Finite Group Databasebeta

← #199068

Subgroups — #199068

Order 512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic
3#21329C211yesmaximal, characteristic
4#17708C211yesmaximal, characteristic
5C2×#1518C211yesmaximal, characteristic
6#9298C214yesmaximal, complemented
7#9297C214yesmaximal, complemented
8C2×#1286C211yesmaximal, characteristic
9#7854C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3249—22no—
13#3230C412yescomplemented
14#3230C2212yes—
15#3209C412yescomplemented
16#3209C2212yes—
17#3189—22no—
18#3170C414yescomplemented
19#3104—22no—
20#3104—22no—
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24#2368—22no—
25#2368—22no—
26#2355—22no—
27#2355—22no—
28#2331C2212yes—
29#2331C2212yes—
30#2317—21no—
31C4×#125—22no—
32C2×#382C2211yescharacteristic
33#2274—21no—
34C4×#117C2211yescharacteristic
35#1995C2214yes—
36#1994—22no—
37#1733—21no—
38#1733—21no—
39#1550—22no—
40#1549C2212yes—
41#1548C2212yes—
42#1542—22no—
43#1518—22no—
44#1518C2214yes—
45C2×#328—22no—
46#1286C2214yes—
47#1274C2214yes—
48C22×M16—21no—
49#533—21no—
50#533—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#533C2×C412yes—
60#533C2×C412yes—
61#533C2×C412yes—
62#533D412yes—
63#533D412yes—
64#533C2312yes—
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×#125—21no—
70C2×#125—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8D412yes—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8D412yes—
79C2×#117—21no—
80C2×#117—21no—
81C2×#117D411yescharacteristic
82C2×#117D411yescharacteristic
83C22×C4⋊C4—21no—
84C22×C42C2311yescharacteristic
85#500C2×C414yescomplemented
86#494C2×C414yescomplemented
87#487C2×C414yescomplemented
88#487C2×C414yescomplemented
89#484—24no—
90#484—24no—
91#477—24no—
92#477—24no—
93#467—28no—
94#462—24no—
95#462—24no—
96#451—24no—
97#451—24no—
98C4×Q16—28no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431C2×C412yes—
105#431C2312yes—
106#431C2312yes—
107#423—21no—
108#423—21no—
109#423—21no—
110#423C2312yes—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2×C412yes—
119#391—24no—
120#391—24no—
121#389—22no—
122#389—22no—
123#387—24no—
124#387—24no—
125#387—24no—
126#387—24no—
127#385—24no—
128#384—22no—
129#382—22no—
130#382—21no—
131#382—21no—
132#382D412yes—
133#382C2312yes—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145#376C2312yes—
146C2×#95—21no—
147C2×#95D412yes—
148C43D412yes—
149#339—24no—
150#339—24no—
151#333—24no—
152#332—24no—
153#328—28no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8C2×D412yes—
158C23×C4—22no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C2×Q16—28no—
163C2×M16—22no—
164C2×M16—41no—
165C2×M16—42no—
166C2×M16—42no—
167C22×C8—22no—
168C22×C8—41no—
169C22×C8—41no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8C22×C412yes—
181C4⋊Q8C22×C412yes—
182C4⋊Q8C22×C412yes—
183C4⋊Q8C22×C412yes—
184C4⋊Q8C2×D412yes—
185C4⋊Q8C2×D412yes—
186#125—22no—
187#125—22no—
188#125—22no—
189#125—22no—
190#125—22no—
191#125—22no—
192#125—41no—
193#125—41no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208C22⋊Q8C22⋊C414yes—
209C22⋊Q8C22⋊C414yes—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—41no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8—41no—
224C4×Q8C22⋊C414yes—
225C4×Q8C22⋊C414yes—
226#117—22no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—41no—
236#117—41no—
237#117Q1614yescomplemented
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
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—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4C2×D412yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—41no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—41no—
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42#4111yescharacteristic
