Finite Group Databasebeta

← #16848799

Subgroups — #16848799

Order 3968.

496 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 209 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×#11758521C211yesmaximal, characteristic
2#14520956C211yesmaximal, characteristic
3#14520953C212yesmaximal
4#14520946C212yesmaximal
5#14520944C212yesmaximal
6C4×Dic31⋊C4C212yesmaximal
7#14520900C212yesmaximal
8#14520900C212yesmaximal
9C31×#391C211yesmaximal, characteristic
10C22×Q8×C31C2211yescharacteristic
11C22×Dic62C2211yescharacteristic
12C2×C4⋊C4×C31C2211yescharacteristic
13C2×C42×C31C2211yescharacteristic
14#11758521C418yescomplemented
15C2×C4⋊Dic31C2211yescharacteristic
16C2×Dic31⋊C4C2212yes—
17C2×Dic31⋊C4C2212yes—
18C2×Dic31⋊C4C2212yes—
19C2×Dic31⋊C4C2212yes—
20C2×Dic31⋊C4C2212yes—
21C2×C4×Dic31C2212yes—
22C2×C4×Dic31C2212yes—
23C2×C4×Dic31C2211yescharacteristic
24C2×C4×Dic31C2212yes—
25C2×C4×Dic31C2211yescharacteristic
26C31×#95C2212yes—
27C31×#95C2212yes—
28C31⋊#95C2212yes—
29C31⋊#95C2212yes—
30C31⋊#95C2212yes—
31C31⋊#95C2212yes—
32C2×Q8×C31—41no—
33C2×Q8×C31—41no—
34C2×Q8×C31C2×C414yes—
35C22×C4×C31C2312yes—
36C22×C4×C31C2311yescharacteristic
37C22×C4×C31C2311yescharacteristic
38C22×C4×C31C2311yescharacteristic
39C22×C4×C31C2312yes—
40C22×Dic31C2312yes—
41C22×Dic31C2312yes—
42C22×Dic31C2312yes—
43C22×Dic31C2312yes—
44C2×Dic62—41no—
45C2×Dic62—41no—
46C2×Dic62C2×C414yes—
47C4⋊C4×C31—41no—
48C42×C31—41no—
49C4⋊Dic31—41no—
50Dic31⋊C4—24no—
51Dic31⋊C4—42no—
52Dic31⋊C4C2×C418yes—
53Dic31⋊C4C2×C418yes—
54Dic31⋊C4Q818yescomplemented
55C4×Dic31—24no—
56C4×Dic31—24no—
57C4×Dic31—41no—
58C4×Dic31—42no—
59C4×Dic31C2×C414yes—
60C23×C31C2411yescharacteristic
61Q8×C31—44no—
62C2×C4×C31—22no—
63C2×C4×C31—22no—
64C2×C4×C31—21no—
65C2×C4×C31—21no—
66C2×C4×C31—21no—
67C2×C4×C31—21no—
68C2×C4×C31—21no—
69C2×C4×C31—21no—
70C2×C4×C31—22no—
71C2×C4×C31—22no—
72C2×C4×C31—22no—
73C2×C4×C31C22×C412yes—
74C2×C4×C31C22×C412yes—
75C2×C4×C31C22×C412yes—
76C2×C4×C31C2×Q814yes—
77C2×Dic31—22no—
78C2×Dic31—22no—
79C2×Dic31—22no—
80C2×Dic31—22no—
81C2×Dic31—22no—
82C2×Dic31—22no—
83C2×Dic31—22no—
84C2×Dic31C22×C414yes—
85C2×Dic31C22×C414yes—
86C2×Dic31C2×Q814yes—
87C2×Dic31C2×Q814yes—
88C2×Dic31#4114yes—
89Dic62—44no—
90#2369—311nomaximal, Sylow
91C22×C31C23×C411yescharacteristic
92C22×C31C22×Q812yes—
93C22×C31C2×#4112yes—
94C22×C31H(5,𝔽2)11yescharacteristic
95C22×C31H(5,𝔽2)11yescharacteristic
96C4×C31—22no—
97C4×C31—22no—
98C4×C31—22no—
99C4×C31—24no—
100C4×C31—42no—
101Dic31—24no—
102Dic31—24no—
103Dic31—42no—
104Dic31C4×Q818yes—
