Finite Group Databasebeta

← #68137

Subgroups — #68137

Order 384.

246 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 53 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#5510C211yesmaximal, characteristic
2#4986C211yesmaximal, characteristic, complemented
3#4735C212yesmaximal, complemented
4C3×#324C211yesmaximal, characteristic, complemented
5#4470C212yesmaximal, complemented
6#1616—31nomaximal, Sylow
7C2×#235—21no—
8C2×C3×C4⋊C4C2211yescharacteristic
9#930—21no—
10#930C2212yes—
11C2×C4⋊Dic3C2211yescharacteristic
12#897—22no—
13#895—22no—
14C3×C22⋊C8C2212yes—
15C3⋊#95—21no—
16C3⋊#95—21no—
17C3⋊#95C2211yescharacteristic
18#821—22no—
19#817—22no—
20#537—31no—
21#481—32no—
22#400—31no—
23#327—32no—
24#324D311yescharacteristic, complemented
25C22×C3×C4D411yescharacteristic
26C22×C3×C4D411yescharacteristic
27C22×C3×C4C2311yescharacteristic
28C2×#67—22no—
29C22×Dic3—21no—
30C22×Dic3—21no—
31#235—42no—
32#235—42no—
33C2×D12—21no—
34C2×D12—21no—
35C2×D12D412yes—
36C2×C4×D3—22no—
37C2×Dic6—21no—
38C2×C3×C8—22no—
39C3×C4⋊C4—22no—
40(C2×D3)⋊C4—22no—
41(C2×D3)⋊C4—22no—
42C4⋊Dic3—21no—
43C4⋊Dic3—22no—
44C4⋊Dic3D412yes—
45C2×#41—61no—
46C22⋊Q8—62no—
47#121—32no—
48#121—62no—
49#121—61no—
50C2×C4⋊C4—31no—
51C2×C4⋊C4C2×D311yescharacteristic
52#106—62no—
53#102—62no—
54C22⋊C8C2×D312yes—
55#95—31no—
56#95—61no—
57#95—61no—
58C23×C3C2×D411yescharacteristic
59C22×D3—22no—
60C2×C3×C4—21no—
61C2×C3×C4—21no—
62C2×C3×C4—21no—
63C2×C3×C4—41no—
64C2×C3×C4—41no—
65C2×C3×C4SD1612yes—
66C2×C3×C4C2×D412yes—
67#67—42no—
68#67—42no—
69C2×Dic3—22no—
70C2×Dic3—41no—
71C2×Dic3—41no—
72C2×Dic3—42no—
73C2×Dic3—41no—
74C2×Dic3—41no—
75D12—42no—
76D12—42no—
77D12—81no—
78C4×D3—44no—
79Dic6—42no—
80C3×C8—42no—
81#41—122no—
82#41—122no—
83C2×Q8—61no—
84C2×D4—32no—
85C2×D4—61no—
86C2×D4—61no—
87C2×D4—62no—
88C22×C4—62no—
89C22×C4—61no—
90C22×C4—61no—
91C22×C4D1211yescharacteristic
92C22×C4D1211yescharacteristic
93C22×C4C22×D311yescharacteristic
94C2×C8—22no—
95C4⋊C4—22no—
96C4⋊C4—32no—
97C4⋊C4—61no—
98C4⋊C4—62no—
99C22⋊C4—62no—
100C22⋊C4—62no—
101C22×C3—21no—
102C22×C3—21no—
103C22×C3#12011yescharacteristic
104C22×C3C2×SD1611yescharacteristic
105C22×C3AGL(1,ℤ/8)11yescharacteristic
106C2×D3—42no—
107C2×D3—42no—
108C2×D3—42no—
109C3×C4—22no—
110C3×C4—22no—
111C3×C4—41no—
112Dic3—42no—
113Dic3—81no—
114Dic3—81no—
115C23—62no—
116C23C2×D1211yescharacteristic
117Q8—122no—
118D4—122no—
119D4—122no—
120D4—122no—
121D4—122no—
122D4—241no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—62no—
129C2×C4—124no—
130C2×C4—121no—
131C2×C4—121no—
132C2×C4—122no—
133C2×C4—121no—
134C2×C4—121no—
135C2×C4#20412yes—
136C2×C4D3×D412yes—
137C8—42no—
138C2×C3—21no—
139C2×C3—21no—
140C2×C3#45011yescharacteristic
141C2×C3#45411yescharacteristic
142C2×C3#45811yescharacteristic
143D3—82no—
144C22—21no—
145C22—21no—
146C22—122no—
147C22—122no—
148C22—122no—
149C22#88211yescharacteristic
150C22C2×#20411yescharacteristic
151C22#90811yescharacteristic
152C4—22no—
153C4—22no—
154C4—41no—
155C4—122no—
156C4—241no—
157C4—241no—
158C3#161611yescharacteristic, complemented, Sylow
159C2—21no—
160C2—21no—
161C2—242no—
162C2#473111yescharacteristic
163C2#474211yescharacteristic
164C2#475311yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3#120
prime radical#5510C2
hypercenter#324D3
hypocenterC3#1616
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #4986C2semidirect
3. #4735C2semidirect
4. C3×#324C2semidirect
5. #4470C2semidirect
24. #324D3semidirect
158. C3#1616semidirect

Maximal subgroups

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

subgroupindex
#55102
#49862
#47352
C3×#3242
#44702
#16163