Finite Group Databasebeta

← #496579

Subgroups — #496579

Order 512.

876 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35670C211yesmaximal, characteristic
2#20730C211yesmaximal, characteristic
3#20639C211yesmaximal, characteristic
4#13303C212yesmaximal, complemented
5#13227C212yesmaximal, complemented
6#13094C212yesmaximal
7#13094C212yesmaximal
8#12804C212yesmaximal, complemented
9#12802C212yesmaximal, complemented
10C2×#541C411yescharacteristic
11C2×#519C2211yescharacteristic
12#2921—21no—
13#2905—21no—
14C22×#101C2212yes—
15#2565C411yescharacteristic
16#2565C2211yescharacteristic
17#2546C411yescharacteristic
18#2546C2211yescharacteristic
19#2499C411yescharacteristic
20#2499C411yescharacteristic
21#2478—21no—
22#2478—21no—
23#2478—21no—
24#2478—21no—
25#2462—21no—
26#2462—21no—
27#2462—21no—
28#2462—21no—
29#2434C2211yescharacteristic
30#2434C2211yescharacteristic
31#2327C411yescharacteristic
32#2327C2211yescharacteristic
33C4×C4⋊Q8C411yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35#2298C411yescharacteristic
36C2×#356C2214yes—
37C2×#356C2214yes—
38#1822—21no—
39#1801—21no—
40#1801—21no—
41#1801—21no—
42#1801C2212yes—
43#1801C2212yes—
44#1801C2212yes—
45#1784C2212yes—
46C4×#101—21no—
47C4×#101C2212yes—
48#1742—21no—
49#1394C2214yes—
50#1394C2214yes—
51C23×Q8C2311yescharacteristic
52C22×M16—22no—
53#541—44no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×#126—21no—
60C2×#126—21no—
61C2×#125—21no—
62C2×#125—21no—
63C2×#125C2×C411yescharacteristic
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×C22⋊Q8C2×C411yescharacteristic
66#519—44no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117D411yescharacteristic
80C2×#117D411yescharacteristic
81C2×#117D411yescharacteristic
82C2×#117D411yescharacteristic
83#433—44no—
84#424—44no—
85#423—22no—
86#423—22no—
87#423—22no—
88C2×#101—22no—
89C2×#101—22no—
90C2×#101—44no—
91C2×#101C2314yes—
92C2×#101C2314yes—
93#407—22no—
94C4×M16—22no—
95#398—21no—
96#398—21no—
97#398—21no—
98#398—21no—
99#398—21no—
100#398—21no—
101#398—21no—
102#398—21no—
103#397—21no—
104#397—21no—
105#397—21no—
106#397—21no—
107#395—21no—
108#395—21no—
109#395—21no—
110#395—21no—
111#391C2×C411yescharacteristic
112#391C2×C411yescharacteristic
113#391C2×C411yescharacteristic
114#391C2×C411yescharacteristic
115#391C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#391C2311yescharacteristic
118#391C2311yescharacteristic
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#388—21no—
128#388—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#385—21no—
148#385—21no—
149#385—21no—
150#385—21no—
151#384C2×C411yescharacteristic
152#384C2×C411yescharacteristic
153#382C2×C411yescharacteristic
154#382C2×C411yescharacteristic
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4C2×C411yescharacteristic
160C4×C4⋊C4C2×C411yescharacteristic
161C4×C22⋊C4C2×C411yescharacteristic
162C4×C22⋊C4C2×C411yescharacteristic
163#376C2×C411yescharacteristic
164C43C2×C411yescharacteristic
165#356—44no—
166#356—44no—
167#343—22no—
168#343—22no—
169#343—22no—
170#343—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×Q8C22×C411yescharacteristic
178C22×Q8C2×D411yescharacteristic
179C22×Q8C2×D411yescharacteristic
180C22×Q8C2411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C2×M16—24no—
185C2×M16—24no—
186C2×M16—42no—
187C2×M16—44no—
188C2×M16—44no—
189C22×C8—42no—
190C4⋊Q8—22no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—42no—
195C4⋊Q8—42no—
196C4⋊Q8—42no—
197#126—44no—
198#126—44no—
199#125—42no—
200#125—44no—
201#125—44no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×Q8—42no—
212C4×Q8—42no—
213C4×Q8—42no—
214#117—24no—
215#117—24no—
216#117—24no—
217#117—42no—
218#117—42no—
219#117—42no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C4⋊C8—44no—
285C4⋊C8—44no—
286C4⋊C8—44no—
287#101—24no—
288#101—24no—
289#101—24no—
290#101—24no—
291#101—44no—
292#101—44no—
293C22⋊C8—44no—
294#97—42no—
295C4×C8—42no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322C24C2×C22⋊C411yescharacteristic
323C2×Q8—22no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8H(5,𝔽2)12yes—
344C2×Q82−1+412yes—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377M16—44no—
378M16—44no—
379M16—44no—
380M16—44no—
381M16—84no—
382C2×C8—44no—
383C2×C8—44no—
384C2×C8—84no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4#9914yes—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C23—21no—
452C23—21no—
453C23—41no—
454C23—41no—
455C23C2×#9912yes—
456C23C22×C22⋊C411yescharacteristic
457Q8—42no—
458Q8—42no—
459Q8—44no—
460Q8—44no—
461Q8—42no—
462Q8—42no—
463Q8—82no—
464Q8—82no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4C2×#9912yes—
522C2×C4C2×#9912yes—
523C2×C4#41012yes—
524C2×C4#52012yes—
525C2×C4#53511yescharacteristic
526C2×C4#53611yescharacteristic
527C2×C4#53712yes—
528C8—84no—
529C8—84no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—81no—
536C22—81no—
537C22#231111yescharacteristic
538C22C22×#9911yescharacteristic
539C22C2×#41011yescharacteristic
540C4—22no—
541C4—22no—
542C4—21no—
543C4—22no—
544C4—21no—
545C4—22no—
546C4—22no—
547C4—42no—
548C4—42no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C2—21no—
564C2—21no—
565C2—42no—
566C2#2187411yescharacteristic
567C2#3337211yescharacteristic
568C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13303C2semidirect
5. #13227C2semidirect
8. #12804C2semidirect
9. #12802C2semidirect

Maximal subgroups

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

subgroupindex
#356702
#207302
#206392
#133032
#132272
#130942
#128042
#128022