Finite Group Databasebeta

← #6563360

Subgroups — #6563360

Order 512.

1437 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36314C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#30093C212yesmaximal, complemented
4#29937C211yesmaximal, characteristic
5#29390C212yesmaximal, complemented
6#29317C212yesmaximal, complemented
7#27474C214yesmaximal
8#27180C214yesmaximal
9#26507C212yesmaximal, complemented
10#25811C211yesmaximal, characteristic
11#25454C212yesmaximal, complemented
12#24302C211yesmaximal, characteristic
13#23782C212yesmaximal, complemented
14#23757C211yesmaximal, characteristic
15#23163C212yesmaximal, complemented
16#22971C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#498C2212yes—
20C2×#477C2212yes—
21#2864C2212yes—
22#2857C2211yescharacteristic
23#2571C2211yescharacteristic
24#2568C2211yescharacteristic
25#2533—22no—
26#2530—22no—
27#2511—21no—
28#2499—21no—
29#2495C2212yes—
30#2421C2212yes—
31#2369C2212yes—
32#2369C2212yes—
33#2365—21no—
34#2365—21no—
35#2365C2211yescharacteristic
36#2362—21no—
37#2356—21no—
38Q8×C4⋊C4C2211yescharacteristic
39#2345C2214yes—
40Q8×C22⋊C4—22no—
41#2331—22no—
42#2316—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C22⋊Q8C2211yescharacteristic
46#2295C2212yes—
47#2285C2211yescharacteristic
48#2274—21no—
49C4×#117C2211yescharacteristic
50#2101C2214yes—
51#2100C2211yescharacteristic
52#2100C2212yes—
53#2095C2211yescharacteristic
54#2086C2214yes—
55#2085C2214yes—
56#2084C2212yes—
57#2083C2211yescharacteristic
58#2077C2214yes—
59#2076C2212yes—
60#2072C2211yescharacteristic
61#2066C2214yes—
62#2058C2212yes—
63#2058C2212yes—
64#2057C2212yes—
65#2036C2212yes—
66#2024C2212yes—
67#1990C2212yes—
68#1988C2212yes—
69#1987C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1948C2212yes—
73#1944C2214yes—
74#1943C2214yes—
75#1939C2212yes—
76#1927C2214yes—
77#1924C2214yes—
78#1920C2214yes—
79#1920C2214yes—
80#1919C2214yes—
81#1918C2214yes—
82#1872C2212yes—
83#1865C2214yes—
84#1853C2214yes—
85#1852C2212yes—
86#1839C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89C4×#107C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92#1763C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×C4×Q8—21no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2312yes—
109C2×#117—21no—
110C2×#117D411yescharacteristic
111C2×#117D411yescharacteristic
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115#498—28no—
116#477—28no—
117C2×#107C2314yes—
118C2×#107C2312yes—
119C2×#107C2312yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2314yes—
125C2×#103C2314yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×#97C2314yes—
131C2×C4×C8C2312yes—
132#400—22no—
133#400—22no—
134#400—22no—
135#400—22no—
136#397—22no—
137#397—22no—
138#391—21no—
139#391—21no—
140#391—21no—
141#391—22no—
142#391—22no—
143#391—21no—
144#391—24no—
145#391—24no—
146#391C2312yes—
147#391C2311yescharacteristic
148#389—21no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—21no—
154#389—22no—
155#389C2314yes—
156#389C2314yes—
157#389C2311yescharacteristic
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#388—22no—
163#388—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—21no—
171#387—24no—
172#387—24no—
173#387—22no—
174#387—22no—
175#385—21no—
176#384—21no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#383—22no—
181#382—21no—
182#382—21no—
183#382C2312yes—
184#382C2312yes—
185#380—21no—
186#380—21no—
187#380—24no—
188#380C2311yescharacteristic
189#380C2312yes—
190#380C2312yes—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4—21no—
209C4×C22⋊C4—24no—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4C2312yes—
212#376—22no—
213#376D412yes—
214C43C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#336C2312yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2314yes—
230#336C2314yes—
231C22×Q8—28no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8—82no—
