Finite Group Databasebeta

← #6498084

Subgroups — #6498084

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2C2×#2295C211yesmaximal, characteristic
3#33147C212yesmaximal
4#32954C212yesmaximal
5#29983C212yesmaximal, complemented
6#29866C212yesmaximal, complemented
7#29575C214yesmaximal, complemented
8#29390C214yesmaximal, complemented
9#29297C212yesmaximal
10#29105C212yesmaximal
11#27611C212yesmaximal
12#27583C212yesmaximal
13#22112C214yesmaximal, complemented
14#20777C211yesmaximal, characteristic
15C2×#533—21no—
16C22×C4×Q8C2211yescharacteristic
17C2×#475C2214yes—
18#2868C2211yescharacteristic
19#2868C2211yescharacteristic
20#2862—21no—
21#2859—21no—
22#2859C2212yes—
23#2857—21no—
24#2857—21no—
25#2857C2212yes—
26#2857C2212yes—
27C4×C4⋊Q8—21no—
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8C2212yes—
30C4×C4⋊Q8C2212yes—
31#2295—21no—
32#2295—21no—
33#2295—21no—
34#2295—21no—
35#2295C2212yes—
36#2295C2212yes—
37#2295C2212yes—
38#2295C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#387—21no—
44C2×#387—21no—
45#2285—21no—
46#2285C2212yes—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×#376—21no—
50C2×C43C2211yescharacteristic
51#2099C2214yes—
52#2096C2214yes—
53#2084C2214yes—
54#2084C2214yes—
55#2072C2214yes—
56#2072C2214yes—
57#2058C2214yes—
58#2024C2214yes—
59#1988C2214yes—
60#1948C2214yes—
61#1941C2218yes—
62#1933C2218yes—
63#1927C2218yes—
64#1920C2218yes—
65#1919C2218yes—
66#1865C2218yes—
67#1853C2214yes—
68#1853C2214yes—
69#1839C2212yes—
70#1831C2212yes—
71#1824C2214yes—
72#1817C2212yes—
73#1803C2214yes—
74#1797C2212yes—
75#1787C2214yes—
76C4×C4⋊C8C2212yes—
77C4×C22⋊C8C2212yes—
78#1744C2212yes—
79C23×Q8C2311yescharacteristic
80#533—48no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×#125—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C4×Q8—28no—
91C2×C4×Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×#117—22no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C42—21no—
105C22×C42—21no—
106C22×C42D411yescharacteristic
107C22×C42D411yescharacteristic
108C22×C42D411yescharacteristic
109C22×C42D411yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#475—216no—
114C2×#106C2318yes—
115C2×#106C2318yes—
116C2×C4⋊C8C2314yes—
117C2×#103C2318yes—
118C2×#103C2318yes—
119C2×C22⋊C8C2314yes—
120C2×C4×C8C2314yes—
121#391—22no—
122#391—22no—
123#391—22no—
124#391—22no—
125#391—21no—
126#391—21no—
127#391—21no—
128#391—21no—
129#391C2312yes—
130#391C2312yes—
131#391C2312yes—
132#391C2312yes—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—21no—
143#389C2314yes—
144#389C2314yes—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#389C2312yes—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#384—21no—
158#384—21no—
159#384—21no—
160#384—21no—
161#384C2312yes—
162#384C2312yes—
163#384C2312yes—
164#384C2312yes—
165#382—22no—
166#382—22no—
167#380—22no—
168#380—22no—
169#380—22no—
170#380C2314yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4C2314yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183#376—22no—
184#376—22no—
185#376—22no—
186#376—22no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C43D411yescharacteristic
195C43D411yescharacteristic
196C43D411yescharacteristic
197C43D411yescharacteristic
198C43C2311yescharacteristic
199C43C2311yescharacteristic
200C43C2311yescharacteristic
201C43C2311yescharacteristic
202#340C2318yes—
203#340C2318yes—
204#340C2318yes—
205#340C2318yes—
206#336C2314yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210C22×Q8—24no—
211C22×Q8—21no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—24no—
