Finite Group Databasebeta

← #35413364

Subgroups — #35413364

Order 21248.

1606 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 235 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×#19128265C211yesmaximal, characteristic
2#24910040C212yesmaximal, complemented
3#24907913C218yesmaximal, complemented
4#24907538C212yesmaximal, complemented
5#24907381C211yesmaximal, characteristic, complemented
6C83×#1782C211yesmaximal, characteristic, complemented
7C23×D332C2211yescharacteristic
8#19128265—28no—
9C2×#14581788C412yescomplemented
10C2×#14581781C412yescomplemented
11C2×#14581766C414yescomplemented
12C2×#14581760C2214yes—
13C2×#14581751C2211yescharacteristic
14C2×C22⋊C4×D83C2214yes—
15C2×#14581707C2214yes—
16#19127579—28no—
17#19127564C2212yes—
18#19127556C2214yes—
19#19127467C2218yes—
20C22×#41×C83C2211yescharacteristic
21C2×#99×C83C2214yes—
22#381×C83C2212yes—
23C24×D83C2312yes—
24C22×D332—44no—
25C22×D332C2×C412yes—
26C22×D332C2×C412yes—
27C22×D332D418yes—
28C22×C22⋊D83C2×C418yescomplemented
29C22×C4×D83—24no—
30#14581788—28no—
31#14581781—28no—
32#14581766—216no—
33#14581760—28no—
34#14581760—28no—
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×C418yescomplemented
42C22⋊C4×D83—48no—
43C2×C4×Dic83C2×C412yescomplemented
44#14581707—44no—
45C83⋊#95—24no—
46C2×#41×C83—24no—
47C2×#41×C83—44no—
48C22×Q8×C83C2×C411yescharacteristic, complemented
49C22×D4×C83C2×C411yescharacteristic, complemented
50C22×D4×C83C2312yes—
51C23×C4×C83C2×C412yescomplemented
52C23×C4×C83C2311yescharacteristic
53C2×C22⋊C4×C83C2314yes—
54#99×C83—44no—
55#95×C83—24no—
56C23×D83—28no—
57C23×D83—84no—
58C23×D83C22×C414yes—
59C23×D83C2×D414yes—
60C23×D83C2×D414yes—
61C2×D332—44no—
62C2×D332—44no—
63C2×D332—48no—
64C2×D332—48no—
65C2×D332C22⋊C418yes—
66C2×D332C22⋊C418yes—
67C2×C22⋊D83—216no—
68C2×C22⋊D83—88no—
69C22×Dic83—24no—
70C22×Dic83C22×C414yes—
71C2×C4×D83—48no—
72C2×C4×D83—88no—
73C4⋊Dic83—24no—
74C22⋊Dic83—24no—
75C22⋊Dic83—28no—
76(C2×D83)⋊C4—216no—
77(C2×D83)⋊C4—48no—
78(C2×D83)⋊C4—88no—
79C4×Dic83—24no—
80C24×C83C22×C412yes—
81C24×C83C2411yescharacteristic
82#41×C83—416no—
83C2×Q8×C83—22no—
84C2×Q8×C83—42no—
85C2×D4×C83—24no—
86C2×D4×C83—24no—
87C2×D4×C83—24no—
88C2×D4×C83—22no—
89C2×D4×C83—42no—
90C22×C4×C83—24no—
91C22×C4×C83—24no—
92C22×C4×C83—24no—
93C22×C4×C83—42no—
94C22×C4×C83—42no—
95C22×C4×C83C22×C412yes—
96C22×C4×C83C22×C411yescharacteristic
97C22×C4×C83C22×C411yescharacteristic
98C22×C4×C83C2×D414yes—
99C22⋊C4×C83—28no—
100C23×C83—22no—
101C23×C83—22no—
102C23×C83—22no—
103C23×C83—42no—
104C23×C83—44no—
105C23×C83C23×C411yescharacteristic
106C23×C83C22×D412yes—
107C23×C83H(5,𝔽2)14yes—
108C22×D83—24no—
109C22×D83—24no—
110C22×D83—28no—
111C22×D83—28no—
112C22×D83—48no—
113C22×D83—48no—
114C22×D83—48no—
115C22×D83—816no—
116C22×D83C2×C22⋊C418yes—
117Q8×C83—44no—
118D4×C83—44no—
119D4×C83—48no—
120C2×C4×C83—22no—
121C2×C4×C83—22no—
122C2×C4×C83—28no—
123C2×C4×C83—22no—
124C2×C4×C83—22no—
125C2×C4×C83—22no—
126C2×C4×C83—48no—
127C2×C4×C83—48no—
128C2×C4×C83—44no—
129C2×C4×C83C2×C22⋊C412yes—
130C2×C4×C83C2×C22⋊C412yes—
131C22⋊D83—816no—
132C2×Dic83—24no—
133C2×Dic83—44no—
134C2×Dic83—48no—
135C2×Dic83—44no—
136D332—48no—
137D332—48no—
138C4×D83—816no—
139C22×C83—24no—
140C22×C83—22no—
141C22×C83—44no—
142C22×C83—42no—
143C22×C83—44no—
144C22×C83—82no—
145C22×C83—82no—
146C22×C83C22×C22⋊C411yescharacteristic
147C22×C83#51812yes—
148C22×C83#53414yes—
149C2×D83—28no—
150C2×D83—216no—
151C2×D83—48no—
152C2×D83—48no—
153C2×D83—88no—
154C2×D83—88no—
155C4×C83—22no—
156C4×C83—22no—
157C4×C83—42no—
158C4×C83—84no—
159Dic83—44no—
160Dic83—84no—
161#21750—831nomaximal, Sylow
162C2×C83—24no—
163C2×C83—22no—
164C2×C83—82no—
165C2×C83#230811yescharacteristic
166C2×C83#288312yes—
167D83—48no—
168C2×#535—831no—
169#2308—832no—
170#1898—838no—
171#1820—832no—
