Finite Group Databasebeta

← #73515

Subgroups — #73515

Order 384.

568 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 157 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×#930C211yesmaximal, characteristic
2C2×#926C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
4#4961C214yesmaximal, complemented
5#4939C211yesmaximal, characteristic, complemented
6#4939C211yesmaximal, characteristic, complemented
7#4936C211yesmaximal, characteristic, complemented
8#4935C211yesmaximal, characteristic, complemented
9#4670C214yesmaximal, complemented
10#2500—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C22×#67C2212yes—
13C22×D12C2211yescharacteristic
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C42C2211yescharacteristic
16C2×C3×C42C2211yescharacteristic
17C2×#217C2212yes—
18#930—24no—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2214yes—
21C2×(C2×D3)⋊C4C2214yes—
22#926—24no—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊Dic3C2214yes—
27C2×C4×Dic3C2212yes—
28C3×#95C2212yes—
29C3⋊#95C2214yes—
30C2×#121—31no—
31C2×#117—31no—
32#394—34no—
33#390—31no—
34#390—31no—
35#388—34no—
36#384—31no—
37#382—31no—
38C4×C22⋊C4D311yescharacteristic, complemented
39C24×C3C2311yescharacteristic
40C23×D3C2312yes—
41C22×C3×C4—22no—
42C22×C3×C4—22no—
43C22×C3×C4D414yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C2×#67—24no—
49C2×#67—44no—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C2×D12—22no—
54C2×D12—22no—
55C2×D12—22no—
56C3×C22⋊C4—24no—
57C3×C42—22no—
58C3×C42—22no—
59#217—24no—
60(C2×D3)⋊C4—24no—
61(C2×D3)⋊C4—28no—
62(C2×D3)⋊C4—44no—
63C4⋊Dic3—22no—
64C4⋊Dic3—22no—
65C4⋊Dic3—22no—
66C4⋊Dic3—44no—
67C4×Dic3—24no—
68C22×D4—32no—
69C22×D4—31no—
70C23×C4C2×D311yescharacteristic
71#121—64no—
72#117—64no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4—31no—
76C2×C4⋊C4—34no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—34no—
80C2×C22⋊C4—34no—
81C2×C22⋊C4C2×D312yes—
82C2×C42—32no—
83C2×C42C2×D311yescharacteristic
84C2×C42C2×D311yescharacteristic
85#95—34no—
86#95C2×D312yes—
87C23×C3—22no—
88C23×C3—21no—
89C23×C3—21no—
90C23×C3—22no—
91C23×C3C2×D412yes—
92C23×C3C2411yescharacteristic
93C22×D3—22no—
94C22×D3—24no—
95C22×D3—22no—
96C22×D3—22no—
97C22×D3—24no—
98C2×C3×C4—22no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—22no—
102C2×C3×C4—28no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4C2×D412yes—
107C2×C3×C4C2×D412yes—
108C2×C3×C4#4114yes—
109C2×C3×C4#4112yes—
110C2×C3×C4#4112yes—
111#67—48no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—24no—
116C2×Dic3—24no—
117C2×Dic3—24no—
118D12—44no—
119C24—32no—
120C24C22×D311yescharacteristic
121C2×D4—64no—
122C2×D4—62no—
123C2×D4—62no—
124C2×D4—62no—
125C2×D4—124no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—32no—
129C22×C4—32no—
130C22×C4—32no—
131C22×C4D1214yes—
132C22×C4C22×D312yes—
133C22×C4C22×D311yescharacteristic
134C22×C4C22×D312yes—
135C22×C4C22×D311yescharacteristic
136C4⋊C4—62no—
137C4⋊C4—62no—
138C4⋊C4—62no—
139C4⋊C4—124no—
140C22⋊C4—24no—
141C22⋊C4—64no—
142C22⋊C4—64no—
143C22⋊C4—68no—
144C22⋊C4—124no—
145C42—22no—
146C42—22no—
147C42—64no—
148C22×C3—22no—
149C22×C3—22no—
150C22×C3—22no—
151C22×C3—42no—
152C22×C3—42no—
153C22×C3C22×D411yescharacteristic
154C22×C3C2×#4112yes—
155C22×C3C2×#4111yescharacteristic
156C22×C3C2×#4111yescharacteristic
157C22×C3H(5,𝔽2)12yes—
158C2×D3—42no—
159C2×D3—42no—
160C2×D3—42no—
161C2×D3—44no—
162C2×D3—44no—
163C3×C4—22no—
164C3×C4—24no—
165C3×C4—42no—
166C3×C4#12314yes—
167Dic3—42no—
168Dic3—42no—
169Dic3—42no—
170C23—22no—
171C23—21no—
172C23—21no—
173C23—22no—
174C23—62no—
175C23—64no—
176C23—62no—
177C23—62no—
178C23—64no—
179C23C2×D1212yes—
180C23C23×D311yescharacteristic
181D4—128no—
182D4—124no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—28no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—64no—
195C2×C4—64no—
196C2×C4—64no—
197C2×C4C2×D1212yes—
198C2×C4C2×D1212yes—
199C2×C4#23514yes—
200C2×C4#23712yes—
201C2×C4#23712yes—
202C2×C3—42no—
203C2×C3C2×#12311yescharacteristic
204C2×C3#53211yescharacteristic
205C2×C3#53511yescharacteristic
206C2×C3#55312yes—
207C2×C3#55512yes—
208D3—82no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—42no—
213C22—42no—
214C22—122no—
215C22—122no—
216C22—122no—
217C22—124no—
218C22—124no—
219C22C22×D1211yescharacteristic
220C22C2×#23512yes—
221C22C2×#23711yescharacteristic
222C22C2×#23711yescharacteristic
223C22#100912yes—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—122no—
228C4—122no—
229C4—122no—
230C4#88614yes—
231C3#250011yescharacteristic, complemented, Sylow
232C2—42no—
233C2—242no—
234C2#547711yescharacteristic
235C2C2×#88611yescharacteristic
236C2#552511yescharacteristic
237C2#556312yes—
238C2#556512yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×C4×C22⋊C4C2
prime radicalC2×#930C2
hypercenterC4×C22⋊C4D3
hypocenterC3#2500
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#926C2semidirect
3. C3×C4×C22⋊C4C2semidirect
4. #4961C2semidirect
5. #4939C2semidirect
6. #4939C2semidirect
7. #4936C2semidirect
8. #4935C2semidirect
9. #4670C2semidirect
38. C4×C22⋊C4D3semidirect
231. C3#2500semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C2×#9262
C3×C4×C22⋊C42
#49612
#49392
#49362
#49352
#46702
#25003