Finite Group Databasebeta

← #67489

Subgroups — #67489

Order 384.

222 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 35 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
1#4960C211yesmaximal, characteristic
2C3×#342C211yesmaximal, characteristic, complemented
3#4525C211yesmaximal, characteristic, complemented
4#1343—31nomaximal, Sylow
5C22×#67—21no—
6C22×D12C411yescharacteristic, complemented
7C2×C3×C22⋊C4—21no—
8C2×C3×C22⋊C4C2211yescharacteristic
9C2×(C2×D3)⋊C4—21no—
10C3⋊#95C411yescharacteristic, complemented
11#830—21no—
12#392—31no—
13#342D311yescharacteristic, complemented
14#323—31no—
15C24×C3D411yescharacteristic
16C23×D3—21no—
17C22×C3×C4—41no—
18C22×C3×C4C2×C411yescharacteristic
19C22×C3×C4D411yescharacteristic
20C2×#67—42no—
21C2×#67—42no—
22C2×#67—41no—
23C2×#67—41no—
24C22×Dic3—21no—
25C2×D12—41no—
26C2×D12—42no—
27C3×C22⋊C4—22no—
28C3×C22⋊C4—41no—
29C3×C22⋊C4—41no—
30(C2×D3)⋊C4—42no—
31C2×C3⋊C8—41no—
32C22×D4—31no—
33C22×D4—61no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—61no—
36C2×C22⋊C4C2×D311yescharacteristic
37C22⋊C8—61no—
38#95—31no—
39C23×C3—21no—
40C23×C3—21no—
41C23×C3—41no—
42C23×C3—41no—
43C23×C3C22⋊C411yescharacteristic
44C23×C3D811yescharacteristic
45C23×C3SD1611yescharacteristic
46C22×D3—22no—
47C22×D3—41no—
48C22×D3—41no—
49C22×D3—41no—
50C22×D3—41no—
51C22×D3—41no—
52C22×D3—41no—
53C2×C3×C4—21no—
54C2×C3×C4—21no—
55C2×C3×C4—41no—
56C2×C3×C4—41no—
57C2×C3×C4—42no—
58C2×C3×C4—81no—
59C2×C3×C4—81no—
60#67—44no—
61#67—44no—
62C2×Dic3—22no—
63C2×Dic3—41no—
64C2×Dic3—41no—
65D12—42no—
66D12—42no—
67C3⋊C8—81no—
68C24—61no—
69C24#6711yescharacteristic
70C2×D4—121no—
71C2×D4—122no—
72C2×D4—122no—
73C2×D4—122no—
74C2×D4—121no—
75C2×D4—121no—
76C22×C4—41no—
77C22×C4—61no—
78C22×C4C4×D311yescharacteristic
79C22×C4D1211yescharacteristic
80C2×C8—121no—
81C22⋊C4—22no—
82C22⋊C4—41no—
83C22⋊C4—41no—
84C22⋊C4—122no—
85C22×C3—21no—
86C22×C3—21no—
87C22×C3—21no—
88C22×C3—21no—
89C22×C3—41no—
90C22×C3—41no—
91C22×C3—41no—
92C22×C3—41no—
93C22×C3—81no—
94C22×C3#9911yescharacteristic
95C22×C3#10211yescharacteristic
96C22×C3#10411yescharacteristic
97C2×D3—42no—
98C2×D3—42no—
99C2×D3—42no—
100C2×D3—81no—
101C2×D3—81no—
102C2×D3—81no—
103C2×D3—81no—
104C3×C4—41no—
105C3×C4—41no—
106C3×C4—82no—
107Dic3—42no—
108C23—21no—
109C23—21no—
110C23—41no—
111C23—41no—
112C23—62no—
113C23—121no—
114C23—121no—
115C23—121no—
116C23—121no—
117C23—121no—
118C23—121no—
119C23(C2×D3)⋊C411yescharacteristic
120C23#21311yescharacteristic
121C23#21511yescharacteristic
122D4—122no—
123D4—122no—
124D4—124no—
125D4—124no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—62no—
132C2×C4—81no—
133C2×C4—81no—
134C2×C4—121no—
135C2×C4—121no—
136C8—241no—
137C2×C3—21no—
138C2×C3—21no—
139C2×C3—41no—
140C2×C3—41no—
141C2×C3#32711yescharacteristic
142C2×C3AΣL(1,𝔽16)11yescharacteristic
143C2×C3#35311yescharacteristic
144D3—82no—
145C22—21no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—41no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22—122no—
155C22—122no—
156C22—122no—
157C22—241no—
158C22—241no—
159C22—241no—
160C22—241no—
161C22#80511yescharacteristic
162C22#80611yescharacteristic
163C22#80911yescharacteristic
164C4—41no—
165C4—41no—
166C4—82no—
167C4—122no—
168C3#134311yescharacteristic, complemented, Sylow
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—242no—
174C2#445111yescharacteristic
175C2#447111yescharacteristic
176C2#447411yescharacteristic
subgroupG/H
centerC22#806
commutatorC22×C3×C4C2×C4
FrattiniC2×C22⋊C4C2×D3
FittingC3×#342C2
socleC22×C3#99
prime radical#4960C2
cosocleC2×C3×C22⋊C4C22
hypercenter#342D3
hypocenterC3#1343
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C3×#342C2semidirect
3. #4525C2semidirect
6. C22×D12C4semidirect
10. C3⋊#95C4semidirect
13. #342D3semidirect
168. C3#1343semidirect

Maximal subgroups

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

subgroupindex
#49602
C3×#3422
#45252
#13433