Finite Group Databasebeta

← #76485

Subgroups — #76485

Order 384.

516 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 197 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×Q8×Dic3C211yesmaximal, characteristic
2C2×#944C211yesmaximal, characteristic
3C2×#887C211yesmaximal, characteristic
4C3×#391C211yesmaximal, characteristic
5#4977C211yesmaximal, characteristic
6#4932C211yesmaximal, characteristic
7#4929C211yesmaximal, characteristic
8#4655C214yesmaximal
9#4652C214yesmaximal
10#2327—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C42C2211yescharacteristic
15Q8×Dic3—24no—
16#944C418yescomplemented
17C2×C4⋊Dic3C2212yes—
18C2×C4⋊Dic3C2211yescharacteristic
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2214yes—
21C2×C4×Dic3C2212yes—
22C2×C4×Dic3C2212yes—
23C2×C4×Dic3C2211yescharacteristic
24C2×C4×Dic3C2211yescharacteristic
25#887—24no—
26C3×#95C2214yes—
27C3⋊#95C2212yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C3⋊#95C2212yes—
31C2×C4⋊Q8—31no—
32C2×C4×Q8—31no—
33C2×C4×Q8—31no—
34#391—31no—
35#391D311yescharacteristic
36#387—34no—
37#387—34no—
38#382—31no—
39#376—31no—
40C2×C3×Q8—22no—
41C2×C3×Q8—22no—
42C2×C3×Q8C2×C414yes—
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2311yescharacteristic
51C22×Dic3C2311yescharacteristic
52C2×Dic6—22no—
53C2×Dic6—22no—
54C2×Dic6C2×C414yes—
55C3×C4⋊C4—22no—
56C3×C42—22no—
57C4⋊Dic3—22no—
58C4⋊Dic3—24no—
59Dic3⋊C4—24no—
60Dic3⋊C4C2×C4116yes—
61C4×Dic3—24no—
62C4×Dic3—24no—
63C4×Dic3C2×C414yes—
64C4×Dic3D414yes—
65C22×Q8—31no—
66C22×Q8C2×D311yescharacteristic
67C4⋊Q8—38no—
68C4×Q8—64no—
69C4×Q8—64no—
70C2×C4⋊C4—34no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4C2×D311yescharacteristic
75C2×C42—31no—
76C2×C42—31no—
77C2×C42—32no—
78C2×C42—32no—
79C2×C42C2×D311yescharacteristic
80#95—32no—
81#95—32no—
82#95—34no—
83#95—34no—
84#95C2×D314yes—
85C23×C3C2411yescharacteristic
86C3×Q8—28no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—24no—
93C2×C3×C4—22no—
94C2×C3×C4C22×C414yes—
95C2×C3×C4C22×C412yes—
96C2×C3×C4C2×D412yes—
97C2×C3×C4#4112yes—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—22no—
101C2×Dic3—22no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—24no—
106C2×Dic3C22×C414yes—
107C2×Dic3C22×C414yes—
108C2×Dic3C2×D412yes—
109C2×Dic3C2×D412yes—
110Dic6—28no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—34no—
114C2×Q8—62no—
115C2×Q8—62no—
116C2×Q8C4×D314yes—
117C22×C4—32no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—32no—
121C22×C4—32no—
122C22×C4C22×D312yes—
123C22×C4C22×D312yes—
124C22×C4C22×D312yes—
125C22×C4C22×D311yescharacteristic
126C4⋊C4—22no—
127C4⋊C4—316no—
128C4⋊C4—64no—
129C4⋊C4—64no—
130C4⋊C4—62no—
131C42—22no—
132C42—34no—
133C42—34no—
134C42—64no—
135C42—64no—
136C22×C3C23×C411yescharacteristic
137C22×C3C22×D411yescharacteristic
138C22×C3C2×#4111yescharacteristic
139C22×C32−1+412yes—
140C22×C32−1+412yes—
141C3×C4—24no—
142C3×C4—42no—
143C3×C4—42no—
144C3×C4C4×D414yes—
145Dic3—24no—
146Dic3—24no—
147Dic3—22no—
148Dic3—22no—
149Dic3—42no—
150C23C23×D311yescharacteristic
151Q8—28no—
152Q8—68no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—34no—
161C2×C4—34no—
162C2×C4—32no—
163C2×C4—32no—
164C2×C4—62no—
165C2×C4—62no—
166C2×C4—62no—
167C2×C4—62no—
168C2×C4—62no—
169C2×C4—62no—
170C2×C4—62no—
171C2×C4—64no—
172C2×C4C2×C4×D314yes—
173C2×C4C2×C4×D312yes—
174C2×C4D3×D412yes—
175C2×C4#23712yes—
176C2×C3C2×C4×D411yescharacteristic
177C2×C3#51912yes—
178C2×C3#53611yescharacteristic
179C2×C3#53611yescharacteristic
180C2×C3#54112yes—
181C22C22×C4×D311yescharacteristic
182C22C2×D3×D411yescharacteristic
183C22C2×#23711yescharacteristic
184C22#100712yes—
185C22#101012yes—
186C4—24no—
187C4—42no—
188C4—42no—
189C4—62no—
190C4—62no—
191C4—64no—
192C4—64no—
193C4—122no—
194C4#67⋊C414yes—
195C3#232711yescharacteristic, complemented, Sylow
196C2C2×#67⋊C411yescharacteristic
197C2#557212yes—
198C2#562512yes—
199C2#567511yescharacteristic
200C2#572411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#391C2
socle, prime radical, cosocleC23×C3C24
hypercenter#391D3
hypocenterC3#2327
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
16. #944C4semidirect
195. C3#2327semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic32
C2×#9442
C2×#8872
C3×#3912
#49772
#49322
#49292
#46552
#46522
#23273