Finite Group Databasebeta

← #6482296

Subgroups — #6482296

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2C2×#2370C211yesmaximal, characteristic
3#33151C212yesmaximal
4#33150C212yesmaximal
5#29973C212yesmaximal
6#29972C212yesmaximal
7#26525C214yesmaximal, complemented
8#26523C214yesmaximal, complemented
9#26459C214yesmaximal, complemented
10#26457C214yesmaximal, complemented
11#21955C211yesmaximal, characteristic
12#21838C211yesmaximal, characteristic
13#21838C211yesmaximal, characteristic
14#21365C211yesmaximal, characteristic
15#21365C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19#2851C2212yes—
20#2851C2212yes—
21#2646—21no—
22#2646—21no—
23#2571C2212yes—
24#2571C2212yes—
25#2570—22no—
26#2570—22no—
27#2552C2212yes—
28#2552C2212yes—
29#2370—22no—
30#2370—22no—
31#2370C2214yes—
32#2370C2214yes—
33#2317—21no—
34#2317—21no—
35C2×#389—21no—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#387—21no—
39C2×#387—21no—
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42C2×C4×C4⋊C4—21no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47#2101C2214yes—
48#2101C2214yes—
49#2088C2214yes—
50#2088C2214yes—
51#2087C2214yes—
52#2086C2214yes—
53#2059C2214yes—
54#2059C2214yes—
55#1992C2212yes—
56#1992C2212yes—
57#1992C2212yes—
58#1992C2212yes—
59#1989C2214yes—
60#1989C2214yes—
61#1989C2214yes—
62#1989C2214yes—
63#1949C2214yes—
64#1949C2214yes—
65#1948C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1947C2212yes—
69#1940C2214yes—
70#1940C2214yes—
71#1921C2214yes—
72#1920C2214yes—
73#1919C2214yes—
74#1918C2214yes—
75#1853C2212yes—
76#1853C2212yes—
77#1852C2212yes—
78#1852C2212yes—
79#1824C2212yes—
80#1824C2212yes—
81#1823C2212yes—
82#1823C2212yes—
83#1797C2212yes—
84#1797C2212yes—
85#1776C2214yes—
86#1776C2214yes—
87C2×#125—21no—
88C2×#125—21no—
89C2×#125—24no—
90C2×#125Q818yes—
91C2×#125C2312yes—
92C2×#125C2312yes—
93C2×#125C2312yes—
94C2×#125C2312yes—
95C2×#117—21no—
96C2×#117—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4D411yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C42—21no—
108C22×C42—21no—
109C22×C42—21no—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#432—24no—
114#432—24no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2314yes—
122C2×C4⋊C8C2314yes—
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2312yes—
125C2×#97C2312yes—
126C2×#97C2312yes—
127#400—22no—
128#400—22no—
129#400—22no—
130#400—22no—
131#398—22no—
132#398—22no—
133#398—22no—
134#398—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—24no—
140#389—24no—
141#389—22no—
142#389—22no—
143#389C2314yes—
144#389C2314yes—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#389C2312yes—
149#387—24no—
150#387—24no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—24no—
156#387—24no—
157#385—22no—
158#385—22no—
159#385—22no—
160#385—22no—
161#384—22no—
162#384—22no—
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#384C2311yescharacteristic
166#384C2311yescharacteristic
167#382—22no—
168#382—22no—
169#382—21no—
170#382—21no—
171#382C2312yes—
172#382C2312yes—
173#380—21no—
174#380—21no—
175#380C2312yes—
176#380C2312yes—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—24no—
184C4×C4⋊C4—24no—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4D412yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2314yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2411yescharacteristic
227C22×C8C2414yes—
228C22×C8C2414yes—
229#125—28no—
230#125—28no—
231#125—24no—
232#125—24no—
233#125—44no—
234#125—44no—
235#125C2×Q818yes—
236#125C2×Q818yes—
237#117—44no—
238#117—44no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×Q814yes—
280C2×C4⋊C4C2×Q814yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42—82no—
301C2×C42—82no—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×Q814yes—
307C2×C42#4114yes—
308C2×C42C2412yes—
309C2×C42C2412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312#107—28no—
313#107—44no—
314#106—28no—
315#106—44no—
316C4⋊C8—24no—
317C4⋊C8—24no—
318C4⋊C8—44no—
319C4⋊C8—44no—
320C22⋊C8—24no—
321C22⋊C8—24no—
322#97—24no—
323#97—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95C2×D412yes—
343#95C2×D412yes—
344#95C2×D412yes—
345#95C2×D412yes—
346C24C22×D411yescharacteristic
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4C22⋊Q814yes—
386C22×C4C22⋊Q814yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×Q812yes—
397C22×C4C2×#4112yes—
398C22×C4C2511yescharacteristic
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—28no—
408C4⋊C4—28no—
409C4⋊C4—28no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—48no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4C22×Q814yes—
445C4⋊C4C22×Q814yes—
446C4⋊C42−1+414yes—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C42—24no—
450C42—24no—
451C42—42no—
452C42—42no—
453C42—48no—
454C42—44no—
455C42—44no—
456C42—82no—
457C42—82no—
458C42—84no—
459C42C22⋊Q814yes—
460C42C22⋊Q814yes—
461C42C22×Q812yes—
462C42C22×Q812yes—
463C42C2×#4112yes—
464C42C2×#4112yes—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23C2×C22⋊Q812yes—
470C23#57512yes—
471C23#57512yes—
472C23C23×D411yescharacteristic
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—28no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—84no—
531C2×C4C2×C22⋊Q812yes—
532C2×C4C2×C22⋊Q812yes—
533C2×C4C2×C22⋊Q812yes—
534C2×C4C2×C22⋊Q812yes—
535C2×C4C2×C22⋊Q812yes—
536C2×C4C23×Q811yescharacteristic
537C2×C4C22×#4111yescharacteristic
538C2×C4C2×2−1+412yes—
539C2×C4#58512yes—
540C8—44no—
541C8—44no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22#308014yes—
549C22C22×C22⋊Q811yescharacteristic
550C22#348511yescharacteristic
551C22#348611yescharacteristic
552C22C2×#57511yescharacteristic
553C22C2×#57511yescharacteristic
554C22#358511yescharacteristic
555C22#358611yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—44no—
566C4—44no—
567C4—44no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4#346112yes—
574C4#350212yes—
575C2—24no—
576C2#3515411yescharacteristic
577C2C2×#308011yescharacteristic
578C2#6049211yescharacteristic
579C2#6171111yescharacteristic
580C2#6171311yescharacteristic
581C2#6183111yescharacteristic
582C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #26525C2semidirect
8. #26523C2semidirect
9. #26459C2semidirect
10. #26457C2semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×#23702
#331512
#331502
#299732
#299722
#265252
#265232
#264592
#264572
#219552
#218382
#213652