Finite Group Databasebeta

← #7569935

Subgroups — #7569935

Order 512.

1415 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59869C211yesmaximal, characteristic
2#36593C211yesmaximal, characteristic
3#32145C214yesmaximal
4#29219C211yesmaximal, characteristic
5#28792C212yesmaximal
6#28689C211yesmaximal, characteristic
7#28675C212yesmaximal
8#27295C214yesmaximal
9#26445C212yesmaximal
10#26283C212yesmaximal
11#26109C212yesmaximal
12#25840C212yesmaximal
13#23149C211yesmaximal, characteristic
14#22965C212yesmaximal
15#22521C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#475C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#500C2212yes—
20C2×#477C2211yescharacteristic
21C2×#475C2211yescharacteristic
22C2×C8×Q8C2212yes—
23#2868—21no—
24#2868C2211yescharacteristic
25#2809—22no—
26#2571C2212yes—
27#2568—22no—
28#2552—22no—
29#2552—22no—
30#2552C2212yes—
31#2552C2212yes—
32#2546—22no—
33#2546—21no—
34#2546C2212yes—
35#2546C2212yes—
36#2508—22no—
37#2508C2214yes—
38#2369C2212yes—
39#2355—21no—
40#2355C2212yes—
41Q8×C4⋊C4C2212yes—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C42×Q8C2211yescharacteristic
48#2101C2214yes—
49#2099C2214yes—
50#2086C2214yes—
51#2086C2214yes—
52#2085C2214yes—
53#2083C2214yes—
54#2059C2214yes—
55#2059C2214yes—
56#1991C2214yes—
57#1990C2212yes—
58#1990C2212yes—
59#1989C2214yes—
60#1988C2211yescharacteristic
61#1988C2212yes—
62#1988C2212yes—
63#1987C2211yescharacteristic
64#1984C2212yes—
65#1948C2211yescharacteristic
66#1947C2212yes—
67#1947C2212yes—
68#1947C2211yescharacteristic
69#1942C2214yes—
70#1942C2214yes—
71#1939C2212yes—
72#1921C2212yes—
73#1921C2214yes—
74#1920C2214yes—
75#1919C2212yes—
76#1916C2214yes—
77#1865C2212yes—
78#1863C2212yes—
79#1863C2214yes—
80#1863C2214yes—
81#1834C2212yes—
82#1805C2214yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85C4×#107C2212yes—
86C4×#106C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#103C2212yes—
92C4×#103C2214yes—
93#557—28no—
94#557—44no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8D411yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—24no—
105C2×#125C2314yes—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114#500—44no—
115#477Q818yescomplemented
116#475—24no—
117C8×Q8—44no—
118C2×#107C2312yes—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#107C2312yes—
122C2×#106C2312yes—
123C2×#106C2312yes—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×C4×C8C2312yes—
134C2×C4×C8C2312yes—
135C2×C4×C8C2314yes—
136#400—24no—
137#400—24no—
138#398—24no—
139#398—24no—
140#398—24no—
141#398—24no—
142#398—24no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—21no—
147#391—21no—
148#391—24no—
149#391D412yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—21no—
155#389—21no—
156#389—24no—
157#389—24no—
158#389D412yes—
159#389D412yes—
160#389C2312yes—
161#389C2312yes—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#387—24no—
167#387—24no—
168#384—22no—
169#384—22no—
170#384—22no—
171#384—22no—
172#384—21no—
173#384—21no—
174#384D411yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2312yes—
178#384C2312yes—
179#382—22no—
180#382—24no—
181#382C2314yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C43—21no—
200C43C2311yescharacteristic
201#340C2312yes—
202#340C2312yes—
203#340C2312yes—
204#340C2312yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2314yes—
214#340C2312yes—
215#336C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8—21no—
222C22×Q8C2×D412yes—
223C22×Q8C2411yescharacteristic
224C22×C8C2412yes—
225C22×C8C2412yes—
226C22×C8C2412yes—
227C22×C8C2412yes—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230C4⋊Q8—22no—
231C4⋊Q8—24no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—48no—
