Finite Group Databasebeta

← #76453

Subgroups — #76453

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×#938C211yesmaximal, characteristic
2C2×#892C211yesmaximal, characteristic, complemented
3C3×#390C211yesmaximal, characteristic, complemented
4#5221C211yesmaximal, characteristic, complemented
5#4977C211yesmaximal, characteristic, complemented
6#4961C214yesmaximal, complemented
7#4951C214yesmaximal, complemented
8C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
9#4935C211yesmaximal, characteristic, complemented
10#2500—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×#67C2212yes—
13C22×C4×D3C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2214yes—
16C2×C3×C42C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2214yes—
19#938—24no—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2214yes—
25C2×C4×Dic3C2211yescharacteristic
26C2×C4×Dic3C2212yes—
27#892—24no—
28C3⋊#95C2212yes—
29C3⋊#95C2214yes—
30C2×#121—31no—
31C2×#117—31no—
32#394—34no—
33#390—31no—
34#390D311yescharacteristic, complemented
35#388—34no—
36#384—31no—
37#382—31no—
38C4×C22⋊C4—31no—
39C24×C3C2312yes—
40C23×D3C2311yescharacteristic
41C2×C3×D4—22no—
42C2×C3×D4—22no—
43C2×C3×D4—22no—
44C22×C3×C4C2312yes—
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C2×#67—24no—
48C2×#67—44no—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2311yescharacteristic
53C2×C4×D3—22no—
54C2×C4×D3—22no—
55C2×C4×D3D414yes—
56C3×C4⋊C4—22no—
57C3×C22⋊C4—28no—
58C3×C42—22no—
59#217—24no—
60#217—44no—
61(C2×D3)⋊C4—24no—
62(C2×D3)⋊C4—24no—
63C4⋊Dic3—22no—
64C4⋊Dic3—22no—
65C4⋊Dic3—44no—
66C4×Dic3—22no—
67C4×Dic3—24no—
68C22×D4—32no—
69C22×D4C2×D311yescharacteristic
70C23×C4—31no—
71#121—64no—
72#117—64no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4—34no—
76C2×C4⋊C4C2×D311yescharacteristic
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—34no—
81C2×C22⋊C4C2×D314yes—
82C2×C42—32no—
83C2×C42—31no—
84C2×C42C2×D311yescharacteristic
85#95—32no—
86#95—34no—
87C23×C3—22no—
88C23×C3—24no—
89C23×C3—22no—
90C23×C3—22no—
91C23×C3—24no—
92C23×C3C2411yescharacteristic
93C22×D3—22no—
94C22×D3—21no—
95C22×D3—21no—
96C22×D3—22no—
97C22×D3C2×D412yes—
98C3×D4—44no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—24no—
103C2×C3×C4C2×D412yes—
104C2×C3×C4#4112yes—
105C2×C3×C4#4114yes—
106C2×C3×C4#4112yes—
107#67—48no—
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—24no—
115C2×Dic3—24no—
116C2×Dic3—22no—
117C2×Dic3C2×D412yes—
118C4×D3—28no—
119C24—31no—
120C24C22×D312yes—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—64no—
125C2×D4—124no—
126C22×C4—31no—
127C22×C4—34no—
128C22×C4—32no—
129C22×C4—32no—
130C22×C4—32no—
131C22×C4—62no—
132C22×C4—62no—
133C22×C4C22×D312yes—
134C22×C4C22×D312yes—
135C22×C4C22×D311yescharacteristic
136C4⋊C4—22no—
137C4⋊C4—62no—
138C4⋊C4—62no—
139C4⋊C4—124no—
140C22⋊C4—28no—
141C22⋊C4—64no—
142C22⋊C4—64no—
143C22⋊C4—64no—
144C22⋊C4—124no—
145C42—22no—
146C42—64no—
147C42—62no—
148C22×C3—42no—
149C22×C3—42no—
150C22×C3—42no—
151C22×C3—44no—
152C22×C3—44no—
153C22×C3C22×D411yescharacteristic
154C22×C3C2×#4111yescharacteristic
155C22×C3C2×#4111yescharacteristic
156C22×C3C2×#4112yes—
157C22×C3H(5,𝔽2)12yes—
158C2×D3—22no—
159C2×D3—22no—
160C2×D3—22no—
161C2×D3—42no—
162C2×D3—42no—
163C3×C4—24no—
164C3×C4—42no—
165C3×C4#12314yes—
166Dic3—22no—
167Dic3—42no—
168Dic3—42no—
169Dic3—42no—
170C23—22no—
171C23—22no—
172C23—24no—
173C23—22no—
174C23—24no—
175C23—32no—
176C23—61no—
177C23—62no—
178C23—61no—
179C23—62no—
180C23C23×D311yescharacteristic
181D4—44no—
182D4—128no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—32no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—68no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—64no—
197C2×C4—64no—
198C2×C4#23512yes—
199C2×C4D3×D412yes—
200C2×C4#23714yes—
201C2×C4#23712yes—
202C2×C3—82no—
203C2×C3C2×#12311yescharacteristic
204C2×C3#53211yescharacteristic
205C2×C3#53511yescharacteristic
206C2×C3#55312yes—
207C2×C3#55512yes—
208D3—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—44no—
213C22—44no—
214C22—62no—
215C22—62no—
216C22—62no—
217C22—122no—
218C22—122no—
219C22C2×#23511yescharacteristic
220C22C2×D3×D411yescharacteristic
221C22C2×#23712yes—
222C22C2×#23711yescharacteristic
223C22#100912yes—
224C4—24no—
225C4—42no—
226C4—62no—
227C4—122no—
228C4—122no—
229C4—122no—
230C4#88314yes—
231C3#250011yescharacteristic, complemented, Sylow
232C2—82no—
233C2—122no—
234C2C2×#88311yescharacteristic
235C2#556312yes—
236C2#561812yes—
237C2#567411yescharacteristic
238C2#571611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC2×#938C2
hypercenter#390D3
hypocenterC3#2500
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#892C2semidirect
3. C3×#390C2semidirect
4. #5221C2semidirect
5. #4977C2semidirect
6. #4961C2semidirect
7. #4951C2semidirect
8. C4×(C2×D3)⋊C4C2semidirect
9. #4935C2semidirect
34. #390D3semidirect
231. C3#2500semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
C2×#8922
C3×#3902
#52212
#49772
#49612
#49512
C4×(C2×D3)⋊C42
#49352
#25003