Finite Group Databasebeta

← #6561644

Subgroups — #6561644

Order 512.

1392 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36454C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic
3#30366C212yesmaximal, complemented
4#30280C211yesmaximal, characteristic, complemented
5#29355C211yesmaximal, characteristic, complemented
6#29226C211yesmaximal, characteristic, complemented
7#29159C212yesmaximal, complemented
8#28852C214yesmaximal, complemented
9#28706C212yesmaximal
10#28598C212yesmaximal
11#28078C214yesmaximal, complemented
12#27855C212yesmaximal, complemented
13#23838C211yesmaximal, characteristic, complemented
14#23763C212yesmaximal, complemented
15#23221C212yesmaximal, complemented
16#23163C212yesmaximal, complemented
17#20558C211yesmaximal, characteristic
18C2×#498C2212yes—
19C2×#487C2212yes—
20C2×C8×D4C2212yes—
21#2868C2211yescharacteristic
22#2862C2211yescharacteristic
23#2857C2212yes—
24#2855C2212yes—
25#2575—22no—
26#2571—21no—
27#2563C2212yes—
28#2547C2211yescharacteristic
29#2542C2211yescharacteristic
30#2538—21no—
31#2517—22no—
32#2511—21no—
33#2497C2214yes—
34#2492—22no—
35#2489C2212yes—
36#2447—21no—
37#2439—22no—
38#2421C2212yes—
39#2369C2212yes—
40#2367—21no—
41#2365C2211yescharacteristic
42#2365C2211yescharacteristic
43#2355—21no—
44#2355—21no—
45#2354—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#125C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2099C2212yes—
52#2099C2212yes—
53#2096C2212yes—
54#2096C2214yes—
55#2096C2212yes—
56#2085C2212yes—
57#2084C2211yescharacteristic
58#2083C2211yescharacteristic
59#2077C2214yes—
60#2077C2214yes—
61#2072C2212yes—
62#2059C2212yes—
63#2058C2211yescharacteristic
64#2057C2211yescharacteristic
65#2036C2214yes—
66#2036C2214yes—
67#2024C2212yes—
68#1992C2212yes—
69#1988C2212yes—
70#1987C2212yes—
71#1987C2212yes—
72#1981C2212yes—
73#1944C2214yes—
74#1935C2214yes—
75#1935C2214yes—
76#1927C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1865C2214yes—
82#1836C2212yes—
83#1807C2214yes—
84#1807C2214yes—
85#1776C2214yes—
86C4×#107C2212yes—
87C4×#106C2212yes—
88#1770C2212yes—
89#1767C2211yescharacteristic
90#1763C2214yes—
91#1752C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94C2×#336C2212yes—
95C23×C8C2314yes—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125C2311yescharacteristic
103C2×#124—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×#117—22no—
114C22×C42C2312yes—
115#498—28no—
116#487—28no—
117C8×D4—28no—
118C2×#107C2312yes—
119C2×#107C2312yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2312yes—
123C2×#106C2312yes—
124C2×C4⋊C8C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×C22⋊C8C2314yes—
130C2×#97C2312yes—
131C2×#97C2314yes—
132C2×C4×C8C2312yes—
133#400—24no—
134#400—22no—
135#400—22no—
136#399—22no—
137#397—22no—
138#397—22no—
139#397—22no—
140#396—24no—
141#393—24no—
142#391—21no—
143#391—21no—
144#391—21no—
145#391—22no—
146#391—21no—
147#391—21no—
148#391—24no—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#390—22no—
153#390—21no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389C2311yescharacteristic
159#389C2311yescharacteristic
160#389C2312yes—
161#389C2312yes—
162#389C2314yes—
163#388—22no—
164#388—22no—
165#388—24no—
166#387—21no—
167#387—21no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—24no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#382—22no—
184#382—22no—
185#382—21no—
186#382—21no—
187#382—22no—
188#382C2311yescharacteristic
189#381—22no—
190#380—22no—
191#380—22no—
192#380C2314yes—
193#380C2314yes—
194#380C2312yes—
195#380C2312yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215C43C2311yescharacteristic
216#340C2312yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#336C2311yescharacteristic
