Finite Group Databasebeta

← #35314044

Subgroups — #35314044

Order 21248.

462 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 87 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#24912462C211yesmaximal, characteristic
2#24905787C211yesmaximal, characteristic, complemented
3#24905394C214yesmaximal, complemented
4C83×#1467C211yesmaximal, characteristic, complemented
5C2×#14581789C411yescharacteristic
6C2×#14581781C412yes—
7#19127754—21no—
8#19127618—21no—
9#19127618—21no—
10#19127603—21no—
11#19127603—21no—
12#19127602C2212yes—
13C4×C4⋊Dic83C411yescharacteristic
14#19127427—21no—
15#19127427C2212yes—
16C2×C4×Q8×C83C2211yescharacteristic
17#432×C83—21no—
18#324×C83—21no—
19#324×C83C2212yes—
20C22×D332—21no—
21C22×D332—21no—
22C22×D332C2×C412yes—
23#14581789—41no—
24#14581789—41no—
25#14581781—44no—
26C2×C4⋊Dic83—21no—
27C2×(C2×D83)⋊C4—22no—
28C2×(C2×D83)⋊C4—22no—
29C2×(C2×D83)⋊C4—22no—
30C2×(C2×D83)⋊C4—22no—
31C2×C4×Dic83C2×C412yes—
32C4⋊(C83⋊C8)—41no—
33C22⋊(C83⋊C8)—22no—
34C4×C83⋊C8—41no—
35C83⋊#95—21no—
36C22×Q8×C83C2×C411yescharacteristic
37C4×Q8×C83—41no—
38C4×Q8×C83—41no—
39C2×C4⋊C4×C83D411yescharacteristic
40C2×C4⋊C4×C83D411yescharacteristic
41C2×C4⋊C4×C83C2311yescharacteristic
42C2×C42×C83C2×C411yescharacteristic
43C2×C42×C83D411yescharacteristic
44C2×C42×C83D411yescharacteristic
45C4⋊C8×C83—41no—
46C22⋊C8×C83—22no—
47#97×C83—41no—
48C23×D83—22no—
49C2×D332—24no—
50C2×D332—42no—
51C2×D332—42no—
52C2×D332—42no—
53C2×D332—42no—
54C2×D332—42no—
55C2×D332—41no—
56C2×D332—41no—
57C2×D332—82no—
58C22×Dic83—22no—
59C2×C83⋊C8—42no—
60C4⋊Dic83—42no—
61(C2×D83)⋊C4—44no—
62(C2×D83)⋊C4—44no—
63(C2×D83)⋊C4—82no—
64(C2×D83)⋊C4—82no—
65C4×Dic83—42no—
66C2×Q8×C83—22no—
67C2×Q8×C83—22no—
68C2×Q8×C83—41no—
69C22×C4×C83C22⋊C411yescharacteristic
70C22×C4×C83C22⋊C411yescharacteristic
71C22×C4×C83C22⋊C411yescharacteristic
72C22×C4×C83C22⋊C411yescharacteristic
73C22×C4×C83C22×C411yescharacteristic
74C22×C4×C83C2×D411yescharacteristic
75C22×C4×C83C2×D411yescharacteristic
76C2×C8×C83—42no—
77C4⋊C4×C83—41no—
78C4⋊C4×C83—41no—
79C4⋊C4×C83—41no—
80C42×C83—21no—
81C42×C83—21no—
82C42×C83—21no—
83C42×C83—21no—
84C42×C83—21no—
85C42×C83—21no—
86C23×C83C2×C22⋊C411yescharacteristic
87C22×D83—42no—
88C22×D83—42no—
89C22×D83—42no—
90C22×D83—42no—
91C22×D83—42no—
92C22×D83—42no—
93C22×D83—42no—
94Q8×C83—44no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—22no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—41no—
107C2×C4×C83—41no—
108C2×C4×C83—41no—
109C2×C4×C83#9911yescharacteristic
110C2×C4×C83#9911yescharacteristic
111C2×C4×C83#10012yes—
112C2×C4×C83#13112yes—
113C2×Dic83—42no—
114C2×Dic83—42no—
115C2×Dic83—42no—
116D332—84no—
117D332—82no—
118D332—84no—
119D332—82no—
120C8×C83—82no—
121C83⋊C8—82no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83#40811yescharacteristic
125C22×C83C2×#9911yescharacteristic
126C22×C83C2×#10011yescharacteristic
127C2×D83—82no—
128C2×D83—82no—
129C2×D83—82no—
130C2×D83—82no—
131C2×D83—82no—
132C2×D83—82no—
133C2×D83—82no—
134C4×C83—21no—
135C4×C83—22no—
136C4×C83—22no—
137C4×C83—41no—
138C4×C83—41no—
139C4×C83—41no—
140C4×C83—41no—
141C4×C83#35512yes—
142Dic83—82no—
143#11567—831nomaximal, Sylow
144C2×C83—21no—
145C2×C83—21no—
146C2×C83#147111yescharacteristic