172#1782—831no—
173#1782D8311yescharacteristic, complemented
174C83#2175011yescharacteristic, complemented, Sylow
175C22×#41C2×D8311yescharacteristic
176C23×D4—831no—
177#535—1668no—
178C2×#127—832no—
179C2×#124—832no—
180C2×#121—834no—
181C2×#120—834no—
182C2×#117—831no—
183C22×C22⋊C4—834no—
184C2×#99—834no—
185C2×#99C2×D8314yes—
186#386—1668no—
187#381—832no—
188#381C2×D8312yes—
189#379—834no—
190#342—838no—
191C25—832no—
192C2×#41—24no—
193C2×#41—44no—
194C22×Q8C4×D8311yescharacteristic, complemented
195C22×D4—3324no—
196C22×D4—838no—
197C22×D4—832no—
198C22×D4—832no—
199C22×D4—838no—
200C22×D4C4×D8311yescharacteristic, complemented
201C22×D4C22×D8312yes—
202C23×C4—1664no—
203C23×C4C4×D8312yescomplemented
204C23×C4C22×D8311yescharacteristic
205#127—1668no—
206#124—1668no—
207#121—16616no—
208#120—1668no—
209#120—1668no—
210#117—1664no—
211C2×C4⋊C4—832no—
212C2×C22⋊C4—3328no—
213C2×C22⋊C4—832no—
214C2×C22⋊C4—838no—
215C2×C22⋊C4—834no—
216C2×C22⋊C4—1664no—
217C2×C22⋊C4—1668no—
218C2×C22⋊C4C22×D8314yes—
219C2×C42—832no—
220#99—44no—
221#99—3324no—
222#95—24no—
223#95—1664no—
224C24—834no—
225C24—834no—
226C24—834no—
227C24—6644no—
228C24—1668no—
229C24C2×C4×D8312yes—
230C24C23×D8311yescharacteristic
231#41—416no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—3324no—
240C2×D4—3324no—
241C2×D4—3328no—
242C2×D4—3328no—
243C2×D4—838no—
244C2×D4—838no—
245C2×D4—6648no—
246C2×D4—16616no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—3328no—
253C22×C4—834no—
254C22×C4—6648no—
255C22×C4—1664no—
256C22×C4C2×C4×D8312yes—
257C22×C4C2×C4×D8311yescharacteristic
258C22×C4C2×C4×D8311yescharacteristic
259C22×C4D4×D8314yes—
260C4⋊C4—1664no—
261C22⋊C4—28no—
262C22⋊C4—3328no—
263C22⋊C4—6648no—
264C22⋊C4—16616no—
265C22⋊C4—1668no—
266C22⋊C4—1664no—
267C42—1664no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—42no—
272C23—44no—
273C23—3328no—
274C23—3328no—
275C23—3328no—
276C23—838no—
277C23—66416no—
278C23—1664no—
279C23—1664no—
280C23—1668no—
281C23—1668no—
282C23C22×C4×D8311yescharacteristic
283C23C2×D4×D8312yes—
284C23#1458183714yes—
285Q8—44no—
286D4—44no—
287D4—48no—
288D4—3328no—
289D4—3328no—
290D4—66416no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—28no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—48no—
298C2×C4—48no—
299C2×C4—44no—
300C2×C4—3324no—
301C2×C4—3328no—
302C2×C4—3324no—
303C2×C4—66416no—
304C2×C4—1664no—
305C2×C4C22⋊C4×D8312yes—
306C2×C4C22⋊C4×D8312yes—
307C22—24no—
308C22—22no—
309C22—44no—
310C22—44no—
311C22—42no—
312C22—82no—
313C22—82no—
314C22—3328no—
315C22—3328no—
316C22—6648no—
317C22—6648no—
318C22—16616no—
319C22—1668no—
320C22C2×C22⋊C4×D8311yescharacteristic
321C22#1912817412yes—
322C22#1912826114yes—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—84no—
327C4—3324no—
328C4—6644no—
329C2—24no—
330C2—22no—
331C2—82no—
332C2—3328no—
333C2#2491003911yescharacteristic
334C2#2491592212yes—
subgroupG/H
centerC22C2×C22⋊C4×D83
commutatorC23×C83C23×C4
FrattiniC24C23×D83
FittingC83×#1782C2
socleC22×C83C22×C22⋊C4
prime radicalC2×#19128265C2
cosocleC24×C83C24
hypercenter#1782D83
hypocenterC83#21750
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #24910040C2semidirect
3. #24907913C2semidirect
4. #24907538C2semidirect
5. #24907381C2semidirect
6. C83×#1782C2semidirect
9. C2×#14581788C4semidirect
10. C2×#14581781C4semidirect
11. C2×#14581766C4semidirect
28. C22×C22⋊D83C2×C4semidirect
36. C2×C4⋊Dic83C2×C4semidirect
41. C2×(C2×D83)⋊C4C2×C4semidirect
43. C2×C4×Dic83C2×C4semidirect
48. C22×Q8×C83C2×C4semidirect
49. C22×D4×C83C2×C4semidirect
51. C23×C4×C83C2×C4semidirect
173. #1782D83semidirect
174. C83#21750semidirect
194. C22×Q8C4×D83semidirect
200. C22×D4C4×D83semidirect
203. C23×C4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191282652
#249100402
#249079132
#249075382
#249073812
C83×#17822
#2175083