235C4⋊Q8—44no—
236C4⋊Q8—82no—
237C4⋊Q8—84no—
238C4⋊Q8—84no—
239C4⋊Q8C2×Q814yes—
240#125—28no—
241#125—44no—
242#125—416no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—24no—
246C4×Q8—22no—
247C4×Q8—24no—
248C4×Q8—22no—
249C4×Q8—28no—
250C4×Q8—44no—
251C4×Q8—48no—
252C4×Q8—44no—
253C4×Q8—82no—
254C4×Q8C2×Q814yes—
255C4×Q8#4114yes—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2411yescharacteristic
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—24no—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2414yes—
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311#107—24no—
312#107—44no—
313#107—44no—
314#107C2×Q818yes—
315#106—24no—
316#106—24no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—44no—
321C4⋊C8C2×Q814yes—
322#103—24no—
323#103—24no—
324#103—24no—
325#103C2×Q818yes—
326C4×C8—24no—
327C4×C8—24no—
328C4×C8—44no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—22no—
333#95—24no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—22no—
339#95—24no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95C2×D414yes—
344#95C2×D414yes—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—44no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—42no—
358C2×Q8—81no—
359C2×Q8—81no—
360C2×Q8C22×Q812yes—
361C2×Q8C2×#4112yes—
362C2×Q82−1+412yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4C22×D414yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D414yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2511yescharacteristic
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8C22×Q814yes—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4C22×Q814yes—
435C4⋊C4C22×Q812yes—
436C4⋊C4C2×#4112yes—
437C4⋊C42−1+412yes—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—22no—
442C42—22no—
443C42—24no—
444C42—24no—
445C42—21no—
446C42—21no—
447C42—41no—
448C42—48no—
449C42—42no—
450C42—44no—
451C42—44no—
452C42—42no—
453C42—42no—
454C42—84no—
455C42#13514yes—
456C42C22×Q812yes—
457C42C2×#4112yes—
458C42H(5,𝔽2)12yes—
459C422−1+412yes—
460C23C23×D411yescharacteristic
461Q8—22no—
462Q8—22no—
463Q8—22no—
464Q8—48no—
465Q8—84no—
466Q8—88no—
467Q8—84no—
468Q8#55714yes—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4#54612yes—
512C2×C4#54612yes—
513C2×C4D4×Q812yes—
514C2×C4D4×Q812yes—
515C2×C4C2×#13512yes—
516C2×C4C2×#13512yes—
517C2×C4C2×#13512yes—
518C2×C4#57612yes—
519C2×C4#57812yes—
520C2×C4C23×Q811yescharacteristic
521C2×C4C22×#4111yescharacteristic
522C2×C4C2×H(5,𝔽2)11yescharacteristic
523C2×C4C2×2−1+411yescharacteristic
524C2×C4C2×2−1+411yescharacteristic
525C2×C4C2×2−1+411yescharacteristic
526C8—24no—
527C8—42no—
528C8—42no—
529C8—42no—
530C22C2×#54611yescharacteristic
531C22C2×D4×Q811yescharacteristic
532C22#348611yescharacteristic
533C22C22×#13511yescharacteristic
534C22C2×#57611yescharacteristic
535C22C2×#57811yescharacteristic
536C22#358611yescharacteristic
537C4—24no—
538C4—22no—
539C4—22no—
540C4—22no—
541C4—44no—
542C4—42no—
543C4—41no—
544C4—42no—
545C4—44no—
546C4—42no—
547C4—44no—
548C4—82no—
549C4—82no—
550C4—84no—
551C4—82no—
552C4#330514yes—
553C4#338314yes—
554C4C2×#55711yescharacteristic
555C4#350311yescharacteristic
556C4#350911yescharacteristic
557C4#355211yescharacteristic
558C2#5914411yescharacteristic
559C2C2×#330511yescharacteristic
560C2#6121011yescharacteristic
561C2C2×#338311yescharacteristic
562C2#6138111yescharacteristic
563C2#6182211yescharacteristic
564C2#6187011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
115. #477Q8semidirect

Maximal subgroups

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

subgroupindex
#598692
#365932
#321452
#292192
#287922
#286892
#286752
#272952
#264452
#262832
#261092
#258402
#231492
#229652
#225212
C4×#4772
C4×#4752