Finite Group Databasebeta

← #35313625

Subgroups — #35313625

Order 21248.

454 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 109 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×(C22⋊D83)⋊C8C211yesmaximal, characteristic
2C22⋊C16×D83C212yesmaximal, complemented
3#24905402C212yesmaximal, complemented
4#24905387C211yesmaximal, characteristic, complemented
5C83×#1314C211yesmaximal, characteristic, complemented
6C23×C83⋊C8C411yescharacteristic
7C22×C8×D83C2211yescharacteristic
8C2×C4×C22⋊D83C411yescharacteristic
9C2×C16×D83—22no—
10(C22⋊D83)⋊C8—22no—
11(C22⋊D83)⋊C8—22no—
12C2×Dic83⋊C8C2211yescharacteristic
13C2×(C2×D83)⋊C8C411yescharacteristic
14C2×C8×Dic83C411yescharacteristic
15C22⋊(C83⋊C16)C2212yes—
16(C2×D83)⋊C16—22no—
17C2×C22⋊C8×C83C2211yescharacteristic
18C22⋊C16×C83C2212yes—
19C22×C22⋊D83C811yescharacteristic
20C22×C4×D83C2×C411yescharacteristic
21C22×C83⋊C8—21no—
22C22×C83⋊C8—21no—
23C22×C83⋊C8—42no—
24C22×C83⋊C8C2×C411yescharacteristic
25C22×C83⋊C8C2×C411yescharacteristic
26C2×C8×D83—42no—
27C2×C8×D83D412yes—
28C2×C8×D83D412yes—
29C4×C22⋊D83—22no—
30C4×C22⋊D83—22no—
31C2×Dic83⋊C4C811yescharacteristic
32C2×C22⋊Dic83C811yescharacteristic
33C2×(C2×D83)⋊C4C811yescharacteristic
34C2×C4×Dic83C2×C411yescharacteristic
35C2×C83⋊C16—22no—
36C16×D83—44no—
37Dic83⋊C8—22no—
38(C2×D83)⋊C8—22no—
39C8×Dic83—21no—
40C8×Dic83—21no—
41C23×C4×C83C2×C411yescharacteristic
42C22×C8×C83C2×C411yescharacteristic
43C22×C8×C83C2311yescharacteristic
44C2×C16×C83—22no—
45C22⋊C8×C83—22no—
46C23×D83C2×C811yescharacteristic
47C2×C22⋊D83—22no—
48C2×C22⋊D83—22no—
49C2×C22⋊D83C1614yescomplemented
50C22×Dic83C2×C811yescharacteristic
51C22×Dic83C2×C811yescharacteristic
52C22×Dic83C2×C811yescharacteristic
53C2×C4×D83—41no—
54C2×C4×D83—41no—
55C2×C4×D83C22⋊C412yes—
56C2×C4×D83M1612yes—
57C2×C83⋊C8—22no—
58C2×C83⋊C8—21no—
59C2×C83⋊C8—42no—
60C2×C83⋊C8—42no—
61C2×C83⋊C8—41no—
62C2×C83⋊C8C22⋊C411yescharacteristic
63C2×C83⋊C8C22⋊C411yescharacteristic
64C8×D83—24no—
65C8×D83—42no—
66Dic83⋊C4—22no—
67C22⋊Dic83—22no—
68(C2×D83)⋊C4—22no—
69C4×Dic83—21no—
70C4×Dic83—21no—
71C83⋊C16—42no—
72C24×C83C2×C811yescharacteristic
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—41no—
76C22×C4×C83—41no—
77C22×C4×C83C2×C811yescharacteristic
78C22×C4×C83C22×C411yescharacteristic
79C2×C8×C83—21no—
80C2×C8×C83—41no—
81C2×C8×C83—41no—
82C2×C8×C83C2×D412yes—
83C16×C83—42no—
84C23×C83—21no—
85C23×C83—21no—
86C23×C83—41no—
87C23×C83—41no—
88C23×C83C2×C1612yescomplemented
89C23×C83C22×C811yescharacteristic
90C22×D83—41no—
91C22×D83—41no—
92C22×D83C22⋊C812yes—
93C22×D83C2×C1612yes—
94C22×D83M3212yes—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—21no—
98C2×C4×C83—42no—
99C2×C4×C83—41no—
100C2×C4×C83—41no—
101C2×C4×C83—41no—
102C2×C4×C83C2×C22⋊C411yescharacteristic
103C2×C4×C83C2×M1611yescharacteristic
104C22⋊D83—28no—
105C2×Dic83—21no—
106C2×Dic83—21no—
107C2×Dic83—21no—
108C2×Dic83—41no—
109C2×Dic83—41no—
110C2×Dic83C22⋊C812yes—
111C2×Dic83C2×C1612yescomplemented
112C4×D83—22no—
113C4×D83—22no—
114C4×D83—42no—
115C8×C83—22no—
116C8×C83—41no—
117C83⋊C8—21no—
118C83⋊C8—41no—
119C83⋊C8#9912yes—
120C22×C83—21no—
121C22×C83—21no—
122C22×C83—22no—
123C22×C83—42no—
124C22×C83—42no—
125C22×C83—41no—
126C22×C83—41no—
127C22×C83C2×C22⋊C811yescharacteristic
128C22×C83C22×C1611yescharacteristic
129C22×C83C2×M3211yescharacteristic
130C2×D83—22no—
131C2×D83—22no—
132C2×D83—42no—
133C2×D83—42no—
134C2×D83C22⋊C1614yes—
135C4×C83—21no—
136C4×C83—41no—
137C4×C83C2×#9911yescharacteristic
138C4×C83#41311yescharacteristic
139Dic83—22no—
