Finite Group Databasebeta

← #7542164

Subgroups — #7542164

Order 512.

1297 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59869C211yesmaximal, characteristic
2#36598C211yesmaximal, characteristic
3#32357C214yesmaximal
4#29208C211yesmaximal, characteristic
5#29206C211yesmaximal, characteristic
6#28830C212yesmaximal
7#26857C214yesmaximal
8#26511C212yesmaximal
9#26283C212yesmaximal
10#26089C212yesmaximal
11#25472C212yesmaximal
12#25063C212yesmaximal
13#23149C211yesmaximal, characteristic
14#23063C212yesmaximal
15#22727C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#475C211yesmaximal, characteristic
18C2×#477C2211yescharacteristic
19C2×#475C2211yescharacteristic
20C2×C8×Q8C2212yes—
21#2868C2211yescharacteristic
22#2862—21no—
23#2809—22no—
24#2582C2212yes—
25#2571C2212yes—
26#2571C2212yes—
27#2570—22no—
28#2568—22no—
29#2568C2212yes—
30#2546—21no—
31#2546C2212yes—
32#2511—22no—
33#2508—22no—
34#2508C2214yes—
35#2446—22no—
36#2369C2212yes—
37#2365C2212yes—
38#2355—21no—
39#2355C2212yes—
40#2355C2212yes—
41#2355C2212yes—
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C42×Q8C2211yescharacteristic
47#2101C2214yes—
48#2097C2214yes—
49#2088C2214yes—
50#2085C2214yes—
51#2085C2214yes—
52#2066C2214yes—
53#2058C2214yes—
54#2057C2214yes—
55#1990C2212yes—
56#1990C2212yes—
57#1988C2211yescharacteristic
58#1987C2211yescharacteristic
59#1984C2212yes—
60#1949C2214yes—
61#1949C2214yes—
62#1948C2212yes—
63#1948C2212yes—
64#1948C2211yescharacteristic
65#1947C2212yes—
66#1947C2212yes—
67#1947C2211yescharacteristic
68#1941C2214yes—
69#1939C2212yes—
70#1933C2214yes—
71#1928C2214yes—
72#1921C2214yes—
73#1921C2212yes—
74#1920C2214yes—
75#1919C2212yes—
76#1918C2214yes—
77#1916C2214yes—
78#1865C2212yes—
79#1863C2212yes—
80#1846C2214yes—
81#1845C2212yes—
82#1844C2212yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85C4×#107C2214yes—
86C4×#107C2212yes—
87C4×#106C2212yes—
88C8×C4⋊C4C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#103C2212yes—
92C4×#103C2212yes—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8D411yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4×Q8—22no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2312yes—
107#477Q818yescomplemented
108#475—24no—
109C8×Q8—28no—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#107C2312yes—
113C2×#107C2312yes—
114C2×#106C2312yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×#106C2312yes—
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2314yes—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×C4×C8C2312yes—
128C2×C4×C8C2312yes—
129C2×C4×C8C2314yes—
130#400—24no—
131#400—24no—
132#400—24no—
133#400—24no—
134#398—24no—
135#398—24no—
136#398—24no—
137#391—22no—
138#391—22no—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—24no—
147#391D412yes—
148#391C2314yes—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—21no—
154#389—21no—
155#389D412yes—
156#389D412yes—
157#389C2312yes—
158#389C2312yes—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#387—24no—
163#387—24no—
164#387—24no—
165#387—24no—
166#387—24no—
167#384—21no—
168#384—21no—
169#384D411yescharacteristic
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#384C2312yes—
173#384C2312yes—
174#382—22no—
175#382—22no—
176#382—22no—
177#382—24no—
178#382C2314yes—
179#382C2314yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2311yescharacteristic
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2311yescharacteristic
194C4×C4⋊C4C2314yes—
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C43—21no—
199C43C2311yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2312yes—
205#340C2312yes—
206#340C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215#336C2314yes—
216C22×Q8—21no—
217C22×Q8—22no—
218C22×Q8—21no—
219C22×Q8C2×D412yes—
