Finite Group Databasebeta

← #78210

Subgroups — #78210

Order 384.

416 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 129 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×#889C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic
3C3×#400C211yesmaximal, characteristic
4#4985C211yesmaximal, characteristic
5#4979C211yesmaximal, characteristic
6#4658C212yesmaximal
7#4657C212yesmaximal
8#4656C211yesmaximal, characteristic
9#4655C212yesmaximal
10#4654C212yesmaximal
11#4651C211yesmaximal, characteristic
12#2634—31nomaximal, Sylow
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic3C2212yes—
17C2×C4⋊Dic3C2211yescharacteristic
18C2×Dic3⋊C4C2211yescharacteristic
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2211yescharacteristic
21C2×Dic3⋊C4C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2211yescharacteristic
24C2×C4×Dic3C2212yes—
25C2×C4×Dic3C2211yescharacteristic
26#889—24no—
27#887—24no—
28C3×#95C2211yescharacteristic
29C3×#95C2212yes—
30C3×#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×C4×Q8—31no—
40#400—31no—
41#400—32no—
42#400D311yescharacteristic
43#398—32no—
44#395—31no—
45#395—31no—
46#387—32no—
47#387—32no—
48#383—31no—
49C22×C3×C4C2312yes—
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×Dic3C2312yes—
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C2×Dic6—22no—
61C2×Dic6—24no—
62C3×C4⋊C4—22no—
63C3×C4⋊C4—22no—
64C4⋊Dic3—22no—
65C4⋊Dic3—24no—
66Dic3⋊C4—22no—
67Dic3⋊C4—24no—
68Dic3⋊C4—22no—
69Dic3⋊C4—22no—
70Dic3⋊C4—22no—
71Dic3⋊C4—24no—
72C4×Dic3—24no—
73C4×Dic3D414yes—
74C22×Q8—31no—
75#125—64no—
76C4×Q8—64no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—31no—
79C2×C4⋊C4—31no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4C2×D311yescharacteristic
86C2×C4⋊C4C2×D311yescharacteristic
87C2×C42—31no—
88C2×C42—32no—
89#95—32no—
90#95—31no—
91#95—32no—
92#95—32no—
93#95—32no—
94#95—32no—
95#95—32no—
96#95C2×D311yescharacteristic
97#95C2×D312yes—
98#95C2×D312yes—
99C23×C3C2411yescharacteristic
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—21no—
107C2×C3×C4—24no—
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4C2×D412yes—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—22no—
117C2×Dic3—21no—
118C2×Dic3—21no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—21no—
122C2×Dic3—22no—
123C2×Dic3—21no—
124C2×Dic3—22no—
125C2×Dic3C2×D412yes—
126C2×Dic3C2×D412yes—
127C2×Dic3#4114yes—
128Dic6—28no—
129C2×Q8—64no—
130C2×Q8—62no—
131C22×C4—31no—
132C22×C4—31no—
133C22×C4—32no—
134C22×C4—32no—
135C22×C4—31no—
136C22×C4—31no—
137C22×C4C22×D312yes—
138C22×C4C22×D312yes—
139C22×C4C22×D311yescharacteristic
140C22×C4C22×D311yescharacteristic
141C22×C4C22×D311yescharacteristic
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—64no—
145C4⋊C4—62no—
146C4⋊C4—62no—
147C4⋊C4—62no—
148C4⋊C4—62no—
149C4⋊C4—64no—
150C4⋊C4—64no—
151C4⋊C4—62no—
152C42—34no—
153C42—64no—
154C22×C3C22×D411yescharacteristic
155C22×C3C2×#4112yes—
156C22×C32−1+411yescharacteristic
157C22×C32−1+412yes—
158C22×C32−1+411yescharacteristic
159C3×C4—22no—
160C3×C4—41no—
161C3×C4—42no—
162C3×C4—41no—
163C3×C4—42no—
164Dic3—24no—
165Dic3—22no—
166Dic3—22no—
167Dic3—41no—
168Dic3—41no—
169Dic3—42no—
170C23C23×D311yescharacteristic
171Q8—68no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—32no—
184C2×C4—32no—
185C2×C4—34no—
186C2×C4—62no—
187C2×C4—61no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—62no—
194C2×C4—61no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—61no—
199C2×C4D3×D412yes—
200C2×C3#52511yescharacteristic
201C2×C3#53611yescharacteristic
202C2×C3#53611yescharacteristic
203C2×C3#54112yes—
204C2×C3#54112yes—
205C22C2×D3×D411yescharacteristic
206C22#100711yescharacteristic
207C22D3×#4112yes—
208C22#101012yes—
209C22#101011yescharacteristic
210C4—22no—
211C4—41no—
212C4—42no—
213C4—42no—
214C4—41no—
215C4—64no—
216C4—62no—
217C4—62no—
218C4—121no—
219C4—122no—
220C4—121no—
221C3#263411yescharacteristic, complemented, Sylow
222C2#562811yescharacteristic
223C2#565611yescharacteristic
224C2#567511yescharacteristic
225C2#569612yes—
226C2#570112yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#400C2
hypercenter#400D3
hypocenterC3#2634
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
221. C3#2634semidirect

Maximal subgroups

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

subgroupindex
C2×#8892
C2×#8872
C3×#4002
#49852
#49792
#46582
#46572
#46562
#46552
#46542
#46512
#26343