147C2×C83#212811yescharacteristic
148C2×C83C2×#35511yescharacteristic
149D83—162no—
150#2548—831no—
151#1467D8311yescharacteristic, complemented
152#1461—831no—
153#1318—834no—
154C83#1156711yescharacteristic, complemented, Sylow
155C2×#127—831no—
156C2×#124—832no—
157C2×C4×Q8C2×D8311yescharacteristic
158#432—21no—
159#431—1661no—
160#394—1661no—
161#394—1661no—
162#390—832no—
163#390—1661no—
164#390—1661no—
165C4×C4⋊C4—831no—
166#324—21no—
167#324—832no—
168#324—1661no—
169#324C2×D8312yes—
170C22×Q8C4×D8311yescharacteristic
171C22×D4—832no—
172C22×D4—1661no—
173C22×D4—1661no—
174#127—3321no—
175#127—3321no—
176#124—3324no—
177C4×Q8—41no—
178C4×Q8—41no—
179C2×C4⋊C4—1661no—
180C2×C4⋊C4D33211yescharacteristic
181C2×C4⋊C4C22⋊D8311yescharacteristic
182C2×C4⋊C4C22×D8311yescharacteristic
183C2×C22⋊C4—1662no—
184C2×C22⋊C4—1662no—
185C2×C22⋊C4—1662no—
186C2×C22⋊C4—1662no—
187C2×C42—832no—
188C2×C42C4×D8311yescharacteristic
189C2×C42D33211yescharacteristic
190C2×C42C22⋊D8311yescharacteristic
191C4⋊C8—41no—
192C4⋊C8—3321no—
193C22⋊C8—22no—
194C22⋊C8—1662no—
195#97—41no—
196C4×C8—3321no—
197#95—1661no—
198C24—1662no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C2×D4—3322no—
203C2×D4—3322no—
204C2×D4—3322no—
205C2×D4—3322no—
206C2×D4—3322no—
207C2×D4—3321no—
208C2×D4—3321no—
209C2×D4—6642no—
210C2×D4—1664no—
211C22×C4—1662no—
212C22×C4(C2×D83)⋊C411yescharacteristic
213C22×C4(C2×D83)⋊C411yescharacteristic
214C22×C4(C2×D83)⋊C411yescharacteristic
215C22×C4(C2×D83)⋊C411yescharacteristic
216C22×C4C2×C4×D8311yescharacteristic
217C22×C4C2×C22⋊D8311yescharacteristic
218C22×C4C2×D33211yescharacteristic
219C2×C8—42no—
220C2×C8—3322no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—3322no—
225C22⋊C4—3324no—
226C22⋊C4—3324no—
227C22⋊C4—6642no—
228C22⋊C4—6642no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—3322no—
236C23—3322no—
237C23—3322no—
238C23—3322no—
239C23—3322no—
240C23—3322no—
241C23—3322no—
242C23—3322no—
243C23C2×(C2×D83)⋊C411yescharacteristic
244Q8—44no—
245D4—6644no—
246D4—6642no—
247D4—6644no—
248D4—6642no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—3322no—
264C2×C4—3322no—
265C2×C4—3322no—
266C2×C4#1458170611yescharacteristic
267C2×C4#1458170611yescharacteristic
268C2×C4#1458170812yes—
269C2×C4#1458180012yes—
270C8—82no—
271C8—6642no—
272C22—21no—
273C22—21no—
274C22—6642no—
275C22—6642no—
276C22—6642no—
277C22—6642no—
278C22—6642no—
279C22—6642no—
280C22—6642no—
281C22#1912766911yescharacteristic
282C22C2×#1458170611yescharacteristic
283C22C2×#1458170811yescharacteristic
284C4—21no—
285C4—22no—
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—6642no—
292C4#1912749712yes—
293C2—21no—
294C2—21no—
295C2—13282no—
296C2#2490582411yescharacteristic
297C2#2490913411yescharacteristic
298C2C2×#1912749711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1467C2
socleC22×C83C2×#99
prime radicalC2×#14581789C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1467D83
hypocenterC83#11567
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #24905787C2semidirect
3. #24905394C2semidirect
4. C83×#1467C2semidirect
151. #1467D83semidirect
154. C83#11567semidirect

Maximal subgroups

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

subgroupindex
#249124622
#249057872
#249053942
C83×#14672
#1156783