Finite Group Databasebeta

← #6560735

Subgroups — #6560735

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36560C211yesmaximal, characteristic
2#36285C211yesmaximal, characteristic
3#30377C212yesmaximal
4#30228C211yesmaximal, characteristic
5#29575C212yesmaximal
6#29512C212yesmaximal
7#29313C211yesmaximal, characteristic
8#29013C214yesmaximal
9#28512C211yesmaximal, characteristic
10#27990C214yesmaximal
11#25472C212yesmaximal
12#25472C212yesmaximal
13#23816C211yesmaximal, characteristic
14#23776C212yesmaximal
15#23062C212yesmaximal
16#22964C212yesmaximal
17#20569C211yesmaximal, characteristic
18C2×#557—21no—
19C2×#475C2212yes—
20C2×#475C2212yes—
21C2×C8×Q8C2212yes—
22#2868C2211yescharacteristic
23#2863C2212yes—
24#2859C2211yescharacteristic
25#2859C2212yes—
26#2606—22no—
27#2590—22no—
28#2582C2211yescharacteristic
29#2582C2211yescharacteristic
30#2570—21no—
31#2568—21no—
32#2565—22no—
33#2565—21no—
34#2546—21no—
35#2546C2212yes—
36#2507C2212yes—
37#2506—22no—
38#2506C2214yes—
39#2446C2212yes—
40#2370—21no—
41#2365—21no—
42#2365C2212yes—
43#2356—21no—
44#2355C2211yescharacteristic
45Q8×C4⋊C4C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49#2295C2212yes—
50C42×Q8C2211yescharacteristic
51#2100C2212yes—
52#2099C2212yes—
53#2099C2214yes—
54#2099C2212yes—
55#2096C2212yes—
56#2096C2212yes—
57#2088C2214yes—
58#2088C2214yes—
59#2084C2212yes—
60#2072C2211yescharacteristic
61#2072C2211yescharacteristic
62#2066C2214yes—
63#2066C2214yes—
64#2058C2212yes—
65#2043C2212yes—
66#2024C2211yescharacteristic
67#2024C2211yescharacteristic
68#1990C2212yes—
69#1990C2212yes—
70#1956C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1941C2214yes—
74#1941C2214yes—
75#1941C2214yes—
76#1941C2214yes—
77#1933C2214yes—
78#1933C2214yes—
79#1933C2214yes—
80#1927C2214yes—
81#1920C2214yes—
82#1919C2214yes—
83#1865C2214yes—
84#1844C2212yes—
85C4×#106C2214yes—
86C8×C4⋊C4C2212yes—
87C8×C4⋊C4C2214yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2212yes—
90C4×#103C2212yes—
91C4×#103C2212yes—
92C4×#103C2214yes—
93#1744C2211yescharacteristic
94#1744C2212yes—
95#557—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—24no—
103C2×#125—21no—
104C2×#125—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114#475—28no—
115#475—28no—
116C8×Q8—28no—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2314yes—
122C2×C4⋊C8C2312yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×C4×C8C2314yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2312yes—
132C2×C4×C8C2314yes—
133#400—24no—
134#400—22no—
135#398—22no—
136#398—22no—
137#398—24no—
138#398—24no—
139#398—22no—
140#398—22no—
141#398—22no—
142#391—21no—
143#391—21no—
144#391—22no—
145#391—22no—
146#391—21no—
147#391—22no—
148#391C2311yescharacteristic
149#391C2311yescharacteristic
150#391C2312yes—
151#391C2312yes—
152#391C2312yes—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2314yes—
173#389C2312yes—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—24no—
182#387—24no—
183#387—22no—
184#387—22no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#383—22no—
195#382—21no—
196#382—22no—
197#382—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214#376—22no—
215C43C2311yescharacteristic
216C43C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2411yescharacteristic
236C22×Q8C2411yescharacteristic
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2414yes—
240C4⋊Q8—24no—
241C4⋊Q8—22no—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—41no—
245C4⋊Q8—48no—
246C4⋊Q8—44no—
247C4⋊Q8—44no—
248#125—416no—
249#125—48no—
250#125—44no—
251C4×Q8—24no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—42no—
257C4×Q8—48no—
258C4×Q8—48no—
259C4×Q8—44no—
260C4×Q8SD1618yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2412yes—
317C2×C42C2412yes—
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321#106—28no—
322#106—28no—
323#106—28no—
324#106—44no—
325C4⋊C8—28no—
326C4⋊C8#4118yes—
327#103—24no—
328#103—28no—
329#103—28no—
330#103—24no—
331#103—24no—
332#103—42no—
333C4×C8—28no—
334C4×C8—24no—
335C4×C8—44no—
336C4×C8—42no—
337#95—21no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—21no—
342#95—21no—
343#95—22no—
344#95—22no—
345#95—21no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—42no—
373C2×Q8—44no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8C2×SD1614yes—
378C2×Q82−1+412yes—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2511yescharacteristic
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8C2×#4118yes—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4C2×SD1614yes—
451C4⋊C4C2×SD1618yes—
452C4⋊C42−1+414yes—
453C42—24no—
454C42—24no—
455C42—22no—
456C42—22no—
457C42—21no—
458C42—22no—
459C42—21no—
460C42—42no—
461C42—41no—
462C42—41no—
463C42—42no—
464C42—44no—
465C42—42no—
466C42—42no—
467C42—48no—
468C42—44no—
469C42—82no—
470C42C2×SD1618yes—
471C42C2×SD1614yes—
472C42#13514yes—
473C42C2×#4114yes—
474C23C23×D411yescharacteristic
475Q8—28no—
476Q8—24no—
477Q8—48no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—88no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4#53212yes—
530C2×C4#53212yes—
531C2×C4C22×SD1614yes—
532C2×C4C22×SD1612yes—
533C2×C4C22×SD1612yes—
534C2×C4C22×SD1614yes—
535C2×C4C22×SD1612yes—
536C2×C4C2×#13512yes—
537C2×C4C2×#13512yes—
538C2×C4C2×#13512yes—
539C2×C4C22×#4112yes—
540C2×C4C2×2−1+412yes—
541C2×C4C2×2−1+411yescharacteristic
542C2×C4#58511yescharacteristic
543C8—28no—
544C8—42no—
545C8—42no—
546C22C2×#53211yescharacteristic
547C22#348611yescharacteristic
548C22#348911yescharacteristic
549C22C23×SD1611yescharacteristic
550C22C22×#13511yescharacteristic
551C22#358611yescharacteristic
552C22#358611yescharacteristic
553C4—22no—
554C4—22no—
555C4—22no—
556C4—24no—
557C4—22no—
558C4—22no—
559C4—24no—
560C4—42no—
561C4—42no—
562C4—44no—
563C4—42no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—84no—
571C4#316714yes—
572C4#351512yes—
573C4#352312yes—
574C2#3594911yescharacteristic
575C2C2×#316711yescharacteristic
576C2#6072611yescharacteristic
577C2#6181811yescharacteristic
578C2#6182211yescharacteristic
579C2#6209911yescharacteristic
580C2#6210111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#365602
#362852
#303772
#302282
#295752
#295122
#293132
#290132
#285122
#279902
#254722
#238162
#237762
#230622
#229642
#205692