236C22×Q8—82no—
237C22×Q8C2411yescharacteristic
238C23×C4—22no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C22×C8C2412yes—
243C22×C8C2414yes—
244C22×C8C2412yes—
245C4⋊Q8—28no—
246C4⋊Q8—28no—
247C4⋊Q8—28no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8#4118yes—
251C22⋊Q8—22no—
252C22⋊Q8—41no—
253C22⋊Q8—44no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—42no—
257C4×Q8—44no—
258C4×Q8—84no—
259C4×Q8—84no—
260#117—24no—
261#117—42no—
262#117—44no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4C2×D412yes—
302C2×C22⋊C4C2×D412yes—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2412yes—
318C2×C42C2411yescharacteristic
319C2×C42C2412yes—
320C2×C42C2411yescharacteristic
321#107—24no—
322#107—28no—
323#107—42no—
324#106—28no—
325#106—28no—
326#106—44no—
327C4⋊C8—28no—
328C4⋊C8—24no—
329#103—24no—
330#103—24no—
331#103—28no—
332C22⋊C8—24no—
333C22⋊C8—42no—
334#97—44no—
335C4×C8—24no—
336#95—21no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—21no—
341#95—24no—
342#95—24no—
343#95—21no—
344#95—22no—
345#95—21no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—21no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363C24C22×D411yescharacteristic
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—44no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—44no—
374C2×Q8—82no—
375C2×Q8—82no—
376C2×Q8—816no—
377C2×Q82−1+412yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—41no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4#13714yes—
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2511yescharacteristic
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C82−1+414yes—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—28no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—81no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4C2×#4118yes—
456C22⋊C4—24no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—81no—
462C42—22no—
463C42—22no—
464C42—24no—
465C42—24no—
466C42—22no—
467C42—24no—
468C42—22no—
469C42—22no—
470C42—42no—
471C42—42no—
472C42—44no—
473C42—44no—
474C42—42no—
475C42—44no—
476C42—42no—
477C42—84no—
478C42—84no—
479C42C2×#4114yes—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23C2×#13712yes—
487C23C23×D411yescharacteristic
488Q8—24no—
489Q8—416no—
490Q8—42no—
491Q8—88no—
492Q8—84no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—88no—
546C2×C4#52512yes—
547C2×C4#52512yes—
548C2×C4C2×#13712yes—
549C2×C4C2×#13712yes—
550C2×C4#57512yes—
551C2×C4#57712yes—
552C2×C4C22×#4112yes—
553C2×C4C2×2−1+412yes—
554C2×C4C2×2−1+411yescharacteristic
555C2×C4#58511yescharacteristic
556C8—24no—
557C8—44no—
558C8—42no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22C2×#52511yescharacteristic
567C22#348911yescharacteristic
568C22#349911yescharacteristic
569C22C22×#13711yescharacteristic
570C22C2×#57511yescharacteristic
571C22C2×#57711yescharacteristic
572C22#358511yescharacteristic
573C4—22no—
574C4—22no—
575C4—22no—
576C4—42no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—42no—
581C4—48no—
582C4—42no—
583C4—82no—
584C4—81no—
585C4—82no—
586C4—82no—
587C4—81no—
588C4—81no—
589C4—82no—
590C4#311714yes—
591C4#351612yes—
592C4#352312yes—
593C2—42no—
594C2#3599711yescharacteristic
595C2C2×#311711yescharacteristic
596C2#6057911yescharacteristic
597C2#6211211yescharacteristic
598C2#6214511yescharacteristic
599C2#6247811yescharacteristic
600C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30093C2semidirect
5. #29390C2semidirect
6. #29317C2semidirect
9. #26507C2semidirect
11. #25454C2semidirect
13. #23782C2semidirect
15. #23163C2semidirect
16. #22971C2semidirect

Maximal subgroups

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

subgroupindex
#363142
#342162
#300932
#299372
#293902
#293172
#274742
#271802
#265072
#258112
#254542
#243022
#237822
#237572
#231632
#229712
#226432