Finite Group Databasebeta

← #6481943

Subgroups — #6481943

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3#33150C212yesmaximal
4#33149C211yesmaximal, characteristic
5#33147C211yesmaximal, characteristic
6#29957C212yesmaximal
7#29953C211yesmaximal, characteristic
8#29951C211yesmaximal, characteristic
9#26101C214yesmaximal, complemented
10#26100C214yesmaximal, complemented
11#26035C214yesmaximal, complemented
12#26034C214yesmaximal, complemented
13#21987C212yesmaximal
14#21955C211yesmaximal, characteristic
15#20741C212yesmaximal
16C22×C4×Q8C2211yescharacteristic
17C2×#432C2212yes—
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2851C2212yes—
21#2627—21no—
22#2627—21no—
23#2606—22no—
24#2605—22no—
25#2570C2212yes—
26#2570C2212yes—
27#2550C2212yes—
28#2550C2212yes—
29#2355—22no—
30#2355—22no—
31#2355C2214yes—
32#2355C2214yes—
33#2329—21no—
34#2329—21no—
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#387—21no—
38C2×#387—21no—
39C2×#384C2211yescharacteristic
40#2283C2212yes—
41C2×#382—21no—
42C2×C4×C4⋊C4—21no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2097C2214yes—
47#2096C2212yes—
48#2095C2212yes—
49#2068C2214yes—
50#2068C2214yes—
51#2066C2214yes—
52#2066C2214yes—
53#2026C2214yes—
54#2024C2212yes—
55#2024C2212yes—
56#1989C2214yes—
57#1988C2212yes—
58#1987C2212yes—
59#1949C2214yes—
60#1948C2212yes—
61#1948C2214yes—
62#1947C2212yes—
63#1947C2214yes—
64#1940C2214yes—
65#1940C2214yes—
66#1921C2214yes—
67#1920C2214yes—
68#1919C2214yes—
69#1918C2214yes—
70#1853C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1847C2214yes—
75#1836C2214yes—
76#1824C2212yes—
77#1824C2212yes—
78#1805C2214yes—
79#1805C2214yes—
80#1797C2211yescharacteristic
81#1797C2212yes—
82#1797C2211yescharacteristic
83#1787C2212yes—
84#1787C2212yes—
85#1763C2214yes—
86#1763C2214yes—
87C23×Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×#125C2312yes—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C4×Q8—22no—
94C2×C4×Q8—24no—
95C2×C4×Q8Q818yescomplemented
96C2×C4×Q8C2314yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C42—21no—
106C22×C42—21no—
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111#432—28no—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2314yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2314yes—
119C2×C4⋊C8C2312yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126C2×#97C2314yes—
127#400—22no—
128#400—22no—
129#398—22no—
130#398—22no—
131#398—22no—
132#398—22no—
133#395—22no—
134#395—22no—
135#391—21no—
136#391—21no—
137#391—22no—
138#391—21no—
139#391—21no—
140#391—22no—
141#391—22no—
142#391C2312yes—
143#391C2314yes—
144#391C2312yes—
145#389—21no—
146#389—21no—
147#389—22no—
148#389—22no—
149#389C2312yes—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—24no—
160#387—24no—
161#387—24no—
162#387—24no—
163#385—22no—
164#385—22no—
165#384—22no—
166#384C2311yescharacteristic
167#384C2311yescharacteristic
168#384C2314yes—
169#383—22no—
170#382—24no—
171#382—24no—
172#382C2312yes—
173#382C2312yes—
174#382C2312yes—
175#381—22no—
176#380—21no—
177#380—21no—
178#380C2312yes—
179#380C2312yes—
180C4×C4⋊C4—24no—
181C4×C4⋊C4—24no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4D412yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C22⋊C4—22no—
195#376—22no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95D411yescharacteristic
203C2×#95D411yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215C22×Q8—21no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8C2×Q814yes—
222C22×Q8C2412yes—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—42no—
237C4⋊Q8—42no—
238#125—44no—
239C22⋊Q8—44no—
240C22⋊Q8—44no—
241C4×Q8—216no—
242C4×Q8—28no—
243C4×Q8C2×Q8116yes—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×Q814yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—82no—
294C2×C42—82no—
295C2×C42C2×D412yes—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D412yes—
299C2×C42C2×Q814yes—
300C2×C42#4114yes—
301C2×C42C2412yes—
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2412yes—
305#107—44no—
306#107—44no—
307#106—44no—
308#106—44no—
309C4⋊C8—28no—
310C4⋊C8—44no—
311C4⋊C8—42no—
312C4⋊C8—42no—
313#103—44no—
314#103—44no—
315#103—44no—
316#103—44no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319#97—28no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95C2×D412yes—
344#95C2×D412yes—
345#95C2×D412yes—
346#95C2×D412yes—
347C24C22×D411yescharacteristic
348C2×Q8—24no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—48no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—48no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—81no—
359C2×Q8—81no—
360C2×Q8—81no—
361C2×Q8—81no—
362C2×Q8C22×Q814yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4C22⋊Q814yes—
401C22×C4C22⋊Q814yes—
402C22×C4#13714yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×Q812yes—
413C22×C4C2×#4112yes—
414C22×C4C2511yescharacteristic
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C4⋊C4—216no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—81no—
443C4⋊C4—82no—
444C4⋊C4—81no—
445C4⋊C4—82no—
446C4⋊C4—84no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4C22×Q814yes—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C42—216no—
455C42—24no—
456C42—22no—
457C42—22no—
458C42—41no—
459C42—41no—
460C42—42no—
461C42—82no—
462C42—82no—
463C42—82no—
464C42—82no—
465C42—84no—
466C42C22⋊Q818yes—
467C42C22×Q814yes—
468C42C2×#4114yes—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23C2×C22⋊Q812yes—
474C23C2×#13712yes—
475C23#57512yes—
476C23C23×D411yescharacteristic
477Q8—28no—
478Q8—48no—
479Q8—44no—
480Q8—162no—
481Q8—162no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4C2×C22⋊Q814yes—
541C2×C4C2×C22⋊Q812yes—
542C2×C4C2×C22⋊Q814yes—
543C2×C4C2×#13712yes—
544C2×C4C2×#13712yes—
545C2×C4C23×Q811yescharacteristic
546C2×C4C22×#4111yescharacteristic
547C2×C4#58512yes—
548C2×C4#58512yes—
549C8—44no—
550C8—44no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22#307814yes—
558C22C22×C22⋊Q811yescharacteristic
559C22#348511yescharacteristic
560C22#348711yescharacteristic
561C22C22×#13711yescharacteristic
562C22C2×#57511yescharacteristic
563C22#358511yescharacteristic
564C22#358611yescharacteristic
565C4—22no—
566C4—22no—
567C4—24no—
568C4—24no—
569C4—22no—
570C4—42no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—44no—
575C4—44no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4#345812yes—
583C4#349312yes—
584C2—24no—
585C2#3515111yescharacteristic
586C2C2×#307811yescharacteristic
587C2#6049211yescharacteristic
588C2#6170911yescharacteristic
589C2#6171211yescharacteristic
590C2#6189311yescharacteristic
591C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #26101C2semidirect
10. #26100C2semidirect
11. #26035C2semidirect
12. #26034C2semidirect
95. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#23552
#331502
#331492
#331472
#299572
#299532
#299512
#261012
#261002
#260352
#260342
#219872
#219552
#207412