Finite Group Databasebeta

← #35404147

Subgroups — #35404147

Order 21248.

1602 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1602 conjugacy classes of subgroups — too many to draw. Only its 275 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128175C211yesmaximal, characteristic
2#24910040C212yesmaximal
3#24907540C211yesmaximal, characteristic, complemented
4#24907538C212yesmaximal, complemented
5#24907536C214yesmaximal, complemented
6#342×D83C214yesmaximal, complemented
7C83×#1820C211yesmaximal, characteristic, complemented
8C23×D332C2211yescharacteristic
9#19128175—28no—
10C2×C4×D332C414yescomplemented
11C2×#14581754C414yescomplemented
12C2×#14581751C2211yescharacteristic
13C2×C22⋊C4×D83C2214yes—
14C2×C22⋊C4×D83C2212yes—
15C2×C22⋊C4×D83C2212yes—
16#19127579—28no—
17#19127564C2212yes—
18#19127556C2214yes—
19#19127468C2214yes—
20#19127467C2218yes—
21C2×#117×C83C2211yescharacteristic
22#381×C83C2212yes—
23#342×C83C2214yes—
24C24×D83C2311yescharacteristic
25C24×D83C2311yescharacteristic
26C22×D332—28no—
27C22×D332C2×C412yes—
28C22×D332C2×C412yes—
29C22×D332D418yes—
30C22×C4×D83—24no—
31C22×C4×D83C2×C414yes—
32C22×C4×D83C2×C414yescomplemented
33C4×D332—216no—
34#14581754—216no—
35#14581751—24no—
36C2×C4⋊Dic83C2×C412yescomplemented
37C2×C22⋊Dic83C2314yes—
38C2×C22⋊Dic83C2312yes—
39C2×(C2×D83)⋊C4—24no—
40C2×(C2×D83)⋊C4—28no—
41C2×(C2×D83)⋊C4C2×C414yes—
42C2×(C2×D83)⋊C4C2×C414yescomplemented
43C22⋊C4×D83—28no—
44C22⋊C4×D83—28no—
45C22⋊C4×D83—216no—
46C2×C4×Dic83C2×C412yescomplemented
47C83⋊#95—24no—
48C23×C4×C83C2311yescharacteristic
49#117×C83—24no—
50C2×C4⋊C4×C83C2×C412yescomplemented
51C2×C22⋊C4×C83C2314yes—
52C2×C22⋊C4×C83C2312yes—
53C2×C42×C83C2×C412yescomplemented
54#95×C83—24no—
55C23×D83—24no—
56C23×D83—82no—
57C23×D83—82no—
58C23×D83C22×C412yes—
59C23×D83C22×C412yes—
60C23×D83C2×D414yes—
61C23×D83C2×D414yes—
62C23×D83#4118yes—
63C2×D332—28no—
64C2×D332—28no—
65C2×D332—216no—
66C2×D332—216no—
67C2×D332C22⋊C418yes—
68C2×D332C22⋊C418yes—
69C22×Dic83—24no—
70C22×Dic83C22×C414yes—
71C2×C4×D83—28no—
72C2×C4×D83—28no—
73C2×C4×D83—48no—
74C2×C4×D83—88no—
75C2×C4×D83—84no—
76C2×C4×D83—84no—
77C4⋊Dic83—24no—
78C22⋊Dic83—24no—
79C22⋊Dic83—28no—
80(C2×D83)⋊C4—28no—
81(C2×D83)⋊C4—28no—
82(C2×D83)⋊C4—48no—
83(C2×D83)⋊C4—88no—
84C4×Dic83—24no—
85C24×C83C2411yescharacteristic
86C22×C4×C83—24no—
87C22×C4×C83—24no—
88C22×C4×C83C22×C411yescharacteristic
89C22×C4×C83C22×C411yescharacteristic
90C22×C4×C83C22×C414yes—
91C22×C4×C83C2×D414yes—
92C4⋊C4×C83—24no—
93C22⋊C4×C83—24no—
