Finite Group Databasebeta

← #6560769

Subgroups — #6560769

Order 512.

1362 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2#36299C211yesmaximal, characteristic
3#30385C212yesmaximal, complemented
4#30228C211yesmaximal, characteristic, complemented
5#29572C212yesmaximal
6#29512C212yesmaximal, complemented
7#29320C211yesmaximal, characteristic, complemented
8#29020C214yesmaximal
9#28519C211yesmaximal, characteristic
10#28009C214yesmaximal, complemented
11#25471C212yesmaximal, complemented
12#25470C212yesmaximal
13#23817C211yesmaximal, characteristic
14#23781C212yesmaximal
15#23069C212yesmaximal, complemented
16#22969C212yesmaximal
17#20572C211yesmaximal, characteristic, complemented
18C2×#477C2212yes—
19C2×#476C2212yes—
20C2×C8×Q8C2212yes—
21#2868C2211yescharacteristic
22#2863C2212yes—
23#2859C2211yescharacteristic
24#2859C2212yes—
25#2582—21no—
26#2582C2211yescharacteristic
27#2567C2212yes—
28#2552—21no—
29#2550—22no—
30#2538C2211yescharacteristic
31#2515—22no—
32#2509C2212yes—
33#2507C2214yes—
34#2495—21no—
35#2475—22no—
36#2451—22no—
37#2447C2212yes—
38#2421—21no—
39#2369C2212yes—
40#2360—21no—
41#2355—21no—
42#2355C2211yescharacteristic
43Q8×C4⋊C4—21no—
44D4×C4⋊C4C2211yescharacteristic
45#2344—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#124C2211yescharacteristic
49#2293C2212yes—
50C42×Q8C2211yescharacteristic
51#2101C2212yes—
52#2099C2212yes—
53#2099C2214yes—
54#2099C2212yes—
55#2096C2212yes—
56#2093C2212yes—
57#2088C2214yes—
58#2087C2214yes—
59#2083C2212yes—
60#2072C2211yescharacteristic
61#2071C2211yescharacteristic
62#2068C2214yes—
63#2065C2214yes—
64#2058C2212yes—
65#2042C2212yes—
66#2024C2211yescharacteristic
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1990C2212yes—
70#1959C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1944C2214yes—
74#1944C2214yes—
75#1942C2214yes—
76#1941C2214yes—
77#1935C2214yes—
78#1934C2214yes—
79#1933C2214yes—
80#1927C2214yes—
81#1920C2214yes—
82#1918C2214yes—
83#1862C2214yes—
84#1847C2212yes—
85#1776C2214yes—
86#1770C2212yes—
87#1770C2214yes—
88#1767C2211yescharacteristic
89#1767C2212yes—
90#1762C2214yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1744C2211yescharacteristic
94#1744C2212yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125—21no—
100C2×#125—22no—
101C2×#124C2311yescharacteristic
102C2×C22⋊Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—22no—
105C2×C4×Q8D411yescharacteristic
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×D4C2311yescharacteristic
110C2×C4×D4C2312yes—
111C2×#117—22no—
112C22×C4⋊C4—21no—
113#477—28no—
114#476—28no—
115C8×Q8—28no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2314yes—
121C2×C4⋊C8C2312yes—
122C2×#103C2314yes—
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#102C2312yes—
126C2×#102C2314yes—
127C2×#102C2312yes—
128C2×#97C2312yes—
129C2×#97C2314yes—
130C2×C4×C8C2314yes—
131C2×C4×C8C2312yes—
132#400—22no—
133#400—22no—
134#398—24no—
135#398—22no—
136#398—24no—
137#397—22no—
138#397—22no—
139#396—22no—
140#395—24no—
141#391—22no—
142#391—21no—
143#391—22no—
144#391—21no—
145#391—24no—
146#391C2312yes—
147#391C2311yescharacteristic
148#391C2312yes—
149#390C2312yes—
150#390C2311yescharacteristic
151#389—22no—
152#389—21no—
153#389—22no—
154#389—21no—
155#389—22no—
156#389—22no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—24no—
161#389C2314yes—
162#389C2312yes—
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#389C2312yes—
170#388—22no—
171#388—21no—
172#388—22no—
173#388—21no—
174#387—24no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#385—22no—
181#385—22no—
182#384—22no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#383—22no—
191#382—21no—
192#382—22no—
193#382—21no—
194#382C2311yescharacteristic
195#381—22no—
196#380—21no—
197#380—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C43C2312yes—
215C43C2311yescharacteristic
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#336C2312yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8C2×D411yescharacteristic
