Finite Group Databasebeta

← #16854938

Subgroups — #16854938

Order 3968.

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

normal subgroups only

This group has 994 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×#11758509C212yesmaximal, complemented
2#14521560C211yesmaximal, characteristic, complemented
3#14521200C214yesmaximal, complemented
4#14521183C214yesmaximal, complemented
5#14521054C212yesmaximal, complemented
6#14521013C211yesmaximal, characteristic, complemented
7C23×D4×C31C211yesmaximal, characteristic, complemented
8C25×C31—21no—
9C22×D4×C31—28no—
10C22×D4×C31—22no—
11C22×D4×C31—21no—
12C22×D4×C31C2214yes—
13C22×D4×C31C2212yes—
14C23×C4×C31C2211yescharacteristic
15C2×#92495C2211yescharacteristic, complemented
16#11758530—21no—
17C2×#92480C2214yescomplemented
18#11758509—28no—
19C2×#92479C2214yescomplemented
20#11758507C2212yescomplemented
21#11758506—21no—
22C4×C22⋊D31—22no—
23#11758503C2211yescharacteristic, complemented
24#11758501C2212yescomplemented
25C2×#92474C2212yescomplemented
26#11758420C2218yescomplemented
27C22⋊(C31⋊C8)C2214yescomplemented
28C24×C31—28no—
29C24×C31—24no—
30C24×C31—22no—
31C24×C31—48no—
32C24×C31D411yescharacteristic
33C2×D4×C31—28no—
34C2×D4×C31—216no—
35C2×D4×C31—216no—
36C2×D4×C31—24no—
37C2×D4×C31—22no—
38C2×D4×C31—24no—
39C2×D4×C31D418yescomplemented
40C2×D4×C31C2314yes—
41C22×C4×C31—24no—
42C22×C4×C31D412yes—
43C22×C4×C31D411yescharacteristic
44C22×C4×C31C2312yes—
45C22×C4×C31C2311yescharacteristic
46C2×C22⋊D31—22no—
47C2×C22⋊D31—21no—
48#92495—24no—
49C2×D124C2311yescharacteristic, complemented
50C2×C4×D31—21no—
51C2×Dic62C2311yescharacteristic, complemented
52C22⋊Dic31—22no—
53C22⋊Dic31—21no—
54#92480—28no—
55#92479—28no—
56(C2×D31)⋊C4—22no—
57C4⋊Dic31C2312yescomplemented
58Dic31⋊C4—22no—
59C4×Dic31—21no—
60#92474—24no—
61C2×C31⋊C8C2314yescomplemented
62C23×C31—22no—
63C23×C31—24no—
64C23×C31—216no—
65C23×C31—28no—
66C23×C31—24no—
67C23×C31—432no—
68C23×C31—48no—
69C23×C31—48no—
70C23×C31—48no—
71C23×C31C2×D411yescharacteristic
72C23×C31C2×D412yes—
73C22×D31—21no—
74D4×C31—216no—
75D4×C31—28no—
76D4×C31—24no—
77D4×C31C2×D418yes—
78C2×C4×C31—22no—
79C2×C4×C31—24no—
80C2×C4×C31—24no—
81C2×C4×C31C2×D412yes—
82C2×C4×C31C2×D411yescharacteristic
83C2×C4×C31C2×D414yes—
84C2×C4×C31C2411yescharacteristic
85C22⋊D31—42no—
86C22⋊D31—44no—
87C2×Dic31—22no—
88C2×Dic31—21no—
89D124—22no—
90C4×D31—42no—
91Dic62—22no—
92C31⋊C8—24no—
93#3003—311nomaximal, Sylow
94C22×C31—28no—
95C22×C31—22no—
96C22×C31—22no—
97C22×C31—22no—
98C22×C31—28no—
99C22×C31—416no—
100C22×C31—432no—
101C22×C31—48no—
102C22×C31#12012yes—
103C22×C31AGL(1,ℤ/8)14yescomplemented
104C22×C31C22×D411yescharacteristic
105C2×D31—41no—
106C2×D31—42no—
107C4×C31—22no—
108C4×C31#12012yes—
109C4×C31C22×D412yes—
110Dic31—41no—
111Dic31—42no—
112C23×D4D3111yescharacteristic, complemented
113C2×AGL(1,ℤ/8)—312no—
114#525—311no—
115#450—314no—
116#447—314no—
117#418—312no—
118#407—311no—
119C2×C31—22no—
120C2×C31—24no—
121C2×C31—48no—