140Dic83—22no—
141Dic83—41no—
142#11465—831nomaximal, Sylow
143C2×C83—21no—
144C2×C83—21no—
145C2×C83—42no—
146C2×C83#145911yescharacteristic
147C2×C83C2×C22⋊C1611yescharacteristic
148C2×C83#211511yescharacteristic
149D83—24no—
150C2×C8×D4—831no—
151C2×C22⋊C16—832no—
152#1322—832no—
153#1314—831no—
154#1314D8311yescharacteristic, complemented
155C83#1146511yescharacteristic, complemented, Sylow
156C23×C8—831no—
157C23×C8—831no—
158C2×C4×D4—831no—
159C22×C16—1662no—
160C8×D4—1662no—
161C8×D4—1662no—
162C2×C4⋊C8—831no—
163C2×C22⋊C8—831no—
164C2×C22⋊C8C2×D8311yescharacteristic
165C2×C4×C8—831no—
166C22⋊C16—832no—
167C22⋊C16—1662no—
168C22⋊C16C2×D8312yes—
169C22×D4—831no—
170C23×C4—831no—
171C23×C4C4×D8311yescharacteristic
172C22×C8—3322no—
173C22×C8—3322no—
174C22×C8—832no—
175C22×C8—831no—
176C22×C8—831no—
177C22×C8—832no—
178C22×C8—1661no—
179C22×C8—1661no—
180C22×C8C4×D8311yescharacteristic
181C22×C8C22×D8311yescharacteristic
182C4×D4—1662no—
183C4×D4—1662no—
184C2×C4⋊C4—831no—
185C2×C22⋊C4—831no—
186C2×C22⋊C4—831no—
187C2×C42—831no—
188C2×C16—22no—
189C2×C16—3324no—
190C2×C16—1662no—
191C4⋊C8—1662no—
192C22⋊C8—22no—
193C22⋊C8—1662no—
194C4×C8—1661no—
195C4×C8—1661no—
196C24—831no—
197C24C8×D8311yescharacteristic
198C2×D4—834no—
199C2×D4—1662no—
200C2×D4—1662no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—3321no—
206C22×C4—3321no—
207C22×C4—831no—
208C22×C4—832no—
209C22×C4—831no—
210C22×C4—831no—
211C22×C4—832no—
212C22×C4C8×D8311yescharacteristic
213C22×C4C2×C4×D8311yescharacteristic
214C2×C8—21no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—3322no—
218C2×C8—3322no—
219C2×C8—3322no—
220C2×C8—3321no—
221C2×C8—831no—
222C2×C8—831no—
223C2×C8—1664no—
224C2×C8—1661no—
225C2×C8—1662no—
226C2×C8D4×D8312yes—
227C4⋊C4—1662no—
228C22⋊C4—1662no—
229C22⋊C4—1662no—
230C42—1661no—
231C42—1661no—
232C16—42no—
233C16—3322no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—3321no—
239C23—3321no—
240C23—832no—
241C23—832no—
242C23—832no—
243C23C16×D8312yescomplemented
244C23C2×C8×D8311yescharacteristic
245D4—1668no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—3322no—
254C2×C4—3321no—
255C2×C4—3321no—
256C2×C4—832no—
257C2×C4—832no—
258C2×C4—1661no—
259C2×C4—1661no—
260C2×C4—1661no—
261C2×C4—1662no—
262C2×C4—1662no—
263C2×C4C22⋊C4×D8311yescharacteristic
264C2×C4M16×D8311yescharacteristic
265C8—22no—
266C8—41no—
267C8—3321no—
268C8—832no—
269C8—1661no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—3322no—
278C22—3322no—
279C22—834no—
280C22—1662no—
281C22—1662no—
282C22C22⋊C8×D8311yescharacteristic
283C22C2×C16×D8311yescharacteristic
284C22M32×D8311yescharacteristic
285C4—21no—
286C4—41no—
287C4—3321no—
288C4—1662no—
289C4—1662no—
290C4#99×D8311yescharacteristic
291C4#1912768211yescharacteristic
292C2—21no—
293C2—21no—
294C2—42no—
295C2—1664no—
296C2#2490577811yescharacteristic
297C2C22⋊C16×D8311yescharacteristic
298C2#2490908211yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D83
commutatorC22×C83C22×C16
FrattiniC22×C8C22×D83
FittingC83×#1314C2
socleC22×C83C2×C22⋊C8
prime radicalC22×C22⋊D83C8
cosocleC22×C8×C83C23
hypercenter#1314D83
hypocenterC83#11465
LCSG ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. C22⋊C16×D83C2semidirect
3. #24905402C2semidirect
4. #24905387C2semidirect
5. C83×#1314C2semidirect
49. C2×C22⋊D83C16semidirect
88. C23×C83C2×C16semidirect
111. C2×Dic83C2×C16semidirect
154. #1314D83semidirect
155. C83#11465semidirect
243. C23C16×D83semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D83)⋊C82
C22⋊C16×D832
#249054022
#249053872
C83×#13142
#1146583