215C22×Q8—42no—
216C22×Q8C2412yes—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2411yescharacteristic
228C22×C8C2418yes—
229C4⋊Q8—28no—
230C4⋊Q8—44no—
231C4⋊Q8—44no—
232C4⋊Q8—44no—
233C4⋊Q8—44no—
234C4⋊Q8—44no—
235#125—48no—
236C22⋊Q8—48no—
237C22⋊Q8—48no—
238C4×Q8—28no—
239C4×Q8—28no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—48no—
244C4×Q8—48no—
245C4×Q8#4118yes—
246#117—48no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4—24no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42SD1618yes—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322#106—216no—
323#106—48no—
324C4⋊C8—28no—
325#103—216no—
326#103—48no—
327C22⋊C8—28no—
328C4×C8—44no—
329#95—22no—
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#95—22no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347C24C22×D411yescharacteristic
348C2×Q8—24no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—416no—
353C2×Q8—42no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—42no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8C2×#4114yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4C2×SD1614yes—
397C22×C4C2×SD1614yes—
398C22×C4C2×SD1614yes—
399C22×C4#13714yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C42−1+414yes—
414C22×C4C2511yescharacteristic
415C2×C8—28no—
416C2×C8—28no—
417C2×C8—28no—
418C4⋊C4—24no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—22no—
422C4⋊C4—28no—
423C4⋊C4—24no—
424C4⋊C4—28no—
425C4⋊C4—24no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4C2×#4114yes—
445C22⋊C4—48no—
446C42—28no—
447C42—28no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—44no—
472C42—48no—
473C42—44no—
474C42C2×SD1614yes—
475C42C2×SD1614yes—
476C42C2×SD1614yes—
477C42C2×SD1614yes—
478C42C2×#4114yes—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23#53612yes—
484C23C22×SD1612yes—
485C23C2×#13712yes—
486C23C23×D411yescharacteristic
487Q8—28no—
488Q8—48no—
489Q8—44no—
490Q8—88no—
491Q8—88no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—28no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—48no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4#52512yes—
548C2×C4#52512yes—
549C2×C4#53612yes—
550C2×C4C22×SD1612yes—
551C2×C4C22×SD1612yes—
552C2×C4C22×SD1612yes—
553C2×C4C22×SD1612yes—
554C2×C4C22×SD1612yes—
555C2×C4C22×SD1612yes—
556C2×C4C2×#13712yes—
557C2×C4C2×#13712yes—
558C2×C4#57512yes—
559C2×C4#57512yes—
560C2×C4C22×#4112yes—
561C2×C4C2×2−1+412yes—
562C2×C4#58512yes—
563C8—48no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22#322914yes—
571C22C2×#52511yescharacteristic
572C22C2×#53611yescharacteristic
573C22#344911yescharacteristic
574C22C23×SD1611yescharacteristic
575C22C22×#13711yescharacteristic
576C22C2×#57511yescharacteristic
577C22C2×#57511yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—22no—
587C4—22no—
588C4—41no—
589C4—44no—
590C4—41no—
591C4—44no—
592C4—84no—
593C4—82no—
594C4—84no—
595C4—82no—
596C4#310314yes—
597C4#349514yes—
598C2—24no—
599C2#3532511yescharacteristic
600C2C2×#310311yescharacteristic
601C2#6056211yescharacteristic
602C2C2×#322911yescharacteristic
603C2#6086611yescharacteristic
604C2#6094111yescharacteristic
605C2#6096011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #29983C2semidirect
6. #29866C2semidirect
7. #29575C2semidirect
8. #29390C2semidirect
13. #22112C2semidirect

Maximal subgroups

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

subgroupindex
#366452
C2×#22952
#331472
#329542
#299832
#298662
#295752
#293902
#292972
#291052
#276112
#275832
#221122
#207772