122C2×C31C2×#12011yescharacteristic
123C2×C31C2×AGL(1,ℤ/8)12yes—
124D31—81no—
125C25—21no—
126C2×#41—311no—
127C22×D4—28no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4C2×D3114yes—
131C22×D4C2×D3112yes—
132C23×C4C2×D3111yescharacteristic
133AGL(1,ℤ/8)—628no—
134C2×SD16—314no—
135C2×D8—314no—
136C2×M16—312no—
137#123—621no—
138C22⋊Q8—312no—
139#121—312no—
140#120—621no—
141C4×D4—622no—
142#117—311no—
143#102—318no—
144C22⋊C8—314no—
145C31#300311yescharacteristic, complemented, Sylow
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D3111yescharacteristic
151#41—624no—
152C2×Q8—311no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—311no—
160C2×D4—622no—
161C2×D4—621no—
162C2×D4C22⋊D3118yescomplemented
163C2×D4C22×D3114yes—
164C22×C4—24no—
165C22×C4—621no—
166C22×C4C22⋊D3112yes—
167C22×C4C22⋊D3111yescharacteristic
168C22×C4C22×D3112yes—
169C22×C4C22×D3111yescharacteristic
170SD16—628no—
171D8—628no—
172M16—624no—
173C2×C8—314no—
174C4⋊C4—312no—
175C4⋊C4—622no—
176C22⋊C4—621no—
177C22⋊C4—622no—
178C22⋊C4—622no—
179C42—621no—
180C23—24no—
181C23—22no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—621no—
190C23C2×C22⋊D3112yes—
191C23C2×C22⋊D3111yescharacteristic
192Q8—622no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—622no—
197D4—1244no—
198D4—1242no—
199D4C2×C22⋊D3118yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—622no—
204C2×C4—621no—
205C2×C4—1242no—
206C2×C4C2×C22⋊D3112yes—
207C2×C4C2×C22⋊D3111yescharacteristic
208C2×C4C2×C22⋊D3114yes—
209C2×C4C23×D3111yescharacteristic
210C8—624no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—432no—
217C22—416no—
218C22—48no—
219C22—1241no—
220C22—1242no—
221C22#1175850914yescomplemented
222C22#1175853012yes—
223C22C22×C22⋊D3111yescharacteristic
224C4—22no—
225C4—1241no—
226C4—1242no—
227C4#1175853012yes—
228C4C22×C22⋊D3112yes—
229C2—24no—
230C2—22no—
231C2—48no—
232C2—2481no—
233C2C2×#1175853011yescharacteristic
234C2C2×#1175850912yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC23×D4×C31C2
socleC22×C31C22×D4
hypercenterC23×D4D31
hypocenterC31#3003
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#11758509C2semidirect
2. #14521560C2semidirect
3. #14521200C2semidirect
4. #14521183C2semidirect
5. #14521054C2semidirect
6. #14521013C2semidirect
7. C23×D4×C31C2semidirect
15. C2×#92495C22semidirect
17. C2×#92480C22semidirect
19. C2×#92479C22semidirect
20. #11758507C22semidirect
23. #11758503C22semidirect
24. #11758501C22semidirect
25. C2×#92474C22semidirect
26. #11758420C22semidirect
27. C22⋊(C31⋊C8)C22semidirect
39. C2×D4×C31D4semidirect
49. C2×D124C23semidirect
51. C2×Dic62C23semidirect
57. C4⋊Dic31C23semidirect
61. C2×C31⋊C8C23semidirect
103. C22×C31AGL(1,ℤ/8)semidirect
112. C23×D4D31semidirect
145. C31#3003semidirect
162. C2×D4C22⋊D31semidirect
221. C22#11758509semidirect

Maximal subgroups

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

subgroupindex
C2×#117585092
#145215602
#145212002
#145211832
#145210542
#145210132
C23×D4×C312
#300331