Finite Group Databasebeta

← #76654

Subgroups — #76654

Order 384.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×#946C212yesmaximal, complemented
2C3×#391C211yesmaximal, characteristic, complemented
3#4991C211yesmaximal, characteristic
4#4981C211yesmaximal, characteristic, complemented
5#4940C212yesmaximal, complemented
6#4674C212yesmaximal, complemented
7#4668C212yesmaximal, complemented
8#4665C212yesmaximal
9#4659C212yesmaximal, complemented
10#2728—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×D12C2211yescharacteristic
13C22×C4×D3C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C42C2211yescharacteristic
16#946—28no—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×C4⋊Dic3C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C3×#95C2212yes—
27C3×#95C2212yes—
28C3⋊#95C2211yescharacteristic
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2211yescharacteristic
33C2×C22⋊Q8—32no—
34#400—32no—
35#398—31no—
36#397—32no—
37#394—31no—
38#391D311yescharacteristic, complemented
39#388—32no—
40#386—32no—
41#385—32no—
42C23×D3C2312yes—
43C2×C3×Q8—41no—
44C2×C3×Q8—41no—
45C2×C3×Q8D414yescomplemented
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C2×D12—24no—
55C2×D12—41no—
56C2×C4×D3—24no—
57C2×C4×D3—42no—
58C2×C4×D3—42no—
59C3×C4⋊C4—41no—
60C3×C42—41no—
61(C2×D3)⋊C4—24no—
62(C2×D3)⋊C4—24no—
63(C2×D3)⋊C4—24no—
64(C2×D3)⋊C4—24no—
65(C2×D3)⋊C4—42no—
66C4⋊Dic3—24no—
67C4⋊Dic3—41no—
68Dic3⋊C4—24no—
69Dic3⋊C4—24no—
70C22×Q8C2×D311yescharacteristic
71C22×D4—31no—
72C23×C4—32no—
73C22⋊Q8—68no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4C2×D311yescharacteristic
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4—32no—
82C2×C22⋊C4—32no—
83C2×C22⋊C4—32no—
84C2×C42C2×D311yescharacteristic
85#95—32no—
86#95—32no—
87#95—32no—
88#95—31no—
89#95—31no—
90#95C2×D312yes—
91#95C2×D312yes—
92C23×C3C2411yescharacteristic
93C22×D3—22no—
94C22×D3—22no—
95C22×D3—22no—
96C22×D3—22no—
97C22×D3—22no—
98C22×D3—22no—
99C22×D3#4114yes—
100C3×Q8—44no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4C2×D412yes—
113C2×C3×C4C2×D414yes—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123D12—44no—
124C4×D3—48no—
125C24—32no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8D1214yescomplemented
129C2×D4—64no—
130C2×D4—121no—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4—32no—
134C22×C4—64no—
135C22×C4—122no—
136C22×C4—122no—
137C22×C4C22×D311yescharacteristic
138C22×C4C22×D312yes—
139C22×C4C22×D311yescharacteristic
140C22×C4C22×D312yes—
141C22×C4C22×D311yescharacteristic
142C4⋊C4—41no—
143C4⋊C4—64no—
144C4⋊C4—64no—
145C4⋊C4—64no—
146C4⋊C4—121no—
147C22⋊C4—64no—
148C22⋊C4—64no—
149C22⋊C4—64no—
150C22⋊C4—64no—
151C22⋊C4—122no—
152C42—41no—
153C22×C3C22×D411yescharacteristic
154C22×C3C2×#4112yes—
155C22×C3H(5,𝔽2)11yescharacteristic
156C22×C3H(5,𝔽2)11yescharacteristic
157C22×C32−1+412yes—
158C2×D3—24no—
159C2×D3—24no—
160C2×D3—24no—
161C2×D3—42no—
162C2×D3—42no—
163C2×D3—42no—
164C2×D3—42no—
165C3×C4—24no—
166C3×C4—22no—
167C3×C4—42no—
168C3×C4—41no—
169C3×C4—41no—
170Dic3—42no—
171Dic3—42no—
172Dic3—42no—
173C23—34no—
174C23—62no—
175C23—62no—
176C23—62no—
177C23—62no—
178C23—62no—
179C23—62no—
180C23C23×D311yescharacteristic
181Q8—44no—
182D4—124no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—128no—
204C2×C4C2×D1212yes—
205C2×C4C2×D1214yes—
206C2×C3#53411yescharacteristic
207C2×C3#53912yes—
208C2×C3#54812yes—
209C2×C3#55111yescharacteristic
210C2×C3#56311yescharacteristic
211D3—44no—
212C22—64no—
213C22—64no—
214C22—64no—
215C22—122no—
216C22—122no—
217C22—122no—
218C22—122no—
219C22C22×D1211yescharacteristic
220C22#100411yescharacteristic
221C22#100411yescharacteristic
222C22#100712yes—
223C22D3×#4112yes—
224C4—24no—
225C4—22no—
226C4—42no—
227C4—41no—
228C4—41no—
229C4—122no—
230C4—122no—
231C4—122no—
232C3#272811yescharacteristic, complemented, Sylow
233C2—124no—
234C2#549311yescharacteristic
235C2#557612yes—
236C2#563812yes—
237C2#567911yescharacteristic
238C2#573511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#391C2
prime radicalC22×D12C22
hypercenter#391D3
hypocenterC3#2728
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#946C2semidirect
2. C3×#391C2semidirect
4. #4981C2semidirect
5. #4940C2semidirect
6. #4674C2semidirect
7. #4668C2semidirect
9. #4659C2semidirect
38. #391D3semidirect
45. C2×C3×Q8D4semidirect
128. C2×Q8D12semidirect
232. C3#2728semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C3×#3912
#49912
#49812
#49402
#46742
#46682
#46652
#46592
#27283