Finite Group Databasebeta

← #181135

Subgroups — #181135

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#34177C211yesmaximal, characteristic
3#34109C211yesmaximal, characteristic
4#7945C212yesmaximal, complemented
5#7944C212yesmaximal, complemented
6#7943C212yesmaximal
7#7941C212yesmaximal, complemented
8#7940C212yesmaximal, complemented
9C2×#1284C212yesmaximal
10C22×#117C2211yescharacteristic
11#2924—24no—
12#2924C418yescomplemented
13#2921—21no—
14#2921—21no—
15#2920C2212yes—
16#2920C2212yes—
17#2905—21no—
18C2×#407C2212yes—
19#2873—21no—
20#2347—21no—
21#2346—21no—
22#2345—21no—
23#2344—21no—
24#2343—21no—
25#2343—21no—
26#2343—24no—
27#2343C418yescomplemented
28#2342—21no—
29#2342—21no—
30#2315—21no—
31#2313—21no—
32#2313—21no—
33#2298C412yescomplemented
34#2298C412yescomplemented
35#2296C412yescomplemented
36#2283C2212yes—
37#2283C2212yes—
38#2283C2211yescharacteristic
39#2283C2211yescharacteristic
40#2280C2211yescharacteristic
41C2×#381C2212yes—
42#2273—21no—
43C4×#117C412yescomplemented
44C4×#117C2211yescharacteristic
45C4×#117C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47#1284C2214yes—
48#1284C2214yes—
49#1284C2214yes—
50#1284C2214yes—
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C2×#117—21no—
54C2×#117—21no—
55C2×#117C2×C412yes—
56C2×#117C2×C414yes—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C2×#117C2×C411yescharacteristic
61C2×#117C2×C414yes—
62C2×#117D411yescharacteristic
63C2×#117D411yescharacteristic
64C2×#117D414yes—
65C2×#117D411yescharacteristic
66C2×#117D411yescharacteristic
67C2×#117Q814yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42C2×C412yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#433—28no—
78#433—28no—
79#432—22no—
80#432—22no—
81#432C2×C414yescomplemented
82#432C2×C414yescomplemented
83#431—22no—
84#431—22no—
85#431C2×C414yescomplemented
86#431C2×C414yescomplemented
87#424—28no—
88#423—22no—
89#423—22no—
90#407—22no—
91#407—22no—
92#407C2×C414yes—
93#407C2314yes—
94C2×C22⋊C8—22no—
95C2×C22⋊C8C2×C414yes—
96#405—28no—
97C4×M16—22no—
98C4×M16—22no—
99#389—22no—
100#389—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#387—28no—
106#387—28no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#385—28no—
112#385—22no—
113#385—22no—
114#383—22no—
115#383—21no—
116#383—21no—
117#383C2×C414yescomplemented
118#382—22no—
119#382—21no—
120#382—21no—
121#382C2×C414yescomplemented
122#381—22no—
123#381—21no—
124#381—21no—
125#381—22no—
126#381—21no—
127#381C2×C414yes—
128#381C2314yes—
129#380—22no—
130#380—22no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380—21no—
135#379—21no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4C2×C412yescomplemented
142C4×C4⋊C4C2×C412yescomplemented
143C4×C4⋊C4C2×C412yescomplemented
144C4×C4⋊C4C2×C412yescomplemented
145C4×C4⋊C4C2×C414yescomplemented
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4C2×C412yes—
156C4×C22⋊C4C2×C412yes—
157C4×C22⋊C4C2311yescharacteristic
158C4×C22⋊C4C2311yescharacteristic
159C4×C22⋊C4C2311yescharacteristic
160C4×C22⋊C4C2311yescharacteristic
161#376—22no—
162#376—21no—
163#376—21no—
164#376C2×C412yescomplemented