281C2×C42C2411yescharacteristic
282#107—28no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—41no—
289C4⋊C8—41no—
290C4⋊C8—41no—
291#103—28no—
292#103—28no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—41no—
297C4×C8—22no—
298C4×C8—22no—
299C4×C8—41no—
300C4×C8C22×C414yes—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95C2×D414yes—
311C24#12011yescharacteristic
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—82no—
320C2×Q8—82no—
321C2×Q8—82no—
322C2×Q8—82no—
323C2×Q8C2×C22⋊C414yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—44no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4#12311yescharacteristic
350C22×C4#12311yescharacteristic
351C22×C4C2×Q1612yes—
352C22×C4#13512yes—
353C22×C4#13712yes—
354C22×C4#13712yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358Q16—88no—
359M16—42no—
360M16—42no—
361M16—82no—
362C2×C8—24no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—82no—
366C2×C8—82no—
367C2×C8—82no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4C2×C22⋊C414yes—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C42—22no—
406C42—24no—
407C42—24no—
408C42—22no—
409C42—22no—
410C42—42no—
411C42—42no—
412C42—82no—
413C42—82no—
414C42—84no—
415C42—84no—
416C42C2×C22⋊C411yescharacteristic
417C42C2×C22⋊C411yescharacteristic
418C42C4×D411yescharacteristic
419C42C4×D411yescharacteristic
420C42C4×D411yescharacteristic
421C42C4×D411yescharacteristic
422C42#12111yescharacteristic
423C42#12111yescharacteristic
424C42#12111yescharacteristic
425C42#12111yescharacteristic
426C42#12311yescharacteristic
427C42#12311yescharacteristic
428C42C2×Q1614yes—
429C42C23×C411yescharacteristic
430C42C22×D411yescharacteristic
431C42C2×#4111yescharacteristic
432C42C2×#4111yescharacteristic
433C23—41no—
434C23—41no—
435C23#38611yescharacteristic
436C23#38611yescharacteristic
437C23#45112yes—
438C23#45212yes—
439C23C2×#12011yescharacteristic
440Q8—44no—
441Q8—44no—
442Q8—82no—
443Q8—82no—
444Q8—82no—
445Q8—82no—
446Q8—82no—
447Q8—82no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—164no—
480C2×C4#38611yescharacteristic
481C2×C4#38611yescharacteristic
482C2×C4#38611yescharacteristic
483C2×C4#38611yescharacteristic
484C2×C4#38611yescharacteristic
485C2×C4#38611yescharacteristic
486C2×C4C4×Q1612yes—
487C2×C4C4×Q1612yes—
488C2×C4#44112yes—
489C2×C4#44112yes—
490C2×C4#45112yes—
491C2×C4#45212yes—
492C2×C4#46212yes—
493C2×C4#46212yes—
494C2×C4#46412yes—
495C2×C4#46412yes—
496C2×C4C22×C22⋊C411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12111yescharacteristic
501C2×C4C2×#12311yescharacteristic
502C2×C4C22×Q1611yescharacteristic
503C2×C4C2×#13511yescharacteristic
504C2×C4C2×#13711yescharacteristic
505C2×C4C2×#13711yescharacteristic
506C8—44no—
507C8—82no—
508C8—82no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—42no—
516C22—42no—
517C22#187812yes—
518C22#187812yes—
519C22C2×#38611yescharacteristic
520C22C2×#45111yescharacteristic
521C22C2×#45211yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—44no—
531C4—44no—
532C4—44no—
533C4—82no—
534C4—84no—
535C4—82no—
536C4—82no—
537C4#187812yes—
538C4#187812yes—
539C4C2×C4×Q1611yescharacteristic
540C4C2×#44111yescharacteristic
541C4C2×#46211yescharacteristic
542C4C2×#46411yescharacteristic
543C2—21no—
544C2—21no—
545C2—22no—
546C2—22no—
547C2C2×#187811yescharacteristic
548C2#2099811yescharacteristic
549C2#2101211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9298C2semidirect
7. #9297C2semidirect
13. #3230C4semidirect
15. #3209C4semidirect
18. #3170C4semidirect
85. #500C2×C4semidirect
86. #494C2×C4semidirect
87. #487C2×C4semidirect
88. #487C2×C4semidirect
237. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#609432
#342152
#213292
#177082
C2×#15182
#92982
#92972
C2×#12862
#78542