Finite Group Databasebeta

← #16854945

Subgroups — #16854945

Order 3968.

738 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 173 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×#11758519C212yesmaximal, complemented
2#14521560C211yesmaximal, characteristic
3#14521205C214yesmaximal, complemented
4#14521188C214yesmaximal, complemented
5#14521058C212yesmaximal, complemented
6#14521013C211yesmaximal, characteristic, complemented
7C23×Q8×C31C211yesmaximal, characteristic, complemented
8C22×Q8×C31—28no—
9C22×Q8×C31—22no—
10C22×Q8×C31—21no—
11C22×Q8×C31C2214yes—
12C22×Q8×C31C2212yes—
13C23×C4×C31—21no—
14C23×C4×C31C2211yescharacteristic
15C2×#92495C2211yescharacteristic
16#11758530—21no—
17C2×#92482C2214yescomplemented
18#11758519—28no—
19C2×#92481C2214yescomplemented
20#11758507C2212yes—
21#11758506—21no—
22C4×C22⋊D31—22no—
23#11758503C2211yescharacteristic
24#11758501C2212yes—
25C2×#92474C2212yes—
26#11758423C2218yescomplemented
27C22⋊(C31⋊C8)C2214yes—
28C24×C31D411yescharacteristic
29C2×Q8×C31—216no—
30C2×Q8×C31—28no—
31C2×Q8×C31—216no—
32C2×Q8×C31—24no—
33C2×Q8×C31—24no—
34C2×Q8×C31—22no—
35C2×Q8×C31D418yescomplemented
36C2×Q8×C31C2314yes—
37C22×C4×C31—28no—
38C22×C4×C31—24no—
39C22×C4×C31—22no—
40C22×C4×C31—24no—
41C22×C4×C31D412yes—
42C22×C4×C31D411yescharacteristic
43C22×C4×C31C2312yes—
44C22×C4×C31C2311yescharacteristic
45C2×C22⋊D31—22no—
46C2×C22⋊D31—21no—
47#92495—24no—
48C2×D124C2311yescharacteristic
49C2×C4×D31—21no—
50C2×Dic62C2311yescharacteristic
51C22⋊Dic31—22no—
52C22⋊Dic31—21no—
53#92482—28no—
54#92481—28no—
55(C2×D31)⋊C4—22no—
56C4⋊Dic31C2312yes—
57Dic31⋊C4—22no—
58C4×Dic31—21no—
59#92474—24no—
60C2×C31⋊C8C2314yes—
61C23×C31—22no—
62C23×C31—24no—
63C23×C31C2×D411yescharacteristic
64C23×C31C2×D412yes—
65C22×D31—21no—
66Q8×C31—216no—
67Q8×C31—28no—
68Q8×C31—24no—
69Q8×C31C2×D418yes—
70C2×C4×C31—216no—
71C2×C4×C31—28no—
72C2×C4×C31—22no—
73C2×C4×C31—24no—
74C2×C4×C31—24no—
75C2×C4×C31—24no—
76C2×C4×C31C2×D412yes—
77C2×C4×C31C2×D411yescharacteristic
78C2×C4×C31C2×D414yes—
79C2×C4×C31C2411yescharacteristic
80C22⋊D31—42no—
81C22⋊D31—44no—
82C2×Dic31—22no—
83C2×Dic31—21no—
84D124—22no—
85C4×D31—42no—
86Dic62—22no—
87C31⋊C8—24no—
88#3004—311nomaximal, Sylow
89C22×C31—28no—
90C22×C31—22no—
91C22×C31—22no—
92C22×C31—22no—
93C22×C31#12012yes—
94C22×C31#13714yescomplemented
95C22×C31C22×D411yescharacteristic
96C2×D31—41no—
97C2×D31—42no—
98C4×C31—28no—
99C4×C31—22no—
100C4×C31#12012yes—
101C4×C31C22×D412yes—
102Dic31—41no—
103Dic31—42no—
104C23×Q8D3111yescharacteristic, complemented
105C2×#137—312no—
106#525—311no—
107#451—314no—
108#448—314no—
109#419—312no—
110#407—311no—
111C2×C31—22no—
112C2×C31—24no—