220C22×Q8C2411yescharacteristic
221C22×C8C2412yes—
222C22×C8C2412yes—
223C22×C8C2412yes—
224C22×C8C2412yes—
225C4⋊Q8—24no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—42no—
230C4⋊Q8—42no—
231C4⋊Q8—44no—
232C4⋊Q8—84no—
233C4⋊Q8—82no—
234C4⋊Q8C2×Q814yes—
235C4×Q8—24no—
236C4×Q8—22no—
237C4×Q8—24no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—41no—
241C4×Q8—41no—
242C4×Q8—44no—
243C4×Q8—82no—
244C4×Q8—84no—
245C4×Q8C2×Q814yes—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2411yescharacteristic
272C2×C4⋊C4C2411yescharacteristic
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2411yescharacteristic
278C2×C4⋊C4C2412yes—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2414yes—
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297#107—44no—
298#107—42no—
299#107—44no—
300#107C2×Q818yes—
301#106—24no—
302#106—44no—
303#106—44no—
304#106—42no—
305C4⋊C8—28no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8C2×Q814yes—
309C4⋊C8#4114yes—
310#103—24no—
311#103—42no—
312#103—42no—
313#103C2×Q818yes—
314C4×C8—28no—
315C4×C8—24no—
316C4×C8—24no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95C2×D414yes—
336#95C2×D414yes—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—44no—
351C2×Q8—42no—
352C2×Q8—81no—
353C2×Q8—81no—
354C2×Q8C22×Q812yes—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4C22×D414yes—
365C22×C4C22×D414yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2511yescharacteristic
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8C22×Q814yes—
384C2×C8C2×#4114yes—
385C4⋊C4—24no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—21no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—84no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4C22×Q814yes—
425C4⋊C4C22×Q812yes—
426C42—24no—
427C42—22no—
428C42—22no—
429C42—24no—
430C42—24no—
431C42—21no—
432C42—21no—
433C42—41no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—44no—
438C42—42no—
439C42—44no—
440C42—42no—
441C42—84no—
442C42—84no—
443C42#13514yes—
444C42C22×Q812yes—
445C42C2×#4112yes—
446C422−1+412yes—
447C422−1+412yes—
448C23C23×D411yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—48no—
452Q8—41no—
453Q8—41no—
454Q8—84no—
455Q8—88no—
456Q8—84no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4#54712yes—
500C2×C4#54712yes—
501C2×C4D4×Q812yes—
502C2×C4D4×Q812yes—
503C2×C4C2×#13512yes—
504C2×C4C2×#13512yes—
505C2×C4C2×#13512yes—
506C2×C4#57712yes—
507C2×C4#57812yes—
508C2×C4C23×Q811yescharacteristic
509C2×C4C22×#4111yescharacteristic
510C2×C4C2×2−1+411yescharacteristic
511C2×C4C2×2−1+411yescharacteristic
512C2×C4#58511yescharacteristic
513C2×C4#58511yescharacteristic
514C8—24no—
515C8—24no—
516C8—42no—
517C8—42no—
518C22C2×#54711yescharacteristic
519C22C2×D4×Q811yescharacteristic
520C22#348711yescharacteristic
521C22C22×#13511yescharacteristic
522C22C2×#57711yescharacteristic
523C22C2×#57811yescharacteristic
524C22#358511yescharacteristic
525C4—22no—
526C4—22no—
527C4—22no—
528C4—44no—
529C4—44no—
530C4—42no—
531C4—42no—
532C4—44no—
533C4—41no—
534C4—82no—
535C4—81no—
536C4—82no—
537C4—84no—
538C4—81no—
539C4—82no—
540C4#334014yes—
541C4#338314yes—
542C4Q8×#4111yescharacteristic
543C4#350211yescharacteristic
544C4#351511yescharacteristic
545C4#355511yescharacteristic
546C2#5906711yescharacteristic
547C2C2×#334011yescharacteristic
548C2#6128211yescharacteristic
549C2C2×#338311yescharacteristic
550C2#6138211yescharacteristic
551C2#6188511yescharacteristic
552C2#6193411yescharacteristic
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
107. #477Q8semidirect

Maximal subgroups

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

subgroupindex
#598692
#365982
#323572
#292082
#292062
#288302
#268572
#265112
#262832
#260892
#254722
#250632
#231492
#230632
#227272
C4×#4772
C4×#4752