Finite Group Databasebeta

← #74168

Subgroups — #74168

Order 384.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#895C211yesmaximal, characteristic
2C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#382C211yesmaximal, characteristic, complemented
4#4988C211yesmaximal, characteristic, complemented
5#4977C211yesmaximal, characteristic, complemented
6#4939C211yesmaximal, characteristic, complemented
7C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
8#4675C214yesmaximal, complemented
9#4665C214yesmaximal, complemented
10#2504—31nomaximal, Sylow
11C22×D12C2212yes—
12C22×D12C2211yescharacteristic
13C22×C4×D3C2211yescharacteristic
14C22×C4×D3C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C42C2211yescharacteristic
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2214yes—
21C2×(C2×D3)⋊C4C2214yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25#895—24no—
26D3×C4⋊C4—24no—
27C3×#95C2214yes—
28C3⋊#95C2214yes—
29C3⋊#95C2212yes—
30C2×#121—31no—
31C22×C4⋊C4—31no—
32#397—34no—
33#390—31no—
34#390—31no—
35#386—34no—
36#382—31no—
37#382D311yescharacteristic, complemented
38C4×C22⋊C4—31no—
39C23×D3C2312yes—
40C23×D3C2311yescharacteristic
41C22×C3×C4C2312yes—
42C22×C3×C4C2312yes—
43C22×C3×C4C2312yes—
44C22×C3×C4C2311yescharacteristic
45C22×Dic3C2312yes—
46C22×Dic3C2312yes—
47C22×Dic3C2311yescharacteristic
48C2×D12—24no—
49C2×D12—28no—
50C2×D12—22no—
51C2×D12—22no—
52C2×D12—22no—
53C2×C4×D3—22no—
54C2×C4×D3—22no—
55C2×C4×D3—24no—
56C2×C4×D3—44no—
57C2×C4×D3D414yes—
58C3×C4⋊C4—22no—
59C3×C4⋊C4—22no—
60C3×C42—22no—
61(C2×D3)⋊C4—24no—
62(C2×D3)⋊C4—24no—
63(C2×D3)⋊C4—44no—
64(C2×D3)⋊C4—44no—
65C4⋊Dic3—22no—
66Dic3⋊C4—24no—
67C4×Dic3—22no—
68C22×D4—32no—
69C22×D4—31no—
70C23×C4—31no—
71C23×C4—32no—
72#121—64no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—64no—
76C2×C4⋊C4C2×D311yescharacteristic
77C2×C4⋊C4C2×D311yescharacteristic
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—34no—
81C2×C22⋊C4—34no—
82C2×C42—31no—
83C2×C42C2×D311yescharacteristic
84#95—32no—
85#95—34no—
86#95C2×D314yes—
87C23×C3C2411yescharacteristic
88C22×D3—21no—
89C22×D3—21no—
90C22×D3—22no—
91C22×D3—22no—
92C22×D3—22no—
93C22×D3—22no—
94C22×D3—24no—
95C22×D3—24no—
96C22×D3C2×D412yes—
97C22×D3#4114yes—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—24no—
103C2×C3×C4—22no—
104C2×C3×C4—24no—
105C2×C3×C4C2×D412yes—
106C2×C3×C4#4112yes—
107C2×C3×C4#4112yes—
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—24no—
115C2×Dic3C2×D412yes—
116D12—48no—
117D12—44no—
118C4×D3—28no—
119C4×D3—48no—
120C24—31no—
121C24—32no—
122C2×D4—64no—
123C2×D4—68no—
124C2×D4—62no—
125C2×D4—62no—
126C2×D4—62no—
127C22×C4—31no—
128C22×C4—34no—
129C22×C4—32no—
130C22×C4—32no—
131C22×C4—64no—
132C22×C4—62no—
133C22×C4—62no—
134C22×C4—124no—
135C22×C4C22×D312yes—
136C22×C4C22×D312yes—
137C22×C4C22×D312yes—
138C22×C4C22×D311yescharacteristic
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—64no—
142C4⋊C4—62no—
143C22⋊C4—64no—
144C22⋊C4—64no—
145C22⋊C4—124no—
146C22⋊C4—124no—
147C42—22no—
148C42—62no—
149C22×C3C22×D411yescharacteristic
150C22×C3C2×#4111yescharacteristic
151C22×C3C2×#4111yescharacteristic
152C22×C3C2×#4112yes—
153C22×C3H(5,𝔽2)12yes—
154C2×D3—22no—
155C2×D3—22no—
156C2×D3—24no—
157C2×D3—24no—
158C2×D3—42no—
159C2×D3—42no—
160C2×D3—42no—
161C2×D3—44no—
162C2×D3—44no—
163C2×D3#12414yes—
164C3×C4—42no—
165C3×C4—42no—
166C3×C4—42no—
167C3×C4#12314yes—
168Dic3—22no—
169Dic3—42no—
170Dic3—42no—
171C23—32no—
172C23—34no—
173C23—61no—
174C23—61no—
175C23—62no—
176C23—62no—
177C23—62no—
178C23—62no—
179C23—64no—
180C23—64no—
181C23C23×D311yescharacteristic
182D4—128no—
183D4—124no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—32no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—68no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—64no—
200C2×C4—128no—
201C2×C4#23512yes—
202C2×C4D3×D412yes—
203C2×C4#23912yes—
204C2×C3C2×#12311yescharacteristic
205C2×C3C2×#12411yescharacteristic
206C2×C3#53511yescharacteristic
207C2×C3#55312yes—
208C2×C3#55312yes—
209D3—24no—
210D3—82no—
211C22—34no—
212C22—62no—
213C22—64no—
214C22—62no—
215C22—64no—
216C22—122no—
217C22—122no—
218C22—124no—
219C22—122no—
220C22—124no—
221C22C2×#23511yescharacteristic
222C22C2×D3×D411yescharacteristic
223C22C2×#23911yescharacteristic
224C22D3×#4112yes—
225C22#100912yes—
226C4—42no—
227C4—42no—
228C4—42no—
229C4—62no—
230C4—122no—
231C4—122no—
232C4#89414yes—
233C3#250411yescharacteristic, complemented, Sylow
234C2—64no—
235C2—242no—
236C2C2×#89411yescharacteristic
237C2#553012yes—
238C2D3×#12411yescharacteristic
239C2#567411yescharacteristic
240C2#569512yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#382C2
prime radicalC2×#895C2
hypercenter#382D3
hypocenterC3#2504
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C2×D3×C4⋊C4C2semidirect
3. C3×#382C2semidirect
4. #4988C2semidirect
5. #4977C2semidirect
6. #4939C2semidirect
7. C4×(C2×D3)⋊C4C2semidirect
8. #4675C2semidirect
9. #4665C2semidirect
37. #382D3semidirect
233. C3#2504semidirect

Maximal subgroups

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

subgroupindex
C2×#8952
C2×D3×C4⋊C42
C3×#3822
#49882
#49772
#49392
C4×(C2×D3)⋊C42
#46752
#46652
#25043