Finite Group Databasebeta

← #6559823

Subgroups — #6559823

Order 512.

1416 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#36286C211yesmaximal, characteristic
3#30377C212yesmaximal
4#30280C211yesmaximal, characteristic
5#29575C212yesmaximal
6#29512C212yesmaximal
7#29347C211yesmaximal, characteristic
8#29219C211yesmaximal, characteristic
9#28947C214yesmaximal
10#28677C212yesmaximal
11#28639C212yesmaximal
12#28030C214yesmaximal
13#23837C211yesmaximal, characteristic
14#23776C212yesmaximal
15#23214C212yesmaximal
16#23156C212yesmaximal
17#20569C211yesmaximal, characteristic
18C2×#557—21no—
19C2×#498C2212yes—
20C2×#487C2212yes—
21C2×C8×Q8C2212yes—
22#2868C2211yescharacteristic
23#2863C2212yes—
24#2862C2211yescharacteristic
25#2859C2212yes—
26#2605—22no—
27#2588—22no—
28#2581—21no—
29#2570—21no—
30#2568—21no—
31#2568C2211yescharacteristic
32#2567C2211yescharacteristic
33#2565—22no—
34#2546—21no—
35#2546C2212yes—
36#2511C2212yes—
37#2506—22no—
38#2506C2214yes—
39#2451C2212yes—
40#2369C2211yescharacteristic
41#2365—21no—
42#2365C2212yes—
43#2365C2211yescharacteristic
44#2356—21no—
45#2355—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2212yes—
49#2295C2212yes—
50C42×Q8C2211yescharacteristic
51#2099C2212yes—
52#2099C2212yes—
53#2096C2214yes—
54#2096C2212yes—
55#2096C2212yes—
56#2088C2214yes—
57#2088C2214yes—
58#2084C2212yes—
59#2084C2211yescharacteristic
60#2084C2211yescharacteristic
61#2072C2212yes—
62#2066C2214yes—
63#2066C2214yes—
64#2058C2212yes—
65#2058C2211yescharacteristic
66#2058C2211yescharacteristic
67#2024C2212yes—
68#1990C2212yes—
69#1988C2212yes—
70#1988C2212yes—
71#1987C2212yes—
72#1981C2212yes—
73#1941C2214yes—
74#1941C2214yes—
75#1941C2214yes—
76#1933C2214yes—
77#1933C2214yes—
78#1933C2214yes—
79#1933C2214yes—
80#1927C2214yes—
81#1920C2214yes—
82#1919C2214yes—
83#1865C2214yes—
84#1834C2212yes—
85C4×#106C2212yes—
86C4×#106C2212yes—
87C4×#106C2214yes—
88C8×C4⋊C4C2214yes—
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2212yes—
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⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—24no—
105C2×#125—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114#498—28no—
115#487—28no—
116C8×Q8—28no—
117C2×#106C2312yes—
118C2×#106C2312yes—
119C2×#106C2312yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2312yes—
123C2×C4⋊C8C2314yes—
124C2×C4⋊C8C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×C4×C8C2314yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2314yes—
132C2×C4×C8C2312yes—
133#400—22no—
134#400—22no—
135#400—24no—
136#400—22no—
137#398—24no—
138#398—24no—
139#398—22no—
140#395—22no—
141#395—22no—
142#391—21no—
143#391—21no—
144#391—22no—
145#391—22no—
146#391—21no—
147#391—22no—
148#391—21no—
149#391—21no—
150#391—21no—
151#391—22no—
152#391—22no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2311yescharacteristic
166#389C2311yescharacteristic
167#389C2312yes—
168#389C2314yes—
169#389C2312yes—
170#389C2312yes—
171#389C2314yes—
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#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#383—22no—
193#382—21no—
194#382—22no—
195#382—21no—
196#382—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214#376—22no—
215C43C2311yescharacteristic
216C43C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2412yes—
238C22×C8C2414yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C4⋊Q8—24no—
242C4⋊Q8—22no—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—41no—
248C4⋊Q8—48no—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251#125—416no—
252#125—48no—
253C4×Q8—24no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—42no—
257C4×Q8—48no—
258C4×Q8—48no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8SD1618yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2412yes—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2412yes—
315C2×C42C2412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319#106—28no—
320#106—28no—
321#106—24no—
322#106—24no—
323#106—24no—
324#106—42no—
325C4⋊C8—28no—
326C4⋊C8—24no—
327#103—28no—
328#103—28no—
329#103—28no—
330#103—44no—
331C4×C8—28no—
332C4×C8—44no—
333C4×C8—42no—
334C4×C8#4118yes—
335#95—21no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—21no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—21no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—22no—
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—24no—
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—44no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—44no—
375C2×Q8—42no—
376C2×Q8—44no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×Q8—81no—
380C2×Q8C2×SD1614yes—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C2511yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8C2×#4114yes—
409C2×C8C2×#4114yes—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—81no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4C2×SD1614yes—
450C4⋊C4C2×SD1618yes—
451C4⋊C42−1+414yes—
452C4⋊C42−1+412yes—
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—48no—
466C42—44no—
467C42—42no—
468C42—42no—
469C42—82no—
470C42C2×SD1618yes—
471C42C2×SD1614yes—
472C42#13514yes—
473C42C2×#4114yes—
474C23C23×D411yescharacteristic
475Q8—28no—
476Q8—48no—
477Q8—44no—
478Q8—44no—
479Q8—84no—
480Q8—88no—
481Q8—162no—
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—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
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—24no—
544C8—24no—
545C8—44no—
546C22C2×#53211yescharacteristic
547C22#348611yescharacteristic
548C22#348911yescharacteristic
549C22C23×SD1611yescharacteristic
550C22C22×#13511yescharacteristic
551C22#358611yescharacteristic
552C22#358611yescharacteristic
553C4—22no—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—24no—
558C4—22no—
559C4—24no—
560C4—42no—
561C4—44no—
562C4—42no—
563C4—42no—
564C4—82no—
565C4—82no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—82no—
570C4—84no—
571C4#316714yes—
572C4#350712yes—
573C4#350912yes—
574C2#3593811yescharacteristic
575C2C2×#316711yescharacteristic
576C2#6072011yescharacteristic
577C2#6181811yescharacteristic
578C2#6182211yescharacteristic
579C2#6209711yescharacteristic
580C2#6209911yescharacteristic
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
#362862
#303772
#302802
#295752
#295122
#293472
#292192
#289472
#286772
#286392
#280302
#238372
#237762
#232142
#231562
#205692