Finite Group Databasebeta

← #78233

Subgroups — #78233

Order 384.

434 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 129 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×#898C211yesmaximal, characteristic
2C2×#890C211yesmaximal, characteristic, complemented
3C3×#400C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic
5#4981C211yesmaximal, characteristic, complemented
6#4676C212yesmaximal
7#4674C211yesmaximal, characteristic
8#4670C212yesmaximal
9#4657C212yesmaximal, complemented
10#4649C211yesmaximal, characteristic, complemented
11#4647C212yesmaximal, complemented
12#2664—31nomaximal, Sylow
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×(C2×D3)⋊C4C2211yescharacteristic
16C2×(C2×D3)⋊C4C2212yes—
17C2×(C2×D3)⋊C4C2211yescharacteristic
18C2×(C2×D3)⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×C4⋊Dic3C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#898—24no—
31#890—24no—
32C3×#95C2212yes—
33C3×#95C2211yescharacteristic
34C3×#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#126—31no—
39C2×#125—31no—
40#400D311yescharacteristic, complemented
41#399—31no—
42#398—31no—
43#398—32no—
44#397—31no—
45#396—32no—
46#389—32no—
47#388—32no—
48C4×C4⋊C4—31no—
49C23×D3C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2312yes—
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C3×C4⋊C4—22no—
61C3×C4⋊C4—22no—
62(C2×D3)⋊C4—22no—
63(C2×D3)⋊C4—22no—
64(C2×D3)⋊C4—22no—
65(C2×D3)⋊C4—24no—
66(C2×D3)⋊C4—42no—
67C4⋊Dic3—22no—
68C4⋊Dic3—24no—
69C4⋊Dic3—22no—
70Dic3⋊C4—24no—
71Dic3⋊C4—22no—
72Dic3⋊C4—24no—
73Dic3⋊C4—24no—
74Dic3⋊C4—42no—
75C4×Dic3—42no—
76C4×Dic3D414yes—
77#126—64no—
78#125—64no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C4⋊C4C2×D311yescharacteristic
89C2×C22⋊C4—31no—
90C2×C22⋊C4—31no—
91C2×C22⋊C4—31no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C42—31no—
95C2×C42—32no—
96#95—31no—
97#95—32no—
98#95—32no—
99#95C2×D311yescharacteristic
100#95C2×D312yes—
101#95C2×D312yes—
102C23×C3C2411yescharacteristic
103C22×D3—21no—
104C22×D3—22no—
105C22×D3—21no—
106C22×D3—21no—
107C22×D3—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4C2×D412yes—
121C2×Dic3—22no—
122C2×Dic3—21no—
123C2×Dic3—22no—
124C2×Dic3—21no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—21no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3C2×D412yes—
132C2×Dic3C2×D412yes—
133C2×Dic3#4114yes—
134C24—31no—
135C22×C4—32no—
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—32no—
140C22×C4C22×D311yescharacteristic
141C22×C4C22×D312yes—
142C22×C4C22×D311yescharacteristic
143C22×C4C22×D312yes—
144C22×C4C22×D311yescharacteristic
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—64no—
148C4⋊C4—64no—
149C4⋊C4—62no—
150C4⋊C4—62no—
151C4⋊C4—64no—
152C4⋊C4—64no—
153C4⋊C4—62no—
154C4⋊C4—122no—
155C22⋊C4—62no—
156C22⋊C4—62no—
157C22⋊C4—64no—
158C22⋊C4—62no—
159C22⋊C4—122no—
160C42—34no—
161C42—122no—
162C22×C3C22×D411yescharacteristic
163C22×C3C2×#4112yes—
164C22×C3H(5,𝔽2)11yescharacteristic
165C22×C32−1+411yescharacteristic
166C22×C32−1+412yes—
167C2×D3—41no—
168C2×D3—41no—
169C2×D3—41no—
170C2×D3—42no—
171C2×D3—42no—
172C3×C4—22no—
173C3×C4—42no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4—42no—
177Dic3—22no—
178Dic3—22no—
179Dic3—24no—
180Dic3—41no—
181Dic3—42no—
182C23—61no—
183C23—61no—
184C23—62no—
185C23—61no—
186C23—62no—
187C23C23×D311yescharacteristic
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—32no—
201C2×C4—32no—
202C2×C4—34no—
203C2×C4—61no—
204C2×C4—62no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—61no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4D3×D412yes—
214C2×C3#53211yescharacteristic
215C2×C3#53611yescharacteristic
216C2×C3#53711yescharacteristic
217C2×C3#53912yes—
218C2×C3#54112yes—
219D3—81no—
220C22—121no—
221C22—121no—
222C22—122no—
223C22—122no—
224C22—121no—
225C22C2×D3×D411yescharacteristic
226C22#100711yescharacteristic
227C22D3×#4112yes—
228C22#100911yescharacteristic
229C22#101012yes—
230C4—22no—
231C4—41no—
232C4—42no—
233C4—42no—
234C4—41no—
235C4—62no—
236C4—64no—
237C4—62no—
238C4—121no—
239C4—122no—
240C3#266411yescharacteristic, complemented, Sylow
241C2—241no—
242C2#562911yescharacteristic
243C2#564811yescharacteristic
244C2#566811yescharacteristic
245C2#569612yes—
246C2#570712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC23×D3C23
hypercenter#400D3
hypocenterC3#2664
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#890C2semidirect
3. C3×#400C2semidirect
5. #4981C2semidirect
9. #4657C2semidirect
10. #4649C2semidirect
11. #4647C2semidirect
40. #400D3semidirect
240. C3#2664semidirect

Maximal subgroups

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

subgroupindex
C2×#8982
C2×#8902
C3×#4002
#49932
#49812
#46762
#46742
#46702
#46572
#46492
#46472
#26643