Finite Group Databasebeta

← #35302239

Subgroups — #35302239

Order 21248.

518 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 107 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#24910066C211yesmaximal, characteristic
2#24907562C212yesmaximal
3#24905793C211yesmaximal, characteristic
4#24905315C212yesmaximal
5C83×#2484C211yesmaximal, characteristic
6#19127665C2212yes—
7#19127587—24no—
8#19127564—21no—
9#19127561—21no—
10C4×C22⋊Dic83C2212yes—
11#19127460—24no—
12#19127427—22no—
13C2×#117×C83C2211yescharacteristic
14C22×C4⋊C4×C83C2211yescharacteristic
15#400×C83C412yescomplemented
16#398×C83C412yescomplemented
17#387×C83—22no—
18#385×C83—21no—
19#381×C83—21no—
20#381×C83C2211yescharacteristic
21C2×#13050574—22no—
22C2×C4⋊Dic83—22no—
23C2×C22⋊Dic83—22no—
24C2×C4×Dic83D414yes—
25C22⋊(C83⋊C8)—22no—
26C22⋊(C83⋊C8)—22no—
27C83⋊#95—22no—
28C83⋊#95—24no—
29C83⋊#95—22no—
30C23×C4×C83D411yescharacteristic
31C23×C4×C83D411yescharacteristic
32C23×C4×C83C2311yescharacteristic
33#117×C83—22no—
34#117×C83—22no—
35C2×C4⋊C4×C83—22no—
36C2×C4⋊C4×C83—22no—
37C2×C4⋊C4×C83—22no—
38C2×C4⋊C4×C83C2×C411yescharacteristic
39C2×C4⋊C4×C83C2×C411yescharacteristic
40C2×C4⋊C4×C83C2×C412yes—
41C2×C4⋊C4×C83D412yes—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C42×C83—21no—
45#95×C83—22no—
46#95×C83—22no—
47#95×C83—22no—
48#95×C83—21no—
49#95×C83C2×C412yes—
50C22×Dic83—24no—
51#13050574—44no—
52C2×C83⋊C8—42no—
53C4⋊Dic83—82no—
54C22⋊Dic83—44no—
55C4×Dic83—44no—
56C24×C83C2×D411yescharacteristic
57C22×C4×C83—21no—
58C22×C4×C83—21no—
59C22×C4×C83—21no—
60C22×C4×C83—21no—
61C22×C4×C83—22no—
62C22×C4×C83—21no—
63C22×C4×C83—21no—
64C22×C4×C83—42no—
65C22×C4×C83—42no—
66C22×C4×C83—42no—
67C22×C4×C83C22⋊C412yes—
68C22×C4×C83C22⋊C412yes—
69C22×C4×C83C22×C411yescharacteristic
70C22×C4×C83C2×D412yes—
71C22×C4×C83C2×D411yescharacteristic
72C22×C4×C83#4112yes—
73C4⋊C4×C83—24no—
74C4⋊C4×C83—24no—
75C4⋊C4×C83—22no—
76C4⋊C4×C83—22no—
77C4⋊C4×C83—44no—
78C22⋊C4×C83—42no—
79C22⋊C4×C83—41no—
80C22⋊C4×C83—41no—
81C42×C83—42no—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—41no—
85C23×C83C2×C22⋊C411yescharacteristic
86C23×C83#12011yescharacteristic
87C23×C83#12311yescharacteristic
88C2×C4×C83—24no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—41no—
94C2×C4×C83—41no—
95C2×C4×C83—42no—
96C2×C4×C83—42no—
97C2×C4×C83—42no—
98C2×C4×C83—42no—
99C2×C4×C83—42no—
100C2×C4×C83—42no—
101C2×C4×C83—42no—
102C2×C4×C83—42no—
103C2×C4×C83—41no—
104C2×C4×C83—41no—
105C2×C4×C83—42no—
106C2×C4×C83—42no—
107C2×C4×C83—81no—
108C2×C4×C83C4×D412yes—
109C2×C4×C83#12112yes—
110C2×Dic83—44no—
111C2×Dic83—44no—
112C2×Dic83—44no—
113C83⋊C8—82no—
114C22×C83—21no—
115C22×C83—21no—
116C22×C83—21no—
117C22×C83—21no—
118C22×C83—21no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83—44no—
122C22×C83#38611yescharacteristic
123C22×C83#42112yes—
124C22×C83#45812yes—
125C4×C83—22no—
126C4×C83—22no—
127C4×C83—42no—
128C4×C83—42no—
129C4×C83—82no—
130C4×C83—81no—
