Finite Group Databasebeta

← #16855116

Subgroups — #16855116

Order 3968.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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#14521885C211yesmaximal, characteristic
2#14521623C212yesmaximal, complemented
3#14521238C218yesmaximal, complemented
4#14521023C211yesmaximal, characteristic, complemented
5#14521022C212yesmaximal, complemented
6C31×#536C211yesmaximal, characteristic, complemented
7C2×#41×C31C2211yescharacteristic
8C22×Q8×C31C2211yescharacteristic, complemented
9#11758567—22no—
10#11758566—22no—
11#41×D31—24no—
12#11758564—21no—
13#11758561—21no—
14#11758561C2214yes—
15C2×#92495—22no—
16C2×#92495C2212yes—
17C31×C4⋊Q8—21no—
18C31×#124—21no—
19C31×#123C2214yescomplemented
20C31×C22⋊Q8—22no—
21C31×#117C2211yescharacteristic, complemented
22#11758524—22no—
23#11758523—24no—
24#11758521—22no—
25#11758506C2218yescomplemented
26#11758503C2212yescomplemented
27#11758422C2214yescomplemented
28#11758394C2218yescomplemented
29#41×C31—22no—
30#41×C31—22no—
31C2×Q8×C31—22no—
32C2×Q8×C31—22no—
33C2×Q8×C31—21no—
34C2×Q8×C31—21no—
35C2×Q8×C31D411yescharacteristic
36C2×D4×C31D411yescharacteristic
37C2×D4×C31C2312yes—
38C22×C4×C31D412yes—
39C22×C4×C31C2312yes—
40C22×C4×C31C2311yescharacteristic
41C2×C22⋊D31—24no—
42C2×C22⋊D31C2314yes—
43#92499—24no—
44Q8×D31—24no—
45#92497—24no—
46#92497—28no—
47D4×D31—24no—
48D4×D31—28no—
49#92495—24no—
50#92495—24no—
51#92495—22no—
52#92495—24no—
53#92495—22no—
54#92495—22no—
55#92495—22no—
56C2×D124—22no—
57C2×D124D412yes—
58C2×C4×D31—24no—
59C2×C4×D31D414yes—
60C2×Dic62—22no—
61C2×Dic62D412yes—
62C4⋊C4×C31—21no—
63C4⋊C4×C31—24no—
64C22⋊C4×C31—24no—
65C22⋊C4×C31C2312yes—
66C42×C31—21no—
67C22⋊Dic31C2314yes—
68(C2×D31)⋊C4—28no—
69C4⋊Dic31—22no—
70Dic31⋊C4—28no—
71C4×Dic31—22no—
72C23×C31C2×D412yes—
73C23×C31C2411yescharacteristic
74C22×D31—24no—
75C22×D31C2×D414yes—
76Q8×C31—22no—
77Q8×C31—41no—
78Q8×C31—42no—
79Q8×C31—41no—
80D4×C31—24no—
81D4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—24no—
85C2×C4×C31—22no—
86C2×C4×C31—22no—
87C2×C4×C31C2×D412yes—
88C2×C4×C31C2×D412yes—
89C2×C4×C31C2×D412yes—
90C2×C4×C31C2×D411yescharacteristic
91C2×C4×C31C2×D411yescharacteristic
92C22⋊D31—28no—
93C22⋊D31—24no—
94C22⋊D31—24no—
95C2×Dic31—24no—
96C2×Dic31—24no—
97C2×Dic31C2×D414yes—
98D124—24no—
99D124—22no—
100D124—22no—
101C4×D31—28no—
102C4×D31—24no—
103C4×D31—24no—
104Dic62—24no—
105Dic62—22no—
106Dic62—22no—
107#3028—311nomaximal, Sylow
108C22×C31—22no—
109C22×C31—41no—
110C22×C31—42no—
111C22×C31C22×D412yes—
112C22×C31C22×D411yescharacteristic
113C2×D31—24no—
114C2×D31—44no—
115C2×D31—84no—
116C4×C31—22no—
117C4×C31—42no—
118C4×C31—42no—
119C4×C31#12011yescharacteristic
120C4×C31#12012yes—
121C4×C31#12011yescharacteristic
122Dic31—24no—
123Dic31—44no—
124#585—311no—
125#536—312no—
126#536D3111yescharacteristic, complemented
127#458—318no—
128#410—311no—
129#410—312no—
130C2×C31—22no—
131C2×C31—21no—
132C2×C31—42no—
133C2×C31C2×#12011yescharacteristic
134D31—44no—
