Finite Group Databasebeta

← #11808760

Subgroups — #11808760

Order 1152.

1266 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 135 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×C4×GO+(4,𝔽2)C214yesmaximal, complemented
2C4×#3802C212yesmaximal, complemented
3#10593733C211yesmaximal, characteristic, complemented
4#10593455C214yesmaximal, complemented
5#10593408C214yesmaximal, complemented
6C22×GO+(4,𝔽2)C2214yes—
7C2×#3802C2212yes—
8C2×C4×D32C2214yes—
9C2×C4×#157C2212yes—
10C2×#3773C2214yes—
11C2×D32⋊C4C2218yes—
12C4×GO+(4,𝔽2)—216no—
13C42×C3⋊D3C2211yescharacteristic
14C3×C4×D12—22no—
15#63921C2214yes—
16#63902—22no—
17C4×#625—22no—
18#63900C2212yes—
19#63898—22no—
20#63781C2214yes—
21C4×D3×D4—62no—
22C22×D32C2314yes—
23C22×#157C2314yes—
24C2×GO+(4,𝔽2)—216no—
25C2×GO+(4,𝔽2)—28no—
26C2×C4×C3⋊D3C2312yes—
27C2×C4×C3⋊D3C2311yescharacteristic
28C2×C3×D12—22no—
29C2×C3×C4×D3—24no—
30C2×#625—22no—
31C2×#624C2314yes—
32C2×D3×Dic3—24no—
33#3802—28no—
34C4×D32—216no—
35C4×#157D418yescomplemented
36#3773—28no—
37D32⋊C4—216no—
38C32×C42D411yescharacteristic, complemented
39C4×C3⋊Dic3D412yescomplemented
40C4×C3⋊Dic3D411yescharacteristic, complemented
41C3×(C2×D3)⋊C4—24no—
42C3×C4⋊Dic3—22no—
43#3724—22no—
44(C2×D3)⋊Dic3—24no—
45#2397—91nomaximal, Sylow
46C2×D3×D4—62no—
47C22×C4×D3—64no—
48C3×C4×D4—62no—
49D4×Dic3—62no—
50C4×#67—64no—
51#893—62no—
52D3×C4⋊C4—62no—
53#67⋊C4—64no—
54D3×C22⋊C4—64no—
55C4×D12—22no—
56C42×D3—62no—
57C22×C3⋊D3C2411yescharacteristic
58C22×C3×D3—24no—
59C2×D32—24no—
60C2×D32—28no—
61C2×D32—28no—
62C2×D32#4118yes—
63C2×#157—28no—
64C2×#157C2×D418yes—
65GO+(4,𝔽2)—232no—
66C2×C32×C4C2×D412yes—
67C2×C32×C4C2×D411yescharacteristic
68C2×C3⋊Dic3C2×D412yes—
69C2×C3⋊Dic3C2×D411yescharacteristic
70C4×C3⋊D3—24no—
71C4×C3⋊D3C2×D414yes—
72C2×C3×Dic3—24no—
73C3×D12—28no—
74C3×C4×D3—48no—
75#625—28no—
76#624—28no—
77D3×Dic3—48no—
78C2×C4×D4—92no—
79C2×C4×D4—94no—
80#399—94no—
81#392—94no—
82#376—91no—
83C23×D3—64no—
84C2×C3×D4—62no—
85C22×C3×C4—64no—
86C2×#67—64no—
87C22×Dic3—64no—
88D3×D4—616no—
89C2×D12—22no—
90C2×C4×D3—24no—
91C2×C4×D3—62no—
92C2×C4×D3—62no—
93C2×C4×D3—64no—
94C2×C4×D3—62no—
95C2×C4×D3—64no—
96C2×C4×D3—1216no—
97C3×C4⋊C4—62no—
98C3×C22⋊C4—64no—
99C3×C42—22no—
100#217—64no—
101(C2×D3)⋊C4—24no—
102(C2×D3)⋊C4—64no—
103C4⋊Dic3—22no—
104Dic3⋊C4—64no—
105C4×Dic3—62no—
106C4×Dic3—62no—
107C4×Dic3—62no—
108C22×C32C22×D411yescharacteristic
109C2×C3⋊D3C22×D412yes—
110C2×C3⋊D3C2×#4114yes—
111C2×C3×D3—28no—
112C2×C3×D3—44no—
113C2×C3×D3—44no—
114D32—216no—
115D32—48no—
116#157—28no—
117C32×C4—22no—
118C32×C4#12712yes—
119C3⋊Dic3—22no—
120C3⋊Dic3#12712yes—
121C3×Dic3—44no—
122C22×D4—92no—
123C22×D4—94no—
124C23×C4—94no—
125C4×D4—62no—
126C4×D4—182no—
127C4×D4—182no—
128C4×D4—182no—
129C4×D4—1816no—
130C2×C4⋊C4—92no—
