Finite Group Databasebeta

← #77186

Subgroups — #77186

Order 384.

556 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 181 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×#944C212yesmaximal
2C2×#888C212yesmaximal
3C3×#395C211yesmaximal, characteristic
4#4979C211yesmaximal, characteristic
5#4976C212yesmaximal
6#4975C211yesmaximal, characteristic
7#4656C212yesmaximal
8#4649C212yesmaximal
9#4647C212yesmaximal
10#2551—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×Dic6C2212yes—
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16#944—28no—
17C2×C4⋊Dic3C2212yes—
18C2×C4⋊Dic3C2211yescharacteristic
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×C4×Dic3C2212yes—
26C2×C4×Dic3C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28#888—28no—
29C3×#95C2211yescharacteristic
30C3×#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2211yescharacteristic
34C2×C4⋊Q8—32no—
35C2×C4⋊Q8—32no—
36#395—31no—
37#395—32no—
38#395D311yescharacteristic
39#389—32no—
40#389—32no—
41#384—31no—
42C4×C4⋊C4—32no—
43C2×C3×Q8—24no—
44C2×C3×Q8—41no—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C2×Dic6—24no—
55C2×Dic6—24no—
56C2×Dic6—24no—
57C2×Dic6—41no—
58C2×Dic6—42no—
59C3×C4⋊C4—24no—
60C3×C4⋊C4—41no—
61C4⋊Dic3—24no—
62C4⋊Dic3—41no—
63Dic3⋊C4—24no—
64Dic3⋊C4—24no—
65Dic3⋊C4—24no—
66Dic3⋊C4—24no—
67Dic3⋊C4D418yes—
68Dic3⋊C4Q818yescomplemented
69C4×Dic3—24no—
70C4×Dic3—24no—
71C4×Dic3Q814yescomplemented
72C22×Q8—31no—
73C22×Q8—32no—
74C22×Q8C2×D311yescharacteristic
75C4⋊Q8—68no—
76C4⋊Q8—68no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4C2×D312yes—
86C2×C4⋊C4C2×D311yescharacteristic
87C2×C42—31no—
88C2×C42—32no—
89C2×C42—32no—
90#95—32no—
91#95—31no—
92#95—32no—
93#95C2×D312yes—
94#95C2×D311yescharacteristic
95C23×C3C2411yescharacteristic
96C3×Q8—44no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—21no—
100C2×C3×C4—22no—
101C2×C3×C4—21no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4C2×D414yes—
108C2×C3×C4C2×Q812yes—
109C2×C3×C4C2×Q814yes—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3C2×D414yes—
118C2×Dic3C2×D414yes—
119C2×Dic3C2×Q814yes—
120C2×Dic3C2×Q814yes—
121C2×Dic3C2×Q814yes—
122Dic6—44no—
123Dic6—48no—
124C2×Q8—24no—
125C2×Q8—41no—
126C2×Q8—64no—
127C2×Q8—64no—
128C2×Q8—64no—
129C2×Q8—122no—
130C2×Q8—121no—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4—32no—
134C22×C4—32no—
135C22×C4C22×D311yescharacteristic
136C22×C4C22×D311yescharacteristic
137C22×C4C22×D312yes—
138C22×C4C22×D311yescharacteristic
139C22×C4C22×D312yes—
140C4⋊C4—24no—
141C4⋊C4—38no—
142C4⋊C4—38no—
143C4⋊C4—41no—
144C4⋊C4—64no—
145C4⋊C4—64no—
146C4⋊C4—64no—
147C4⋊C4—64no—
148C4⋊C4—64no—
149C4⋊C4—121no—
150C42—34no—
151C42—64no—
152C42—64no—
153C22×C3C22×D412yes—
154C22×C3C22×Q811yescharacteristic
155C22×C3C22×Q812yes—
156C22×C3H(5,𝔽2)11yescharacteristic
157C22×C32−1+411yescharacteristic
158C3×C4—22no—
159C3×C4—24no—
160C3×C4—24no—
161C3×C4—41no—
162C3×C4—41no—
163Dic3—24no—
164Dic3—24no—
165Dic3—42no—
166Dic3C4⋊Q818yescomplemented
167C23C23×D311yescharacteristic
168Q8—44no—
169Q8—128no—
170Q8—124no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—34no—
182C2×C4—34no—
183C2×C4—34no—
184C2×C4—34no—
185C2×C4—34no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4D3×D414yes—
194C2×C4D3×Q812yes—
195C2×C4D3×Q814yes—
196C2×C3C2×C4⋊Q812yes—
197C2×C3#54411yescharacteristic
198C2×C3D4211yescharacteristic
199C2×C3D4×Q812yes—
200C2×C3Q8211yescharacteristic
201C22C2×D3×D412yes—
202C22#100411yescharacteristic
203C22C2×D3×Q812yes—
204C22C2×D3×Q811yescharacteristic
205C22#101011yescharacteristic
206C4—24no—
207C4—24no—
208C4—22no—
209C4—38no—
210C4—41no—
211C4—41no—
212C4—64no—
213C4—64no—
214C4—122no—
215C3#255111yescharacteristic, complemented, Sylow
216C2#559011yescharacteristic
217C2#562311yescharacteristic
218C2#563212yes—
219C2#572211yescharacteristic
220C2D3×C4⋊Q812yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#395C2
hypercenter#395D3
hypocenterC3#2551
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
68. Dic3⋊C4Q8semidirect
71. C4×Dic3Q8semidirect
166. Dic3C4⋊Q8semidirect
215. C3#2551semidirect

Maximal subgroups

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

subgroupindex
C2×#9442
C2×#8882
C3×#3952
#49792
#49762
#49752
#46562
#46492
#46472
#25513