Finite Group Databasebeta

← #74161

Subgroups — #74161

Order 384.

408 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 173 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×#870C211yesmaximal, characteristic
2C3×#382C211yesmaximal, characteristic
3#4978C212yesmaximal
4#4976C212yesmaximal
5#4936C211yesmaximal, characteristic
6#4655C212yesmaximal
7#4654C212yesmaximal
8#4653C212yesmaximal
9#4647C212yesmaximal
10#2382—31nomaximal, Sylow
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C42C2211yescharacteristic
14C2×C4⋊Dic3C2211yescharacteristic
15C2×C4⋊Dic3C2211yescharacteristic
16C2×C4⋊Dic3C2211yescharacteristic
17C2×C4⋊Dic3C2211yescharacteristic
18C2×Dic3⋊C4C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×C4×Dic3C2212yes—
23C2×C4×Dic3C2212yes—
24#870C418yescomplemented
25C3×#95C2212yes—
26C3×#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×#125—31no—
34#389—32no—
35#389—32no—
36#387—32no—
37#387—32no—
38#387—32no—
39#387—32no—
40#382—31no—
41#382D311yescharacteristic
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—41no—
52C3×C4⋊C4—41no—
53C3×C42C2×C414yes—
54C4⋊Dic3—41no—
55C4⋊Dic3—41no—
56C4⋊Dic3C2×C414yes—
57C4⋊Dic3C2×C414yes—
58Dic3⋊C4—42no—
59Dic3⋊C4—42no—
60Dic3⋊C4C2×C418yes—
61Dic3⋊C4C2×C418yes—
62C4×Dic3—42no—
63C4×Dic3—42no—
64#125—38no—
65C2×C4⋊C4—31no—
66C2×C4⋊C4—31no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4C2×D311yescharacteristic
74C2×C4⋊C4C2×D311yescharacteristic
75C2×C42—32no—
76C2×C42—32no—
77C2×C42C2×D311yescharacteristic
78#95—32no—
79#95—32no—
80#95—32no—
81#95—32no—
82#95—32no—
83#95—32no—
84#95C2×D312yes—
85#95C2×D312yes—
86C23×C3C2411yescharacteristic
87C2×C3×C4—21no—
88C2×C3×C4—21no—
89C2×C3×C4—21no—
90C2×C3×C4—21no—
91C2×C3×C4—21no—
92C2×C3×C4—21no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4C22×C412yes—
100C2×C3×C4C22×C412yes—
101C2×C3×C4C22×C412yes—
102C2×Dic3—22no—
103C2×Dic3—22no—
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×Dic3C22×C414yes—
113C2×Dic3C22×C414yes—
114C22×C4—32no—
115C22×C4—32no—
116C22×C4—32no—
117C22×C4—32no—
118C22×C4C22×D312yes—
119C22×C4C22×D312yes—
120C22×C4C22×D311yescharacteristic
121C22×C4C22×D311yescharacteristic
122C22×C4C22×D311yescharacteristic
123C4⋊C4—38no—
124C4⋊C4—38no—
125C4⋊C4—34no—
126C4⋊C4—34no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—122no—
130C4⋊C4—122no—
131C4⋊C4—121no—
132C4⋊C4—121no—
133C42—122no—
134C42—122no—
135C42C4×D314yes—
136C22×C3C23×C411yescharacteristic
137C22×C3H(5,𝔽2)11yescharacteristic
138C22×C3H(5,𝔽2)11yescharacteristic
139C22×C32−1+411yescharacteristic
140C22×C32−1+411yescharacteristic
141C22×C32−1+411yescharacteristic
142C22×C32−1+411yescharacteristic
143C3×C4—22no—
144C3×C4—22no—
145C3×C4—22no—
146C3×C4—42no—
147C3×C4—42no—
148Dic3—24no—
149Dic3—24no—
150Dic3—42no—
151Dic3—42no—
152C23C23×D311yescharacteristic
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—34no—
166C2×C4—34no—
167C2×C4—62no—
168C2×C4—62no—
169C2×C4—62no—
170C2×C4—62no—
171C2×C4—62no—
172C2×C4—62no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4C2×C4×D312yes—
178C2×C4C2×C4×D312yes—
179C2×C4C2×C4×D312yes—
180C2×C3#51911yescharacteristic
181C2×C3#52011yescharacteristic
182C2×C3#52011yescharacteristic
183C2×C3#56211yescharacteristic
184C2×C3#56311yescharacteristic
185C2×C3#56311yescharacteristic
186C2×C3#56411yescharacteristic
187C22C22×C4×D311yescharacteristic
188C22#100411yescharacteristic
189C22#100711yescharacteristic
190C22#100911yescharacteristic
191C22#101011yescharacteristic
192C22#101011yescharacteristic
193C22#101011yescharacteristic
194C4—22no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—64no—
200C4—64no—
201C4—122no—
202C4—122no—
203C3#238211yescharacteristic, complemented, Sylow
204C2#550411yescharacteristic
205C2#551611yescharacteristic
206C2#552311yescharacteristic
207C2#567211yescharacteristic
208C2#568411yescharacteristic
209C2C3⋊#56411yescharacteristic
210C2#569911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#382C2
socle, prime radical, cosocleC23×C3C24
hypercenter#382D3
hypocenterC3#2382
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
24. #870C4semidirect
203. C3#2382semidirect

Maximal subgroups

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

subgroupindex
C2×#8702
C3×#3822
#49782
#49762
#49362
#46552
#46542
#46532
#46472
#23823