Finite Group Databasebeta

← #35313896

Subgroups — #35313896

Order 21248.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#24905801C212yesmaximal
2#24905409C212yesmaximal, complemented
3#24905397C212yesmaximal, complemented
4C83×#2326C211yesmaximal, characteristic
5#19127665—22no—
6#19127427C2214yes—
7#19127425—22no—
8C2×#124×C83C411yescharacteristic
9C2×C4×Q8×C83C411yescharacteristic
10C2×C4×D4×C83C2211yescharacteristic
11#391×C83C2211yescharacteristic
12#390×C83C411yescharacteristic
13#388×C83—22no—
14#385×C83—22no—
15#384×C83C2211yescharacteristic
16#376×C83C411yescharacteristic
17C2×#13050574—42no—
18C22⋊(C83⋊C8)—24no—
19C22⋊(C83⋊C8)—42no—
20C22×Q8×C83C2×C411yescharacteristic
21C22×D4×C83C2×C411yescharacteristic
22C23×C4×C83D412yes—
23#124×C83—22no—
24#124×C83—22no—
25C4×Q8×C83—42no—
26C4×D4×C83—42no—
27C2×C4⋊C4×C83—21no—
28C2×C4⋊C4×C83—22no—
29C2×C4⋊C4×C83C2×C411yescharacteristic
30C2×C4⋊C4×C83C2311yescharacteristic
31C2×C22⋊C4×C83—22no—
32C2×C22⋊C4×C83—22no—
33C2×C22⋊C4×C83D412yes—
34C2×C42×C83—21no—
35C2×C42×C83C2×C411yescharacteristic
36C2×C42×C83C2×C411yescharacteristic
37C2×C42×C83C2×C411yescharacteristic
38#95×C83—21no—
39#95×C83—21no—
40#95×C83—22no—
41#13050574—84no—
42C2×C83⋊C8—44no—
43C24×C83C22⋊C412yes—
44C2×Q8×C83—21no—
45C2×Q8×C83—41no—
46C2×Q8×C83—41no—
47C2×Q8×C83M1612yescomplemented
48C2×D4×C83—21no—
49C2×D4×C83—21no—
50C2×D4×C83—41no—
51C2×D4×C83—41no—
52C22×C4×C83—21no—
53C22×C4×C83—21no—
54C22×C4×C83—22no—
55C22×C4×C83—22no—
56C22×C4×C83—21no—
57C22×C4×C83—21no—
58C22×C4×C83—82no—
59C22×C4×C83C22⋊C411yescharacteristic
60C22×C4×C83C22⋊C411yescharacteristic
61C22×C4×C83C22×C411yescharacteristic
62C22×C4×C83C2×D412yes—
63C4⋊C4×C83—41no—
64C4⋊C4×C83—41no—
65C4⋊C4×C83—42no—
66C4⋊C4×C83—82no—
67C22⋊C4×C83—28no—
68C22⋊C4×C83—42no—
69C22⋊C4×C83—82no—
70C42×C83—22no—
71C42×C83—21no—
72C42×C83—41no—
73C42×C83—42no—
74C42×C83M1612yescomplemented
75C23×C83—22no—
76C23×C83—22no—
77C23×C83—22no—
78C23×C83—42no—
79C23×C83—42no—
80C23×C83C2×C22⋊C411yescharacteristic
81Q8×C83—42no—
82Q8×C83—42no—
83D4×C83—42no—
84D4×C83—42no—
85C2×C4×C83—21no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83—41no—
92C2×C4×C83—44no—
93C2×C4×C83—41no—
94C2×C4×C83—41no—
95C2×C4×C83—41no—
96C2×C4×C83—41no—
97C2×C4×C83—42no—
98C2×C4×C83—41no—
99C2×C4×C83—41no—
100C2×C4×C83—42no—
101C2×C4×C83—41no—
102C2×C4×C83—41no—
103C2×C4×C83—41no—
104C2×C4×C83—82no—
105C2×C4×C83#9912yes—
106C2×C4×C83C2×M1611yescharacteristic
107C2×C4×C83C2×M1611yescharacteristic
108C83⋊C8—84no—
109C22×C83—21no—
110C22×C83—21no—
111C22×C83—44no—
112C22×C83—42no—
113C22×C83—42no—
114C22×C83—42no—
115C22×C83—82no—
116C22×C83#40711yescharacteristic
117C22×C83C2×#9911yescharacteristic
118C22×C83#41311yescharacteristic
119C4×C83—22no—
120C4×C83—21no—
121C4×C83—21no—
122C4×C83—42no—
123C4×C83—42no—
124C4×C83—41no—
125C4×C83—42no—
126C4×C83—81no—
127C4×C83—81no—
128#11535—831nomaximal, Sylow
129C2×C83—21no—
130C2×C83—21no—
131C2×C83—82no—
132C2×C83#146511yescharacteristic
133C2×C83#212711yescharacteristic
134C2×C83#213511yescharacteristic
135#2326D8311yescharacteristic
136#1465—832no—
137#1325—832no—
138#1319—832no—
139C83#1153511yescharacteristic, complemented, Sylow
140C2×#124Dic8311yescharacteristic
141C2×C4×Q8Dic8311yescharacteristic
142C2×C4×D4C2×D8311yescharacteristic
143#407—1662no—
144#391C2×D8311yescharacteristic
145#390Dic8311yescharacteristic
146#388—22no—
147#385—22no—
148#384C2×D8311yescharacteristic
149#376Dic8311yescharacteristic
150#324—834no—
151#323—1662no—
152C22×Q8C2×Dic8311yescharacteristic
153C22×D4C2×Dic8311yescharacteristic
154C23×C4C22⋊D8312yes—
155C2×M16—3322no—
156#124—22no—
157#124—22no—
158C4×Q8—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C2×Dic8311yescharacteristic
163C2×C4⋊C4C22×D8311yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊D8312yes—
167C2×C42—21no—
168C2×C42C2×Dic8311yescharacteristic
169C2×C42C2×Dic8311yescharacteristic
170C2×C42C2×Dic8311yescharacteristic
171C22⋊C8—3322no—
172C22⋊C8—1664no—
173#95—21no—
174#95—21no—
175#95—22no—
176C24C22⋊Dic8312yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8#1305057412yescomplemented
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—82no—
192C22×C4C22⋊Dic8311yescharacteristic
193C22×C4C22⋊Dic8311yescharacteristic
194C22×C4C22×Dic8311yescharacteristic
195C22×C4C2×C22⋊D8312yes—
196M16—6644no—
197C2×C8—3324no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C22⋊C4—28no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42—21no—
206C42—22no—
207C42—41no—
208C42—42no—
209C42#1305057412yescomplemented
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊Dic8311yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4#1458170712yes—
241C2×C4C2×#1305057411yescharacteristic
242C2×C4C2×#1305057411yescharacteristic
243C8—6644no—
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22#1912766511yescharacteristic
252C22C2×#1458170711yescharacteristic
253C22#1912768511yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2490580111yescharacteristic
267C2#2490913211yescharacteristic
268C2#2490915811yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#2326C2
socleC22×C83C2×#99
prime radicalC22×D4×C83C2×C4
cosocleC2×C4⋊C4×C83C23
hypercenter#2326D83
hypocenterC83#11535
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #24905409C2semidirect
3. #24905397C2semidirect
47. C2×Q8×C83M16semidirect
74. C42×C83M16semidirect
139. C83#11535semidirect
180. C2×Q8#13050574semidirect
209. C42#13050574semidirect

Maximal subgroups

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

subgroupindex
#249058012
#249054092
#249053972
C83×#23262
#1153583