165#376C2×C412yescomplemented
166#376C2×C414yescomplemented
167C2×#95—21no—
168C2×#95—22no—
169C2×#95C2×C414yes—
170C43C2×C412yescomplemented
171C43C2×C412yescomplemented
172C25C2×D411yescharacteristic
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—44no—
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C412yes—
178C23×C4C22×C412yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×Q812yes—
189C23×C4C2411yescharacteristic
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—48no—
193C22×C8—24no—
194#117—44no—
195#117—44no—
196#117—44no—
197#117C4214yes—
198#117C4214yes—
199#117C4214yes—
200#117C4214yes—
201#117C22⋊C414yes—
202#117C22⋊C414yes—
203#117C22⋊C414yes—
204#117C22⋊C414yes—
205#117C4⋊C414yes—
206#117C4⋊C414yes—
207#117C4⋊C414yes—
208#117C4⋊C414yes—
209#117C4⋊C414yes—
210#117C4⋊C414yes—
211#117C4⋊C414yes—
212#117C4⋊C414yes—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4C22⋊C414yes—
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C4⋊C414yes—
231C2×C4⋊C4C4⋊C414yes—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4C22⋊C414yes—
244C2×C22⋊C4C22⋊C411yescharacteristic
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C22⋊C4C22⋊C414yes—
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22⋊C411yescharacteristic
250C2×C22⋊C4C22⋊C411yescharacteristic
251C2×C22⋊C4C22⋊C411yescharacteristic
252C2×C22⋊C4C22⋊C411yescharacteristic
253C2×C22⋊C4C4⋊C414yes—
254C2×C22⋊C4C4⋊C414yes—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—44no—
263C2×C42C22×C412yes—
264C2×C42C22×C412yes—
265C2×C42C22×C412yes—
266C2×C42C22×C412yes—
267C2×C42C22×C412yes—
268C2×C42C22×C412yes—
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C412yes—
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D412yes—
281C2×C42C2×Q812yes—
282C2×C42C2×Q812yes—
283C4⋊C8—24no—
284C4⋊C8—24no—
285C4⋊C8—24no—
286C4⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290C22⋊C8C22×C418yes—
291#97—24no—
292#97—24no—
293C4×C8—24no—
294C4×C8—24no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—21no—
312#95—24no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95C22×C418yes—
317C24—24no—
318C24—42no—
319C24—81no—
320C24C2×C22⋊C412yes—
321C24C2×C22⋊C411yescharacteristic
322C24C2×C22⋊C411yescharacteristic
323C24C2×C4⋊C412yes—
324C24C22×D411yescharacteristic
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—48no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—82no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4C2×C4212yes—
351C22×C4C2×C4212yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C4⋊C412yes—
370C22×C4C2×C4⋊C412yes—
371C22×C4C2×C4⋊C412yes—
372C22×C4C2×C4⋊C412yes—
373C22×C4C2×C4⋊C412yes—
374C22×C4C2×C4⋊C412yes—
375C22×C4C2×C4⋊C412yes—
376C22×C4C2×C4⋊C412yes—
377C22×C4C2×C4⋊C412yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C23×C411yescharacteristic
380C22×C4C23×C411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×Q811yescharacteristic
384M16—48no—
385M16—48no—
386C2×C8—28no—
387C2×C8—48no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—84no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—84no—
395C4⋊C4#9514yes—
396C4⋊C4#9514yes—