234C22×Q8C2411yescharacteristic
235C22×D4C2411yescharacteristic
236C23×C4—21no—
237C23×C4—22no—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C22×C8C2414yes—
241C4⋊Q8—24no—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—82no—
245C4⋊Q8—82no—
246#125—84no—
247#125—82no—
248#124—22no—
249#124—41no—
250C22⋊Q8—44no—
251C4×Q8—24no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—42no—
255C4×Q8—44no—
256C4×Q8—84no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—42no—
260#117—84no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—22no—
305C2×C42—22no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42C2×D412yes—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2412yes—
318C2×C42C2412yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322#107—28no—
323#107—44no—
324#106—28no—
325#106—28no—
326C4⋊C8—28no—
327C4⋊C8#4118yes—
328#103—28no—
329#103—24no—
330#103—42no—
331#102—24no—
332#102—28no—
333#102—24no—
334#97—44no—
335#97—42no—
336C4×C8—28no—
337C4×C8—24no—
338#95—22no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—22no—
347#95—24no—
348#95—22no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—22no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361C24—21no—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—42no—
370C2×Q8—44no—
371C2×Q8—42no—
372C2×Q8—82no—
373C2×Q8—82no—
374C2×Q8#13714yes—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D42−1+412yes—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—82no—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2511yescharacteristic
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8C2×#4118yes—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—82no—
443C4⋊C4—84no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C42−1+414yes—
452C22⋊C4—44no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—82no—
460C42—24no—
461C42—22no—
462C42—22no—
463C42—24no—
464C42—22no—
465C42—24no—
466C42—22no—
467C42—24no—
468C42—22no—
469C42—21no—
470C42—22no—
471C42—22no—
472C42—21no—
473C42—42no—
474C42—42no—
475C42—44no—
476C42—82no—
477C42—84no—
478C42C2×#4114yes—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23C23×D411yescharacteristic
487Q8—28no—
488Q8—48no—
489Q8—42no—
490Q8—42no—
491Q8—164no—
492D4—24no—
493D4—42no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—88no—
542C2×C4—84no—
543C2×C4#53212yes—
544C2×C4#53212yes—
545C2×C4C2×#13714yes—
546C2×C4C2×#13712yes—
547C2×C4#57512yes—
548C2×C4#57612yes—
549C2×C4#57712yes—
550C2×C4C22×#4112yes—
551C2×C4C2×2−1+412yes—
552C2×C4C2×2−1+411yescharacteristic
553C2×C4#58511yescharacteristic
554C8—28no—
555C8—42no—
556C8—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22C2×#53211yescharacteristic
565C22#348611yescharacteristic
566C22#348911yescharacteristic
567C22C22×#13711yescharacteristic
568C22C2×#57511yescharacteristic
569C22C2×#57611yescharacteristic
570C22C2×#57711yescharacteristic
571C4—24no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—42no—
576C4—44no—
577C4—41no—
578C4—42no—
579C4—42no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—84no—
584C4—81no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4#318114yes—
589C4#351512yes—
590C4#352312yes—
591C2—82no—
592C2#3594911yescharacteristic
593C2C2×#318111yescharacteristic
594C2#6072211yescharacteristic
595C2#6184111yescharacteristic
596C2#6184411yescharacteristic
597C2#6214311yescharacteristic
598C2#6214711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36563C2semidirect
3. #30385C2semidirect
4. #30228C2semidirect
6. #29512C2semidirect
7. #29320C2semidirect
10. #28009C2semidirect
11. #25471C2semidirect
15. #23069C2semidirect
17. #20572C2semidirect

Maximal subgroups

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

subgroupindex
#365632
#362992
#303852
#302282
#295722
#295122
#293202
#290202
#285192
#280092
#254712
#254702
#238172
#237812
#230692
#229692
#205722