131C4×C83—81no—
132Dic83—84no—
133#9482—831nomaximal, Sylow
134C2×C83—22no—
135C2×C83—22no—
136C2×C83—41no—
137C2×C83#189011yescharacteristic
138C2×C83#189711yescharacteristic
139C2×C83#191211yescharacteristic
140#2484D8311yescharacteristic
141#2313—831no—
142#1826—832no—
143#1463—831no—
144#1284—832no—
145C83#948211yescharacteristic, complemented, Sylow
146C2×#117C2×D8311yescharacteristic
147C22×C4⋊C4C2×D8311yescharacteristic
148#407—832no—
149#400Dic8312yescomplemented
150#398Dic8312yescomplemented
151#387—22no—
152#387—1664no—
153#385—21no—
154#381—21no—
155#381—1661no—
156#381C2×D8311yescharacteristic
157#380—1661no—
158C4×C22⋊C4—832no—
159#339—1664no—
160#324—1662no—
161C23×C4C22⋊D8311yescharacteristic
162C23×C4C22⋊D8311yescharacteristic
163C23×C4C22×D8311yescharacteristic
164C2×M16—1662no—
165#117—22no—
166#117—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—1662no—
171C2×C4⋊C4C2×Dic8311yescharacteristic
172C2×C4⋊C4C2×Dic8311yescharacteristic
173C2×C4⋊C4C2×Dic8312yes—
174C2×C4⋊C4C22⋊D8312yes—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—1662no—
178C2×C42—21no—
179C2×C42—834no—
180C22⋊C8—1662no—
181C22⋊C8—1662no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—21no—
186#95—1662no—
187#95—1664no—
188#95—1662no—
189#95C2×Dic8312yes—
190C24C2×C22⋊D8311yescharacteristic
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—1664no—
202C22×C4C22⋊Dic8312yes—
203C22×C4C22⋊Dic8312yes—
204C22×C4C22×Dic8311yescharacteristic
205C22×C4D4×D8312yes—
206C22×C4C2×C22⋊D8311yescharacteristic
207C22×C4#1305058912yes—
208M16—3324no—
209C2×C8—3322no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—44no—
215C4⋊C4—6642no—
216C22⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—3324no—
220C42—42no—
221C42—3324no—
222C23—22no—
223C23—22no—
224C23—41no—
225C23C2×C22⋊Dic8311yescharacteristic
226C23#1458176011yescharacteristic
227C23#1458177311yescharacteristic
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—3324no—
249C2×C4—3324no—
250C2×C4—3324no—
251C2×C4D4×Dic8312yes—
252C2×C4#1458176412yes—
253C8—6642no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22#1912758111yescharacteristic
263C22#1912771812yes—
264C22#1912789012yes—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—42no—
269C4—82no—
270C4—81no—
271C4—81no—
272C4—6644no—
273C2—22no—
274C2—22no—
275C2—41no—
276C2#2490787711yescharacteristic
277C2#2490791211yescharacteristic
278C2#2490800111yescharacteristic
subgroupG/H
centerC22#19127581
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2484C2
socleC22×C83#386
prime radicalC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenter#2484D83
hypocenterC83#9482
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
15. #400×C83C4semidirect
16. #398×C83C4semidirect
145. C83#9482semidirect
149. #400Dic83semidirect
150. #398Dic83semidirect

Maximal subgroups

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

subgroupindex
#249100662
#249075622
#249057932
#249053152
C83×#24842
#948283