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8C2412yes—
235C22×D4C2×D411yescharacteristic
236C23×C4C2×D412yes—
237C23×C4C2×D412yes—
238C23×C4C2412yes—
239C22×C8—28no—
240C22×C8—44no—
241C22×C8—44no—
242C22×C8C2414yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—42no—
250C4⋊Q8—82no—
251C4⋊Q8—82no—
252#125—22no—
253#125—41no—
254#124—44no—
255C22⋊Q8—24no—
256C22⋊Q8—42no—
257C22⋊Q8—44no—
258C22⋊Q8—84no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—42no—
265#117—84no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
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—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—44no—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#107—24no—
325#107—42no—
326#106—28no—
327#106—28no—
328#106—42no—
329#106—42no—
330C4⋊C8—24no—
331#103—28no—
332#103—28no—
333#103—44no—
334#103—44no—
335C22⋊C8—28no—
336#97—42no—
337#97—44no—
338C4×C8#4118yes—
339#95—21no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—22no—
344#95—21no—
345#95—22no—
346#95—22no—
347#95—21no—
348#95—21no—
349#95—22no—
350#95—24no—
351#95—22no—
352#95—21no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D412yes—
364C24C22×D412yes—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—82no—
377C2×Q8—82no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4#13714yes—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
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—22no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—88no—
418C2×C8C2×#4114yes—
419C2×C8C2×#4114yes—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—84no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C42−1+412yes—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—84no—
464C22⋊C4—82no—
465C22⋊C4—84no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—24no—
473C42—22no—
474C42—42no—
475C42—42no—
476C42—44no—
477C42—42no—
478C42—42no—
479C42—44no—
480C42—84no—
481C42—82no—
482C42C2×#4114yes—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23C2×#13714yes—
490C23C23×D411yescharacteristic
491Q8—44no—
492Q8—44no—
493Q8—84no—
494Q8—84no—
495Q8—164no—
496D4—28no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—48no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—84no—
549C2×C4#53212yes—
550C2×C4#53212yes—
551C2×C4C2×#13712yes—
552C2×C4#57512yes—
553C2×C4#57712yes—
554C2×C4#57712yes—
555C2×C4C22×#4112yes—
556C2×C4C2×2−1+411yescharacteristic
557C2×C4#58512yes—
558C2×C4#58511yescharacteristic
559C8—24no—
560C8—24no—
561C8—44no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22C2×#53211yescharacteristic
570C22#348511yescharacteristic
571C22#348911yescharacteristic
572C22C22×#13711yescharacteristic
573C22C2×#57511yescharacteristic
574C22C2×#57711yescharacteristic
575C22C2×#57711yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—82no—
586C4—81no—
587C4—82no—
588C4—81no—
589C4—84no—
590C4—82no—
591C4—81no—
592C4—82no—
593C4#318114yes—
594C4#350712yes—
595C4#351012yes—
596C2—44no—
597C2#3595911yescharacteristic
598C2C2×#318111yescharacteristic
599C2#6071211yescharacteristic
600C2#6173111yescharacteristic
601C2#6173511yescharacteristic
602C2#6213111yescharacteristic
603C2#6213411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30366C2semidirect
4. #30280C2semidirect
5. #29355C2semidirect
6. #29226C2semidirect
7. #29159C2semidirect
8. #28852C2semidirect
11. #28078C2semidirect
12. #27855C2semidirect
13. #23838C2semidirect
14. #23763C2semidirect
15. #23221C2semidirect
16. #23163C2semidirect

Maximal subgroups

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

subgroupindex
#364542
#361942
#303662
#302802
#293552
#292262
#291592
#288522
#287062
#285982
#280782
#278552
#238382
#237632
#232212
#231632
#205582