94C22⋊C4×C83—28no—
95C42×C83—24no—
96C23×C83—42no—
97C23×C83C23×C411yescharacteristic
98C23×C83C22×D412yes—
99C23×C83C2×#4114yes—
100C22×D83—24no—
101C22×D83—28no—
102C22×D83—44no—
103C22×D83—44no—
104C22×D83—48no—
105C22×D83—48no—
106C22×D83—88no—
107C22×D83—88no—
108C22×D83C2×C22⋊C414yes—
109C22×D83C2×C22⋊C414yes—
110C22×D83#11718yes—
111C22×D83#12318yes—
112C22×D83#12318yes—
113C2×C4×C83—22no—
114C2×C4×C83—22no—
115C2×C4×C83—28no—
116C2×C4×C83—24no—
117C2×C4×C83—44no—
118C2×C4×C83—48no—
119C2×C4×C83—44no—
120C2×C4×C83C2×C22⋊C412yes—
121C2×C4×C83C2×C22⋊C412yes—
122C2×Dic83—24no—
123C2×Dic83—44no—
124C2×Dic83—48no—
125C2×Dic83—44no—
126D332—216no—
127D332—216no—
128C4×D83—88no—
129C4×D83—88no—
130C4×D83—816no—
131C22×C83—22no—
132C22×C83—24no—
133C22×C83—44no—
134C22×C83C22×C22⋊C411yescharacteristic
135C22×C83C2×#11712yes—
136C22×C83C2×#12314yes—
137C2×D83—28no—
138C2×D83—28no—
139C2×D83—24no—
140C2×D83—48no—
141C2×D83—48no—
142C2×D83—48no—
143C2×D83—48no—
144C2×D83—88no—
145C2×D83#38118yes—
146C4×C83—22no—
147C4×C83—22no—
148C4×C83—44no—
149C4×C83—84no—
150Dic83—44no—
151Dic83—84no—
152#20689—831nomaximal, Sylow
153C2×C83—24no—
154C2×C83—22no—
155C2×C83C2×#38111yescharacteristic
156C2×C83#288312yes—
157D83—28no—
158D83—44no—
159C2×#518—831no—
160#2308—832no—
161#1820—832no—
162#1820—831no—
163#1820D8311yescharacteristic, complemented
164#1819—834no—
165C2×#342—834no—
166C83#2068911yescharacteristic, complemented, Sylow
167C23×D4—831no—
168#518—1668no—
169C2×C4×D4—834no—
170C2×C4×D4—834no—
171C2×#117—831no—
172C2×#117C2×D8311yescharacteristic
173C22×C22⋊C4—832no—
174C22×C22⋊C4—832no—
175C22×C22⋊C4—834no—
176#386—1668no—
177#381—832no—
178#381C2×D8312yes—
179#379—834no—
180#342—838no—
181#342—834no—
182#342C2×D8314yes—
183C25—831no—
184C25—831no—
185C22×D4—832no—
186C22×D4—832no—
187C22×D4—838no—
188C22×D4—1668no—
189C23×C4—834no—
190C23×C4—834no—
191C23×C4—1664no—
192C23×C4C22×D8311yescharacteristic
193C4×D4—16616no—
194C4×D4—16616no—
195#117—24no—
196#117—1664no—
197C2×C4⋊C4—832no—
198C2×C4⋊C4C4×D8312yescomplemented
199C2×C22⋊C4—832no—
200C2×C22⋊C4—834no—
201C2×C22⋊C4—834no—
202C2×C22⋊C4—834no—
203C2×C22⋊C4—16616no—
204C2×C22⋊C4—1668no—
205C2×C22⋊C4—1668no—
206C2×C22⋊C4—1664no—
207C2×C22⋊C4—1668no—
208C2×C22⋊C4C22×D8314yes—
209C2×C22⋊C4C22×D8312yes—
210C2×C42—832no—
211C2×C42C4×D8312yescomplemented
212#95—24no—
213#95—1664no—
214C24—832no—
215C24—834no—
216C24—832no—
217C24—834no—
218C24—838no—
219C24—6642no—
220C24—6642no—
221C24—1664no—
222C24C23×D8311yescharacteristic
