Finite Group Databasebeta

← #74367

Subgroups — #74367

Order 384.

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

normal subgroups only

This group has 568 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×#892C211yesmaximal, characteristic, complemented
3C3×#384C211yesmaximal, characteristic, complemented
4#4991C214yesmaximal, complemented
5#4990C214yesmaximal, complemented
6#4988C211yesmaximal, characteristic, complemented
7#4977C211yesmaximal, characteristic, complemented
8#4939C211yesmaximal, characteristic, complemented
9C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
10#2500—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C22×D12C2212yes—
13C22×C4×D3C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2214yes—
17C2×C3×C42C2211yescharacteristic
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2214yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2214yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4×Dic3C2211yescharacteristic
25C2×C4×Dic3C2212yes—
26#895—24no—
27#892—24no—
28C3⋊#95C2212yes—
29C3⋊#95C2214yes—
30C2×#121—31no—
31C2×#117—31no—
32#394—34no—
33#390—31no—
34#390—31no—
35#388—34no—
36#384D311yescharacteristic, complemented
37#382—31no—
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—22no—
50C2×D12—22no—
51C2×D12—22no—
52C2×D12—44no—
53C2×C4×D3—22no—
54C2×C4×D3—22no—
55C2×C4×D3D414yes—
56C3×C4⋊C4—22no—
57C3×C4⋊C4—22no—
58C3×C4⋊C4—44no—
59C3×C42—22no—
60(C2×D3)⋊C4—24no—
61(C2×D3)⋊C4—24no—
62(C2×D3)⋊C4—28no—
63(C2×D3)⋊C4—24no—
64(C2×D3)⋊C4—44no—
65C4⋊Dic3—22no—
66C4×Dic3—22no—
67C4×Dic3—24no—
68C22×D4—31no—
69C22×D4—32no—
70C23×C4—31no—
71#121—64no—
72#117—64no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4C2×D311yescharacteristic
75C2×C4⋊C4C2×D311yescharacteristic
76C2×C4⋊C4C2×D314yes—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—34no—
81C2×C22⋊C4—34no—
82C2×C42—32no—
83C2×C42—31no—
84C2×C42C2×D311yescharacteristic
85#95—32no—
86#95—34no—
87C23×C3C2411yescharacteristic
88C22×D3—22no—
89C22×D3—21no—
90C22×D3—21no—
91C22×D3—22no—
92C22×D3—22no—
93C22×D3—24no—
94C22×D3—22no—
95C22×D3—22no—
96C22×D3—24no—
97C22×D3C2×D412yes—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—24no—
103C2×C3×C4—24no—
104C2×C3×C4C2×D412yes—
105C2×C3×C4#4112yes—
106C2×C3×C4#4114yes—
107C2×C3×C4#4112yes—
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—24no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3C2×D412yes—
116D12—48no—
117D12—44no—
118C4×D3—28no—
119C24—31no—
120C24—32no—
121C2×D4—64no—
122C2×D4—62no—
123C2×D4—62no—
124C2×D4—62no—
125C2×D4—124no—
126C22×C4—31no—
127C22×C4—34no—
128C22×C4—32no—
129C22×C4—32no—
130C22×C4—62no—
131C22×C4—62no—
132C22×C4C22×D312yes—
133C22×C4C22×D312yes—
134C22×C4C22×D312yes—
135C22×C4C22×D311yescharacteristic
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—44no—
139C4⋊C4—62no—
140C22⋊C4—64no—
141C22⋊C4—64no—
142C22⋊C4—64no—
143C22⋊C4—68no—
144C22⋊C4—124no—
145C42—22no—
146C42—64no—
147C42—62no—
148C22×C3C22×D411yescharacteristic
149C22×C3C2×#4111yescharacteristic
150C22×C3C2×#4111yescharacteristic
151C22×C3C2×#4112yes—
152C22×C3H(5,𝔽2)12yes—
153C2×D3—22no—
154C2×D3—22no—
155C2×D3—22no—
156C2×D3—42no—
157C2×D3—42no—
158C2×D3—42no—
159C2×D3—42no—
160C2×D3—42no—
161C2×D3—44no—
162C2×D3—44no—
163C3×C4—24no—
164C3×C4—42no—
165C3×C4—42no—
166C3×C4#12314yes—
167Dic3—22no—
168Dic3—42no—
169Dic3—42no—
170C23—32no—
171C23—61no—
172C23—62no—
173C23—62no—
174C23—61no—
175C23—62no—
176C23—62no—
177C23—64no—
178C23—64no—
179C23—62no—
180C23C23×D311yescharacteristic
181D4—128no—
182D4—124no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—32no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—68no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—64no—
197C2×C4—62no—
198C2×C4#23512yes—
199C2×C4D3×D412yes—
200C2×C4#23914yes—
201C2×C4#23912yes—
202C2×C3C2×#12311yescharacteristic
203C2×C3#53211yescharacteristic
204C2×C3#53511yescharacteristic
205C2×C3#55312yes—
206C2×C3#55512yes—
207D3—42no—
208D3—82no—
209C22—62no—
210C22—62no—
211C22—62no—
212C22—122no—
213C22—122no—
214C22—122no—
215C22—122no—
216C22—124no—
217C22—122no—
218C22—124no—
219C22C2×#23511yescharacteristic
220C22C2×D3×D411yescharacteristic
221C22#100412yes—
222C22C2×#23912yes—
223C22C2×#23911yescharacteristic
224C4—24no—
225C4—42no—
226C4—42no—
227C4—62no—
228C4—122no—
229C4—122no—
230C4#89414yes—
231C3#250011yescharacteristic, complemented, Sylow
232C2—122no—
233C2—242no—
234C2C2×#89411yescharacteristic
235C2#551412yes—
236C2#571511yescharacteristic
237C2#572511yescharacteristic
238C2#573412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#384C2
prime radicalC2×#895C2
hypercenter#384D3
hypocenterC3#2500
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#892C2semidirect
3. C3×#384C2semidirect
4. #4991C2semidirect
5. #4990C2semidirect
6. #4988C2semidirect
7. #4977C2semidirect
8. #4939C2semidirect
9. C4×(C2×D3)⋊C4C2semidirect
36. #384D3semidirect
231. C3#2500semidirect

Maximal subgroups

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

subgroupindex
C2×#8952
C2×#8922
C3×#3842
#49912
#49902
#49882
#49772
#49392
C4×(C2×D3)⋊C42
#25003