Finite Group Databasebeta

← #6568262

Subgroups — #6568262

Order 512.

1503 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#34334C211yesmaximal, characteristic
3#30175C212yesmaximal, complemented
4#29869C211yesmaximal, characteristic, complemented
5#29255C211yesmaximal, characteristic, complemented
6#29111C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#27583C212yesmaximal, complemented
9#27537C214yesmaximal, complemented
10#27117C214yesmaximal, complemented
11#26300C212yesmaximal
12#25888C212yesmaximal
13#23776C211yesmaximal, characteristic, complemented
14#23763C212yesmaximal, complemented
15#23221C212yesmaximal, complemented
16#23069C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic
18C2×D4×Q8C2211yescharacteristic
19C2×#487C2212yes—
20C2×#477C2212yes—
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2571—21no—
24#2563C2212yes—
25#2538—21no—
26#2497C2214yes—
27#2492—22no—
28#2439—22no—
29#2369—21no—
30#2369C2212yes—
31#2368—22no—
32#2365C2211yescharacteristic
33#2360C2212yes—
34#2355—21no—
35#2355C2211yescharacteristic
36#2354—21no—
37#2349—21no—
38D4×C4⋊C4C2211yescharacteristic
39Q8×C22⋊C4C2212yes—
40#2331—21no—
41#2329—22no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C22⋊Q8C2212yes—
45#2295C2211yescharacteristic
46C2×#391C2212yes—
47#2285C2212yes—
48#2275—21no—
49C42×D4C2211yescharacteristic
50#2099C2211yescharacteristic
51#2096C2212yes—
52#2096C2211yescharacteristic
53#2088C2212yes—
54#2083C2212yes—
55#2083C2212yes—
56#2077C2214yes—
57#2077C2214yes—
58#2072C2212yes—
59#2072C2212yes—
60#2072C2212yes—
61#2072C2212yes—
62#2066C2212yes—
63#2058C2211yescharacteristic
64#2036C2214yes—
65#2036C2214yes—
66#2024C2212yes—
67#2024C2211yescharacteristic
68#1990C2212yes—
69#1987C2212yes—
70#1981C2212yes—
71#1954C2212yes—
72#1948C2212yes—
73#1935C2214yes—
74#1935C2214yes—
75#1927C2214yes—
76#1927C2214yes—
77#1924C2214yes—
78#1918C2214yes—
79#1865C2214yes—
80#1865C2214yes—
81#1853C2214yes—
82#1831C2211yescharacteristic
83#1823C2214yes—
84#1817C2212yes—
85#1803C2214yes—
86#1787C2214yes—
87#1787C2214yes—
88#1776C2214yes—
89C4×#106C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C23×Q8C2312yes—
96D4×Q8—24no—
97D4×Q8—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2312yes—
107C2×C4×Q8—21no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C22×C4⋊C4C2312yes—
118C22×C42C2312yes—
119#487—28no—
120#477—28no—
121C2×#107C2314yes—
122C2×#106C2312yes—
123C2×#106C2314yes—
124C2×#106C2312yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138#400—24no—
139#396—24no—
140#393—24no—
141#391—21no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—24no—
148#391C2312yes—
149#391C2311yescharacteristic
150#391C2311yescharacteristic
151#391C2312yes—
152#391C2312yes—
153#390—22no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2311yescharacteristic
164#389C2311yescharacteristic
165#389C2314yes—
166#388—24no—
167#387—21no—
168#387—22no—
169#387—21no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—24no—
174#385—22no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384—21no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#382—22no—
184#382—21no—
185#382—22no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380C2314yes—
190#380C2314yes—
191#380C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4C2312yes—
214#376—21no—
215C2×#95—22no—
216C2×#95—22no—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2314yes—
232C22×Q8—24no—
233C22×Q8—24no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×Q8C2412yes—
240C22×Q8C2411yescharacteristic
241C22×D4C2×D411yescharacteristic
242C23×C4—22no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—28no—
251C4⋊Q8—22no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—42no—
255C4⋊Q8—82no—
256C4⋊Q8—82no—
257C4⋊Q8#4114yes—
258C22⋊Q8—216no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—84no—
262C4×Q8—22no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×Q8#4114yes—
267C4×D4—28no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—44no—
273#117—84no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—44no—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—28no—
332#106—42no—
333#106—44no—
334#106—42no—
335C4⋊C8—24no—
336C4⋊C8—42no—
337#103—28no—
338#103—28no—
339#103—28no—
340#103—44no—
341#103—44no—
342#103—42no—
343#103—42no—
344C22⋊C8—44no—
345C22⋊C8—44no—
346#97—44no—
347C4×C8—24no—
348#95—21no—
349#95—21no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—24no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—24no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24C22×D412yes—
373C2×Q8—28no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—82no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8—82no—
388C2×Q8—88no—
389C2×Q8—88no—
390C2×Q8C2×#4112yes—
391C2×Q8C2×#4114yes—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4#13714yes—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2511yescharacteristic
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C4⋊C4—28no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—84no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4C2×#4112yes—
474C22⋊C4—28no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—44no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C42—24no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—42no—
488C42—42no—
489C42—41no—
490C42—42no—
491C42—44no—
492C42—82no—
493C42—84no—
494C42—82no—
495C42—82no—
496C42#13714yes—
497C42C2×#4112yes—
498C42C2×#4112yes—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23C2×#13714yes—
506C23C23×D411yescharacteristic
507Q8—24no—
508Q8—28no—
509Q8—44no—
510Q8—42no—
511Q8—84no—
512Q8—88no—
513Q8—164no—
514D4—28no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—48no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—84no—
566C2×C4—88no—
567C2×C4—164no—
568C2×C4#52512yes—
569C2×C4#52512yes—
570C2×C4C2×#13712yes—
571C2×C4C2×#13712yes—
572C2×C4C2×#13712yes—
573C2×C4C2×#13712yes—
574C2×C4#57712yes—
575C2×C4C22×#4111yescharacteristic
576C2×C4C22×#4111yescharacteristic
577C2×C4#58512yes—
578C2×C4#58512yes—
579C8—44no—
580C8—42no—
581C8—42no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C2×#52511yescharacteristic
590C22#348511yescharacteristic
591C22#350011yescharacteristic
592C22C22×#13711yescharacteristic
593C22C22×#13711yescharacteristic
594C22C2×#57711yescharacteristic
595C22#358511yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—44no—
607C4—82no—
608C4—81no—
609C4—82no—
610C4—82no—
611C4—81no—
612C4—84no—
613C4#311314yes—
614C4#351012yes—
615C4#352112yes—
616C2—44no—
617C2#3603311yescharacteristic
618C2C2×#311311yescharacteristic
619C2#6059311yescharacteristic
620C2#6172111yescharacteristic
621C2#6173511yescharacteristic
622C2#6254011yescharacteristic
623C2#6258211yescharacteristic
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. #30175C2semidirect
4. #29869C2semidirect
5. #29255C2semidirect
6. #29111C2semidirect
7. #28717C2semidirect
8. #27583C2semidirect
9. #27537C2semidirect
10. #27117C2semidirect
13. #23776C2semidirect
14. #23763C2semidirect
15. #23221C2semidirect
16. #23069C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#343342
#301752
#298692
#292552
#291112
#287172
#275832
#275372
#271172
#263002
#258882
#237762
#237632
#232212
#230692
#228392