112yes—
533C2×C4#45212yes—
534C2×C4#46212yes—
535C2×C4#46212yes—
536C2×C4#46412yes—
537C2×C4#46412yes—
538C2×C4C22×C22⋊C411yescharacteristic
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C2×C4×D411yescharacteristic
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×#12111yescharacteristic
543C2×C4C2×#12311yescharacteristic
544C2×C4C22×Q1611yescharacteristic
545C2×C4C2×#13511yescharacteristic
546C2×C4C2×#13711yescharacteristic
547C2×C4C2×#13711yescharacteristic
548C8—44no—
549C8—82no—
550C8—82no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—42no—
558C22—42no—
559C22#187812yes—
560C22#187812yes—
561C22C2×#38611yescharacteristic
562C22C2×#45111yescharacteristic
563C22C2×#45211yescharacteristic
564C4—24no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—44no—
575C4—44no—
576C4—84no—
577C4—82no—
578C4—82no—
579C4#187812yes—
580C4#187812yes—
581C4#190612yes—
582C4#290112yes—
583C4C2×C4×Q1611yescharacteristic
584C4C2×#44111yescharacteristic
585C4C2×#46211yescharacteristic
586C4C2×#46411yescharacteristic
587C2—21no—
588C2—21no—
589C2—22no—
590C2—22no—
591C2C2×#187811yescharacteristic
592C2#2098011yescharacteristic
593C2C2×#190611yescharacteristic
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
100. #477C2×C4semidirect
103. #462C2×C4semidirect
290. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
#609432
C4×#5332
C2×#19942
#175752
#174862
#92362
#92322
#80952
C2×#12762