Finite Group Databasebeta

← #35300430

Subgroups — #35300430

Order 21248.

478 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#24916101C211yesmaximal, characteristic
2#24913229C211yesmaximal, characteristic
3#24906134C212yesmaximal, complemented
4#24905322C212yesmaximal, complemented
5C83×#1458C211yesmaximal, characteristic
6C2×#14581792C2211yescharacteristic
7C2×#14581771—21no—
8#19127762—24no—
9#19127725C2212yes—
10#19127665—21no—
11C4×#13050574—21no—
12#19127631—21no—
13#19127631C412yes—
14#19127626—21no—
15#19127626C412yes—
16#19127607—21no—
17#19127567C2211yescharacteristic
18#19127446—24no—
19C2×#117×C83C2211yescharacteristic
20#431×C83C2212yes—
21#324×C83—21no—
22#323×C83—21no—
23C22×Dic166—21no—
24C2×#13050574—22no—
25#14581792—22no—
26#14581792—22no—
27#14581771—44no—
28C2×C4⋊Dic83—21no—
29C2×C4⋊Dic83—21no—
30C2×C4⋊Dic83C2×C412yes—
31C2×Dic83⋊C4—22no—
32C2×C22⋊Dic83—22no—
33C4⋊(C83⋊C8)—22no—
34C4⋊(C83⋊C8)D414yes—
35C22⋊(C83⋊C8)—24no—
36#14581703—22no—
37C4×C83⋊C8—22no—
38C83⋊#95—22no—
39C83⋊#95—22no—
40C83⋊#95—21no—
41C83⋊#95C2×C412yes—
42C23×C4×C83D411yescharacteristic
43#117×C83—22no—
44#117×C83—22no—
45C2×C4⋊C4×C83C2×C411yescharacteristic
46C2×C4⋊C4×C83D411yescharacteristic
47C2×C22⋊C4×C83C2×C411yescharacteristic
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C42×C83D411yescharacteristic
50C2×C42×C83C2311yescharacteristic
51C4⋊C8×C83—22no—
52C22⋊C8×C83—22no—
53C4×C8×C83—22no—
54C2×Dic166—41no—
55C2×Dic166—41no—
56C2×Dic166—42no—
57C2×Dic166—42no—
58C22×Dic83—22no—
59C22×Dic83—22no—
60#13050574—42no—
61#13050574—42no—
62C2×C83⋊C8—24no—
63C4⋊Dic83—24no—
64C4⋊Dic83—42no—
65C4⋊Dic83—42no—
66Dic83⋊C4—44no—
67C22⋊Dic83—44no—
68C24×C83C22⋊C411yescharacteristic
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—22no—
72C22×C4×C83—22no—
73C22×C4×C83C22⋊C411yescharacteristic
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22×C411yescharacteristic
77C22×C4×C83C2×D411yescharacteristic
78C22×C4×C83C2×D411yescharacteristic
79C2×C8×C83—42no—
80C4⋊C4×C83—22no—
81C4⋊C4×C83—22no—
82C22⋊C4×C83—22no—
83C22⋊C4×C83—22no—
84C42×C83—21no—
85C42×C83—21no—
86C42×C83C2×D412yes—
87C42×C83#4112yes—
88C23×C83—21no—
89C23×C83—21no—
90C23×C83—21no—
91C23×C83—41no—
92C23×C83—41no—
93C23×C83C2×C22⋊C411yescharacteristic
94C2×C4×C83—21no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—41no—
100C2×C4×C83—41no—
101C2×C4×C83—42no—
102C2×C4×C83—42no—
103C2×C4×C83—41no—
104C2×C4×C83—41no—
105C2×C4×C83C4×D412yes—
106C2×C4×C83#12011yescharacteristic
107C2×C4×C83#12112yes—
108C2×C4×C83#12311yescharacteristic
109C2×C4×C83#13512yes—
110C2×C4×C83AGL(1,ℤ/8)12yes—
111C2×Dic83—42no—
112C2×Dic83—42no—
113C2×Dic83—42no—
114C2×Dic83—42no—
115C2×Dic83—42no—
116C2×Dic83—42no—
117Dic166—82no—
118Dic166—82no—
119C8×C83—82no—
120C83⋊C8—44no—
121C22×C83—21no—
122C22×C83—21no—
123C22×C83—42no—
124C22×C83—41no—
125C22×C83—41no—
126C22×C83—41no—
127C22×C83—81no—
128C22×C83#38611yescharacteristic
129C22×C83#41611yescharacteristic
130C22×C83#41811yescharacteristic
131C4×C83—22no—
132C4×C83—22no—
133C4×C83—41no—
134C4×C83—41no—
135C4×C83—41no—
136C4×C83#44912yes—
137C4×C83#48212yes—
138Dic83—82no—
139Dic83—82no—
140#9196—831nomaximal, Sylow
141C2×C83—21no—
142C2×C83—21no—
143C2×C83—81no—
144C2×C83#187511yescharacteristic