223C2×D4—838no—
224C2×D4—838no—
225C2×D4—1668no—
226C2×D4—1668no—
227C2×D4—16616no—
228C2×D4—16616no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—3328no—
232C22×C4—834no—
233C22×C4—6648no—
234C22×C4—6644no—
235C22×C4—6644no—
236C22×C4—1668no—
237C22×C4—1668no—
238C22×C4—1664no—
239C22×C4C2×C4×D8311yescharacteristic
240C22×C4C2×C4×D8311yescharacteristic
241C22×C4C2×C4×D8314yes—
242C22×C4D4×D8314yes—
243C4⋊C4—24no—
244C4⋊C4—1664no—
245C22⋊C4—24no—
246C22⋊C4—28no—
247C22⋊C4—3328no—
248C22⋊C4—6648no—
249C22⋊C4—1668no—
250C22⋊C4—1668no—
251C22⋊C4—1668no—
252C22⋊C4—1664no—
253C42—24no—
254C42—1664no—
255C23—42no—
256C23—3324no—
257C23—3324no—
258C23—3328no—
259C23—3328no—
260C23—838no—
261C23—838no—
262C23—838no—
263C23—834no—
264C23—834no—
265C23—6648no—
266C23—6648no—
267C23—1664no—
268C23—1668no—
269C23C22×C4×D8311yescharacteristic
270C23C2×D4×D8312yes—
271C23#41×D8314yes—
272D4—16616no—
273D4—16616no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—28no—
277C2×C4—24no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—3324no—
282C2×C4—3328no—
283C2×C4—3324no—
284C2×C4—6648no—
285C2×C4—6648no—
286C2×C4—66416no—
287C2×C4—1664no—
288C2×C4C22⋊C4×D8312yes—
289C2×C4C22⋊C4×D8312yes—
290C22—22no—
291C22—24no—
292C22—44no—
293C22—3328no—
294C22—3328no—
295C22—3328no—
296C22—3328no—
297C22—838no—
298C22—6648no—
299C22—1664no—
300C22—1668no—
301C22—1668no—
302C22C2×C22⋊C4×D8311yescharacteristic
303C22#117×D8312yes—
304C22#123×D8314yes—
305C4—22no—
306C4—22no—
307C4—44no—
308C4—84no—
309C4—3324no—
310C4—6644no—
311C2—24no—
312C2—22no—
313C2—3324no—
314C2—1668no—
315C2#381×D8311yescharacteristic
316C2#2491592212yes—
subgroupG/H
centerC22C2×C22⋊C4×D83
commutatorC23×C83C23×C4
FrattiniC24C23×D83
FittingC83×#1820C2
socleC22×C83C22×C22⋊C4
prime radicalC23×D332C22
cosocleC24×C83C24
hypercenter#1820D83
hypocenterC83#20689
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
3. #24907540C2semidirect
4. #24907538C2semidirect
5. #24907536C2semidirect
6. #342×D83C2semidirect
7. C83×#1820C2semidirect
10. C2×C4×D332C4semidirect
11. C2×#14581754C4semidirect
32. C22×C4×D83C2×C4semidirect
36. C2×C4⋊Dic83C2×C4semidirect
42. C2×(C2×D83)⋊C4C2×C4semidirect
46. C2×C4×Dic83C2×C4semidirect
50. C2×C4⋊C4×C83C2×C4semidirect
53. C2×C42×C83C2×C4semidirect
163. #1820D83semidirect
166. C83#20689semidirect
198. C2×C4⋊C4C4×D83semidirect
211. C2×C42C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191281752
#249100402
#249075402
#249075382
#249075362
#342×D832
C83×#18202
#2068983