Finite Group Databasebeta

← #76129

Subgroups — #76129

Order 384.

416 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 141 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×#889C212yesmaximal
2C3×#389C211yesmaximal, characteristic
3#4983C212yesmaximal
4#4982C212yesmaximal
5#4976C212yesmaximal
6#4928C211yesmaximal, characteristic
7#4928C211yesmaximal, characteristic
8#4657C212yesmaximal
9#4654C212yesmaximal
10#2726—31nomaximal, Sylow
11C2×C3×C4⋊C4C2212yes—
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C42C2211yescharacteristic
15C2×C4⋊Dic3C2211yescharacteristic
16C2×C4⋊Dic3C2211yescharacteristic
17C2×C4⋊Dic3C2211yescharacteristic
18C2×C4⋊Dic3C2211yescharacteristic
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×C4×Dic3C2212yes—
26#889—28no—
27C3×#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×#125—32no—
34#398—32no—
35#398—32no—
36#398—32no—
37#389—32no—
38#389—31no—
39#389—31no—
40#389D311yescharacteristic
41#387—32no—
42C22×C3×C4C2312yes—
43C22×C3×C4C2312yes—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C3×C4⋊C4—24no—
52C3×C4⋊C4—22no—
53C3×C4⋊C4—41no—
54C3×C42—41no—
55C4⋊Dic3—41no—
56C4⋊Dic3—41no—
57C4⋊Dic3D414yes—
58C4⋊Dic3Q814yes—
59Dic3⋊C4—24no—
60Dic3⋊C4—24no—
61Dic3⋊C4—24no—
62Dic3⋊C4—24no—
63Dic3⋊C4—24no—
64Dic3⋊C4—42no—
65C4×Dic3—24no—
66#125—68no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—31no—
76C2×C4⋊C4—31no—
77C2×C4⋊C4C2×D312yes—
78C2×C4⋊C4C2×D311yescharacteristic
79C2×C4⋊C4C2×D311yescharacteristic
80C2×C42—32no—
81C2×C42C2×D311yescharacteristic
82#95—32no—
83#95—32no—
84#95—32no—
85#95—32no—
86#95—32no—
87#95C2×D312yes—
88C23×C3C2411yescharacteristic
89C2×C3×C4—21no—
90C2×C3×C4—21no—
91C2×C3×C4—21no—
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4—21no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4C2×D412yes—
102C2×C3×C4C2×Q812yes—
103C2×C3×C4#4112yes—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3C2×D414yes—
115C2×Dic3C2×Q814yes—
116C22×C4—32no—
117C22×C4—32no—
118C22×C4—32no—
119C22×C4—32no—
120C22×C4C22×D312yes—
121C22×C4C22×D312yes—
122C22×C4C22×D311yescharacteristic
123C22×C4C22×D311yescharacteristic
124C22×C4C22×D311yescharacteristic
125C4⋊C4—22no—
126C4⋊C4—24no—
127C4⋊C4—34no—
128C4⋊C4—34no—
129C4⋊C4—41no—
130C4⋊C4—64no—
131C4⋊C4—64no—
132C4⋊C4—64no—
133C4⋊C4—64no—
134C4⋊C4—64no—
135C4⋊C4—122no—
136C4⋊C4—121no—
137C4⋊C4—121no—
138C42—41no—
139C42—64no—
140C22×C3C22×D411yescharacteristic
141C22×C3C22×Q811yescharacteristic
142C22×C3C2×#4111yescharacteristic
143C22×C3H(5,𝔽2)11yescharacteristic
144C22×C32−1+411yescharacteristic
145C22×C32−1+411yescharacteristic
146C22×C32−1+411yescharacteristic
147C3×C4—22no—
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—42no—
151C3×C4—42no—
152Dic3—24no—
153Dic3—24no—
154Dic3—42no—
155Dic3—42no—
156C23C23×D311yescharacteristic
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—34no—
170C2×C4—34no—
171C2×C4—62no—
172C2×C4—62no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4#23512yes—
182C2×C4D3×D412yes—
183C2×C4D3×Q812yes—
184C2×C3#53711yescharacteristic
185C2×C3#53911yescharacteristic
186C2×C3#54411yescharacteristic
187C2×C3#54711yescharacteristic
188C2×C3D4×Q811yescharacteristic
189C2×C3#55711yescharacteristic
190C2×C3#56411yescharacteristic
191C22C2×#23511yescharacteristic
192C22C2×D3×D411yescharacteristic
193C22#100411yescharacteristic
194C22C2×D3×Q811yescharacteristic
195C22#100711yescharacteristic
196C22#101011yescharacteristic
197C22#101011yescharacteristic
198C4—22no—
199C4—22no—
200C4—22no—
201C4—42no—
202C4—42no—
203C4—64no—
204C4—64no—
205C4—122no—
206C4—122no—
207C3#272611yescharacteristic, complemented, Sylow
208C2#551511yescharacteristic
209C2#555111yescharacteristic
210C2#556711yescharacteristic
211C2#559711yescharacteristic
212C2#562311yescharacteristic
213C2C3⋊#56411yescharacteristic
214C2#572711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#389C2
hypercenter#389D3
hypocenterC3#2726
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
207. C3#2726semidirect

Maximal subgroups

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

subgroupindex
C2×#8892
C3×#3892
#49832
#49822
#49762
#49282
#46572
#46542
#27263