Finite Group Databasebeta

← #67645

Subgroups — #67645

Order 384.

222 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 37 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
1C3×#392C211yesmaximal, characteristic
2#4526C211yesmaximal, characteristic, complemented
3#4525C211yesmaximal, characteristic, complemented
4#1343—31nomaximal, Sylow
5C22×C3×D4—21no—
6C22×C3×D4C411yescharacteristic, complemented
7C2×C3×C22⋊C4—21no—
8C2×C3×C22⋊C4C2211yescharacteristic
9C2×#217—21no—
10C3×#95C411yescharacteristic, complemented
11#830—21no—
12#392D311yescharacteristic
13#342—31no—
14#323—31no—
15C24×C3—21no—
16C24×C3D411yescharacteristic
17C2×C3×D4—41no—
18C2×C3×D4—42no—
19C2×C3×D4—42no—
20C2×C3×D4—41no—
21C2×C3×D4—41no—
22C2×C3×D4—42no—
23C22×C3×C4—21no—
24C22×C3×C4C2×C411yescharacteristic
25C22×C3×C4D411yescharacteristic
26C22×Dic3—41no—
27C3×C22⋊C4—41no—
28C3×C22⋊C4—42no—
29#217—22no—
30#217—41no—
31C2×C3⋊C8—41no—
32C22×D4—21no—
33C22×D4Dic311yescharacteristic, complemented
34C2×C22⋊C4—21no—
35C2×C22⋊C4—61no—
36C2×C22⋊C4C2×D311yescharacteristic
37C22⋊C8—61no—
38#95Dic311yescharacteristic, complemented
39C23×C3—21no—
40C23×C3—21no—
41C23×C3—22no—
42C23×C3—41no—
43C23×C3—41no—
44C23×C3—41no—
45C23×C3—41no—
46C23×C3—41no—
47C23×C3—41no—
48C23×C3—41no—
49C23×C3—41no—
50C23×C3C22⋊C411yescharacteristic
51C23×C3D811yescharacteristic
52C23×C3SD1611yescharacteristic
53C3×D4—42no—
54C3×D4—44no—
55C3×D4—44no—
56C3×D4—42no—
57C2×C3×C4—22no—
58C2×C3×C4—21no—
59C2×C3×C4—21no—
60C2×C3×C4—41no—
61C2×C3×C4—41no—
62C2×C3×C4—41no—
63C2×C3×C4—41no—
64C2×Dic3—42no—
65C2×Dic3—81no—
66C2×Dic3—81no—
67C3⋊C8—81no—
68C24—21no—
69C24#6711yescharacteristic
70C2×D4—41no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—42no—
76C22×C4—21no—
77C22×C4—121no—
78C22×C4C2×Dic311yescharacteristic
79C22×C4#6711yescharacteristic
80C2×C8—121no—
81C22⋊C4—41no—
82C22⋊C4—42no—
83C22⋊C4—62no—
84C22⋊C4—121no—
85C22×C3—21no—
86C22×C3—21no—
87C22×C3—21no—
88C22×C3—21no—
89C22×C3—41no—
90C22×C3—41no—
91C22×C3—42no—
92C22×C3—42no—
93C22×C3—41no—
94C22×C3—41no—
95C22×C3—42no—
96C22×C3—81no—
97C22×C3—81no—
98C22×C3—81no—
99C22×C3—81no—
100C22×C3—81no—
101C22×C3#9911yescharacteristic
102C22×C3#10211yescharacteristic
103C22×C3#10411yescharacteristic
104C3×C4—42no—
105C3×C4—41no—
106C3×C4—41no—
107Dic3—82no—
108C23—21no—
109C23—21no—
110C23—22no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23#21311yescharacteristic
120C23#21411yescharacteristic
121C23#21711yescharacteristic
122D4—42no—
123D4—44no—
124D4—44no—
125D4—42no—
126C2×C4—22no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—122no—
134C2×C4—241no—
135C2×C4—241no—
136C8—241no—
137C2×C3—21no—
138C2×C3—21no—
139C2×C3—41no—
140C2×C3—41no—
141C2×C3—82no—
142C2×C3#32711yescharacteristic
143C2×C3AΣL(1,𝔽16)11yescharacteristic
144C2×C3#35311yescharacteristic
145C22—21no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—41no—
150C22—41no—
151C22—42no—
152C22—42no—
153C22—41no—
154C22—41no—
155C22—42no—
156C22—81no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22—81no—
161C22#83211yescharacteristic
162C22#83411yescharacteristic
163C22#83711yescharacteristic
164C4—42no—
165C4—41no—
166C4—41no—
167C4—242no—
168C3#134311yescharacteristic, complemented, Sylow
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—82no—
174C2#453611yescharacteristic
175C2#453711yescharacteristic
176C2#454511yescharacteristic
subgroupG/H
centerC22#834
commutatorC22×C3×C4C2×C4
FrattiniC2×C22⋊C4C2×D3
Fitting, prime radicalC3×#392C2
socleC22×C3#99
cosocleC2×C3×C22⋊C4C22
hypercenter#392D3
hypocenterC3#1343
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #392

Complements

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

NH ≅ G/Ntype
2. #4526C2semidirect
3. #4525C2semidirect
6. C22×C3×D4C4semidirect
10. C3×#95C4semidirect
33. C22×D4Dic3semidirect
38. #95Dic3semidirect
168. C3#1343semidirect

Maximal subgroups

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

subgroupindex
C3×#3922
#45262
#45252
#13433