Finite Group Databasebeta

← #74253

Subgroups — #74253

Order 384.

448 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C3×#382C211yesmaximal, characteristic, complemented
2#4991C211yesmaximal, characteristic
3#4989C212yesmaximal, complemented
4#4986C211yesmaximal, characteristic, complemented
5#4940C212yesmaximal, complemented
6#4675C212yesmaximal, complemented
7#4668C212yesmaximal, complemented
8#4665C212yesmaximal
9#4658C212yesmaximal, complemented
10#2782—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C22×C4×D3C2212yes—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C42C2211yescharacteristic
16C2×(C2×D3)⋊C4C2212yes—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C3×#95C2212yes—
25C3×#95C2212yes—
26C3⋊#95C2211yescharacteristic
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2212yes—
32#400—31no—
33#400—32no—
34#397—32no—
35#394—31no—
36#388—32no—
37#386—32no—
38#385—32no—
39#385—32no—
40#382D311yescharacteristic, complemented
41C23×D3C2312yes—
42C22×C3×C4C2311yescharacteristic
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C2×D12—24no—
51C2×D12—41no—
52C2×C4×D3—42no—
53C2×C4×D3—42no—
54C2×C4×D3—42no—
55C3×C4⋊C4—41no—
56C3×C4⋊C4—41no—
57C3×C42—22no—
58(C2×D3)⋊C4—24no—
59(C2×D3)⋊C4—24no—
60(C2×D3)⋊C4—24no—
61(C2×D3)⋊C4—42no—
62(C2×D3)⋊C4—42no—
63C4⋊Dic3—41no—
64Dic3⋊C4—42no—
65Dic3⋊C4—42no—
66C22×D4—31no—
67C23×C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4C2×D311yescharacteristic
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C22⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4—32no—
78C2×C42C2×D311yescharacteristic
79#95—32no—
80#95—32no—
81#95—32no—
82#95—32no—
83#95—31no—
84#95—31no—
85#95C2×D312yes—
86#95C2×D312yes—
87C23×C3C2411yescharacteristic
88C22×D3—22no—
89C22×D3—22no—
90C22×D3—22no—
91C22×D3—22no—
92C22×D3—22no—
93C22×D3—22no—
94C22×D3#4114yes—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—21no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—21no—
107C2×C3×C4#4112yes—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117D12—44no—
118C4×D3—48no—
119C24—32no—
120C2×D4—64no—
121C2×D4—121no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4—122no—
126C22×C4—122no—
127C22×C4—122no—
128C22×C4C22×D311yescharacteristic
129C22×C4C22×D312yes—
130C22×C4C22×D311yescharacteristic
131C22×C4C22×D312yes—
132C22×C4C22×D311yescharacteristic
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—122no—
136C4⋊C4—122no—
137C4⋊C4—121no—
138C22⋊C4—64no—
139C22⋊C4—64no—
140C22⋊C4—64no—
141C22⋊C4—122no—
142C22⋊C4—122no—
143C42—22no—
144C22×C3C2×#4111yescharacteristic
145C22×C3C2×#4112yes—
146C22×C3H(5,𝔽2)11yescharacteristic
147C22×C3H(5,𝔽2)12yes—
148C22×C32−1+411yescharacteristic
149C2×D3—24no—
150C2×D3—24no—
151C2×D3—24no—
152C2×D3—42no—
153C2×D3—42no—
154C2×D3—42no—
155C2×D3—42no—
156C3×C4—22no—
157C3×C4—41no—
158C3×C4—41no—
159C3×C4—42no—
160C3×C4—42no—
161Dic3—42no—
162Dic3—42no—
163Dic3—42no—
164C23—34no—
165C23—62no—
166C23—62no—
167C23—62no—
168C23—62no—
169C23—62no—
170C23—62no—
171C23C23×D311yescharacteristic
172D4—124no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—128no—
195C2×C4#23512yes—
196C2×C3#53912yes—
197C2×C3#53911yescharacteristic
198C2×C3#55111yescharacteristic
199C2×C3#55312yes—
200C2×C3H(3,𝔽4)11yescharacteristic
201D3—44no—
202C22—64no—
203C22—64no—
204C22—64no—
205C22—122no—
206C22—122no—
207C22—122no—
208C22—122no—
209C22C2×#23511yescharacteristic
210C22#100411yescharacteristic
211C22#100711yescharacteristic
212C22D3×#4112yes—
213C22#100912yes—
214C4—22no—
215C4—41no—
216C4—41no—
217C4—42no—
218C4—42no—
219C4—122no—
220C4—122no—
221C4—122no—
222C3#278211yescharacteristic, complemented, Sylow
223C2—124no—
224C2#551511yescharacteristic
225C2#553012yes—
226C2#567911yescharacteristic
227C2#568311yescharacteristic
228C2#569212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#382C2
prime radicalC22×D12C22
hypercenter#382D3
hypocenterC3#2782
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C3×#382C2semidirect
3. #4989C2semidirect
4. #4986C2semidirect
5. #4940C2semidirect
6. #4675C2semidirect
7. #4668C2semidirect
9. #4658C2semidirect
40. #382D3semidirect
222. C3#2782semidirect

Maximal subgroups

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

subgroupindex
C3×#3822
#49912
#49892
#49862
#49402
#46752
#46682
#46652
#46582
#27823