Finite Group Databasebeta

← #16849000

Subgroups — #16849000

Order 3968.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#11758464C211yesmaximal, characteristic, complemented
2C2×#11758470C211yesmaximal, characteristic
3#14520987C211yesmaximal, characteristic, complemented
4#14520978C212yesmaximal, complemented
5#14520967C211yesmaximal, characteristic
6#14520966C211yesmaximal, characteristic
7#14520962C212yesmaximal, complemented
8#14520896C212yesmaximal, complemented
9#14520896C212yesmaximal, complemented
10#14520894C211yesmaximal, characteristic, complemented
11C31×#400C211yesmaximal, characteristic, complemented
12C22×D124C2211yescharacteristic
13C22×C4×D31C2212yes—
14C22×Dic62C2211yescharacteristic
15C2×C4⋊C4×C31C2211yescharacteristic
16C2×C4⋊C4×C31C2211yescharacteristic
17C2×(C2×D31)⋊C4C2212yes—
18C2×(C2×D31)⋊C4C2212yes—
19C2×(C2×D31)⋊C4C2212yes—
20C2×(C2×D31)⋊C4C2212yes—
21C2×(C2×D31)⋊C4C2212yes—
22C2×Dic31⋊C4C2211yescharacteristic
23C2×Dic31⋊C4C2211yescharacteristic
24C2×C4×Dic31C2212yes—
25C2×C4×Dic31C2211yescharacteristic
26C2×C4×Dic31C2211yescharacteristic
27C2×C4×Dic31C2211yescharacteristic
28#11758470—24no—
29#11758464—24no—
30C31×#95C2211yescharacteristic
31C31×#95C2212yes—
32C31×#95C2212yes—
33C31⋊#95C2212yes—
34C31⋊#95C2211yescharacteristic
35C31⋊#95C2212yes—
36C31⋊#95C2212yes—
37C23×D31C2312yes—
38C22×C4×C31C2312yes—
39C22×C4×C31C2312yes—
40C22×C4×C31C2311yescharacteristic
41C22×C4×C31C2311yescharacteristic
42C22×C4×C31C2311yescharacteristic
43C22×Dic31C2312yes—
44C22×Dic31C2311yescharacteristic
45C22×Dic31C2311yescharacteristic
46C22×Dic31C2311yescharacteristic
47C22×Dic31C2311yescharacteristic
48C2×D124—24no—
49C2×D124—22no—
50C2×C4×D31—24no—
51C2×C4×D31—24no—
52C2×C4×D31—24no—
53C2×Dic62—22no—
54C2×Dic62—22no—
55C2×Dic62—22no—
56C4⋊C4×C31—22no—
57C4⋊C4×C31—22no—
58(C2×D31)⋊C4—24no—
59(C2×D31)⋊C4—24no—
60(C2×D31)⋊C4—24no—
61(C2×D31)⋊C4—24no—
62(C2×D31)⋊C4—24no—
63Dic31⋊C4—22no—
64Dic31⋊C4—22no—
65C4×Dic31—22no—
66C4×Dic31—22no—
67C4×Dic31—24no—
68C4×Dic31D414yes—
69C23×C31C2411yescharacteristic
70C22×D31—22no—
71C22×D31—22no—
72C22×D31—22no—
73C22×D31—22no—
74C22×D31—22no—
75C22×D31—22no—
76C22×D31#4114yes—
77C2×C4×C31—21no—
78C2×C4×C31—21no—
79C2×C4×C31—22no—
80C2×C4×C31—21no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31—22no—
87C2×C4×C31—22no—
88C2×C4×C31C2×D412yes—
89C2×C4×C31#4112yes—
90C2×Dic31—21no—
91C2×Dic31—21no—
92C2×Dic31—21no—
93C2×Dic31—21no—
94C2×Dic31—21no—
95C2×Dic31—22no—
96C2×Dic31—22no—
97C2×Dic31—22no—
98C2×Dic31—22no—
99C2×Dic31C2×D412yes—
100C2×Dic31C2×D412yes—
101C2×Dic31#4112yes—
102C2×Dic31#4114yes—
103D124—28no—
104C4×D31—48no—
105Dic62—28no—
106#2398—311nomaximal, Sylow
107C22×C31C22×D411yescharacteristic
108C22×C31C2×#4111yescharacteristic
109C22×C31C2×#4112yes—
110C22×C31C2×#4111yescharacteristic
111C22×C31C2×#4112yes—
112C2×D31—24no—
113C2×D31—24no—
114C2×D31—24no—
115C2×D31—42no—
116C2×D31—42no—
117C2×D31—42no—
118C2×D31—42no—
119C4×C31—22no—
120C4×C31—22no—
121C4×C31—41no—
122C4×C31—42no—
123C4×C31—42no—
124Dic31—22no—
125Dic31—22no—
126Dic31—41no—