145C2×C83#190911yescharacteristic
146C2×C83#191511yescharacteristic
147#2921—831no—
148#2621—831no—
149#1546—832no—
150#1458D8311yescharacteristic
151#1286—832no—
152C83#919611yescharacteristic, complemented, Sylow
153C2×C4⋊Q8—831no—
154C2×C22⋊Q8—1661no—
155C2×#117C2×D8311yescharacteristic
156#433—1664no—
157#431C2×D8312yes—
158#423—832no—
159#407—1661no—
160C4×M16—1661no—
161#398—832no—
162#398—1661no—
163#397—832no—
164#397—1661no—
165#391—1661no—
166#382—831no—
167#332—1664no—
168#324—21no—
169#323—21no—
170C22×Q8—1661no—
171C23×C4D33211yescharacteristic
172C2×M16—1662no—
173C4⋊Q8—1662no—
174C4⋊Q8—1662no—
175C22⋊Q8—3324no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4—832no—
179C2×C4⋊C4—1662no—
180C2×C4⋊C4—1661no—
181C2×C4⋊C4—1661no—
182C2×C4⋊C4C4×D8311yescharacteristic
183C2×C4⋊C4C22⋊D8311yescharacteristic
184C2×C22⋊C4—1662no—
185C2×C22⋊C4C4×D8311yescharacteristic
186C2×C22⋊C4C22⋊D8311yescharacteristic
187C2×C42D33211yescharacteristic
188C2×C42C22×D8311yescharacteristic
189C4⋊C8—22no—
190C4⋊C8—834no—
191C4⋊C8—1662no—
192C22⋊C8—22no—
193C22⋊C8—1664no—
194#97—1662no—
195C4×C8—22no—
196C4×C8—1662no—
197#95—832no—
198#95—1662no—
199#95—1662no—
200#95—1661no—
201C24(C2×D83)⋊C411yescharacteristic
202C2×Q8—3321no—
203C2×Q8—3321no—
204C2×Q8—3322no—
205C2×Q8—3322no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—1662no—
211C22×C4—1662no—
212C22×C4(C2×D83)⋊C411yescharacteristic
213C22×C4(C2×D83)⋊C411yescharacteristic
214C22×C4(C2×D83)⋊C411yescharacteristic
215C22×C4C2×C4×D8311yescharacteristic
216C22×C4C2×C22⋊D8311yescharacteristic
217C22×C4C2×D33211yescharacteristic
218M16—3322no—
219M16—3322no—
220C2×C8—42no—
221C2×C8—1664no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—3324no—
225C4⋊C4—3322no—
226C4⋊C4—3322no—
227C4⋊C4—1664no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—3324no—
231C42—21no—
232C42—21no—
233C42D4×D8312yes—
234C42#1305058912yes—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23C2×(C2×D83)⋊C411yescharacteristic
241Q8—6642no—
242Q8—6642no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—3322no—
255C2×C4—3322no—
256C2×C4—3322no—
257C2×C4—3322no—
258C2×C4—3322no—
259C2×C4—3322no—
260C2×C4(C22⋊D83)⋊C412yes—
261C2×C4#1458176211yescharacteristic
262C2×C4#1458176412yes—
263C2×C4#1458177611yescharacteristic
264C2×C4#1458181412yes—
265C2×C4#1458182012yes—
266C8—82no—
267C8—3324no—
268C22—21no—
269C22—21no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22#1912757811yescharacteristic
276C22#1912769811yescharacteristic
277C22#1912770711yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—6642no—
284C4—6642no—
285C4#1912784812yes—
286C4#1912802612yes—
287C2—21no—
288C2—21no—
289C2—81no—
290C2#2490778311yescharacteristic
291C2#2490798311yescharacteristic
292C2#2490801311yescharacteristic
subgroupG/H
centerC22#19127578
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1458C2
socleC22×C83#386
prime radicalC24×C83C22⋊C4
cosocleC2×C42×C83C23
hypercenter#1458D83
hypocenterC83#9196
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #24906134C2semidirect
4. #24905322C2semidirect
152. C83#9196semidirect

Maximal subgroups

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

subgroupindex
#249161012
#249132292
#249061342
#249053222
C83×#14582
#919683