105C2×C4⋊Q8—311no—
106#391—311no—
107#391—312no—
108#391D3111yescharacteristic
109#389—312no—
110#389—312no—
111C4×C4⋊C4—312no—
112C4×C4⋊C4—312no—
113C4×C4⋊C4—312no—
114C2×C31C2×C4×Q812yes—
115C2×C31#51811yescharacteristic
116C2×C31#55412yes—
117C2×C31#55511yescharacteristic
118C2×C31Q8211yescharacteristic
119C22×Q8—311no—
120C22×Q8C2×D3111yescharacteristic
121C4⋊Q8—318no—
122C2×C4⋊C4—312no—
123C2×C4⋊C4—312no—
124C2×C4⋊C4—312no—
125C2×C4⋊C4—312no—
126C2×C4⋊C4—312no—
127C2×C4⋊C4—311no—
128C2×C4⋊C4C2×D3111yescharacteristic
129C2×C42—311no—
130C2×C42—312no—
131C2×C42—312no—
132C2×C42—311no—
133C2×C42—312no—
134C2×C42C2×D3111yescharacteristic
135#95—312no—
136#95—312no—
137#95—312no—
138#95—312no—
139#95C2×D3112yes—
140#95C2×D3112yes—
141C31#236911yescharacteristic, complemented, Sylow
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—314no—
145C2×Q8—1241no—
146C2×Q8—1241no—
147C2×Q8C4×D3114yes—
148C22×C4—312no—
149C22×C4—312no—
150C22×C4—312no—
151C22×C4—312no—
152C22×C4C22×D3112yes—
153C22×C4C22×D3111yescharacteristic
154C22×C4C22×D3111yescharacteristic
155C22×C4C22×D3111yescharacteristic
156C22×C4C22×D3112yes—
157C4⋊C4—41no—
158C4⋊C4—318no—
159C4⋊C4—318no—
160C4⋊C4—318no—
161C4⋊C4—624no—
162C4⋊C4—1242no—
163C4⋊C4—1241no—
164C42—41no—
165C42—314no—
166C42—624no—
167C42—624no—
168C42—1242no—
169C42—1241no—
170C23C23×D3111yescharacteristic
171Q8—44no—
172Q8—1244no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—314no—
185C2×C4—314no—
186C2×C4—314no—
187C2×C4—314no—
188C2×C4—314no—
189C2×C4—622no—
190C2×C4—622no—
191C2×C4—622no—
192C2×C4—622no—
193C2×C4—622no—
194C2×C4—622no—
195C2×C4—622no—
196C2×C4C2×C4×D3112yes—
197C2×C4C2×C4×D3112yes—
198C2×C4C2×C4×D3112yes—
199C2×C4Q8×D3114yes—
200C22C22×C4×D3111yescharacteristic
201C22#1175856111yescharacteristic
202C22#1175856111yescharacteristic
203C22C2×Q8×D3112yes—
204C22#41×D3112yes—
205C4—24no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—318no—
211C4—624no—
212C4—624no—
213C4—1242no—
214C2C4×Q8×D3112yes—
215C2#1452152211yescharacteristic
216C2#1452175312yes—
217C2#1452176111yescharacteristic
218C2#1452177911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#391C2
socle, prime radical, cosocleC23×C31C24
hypercenter#391D31
hypocenterC31#2369
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
14. #11758521C4semidirect
54. Dic31⋊C4Q8semidirect
141. C31#2369semidirect

Maximal subgroups

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

subgroupindex
C2×#117585212
#145209562
#145209532
#145209462
#145209442
C4×Dic31⋊C42
#145209002
C31×#3912
#236931