1352−1+4—622no—
1362−1+4—621no—
137H(5,𝔽2)—314no—
138H(5,𝔽2)—622no—
139H(5,𝔽2)—621no—
140C2×#41—312no—
141C2×#41—624no—
142C2×#41—622no—
143C2×#41C2×D3111yescharacteristic
144C22×Q8C2×D3111yescharacteristic, complemented
145C4⋊Q8—21no—
146C4⋊Q8—622no—
147#124—21no—
148#124—622no—
149#123—318no—
150#123C2×D3114yescomplemented
151C22⋊Q8—22no—
152C22⋊Q8—624no—
153#117—312no—
154#117C2×D3111yescharacteristic, complemented
155#99—314no—
156#99—318no—
157C31#302811yescharacteristic, complemented, Sylow
158#41—22no—
159#41—22no—
160#41—624no—
161#41—624no—
162#41—628no—
163#41—624no—
164#41—624no—
165#41—622no—
166#41—622no—
167#41—622no—
168#41—622no—
169#41—624no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—312no—
175C2×Q8—622no—
176C2×Q8—624no—
177C2×Q8C22⋊D3111yescharacteristic
178C2×D4—314no—
179C2×D4—312no—
180C2×D4—622no—
181C2×D4—628no—
182C2×D4—624no—
183C2×D4—624no—
184C2×D4C22⋊D3111yescharacteristic
185C2×D4C22×D3112yes—
186C22×C4—314no—
187C22×C4—624no—
188C22×C4C22⋊D3112yes—
189C22×C4C22×D3112yes—
190C22×C4C22×D3111yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—24no—
193C4⋊C4—628no—
194C4⋊C4—622no—
195C22⋊C4—24no—
196C22⋊C4—314no—
197C22⋊C4—628no—
198C22⋊C4C22×D3112yes—
199C42—21no—
200C42—622no—
201C23—314no—
202C23—624no—
203C23C2×C22⋊D3112yes—
204C23C23×D3111yescharacteristic
205Q8—22no—
206Q8—41no—
207Q8—41no—
208Q8—42no—
209Q8—624no—
210Q8—622no—
211Q8—622no—
212D4—24no—
213D4—22no—
214D4—624no—
215D4—628no—
216D4—624no—
217D4—624no—
218D4—622no—
219D4—622no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—314no—
226C2×C4—628no—
227C2×C4—624no—
228C2×C4—624no—
229C2×C4—624no—
230C2×C4—624no—
231C2×C4D4×D3112yes—
232C2×C4D4×D3112yes—
233C2×C4C2×C22⋊D3112yes—
234C2×C4C2×C22⋊D3111yescharacteristic
235C2×C4C2×C22⋊D3111yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—624no—
240C22—1244no—
241C22—2484no—
242C22C2×D4×D3112yes—
243C22C22×C22⋊D3111yescharacteristic
244C4—22no—
245C4—42no—
246C4—42no—
247C4—624no—
248C4—1244no—
249C4#1175851411yescharacteristic
250C4#1175851411yescharacteristic
251C4#1175851412yes—
252C2—21no—
253C2—22no—
254C2—42no—
255C2—1244no—
256C2C2×#1175851411yescharacteristic
subgroupG/H
centerC2C2×#11758514
commutator, cosocleC23×C31C24
FrattiniC23C23×D31
FittingC31×#536C2
socleC2×C31C2×#120
prime radical#14521885C2
hypercenter#536D31
hypocenterC31#3028
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #14521623C2semidirect
3. #14521238C2semidirect
4. #14521023C2semidirect
5. #14521022C2semidirect
6. C31×#536C2semidirect
8. C22×Q8×C31C22semidirect
19. C31×#123C22semidirect
21. C31×#117C22semidirect
25. #11758506C22semidirect
26. #11758503C22semidirect
27. #11758422C22semidirect
28. #11758394C22semidirect
126. #536D31semidirect
144. C22×Q8C2×D31semidirect
150. #123C2×D31semidirect
154. #117C2×D31semidirect
157. C31#3028semidirect

Maximal subgroups

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

subgroupindex
#145218852
#145216232
#145212382
#145210232
#145210222
C31×#5362
#302831