Finite Group Databasebeta

← #6561628

Subgroups — #6561628

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36476C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic
3#30367C212yesmaximal, complemented
4#30279C211yesmaximal, characteristic, complemented
5#29347C211yesmaximal, characteristic, complemented
6#29218C211yesmaximal, characteristic, complemented
7#29166C212yesmaximal, complemented
8#28871C214yesmaximal, complemented
9#28711C212yesmaximal
10#28608C212yesmaximal
11#28088C214yesmaximal, complemented
12#27864C212yesmaximal, complemented
13#23837C211yesmaximal, characteristic, complemented
14#23762C212yesmaximal, complemented
15#23221C212yesmaximal, complemented
16#23162C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic
18C2×#500C2212yes—
19C2×#487C2212yes—
20C2×#436C2212yes—
21#2868C2211yescharacteristic
22#2862C2211yescharacteristic
23#2857C2212yes—
24#2855C2212yes—
25#2607—22no—
26#2589—22no—
27#2580—21no—
28#2571—21no—
29#2570—21no—
30#2563C2211yescharacteristic
31#2563C2212yes—
32#2542C2211yescharacteristic
33#2538—21no—
34#2497C2214yes—
35#2492—22no—
36#2492C2212yes—
37#2439—22no—
38#2423C2212yes—
39#2369C2212yes—
40#2369C2211yescharacteristic
41#2365C2211yescharacteristic
42#2357—21no—
43#2355—21no—
44#2355—21no—
45#2354—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2100C2212yes—
52#2100C2212yes—
53#2095C2212yes—
54#2095C2214yes—
55#2095C2212yes—
56#2084C2212yes—
57#2084C2211yescharacteristic
58#2084C2211yescharacteristic
59#2075C2214yes—
60#2075C2214yes—
61#2072C2212yes—
62#2058C2211yescharacteristic
63#2058C2212yes—
64#2058C2211yescharacteristic
65#2035C2214yes—
66#2035C2214yes—
67#2024C2212yes—
68#1990C2212yes—
69#1987C2212yes—
70#1987C2212yes—
71#1987C2212yes—
72#1981C2212yes—
73#1943C2214yes—
74#1935C2214yes—
75#1933C2214yes—
76#1927C2214yes—
77#1918C2214yes—
78#1918C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1865C2214yes—
82#1834C2212yes—
83#1809C2214yes—
84#1809C2214yes—
85#1776C2214yes—
86C4×#106C2212yes—
87C4×#106C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×M16C2314yes—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×D4D411yescharacteristic
110C2×C4×D4D411yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C2×#117—22no—
113C22×C42C2312yes—
114#500—28no—
115#487—28no—
116#436—28no—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106C2312yes—
120C2×#106C2312yes—
121C2×#106C2312yes—
122C2×#106C2312yes—
123C2×C4⋊C8C2312yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2314yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2312yes—
132#400—22no—
133#400—24no—
134#399—22no—
135#399—22no—
136#398—22no—
137#396—24no—
138#393—22no—
139#393—24no—
140#393—22no—
141#391—21no—
142#391—21no—
143#391—21no—
144#391—22no—
145#391—21no—
146#391—22no—
147#391—22no—
148#391—24no—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#390—21no—
153#390—22no—
154#390—21no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389C2311yescharacteristic
162#389C2311yescharacteristic
163#389C2312yes—
164#389C2312yes—
165#389C2314yes—
166#388—22no—
167#388—22no—
168#388—24no—
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#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#382—21no—
185#382—22no—
186#382—21no—
187#382—22no—
188#382—21no—
189#381—22no—
190#380—22no—
191#380—22no—
192#380C2312yes—
193#380C2312yes—
194#380C2314yes—
195#380C2314yes—
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⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
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×Q8—21no—
235C22×Q8C2412yes—
236C22×D4C2×D411yescharacteristic
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2412yes—
240C2×M16—28no—
241C2×M16—44no—
242C2×M16—44no—
243C22×C8C2414yes—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C4⋊Q8—22no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—41no—
252C4⋊Q8—42no—
253C4⋊Q8—82no—
254C4⋊Q8—82no—
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—24no—
265#117—84no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
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—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—21no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—44no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325#107—28no—
326#107—28no—
327#106—24no—
328#106—42no—
329#106—42no—
330#106—42no—
331C4⋊C8—24no—
332#103—28no—
333#103—28no—
334#103—44no—
335#103—44no—
336C22⋊C8—28no—
337#97—44no—
338C4×C8—42no—
339C4×C8#4118yes—
340#95—21no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—22no—
345#95—21no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—22no—
352#95—21no—
353#95—22no—
354#95—21no—
355#95—22no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363C24C22×D412yes—
364C2×Q8—24no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—44no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—81no—
376C2×Q8—82no—
377C2×Q8—82no—
378C2×Q8—81no—
379C2×Q8—81no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C2511yescharacteristic
412M16—88no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8C2×#4114yes—
421C2×C8C2×#4114yes—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—81no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—84no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C42−1+412yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—44no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4—84no—
467C22⋊C4—82no—
468C22⋊C4—84no—
469C42—21no—
470C42—21no—
471C42—24no—
472C42—22no—
473C42—42no—
474C42—42no—
475C42—41no—
476C42—41no—
477C42—42no—
478C42—42no—
479C42—44no—
480C42—82no—
481C42—84no—
482C42—82no—
483C42#13714yes—
484C42#13714yes—
485C42C2×#4114yes—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23C23×D411yescharacteristic
494Q8—44no—
495Q8—44no—
496Q8—84no—
497Q8—84no—
498Q8—164no—
499Q8—162no—
500D4—44no—
501C2×C4—21no—
502C2×C4—21no—
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—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—48no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4#53212yes—
552C2×C4#53212yes—
553C2×C4C2×#13712yes—
554C2×C4C2×#13712yes—
555C2×C4C2×#13712yes—
556C2×C4C2×#13712yes—
557C2×C4C2×#13712yes—
558C2×C4C2×#13712yes—
559C2×C4C22×#4112yes—
560C2×C4C2×2−1+411yescharacteristic
561C2×C4#58512yes—
562C2×C4#58511yescharacteristic
563C8—24no—
564C8—24no—
565C8—44no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C2×#53211yescharacteristic
574C22#348511yescharacteristic
575C22#348911yescharacteristic
576C22C22×#13711yescharacteristic
577C22C22×#13711yescharacteristic
578C22#358511yescharacteristic
579C22#358511yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—82no—
592C4—84no—
593C4—82no—
594C4—82no—
595C4—81no—
596C4—82no—
597C4#318514yes—
598C4#350712yes—
599C4#351012yes—
600C2—82no—
601C2#3595911yescharacteristic
602C2C2×#318511yescharacteristic
603C2#6071911yescharacteristic
604C2#6171911yescharacteristic
605C2#6172111yescharacteristic
606C2#6210811yescharacteristic
607C2#6211411yescharacteristic
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. #30367C2semidirect
4. #30279C2semidirect
5. #29347C2semidirect
6. #29218C2semidirect
7. #29166C2semidirect
8. #28871C2semidirect
11. #28088C2semidirect
12. #27864C2semidirect
13. #23837C2semidirect
14. #23762C2semidirect
15. #23221C2semidirect
16. #23162C2semidirect

Maximal subgroups

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

subgroupindex
#364762
#361942
#303672
#302792
#293472
#292182
#291662
#288712
#287112
#286082
#280882
#278642
#238372
#237622
#232212
#231622
#205682