127Dic31—42no—
128Dic31#12414yes—
129C2×C4×Q8—311no—
130C2×C4×D4—311no—
131#400—311no—
132#400D3111yescharacteristic, complemented
133#397—312no—
134#394—311no—
135#394—311no—
136#393—312no—
137C4×C22⋊C4—312no—
138C4×C22⋊C4—312no—
139#376—311no—
140C2×C31#52511yescharacteristic
141C2×C31#52511yescharacteristic
142C2×C31C2×#12411yescharacteristic
143C2×C31#52912yes—
144C2×C31#52912yes—
145D31—44no—
146C22×Q8—311no—
147C22×D4—311no—
148C23×C4—312no—
149C4×Q8—624no—
150C4×D4—624no—
151C2×C4⋊C4—311no—
152C2×C4⋊C4—311no—
153C2×C4⋊C4C2×D3111yescharacteristic
154C2×C4⋊C4C2×D3111yescharacteristic
155C2×C22⋊C4—312no—
156C2×C22⋊C4—312no—
157C2×C22⋊C4—312no—
158C2×C22⋊C4—312no—
159C2×C22⋊C4—312no—
160C2×C42—311no—
161C2×C42—311no—
162C2×C42—311no—
163C2×C42—312no—
164#95—312no—
165#95—311no—
166#95—312no—
167#95—312no—
168#95C2×D3111yescharacteristic
169#95C2×D3112yes—
170#95C2×D3112yes—
171C31#239811yescharacteristic, complemented, Sylow
172C24—312no—
173C2×Q8—622no—
174C2×Q8—622no—
175C2×Q8—622no—
176C2×D4—624no—
177C2×D4—622no—
178C22×C4—311no—
179C22×C4—311no—
180C22×C4—311no—
181C22×C4—312no—
182C22×C4—311no—
183C22×C4—624no—
184C22×C4—624no—
185C22×C4—624no—
186C22×C4C22×D3112yes—
187C22×C4C22×D3112yes—
188C22×C4C22×D3111yescharacteristic
189C22×C4C22×D3111yescharacteristic
190C22×C4C22×D3111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—622no—
194C4⋊C4—622no—
195C22⋊C4—624no—
196C22⋊C4—624no—
197C22⋊C4—624no—
198C22⋊C4—624no—
199C22⋊C4—624no—
200C42—314no—
201C42—624no—
202C42—622no—
203C42—622no—
204C23—314no—
205C23—622no—
206C23—622no—
207C23—622no—
208C23—622no—
209C23—622no—
210C23—622no—
211C23C23×D3111yescharacteristic
212Q8—628no—
213D4—628no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—312no—
226C2×C4—314no—
227C2×C4—312no—
228C2×C4—312no—
229C2×C4—621no—
230C2×C4—621no—
231C2×C4—621no—
232C2×C4—621no—
233C2×C4—621no—
234C2×C4—622no—
235C2×C4—622no—
236C2×C4—622no—
237C2×C4—622no—
238C2×C4—1248no—
239C2×C4D4×D3112yes—
240C2×C4#9249912yes—
241C22—624no—
242C22—624no—
243C22—624no—
244C22—1242no—
245C22—1242no—
246C22—1242no—
247C22—1242no—
248C22C2×D4×D3111yescharacteristic
249C22C2×#9249911yescharacteristic
250C22#41×D3112yes—
251C22#41×D3111yescharacteristic
252C22#41×D3112yes—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—314no—
259C4—622no—
260C4—622no—
261C4—1241no—
262C4—1242no—
263C2—1244no—
264C2#1452155711yescharacteristic
265C2#1452156211yescharacteristic
266C2#124×D3111yescharacteristic
267C2#1452158212yes—
268C2#1452158412yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#400C2
prime radicalC22×D124C22
hypercenter#400D31
hypocenterC31#2398
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#11758464C2semidirect
3. #14520987C2semidirect
4. #14520978C2semidirect
7. #14520962C2semidirect
8. #14520896C2semidirect
9. #14520896C2semidirect
10. #14520894C2semidirect
11. C31×#400C2semidirect
132. #400D31semidirect
171. C31#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#117584642
C2×#117584702
#145209872
#145209782
#145209672
#145209662
#145209622
#145208962
#145208942
C31×#4002
#239831