113C2×C31C2×#12011yescharacteristic
114C2×C31C2×#13712yes—
115D31—81no—
116C2×#41—311no—
117C22×Q8—28no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8C2×D3114yes—
121C22×Q8C2×D3112yes—
122C23×C4—21no—
123C23×C4C2×D3111yescharacteristic
124#137—628no—
125C2×Q16—314no—
126C2×SD16—314no—
127C2×M16—312no—
128#123—621no—
129C22⋊Q8—312no—
130#121—312no—
131#120—621no—
132C4×D4—622no—
133#117—311no—
134#103—318no—
135C22⋊C8—314no—
136C31#300411yescharacteristic, complemented, Sylow
137C24C22⋊D3111yescharacteristic
138#41—624no—
139C2×Q8—216no—
140C2×Q8—28no—
141C2×Q8—216no—
142C2×Q8—24no—
143C2×Q8—24no—
144C2×Q8—22no—
145C2×Q8—311no—
146C2×Q8C22⋊D3118yescomplemented
147C2×Q8C22×D3114yes—
148C2×D4—311no—
149C2×D4—622no—
150C2×D4—621no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—621no—
156C22×C4C22⋊D3112yes—
157C22×C4C22⋊D3111yescharacteristic
158C22×C4C22×D3112yes—
159C22×C4C22×D3111yescharacteristic
160Q16—628no—
161SD16—628no—
162M16—624no—
163C2×C8—314no—
164C4⋊C4—312no—
165C4⋊C4—622no—
166C22⋊C4—621no—
167C22⋊C4—622no—
168C22⋊C4—622no—
169C42—621no—
170C23—24no—
171C23—22no—
172C23—621no—
173C23C2×C22⋊D3112yes—
174C23C2×C22⋊D3111yescharacteristic
175Q8—216no—
176Q8—28no—
177Q8—24no—
178Q8—622no—
179Q8C2×C22⋊D3118yes—
180D4—622no—
181D4—1244no—
182D4—1242no—
183C2×C4—216no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—622no—
190C2×C4—621no—
191C2×C4—1242no—
192C2×C4C2×C22⋊D3112yes—
193C2×C4C2×C22⋊D3111yescharacteristic
194C2×C4C2×C22⋊D3114yes—
195C2×C4C23×D3111yescharacteristic
196C8—624no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—1241no—
202C22—1242no—
203C22#1175851914yescomplemented
204C22#1175853012yes—
205C22C22×C22⋊D3111yescharacteristic
206C4—28no—
207C4—22no—
208C4—1241no—
209C4—1242no—
210C4#1175853012yes—
211C4C22×C22⋊D3112yes—
212C2—24no—
213C2—22no—
214C2—2481no—
215C2C2×#1175853011yescharacteristic
216C2C2×#1175851912yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC23×Q8×C31C2
socleC22×C31C22×D4
prime radical#14521560C2
hypercenterC23×Q8D31
hypocenterC31#3004
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11758519C2semidirect
3. #14521205C2semidirect
4. #14521188C2semidirect
5. #14521058C2semidirect
6. #14521013C2semidirect
7. C23×Q8×C31C2semidirect
17. C2×#92482C22semidirect
19. C2×#92481C22semidirect
26. #11758423C22semidirect
35. C2×Q8×C31D4semidirect
94. C22×C31#137semidirect
104. C23×Q8D31semidirect
136. C31#3004semidirect
146. C2×Q8C22⋊D31semidirect
203. C22#11758519semidirect

Maximal subgroups

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

subgroupindex
C2×#117585192
#145215602
#145212052
#145211882
#145210582
#145210132
C23×Q8×C312
#300431