131C2×C4⋊C4—94no—
132C2×C22⋊C4—94no—
133C2×C22⋊C4—98no—
134C2×C42—92no—
135C2×C42—91no—
136#95—94no—
137C23×C3—64no—
138C22×D3—24no—
139C22×D3—68no—
140C22×D3—62no—
141C22×D3—68no—
142C22×D3—128no—
143C22×D3—128no—
144C3×D4—68no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—64no—
149C2×C3×C4—128no—
150#67—616no—
151C2×Dic3—24no—
152C2×Dic3—62no—
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3—128no—
156D12—28no—
157C4×D3—48no—
158C4×D3—68no—
159C4×D3—68no—
160C4×D3—128no—
161C4×D3—124no—
162C4×D3—128no—
163C2×C32#52512yes—
164C2×C32C2×#12711yescharacteristic
165C3⋊D3#52514yes—
166C3×D3—48no—
167C24—94no—
168C2×D4—62no—
169C2×D4—182no—
170C2×D4—188no—
171C2×D4—188no—
172C2×D4—1816no—
173C22×C4—64no—
174C22×C4—94no—
175C22×C4—91no—
176C22×C4—94no—
177C22×C4—92no—
178C22×C4—184no—
179C22×C4—1816no—
180C4⋊C4—62no—
181C4⋊C4—182no—
182C4⋊C4—188no—
183C22⋊C4—64no—
184C22⋊C4—184no—
185C22⋊C4—1816no—
186C42—98no—
187C42—92no—
188C42—91no—
189C42GO+(4,𝔽2)11yescharacteristic, complemented
190C22×C3—22no—
191C22×C3—68no—
192C22×C3—124no—
193C22×C3—124no—
194C2×D3—28no—
195C2×D3—44no—
196C2×D3—44no—
197C2×D3—64no—
198C2×D3—64no—
199C2×D3—64no—
200C2×D3—1216no—
201C2×D3—1216no—
202C3×C4—24no—
203C3×C4—24no—
204C3×C4—42no—
205C3×C4—124no—
206Dic3—44no—
207Dic3—64no—
208Dic3—64no—
209Dic3—122no—
210C32#239711yescharacteristic, complemented, Sylow
211C23—64no—
212C23—98no—
213C23—91no—
214C23—184no—
215C23—188no—
216C23—188no—
217D4—68no—
218D4—188no—
219D4—1832no—
220C2×C4—64no—
221C2×C4—91no—
222C2×C4—94no—
223C2×C4—98no—
224C2×C4—92no—
225C2×C4—128no—
226C2×C4—184no—
227C2×C4—188no—
228C2×C4—188no—
229C2×C4—368no—
230C2×C4C2×GO+(4,𝔽2)12yes—
231C2×C4C2×GO+(4,𝔽2)11yescharacteristic
232C2×C3—22no—
233C2×C3—22no—
234C2×C3—22no—
235C2×C3—128no—
236D3—48no—
237D3—68no—
238C22—68no—
239C22—92no—
240C22—94no—
241C22—124no—
242C22—124no—
243C22—1816no—
244C22—368no—
245C22C22×GO+(4,𝔽2)11yescharacteristic
246C4—22no—
247C4—92no—
248C4—124no—
249C4—188no—
250C4—182no—
251C4#6423112yes—
252C3—22no—
253C2—94no—
254C2—128no—
255C2C2×#6423111yescharacteristic
256C2#1059651012yes—
subgroupG/H
centerC2×C4C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C24
FrattiniC22C22×GO+(4,𝔽2)
FittingC32×C42D4
socleC22×C32C22×D4
hypercenterC42GO+(4,𝔽2)
hypocenterC32#2397
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×GO+(4,𝔽2)C2semidirect
2. C4×#3802C2semidirect
3. #10593733C2semidirect
4. #10593455C2semidirect
5. #10593408C2semidirect
35. C4×#157D4semidirect
38. C32×C42D4semidirect
39. C4×C3⋊Dic3D4semidirect
40. C4×C3⋊Dic3D4semidirect
189. C42GO+(4,𝔽2)semidirect
210. C32#2397semidirect

Maximal subgroups

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

subgroupindex
C2×C4×GO+(4,𝔽2)2
C4×#38022
#105937332
#105934552
#105934082
#23979