Finite Group Databasebeta

← #77816

Subgroups — #77816

Order 384.

476 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 139 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×#892C211yesmaximal, characteristic
2C2×#890C211yesmaximal, characteristic, complemented
3C3×#398C211yesmaximal, characteristic, complemented
4#4992C211yesmaximal, characteristic
5#4989C212yesmaximal
6#4986C211yesmaximal, characteristic, complemented
7#4984C212yesmaximal, complemented
8#4677C211yesmaximal, characteristic
9#4670C212yesmaximal
10#4654C212yesmaximal, complemented
11#4648C211yesmaximal, characteristic, complemented
12#2611—31nomaximal, Sylow
13C22×C4×D3C2211yescharacteristic
14C2×C3×C4⋊C4C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×(C2×D3)⋊C4C2211yescharacteristic
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29#892—24no—
30#890—24no—
31C3×#95C2211yescharacteristic
32C3×#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×#117—31no—
40#400—31no—
41#399—31no—
42#398D311yescharacteristic, complemented
43#397—31no—
44#389—32no—
45#388—32no—
46#387—32no—
47#385—32no—
48#376—31no—
49C23×D3C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×C4×D3—22no—
61C2×C4×D3—22no—
62C2×C4×D3—22no—
63C3×C4⋊C4—22no—
64C3×C4⋊C4—22no—
65C3×C4⋊C4—24no—
66(C2×D3)⋊C4—24no—
67(C2×D3)⋊C4—22no—
68(C2×D3)⋊C4—24no—
69(C2×D3)⋊C4—22no—
70C4⋊Dic3—22no—
71C4⋊Dic3—24no—
72C4⋊Dic3—24no—
73C4⋊Dic3—22no—
74Dic3⋊C4—24no—
75C4×Dic3—24no—
76C4×Dic3—22no—
77C4×Dic3D414yes—
78C23×C4—31no—
79#125—64no—
80#117—64no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4C2×D311yescharacteristic
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C4⋊C4C2×D312yes—
89C2×C22⋊C4—31no—
90C2×C22⋊C4—31no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C42—31no—
94C2×C42—31no—
95C2×C42—32no—
96#95—32no—
97#95—31no—
98#95—32no—
99#95—32no—
100#95—32no—
101#95C2×D311yescharacteristic
102#95C2×D312yes—
103C23×C3C2411yescharacteristic
104C22×D3—21no—
105C22×D3—22no—
106C22×D3—21no—
107C22×D3—22no—
108C22×D3#4112yes—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4C2×D412yes—
120C2×C3×C4#4112yes—
121C2×C3×C4#4114yes—
122C2×Dic3—22no—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3C2×D412yes—
134C2×Dic3C2×D412yes—
135C4×D3—28no—
136C24—31no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—32no—
140C22×C4—31no—
141C22×C4—32no—
142C22×C4—62no—
143C22×C4—62no—
144C22×C4—62no—
145C22×C4C22×D311yescharacteristic
146C22×C4C22×D312yes—
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D312yes—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—24no—
153C4⋊C4—62no—
154C4⋊C4—64no—
155C4⋊C4—64no—
156C4⋊C4—62no—
157C4⋊C4—64no—
158C22⋊C4—62no—
159C22⋊C4—64no—
160C22⋊C4—62no—
161C22⋊C4—64no—
162C42—34no—
163C42—62no—
164C42—64no—
165C22×C3C22×D411yescharacteristic
166C22×C3C2×#4111yescharacteristic
167C22×C3C2×#4112yes—
168C22×C3C2×#4111yescharacteristic
169C22×C32−1+412yes—
170C2×D3—22no—
171C2×D3—24no—
172C2×D3—41no—
173C2×D3—41no—
174C2×D3—42no—
175C3×C4—22no—
176C3×C4—22no—
177C3×C4—24no—
178C3×C4—42no—
179C3×C4—41no—
180Dic3—22no—
181Dic3—22no—
182Dic3—41no—
183Dic3—42no—
184Dic3—42no—
185C23—32no—
186C23—61no—
187C23—61no—
188C23—62no—
189C23—62no—
190C23C23×D311yescharacteristic
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—32no—
202C2×C4—32no—
203C2×C4—68no—
204C2×C4—61no—
205C2×C4—62no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4D3×D412yes—
216C2×C4#23712yes—
217C2×C4#23914yes—
218C2×C3#52511yescharacteristic
219C2×C3#53211yescharacteristic
220C2×C3#53611yescharacteristic
221C2×C3#55212yes—
222C2×C3#55212yes—
223D3—42no—
224C22—64no—
225C22—62no—
226C22—122no—
227C22—121no—
228C22—121no—
229C22C2×D3×D411yescharacteristic
230C22C2×#23711yescharacteristic
231C22C2×#23912yes—
232C22D3×#4111yescharacteristic
233C22#101012yes—
234C4—24no—
235C4—22no—
236C4—22no—
237C4—41no—
238C4—42no—
239C4—62no—
240C4—62no—
241C4—122no—
242C4—121no—
243C4—122no—
244C3#261111yescharacteristic, complemented, Sylow
245C2—122no—
246C2#560611yescharacteristic
247C2#564212yes—
248C2#564811yescharacteristic
249C2#569412yes—
250C2#572511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radicalC23×D3C23
hypercenter#398D3
hypocenterC3#2611
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#890C2semidirect
3. C3×#398C2semidirect
6. #4986C2semidirect
7. #4984C2semidirect
10. #4654C2semidirect
11. #4648C2semidirect
42. #398D3semidirect
244. C3#2611semidirect

Maximal subgroups

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

subgroupindex
C2×#8922
C2×#8902
C3×#3982
#49922
#49892
#49862
#49842
#46772
#46702
#46542
#46482
#26113