Finite Group Databasebeta

← #214663

Subgroups — #214663

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34266C211yesmaximal, characteristic
2#17368C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#9110C212yesmaximal, complemented
5#9108C212yesmaximal, complemented
6#8968C212yesmaximal, complemented
7#8966C212yesmaximal, complemented
8#8735C212yesmaximal, complemented
9#8303C212yesmaximal, complemented
10C2×#536C411yescharacteristic
11C2×#520C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#2973C2212yes—
14#2972C2212yes—
15#2367—21no—
16#2367—21no—
17#2365—21no—
18#2365—21no—
19#2362—21no—
20#2362—21no—
21#2360—21no—
22#2360—21no—
23#2322C411yescharacteristic
24#2322C411yescharacteristic
25#2322C2211yescharacteristic
26#2322C2211yescharacteristic
27#2310C411yescharacteristic
28C4×#123C411yescharacteristic
29C4×#123C411yescharacteristic
30C4×#123C2211yescharacteristic
31C4×#123C2211yescharacteristic
32#2297C411yescharacteristic
33#2296C411yescharacteristic
34#2274C2211yescharacteristic
35#1817C2214yes—
36#1467—21no—
37#1465—21no—
38#1464—21no—
39#1463—21no—
40#1461—21no—
41#1459—21no—
42#1459C2212yes—
43#1459C2212yes—
44#1458—21no—
45C2×#324—21no—
46C2×#324C2212yes—
47C2×#324C2212yes—
48C2×#324C2214yes—
49#1282C2214yes—
50#1282C2214yes—
51C22×#41C2×C411yescharacteristic
52C23×Q8C2×C411yescharacteristic
53#536—24no—
54#536—24no—
55C2×C4⋊Q8—21no—
56C2×#124—21no—
57C2×#123C2×C411yescharacteristic
58C2×#123C2×C411yescharacteristic
59C2×#123C2×C411yescharacteristic
60C2×#123C2×C411yescharacteristic
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63#520—44no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—44no—
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117D411yescharacteristic
81C2×#117D411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C42C2×C411yescharacteristic
88C22×C42C2×C411yescharacteristic
89C22×C42C2311yescharacteristic
90#435—44no—
91C8×D4—44no—
92#432—22no—
93#431—22no—
94#407—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8C2314yes—
97C2×C22⋊C8C2314yes—
98#391—21no—
99#391—21no—
100#391—21no—
101#391—21no—
102#391—21no—
103#390—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#388—21no—
111#388—21no—
112#388—21no—
113#388—21no—
114#388—21no—
115#388—21no—
116#388—21no—
117#388—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#387—21no—
125#387—21no—
126#386C2×C411yescharacteristic
127#386C2×C411yescharacteristic
128#386C2×C411yescharacteristic
129#386C2×C411yescharacteristic
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385C2×C411yescharacteristic
135#385C2×C411yescharacteristic
136#385C2×C411yescharacteristic
137#385C2×C411yescharacteristic
138#381C2×C411yescharacteristic
139#381C2×C411yescharacteristic
140#381C2×C411yescharacteristic
141#381C2×C411yescharacteristic
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4C2×C411yescharacteristic
159C4×C22⋊C4C2×C411yescharacteristic
160C4×C22⋊C4C2311yescharacteristic
161C4×C22⋊C4C2311yescharacteristic
162#336—44no—
163#324—22no—
164#324—22no—
165#324—22no—
166#324—22no—
167#324—22no—
168#324—22no—
169#324—44no—
170#323—22no—
171#323—22no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—24no—
175C2×#41—42no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—42no—
182C22×Q8C22⋊C411yescharacteristic
183C22×D4C22⋊C411yescharacteristic
184C22×D4C22×C411yescharacteristic
185C22×D4C22×C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2411yescharacteristic
199C2×M16—42no—
200C22×C8—24no—
201C22×C8—24no—
202C22×C8—42no—
203C4⋊Q8—28no—
204#124—28no—
205#123—22no—
206#123—22no—
207#123—22no—
208#123—22no—
209#123—22no—
210#123—22no—
211#123M1614yescomplemented
212#123M1614yescomplemented
213C22⋊Q8—28no—
214C22⋊Q8—28no—
215C4×Q8—42no—
216C4×Q8—42no—
217C4×Q8—44no—
218C4×Q8—44no—
219C4×Q8—44no—
220C4×Q8—44no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—42no—
227#117—22no—
228#117—22no—
229#117—24no—
230#117—42no—
231#117—42no—
232#117—42no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22×C411yescharacteristic
273C2×C22⋊C4C22×C411yescharacteristic
274C2×C22⋊C4C22×C411yescharacteristic
275C2×C22⋊C4C22×C411yescharacteristic
276C2×C22⋊C4C22×C411yescharacteristic
277C2×C22⋊C4C22×C411yescharacteristic
278C2×C22⋊C4C2×D411yescharacteristic
279C2×C22⋊C4C2×D411yescharacteristic
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—24no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42C22⋊C411yescharacteristic
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22⋊C411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C4⋊C8—42no—
302C4⋊C8—42no—
303C22⋊C8—24no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306C22⋊C8—24no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309#97—42no—
310C4×C8—42no—
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—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C24C23×C411yescharacteristic
334#41—48no—
335#41—48no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—41no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—41no—
356C2×D4C2×M1612yes—
357C2×D4C2×M1612yes—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×M1612yes—
407C22×C4C2×M1612yes—
408C22×C4#13114yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4H(5,𝔽2)14yes—
412M16—162no—
413C2×C8—44no—
414C2×C8—44no—
415C2×C8—84no—
416C2×C8—84no—
417C2×C8—162no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—81no—
459C22⋊C4—81no—
460C22⋊C4C2×M1612yes—
461C22⋊C4C2×M1612yes—
462C42—24no—
463C42—22no—
464C42—22no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—81no—
477C42—81no—
478C42—81no—
479C42—81no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23#40712yes—
492C23C22×C22⋊C411yescharacteristic
493C23#51812yes—
494C23C22×M1611yescharacteristic
495C23C22×M1611yescharacteristic
496C23C2×#13112yes—
497Q8—44no—
498Q8—48no—
499Q8—48no—
500D4—24no—
501D4—24no—
502D4—44no—
503D4—42no—
504D4—42no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—161no—
568C2×C4—161no—
569C2×C4#40712yes—
570C2×C4#40812yes—
571C2×C4#40812yes—
572C2×C4#53412yes—
573C2×C4#53412yes—
574C8—84no—
575C8—84no—
576C22—21no—
577C22—22no—
578C22—21no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—81no—
592C22—81no—
593C22#230811yescharacteristic
594C22C2×#40711yescharacteristic
595C22C2×#40811yescharacteristic
596C22#297412yes—
597C22#297412yes—
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—81no—
618C2—21no—
619C2—21no—
620C2—22no—
621C2—22no—
622C2—42no—
623C2—81no—
624C2#2174011yescharacteristic
625C2#2177711yescharacteristic
626C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9110C2semidirect
5. #9108C2semidirect
6. #8968C2semidirect
7. #8966C2semidirect
8. #8735C2semidirect
9. #8303C2semidirect
211. #123M16semidirect
212. #123M16semidirect

Maximal subgroups

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

subgroupindex
#342662
#173682
#173512
#91102
#91082
#89682
#89662
#87352
#83032