397C4⋊C4#9514yes—
398C4⋊C4#9514yes—
399C4⋊C4#9514yes—
400C4⋊C4#9514yes—
401C4⋊C4#9514yes—
402C4⋊C4#9514yes—
403C22⋊C4—24no—
404C22⋊C4—44no—
405C22⋊C4—48no—
406C22⋊C4—42no—
407C22⋊C4—48no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—82no—
412C22⋊C4#9514yes—
413C22⋊C4#9514yes—
414C22⋊C4#9514yes—
415C22⋊C4#9514yes—
416C22⋊C4#9514yes—
417C22⋊C4#9514yes—
418C22⋊C4#9514yes—
419C22⋊C4#9514yes—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—44no—
425C42—44no—
426C42—44no—
427C42—84no—
428C42—82no—
429C42—82no—
430C42—84no—
431C42C2×C4212yes—
432C42C2×C4212yes—
433C42C2×C4212yes—
434C42C2×C4212yes—
435C42C2×C22⋊C412yes—
436C42C2×C22⋊C412yes—
437C42C2×C22⋊C412yes—
438C42C2×C22⋊C412yes—
439C42C2×C4⋊C412yes—
440C42C2×C4⋊C412yes—
441C42C2×C4⋊C412yes—
442C42C2×C4⋊C412yes—
443C42C2×C4⋊C412yes—
444C42C2×C4⋊C412yes—
445C42C2×C4⋊C412yes—
446C42C2×C4⋊C412yes—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—24no—
451C23—22no—
452C23—24no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—88no—
462C23C2×#9512yes—
463C23C2×#9512yes—
464C23#41014yes—
465C23C22×C22⋊C411yescharacteristic
466C23C22×C22⋊C411yescharacteristic
467C23C22×C4⋊C411yescharacteristic
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—48no—
517C2×C4—88no—
518C2×C4—168no—
519C2×C4C2×#9512yes—
520C2×C4C2×#9512yes—
521C2×C4C2×#9512yes—
522C2×C4C2×#9512yes—
523C2×C4C2×#9512yes—
524C2×C4C2×#9512yes—
525C2×C4C2×#9512yes—
526C2×C4C2×#9512yes—
527C2×C4C2×#9512yes—
528C2×C4C2×#9512yes—
529C2×C4C22×C4211yescharacteristic
530C2×C4C22×C22⋊C411yescharacteristic
531C2×C4C22×C4⋊C411yescharacteristic
532C2×C4C22×C4⋊C411yescharacteristic
533C8—48no—
534C22—21no—
535C22—21no—
536C22—22no—
537C22—21no—
538C22—22no—
539C22—21no—
540C22—21no—
541C22—22no—
542C22—21no—
543C22—22no—
544C22—21no—
545C22—22no—
546C22—44no—
547C22—48no—
548C22—44no—
549C22—82no—
550C22—84no—
551C22#174014yes—
552C22C22×#9511yescharacteristic
553C22C2×#41012yes—
554C22#290112yes—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—22no—
567C4—22no—
568C4—21no—
569C4—44no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—88no—
574C2—22no—
575C2—22no—
576C2—24no—
577C2—41no—
578C2—42no—
579C2#2033811yescharacteristic
580C2#2034311yescharacteristic
581C2C2×#174011yescharacteristic
subgroupG/H
center, socleC22C22×#95
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7945C2semidirect
5. #7944C2semidirect
7. #7941C2semidirect
8. #7940C2semidirect
12. #2924C4semidirect
27. #2343C4semidirect
33. #2298C4semidirect
34. #2298C4semidirect
35. #2296C4semidirect
43. C4×#117C4semidirect
81. #432C2×C4semidirect
82. #432C2×C4semidirect
85. #431C2×C4semidirect
86. #431C2×C4semidirect
117. #383C2×C4semidirect
121. #382C2×C4semidirect
141. C4×C4⋊C4C2×C4semidirect
142. C4×C4⋊C4C2×C4semidirect
143. C4×C4⋊C4C2×C4semidirect
144. C4×C4⋊C4C2×C4semidirect
145. C4×C4⋊C4C2×C4semidirect
164. #376C2×C4semidirect
165. #376C2×C4semidirect
166. #376C2×C4semidirect
170. C43C2×C4semidirect
171. C43C2×C4semidirect

Maximal subgroups

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

subgroupindex
#601242
#341772
#341092
#79452
#79442
#79432
#79412
#79402
C2×#12842