Finite Group Databasebeta

← #68135

Subgroups — #68135

Order 384.

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

normal subgroups only

This group has 254 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#4981C211yesmaximal, characteristic, complemented
3#4736C212yesmaximal, complemented
4C3×#324C211yesmaximal, characteristic, complemented
5#4470C212yesmaximal, complemented
6#1610—31nomaximal, Sylow
7C2×#235—21no—
8C2×C3×C4⋊C4C2211yescharacteristic
9#930—21no—
10#930C2212yes—
11C2×C4⋊Dic3C2211yescharacteristic
12C2×Dic3⋊C4—21no—
13#897—22no—
14#895—22no—
15C3×C22⋊C8C2212yes—
16C3⋊#95—21no—
17C3⋊#95C2211yescharacteristic
18#821—22no—
19#818—22no—
20#537—31no—
21#480—32no—
22#398—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—
45Dic3⋊C4—42no—
46C2×#41—61no—
47C22⋊Q8—62no—
48#121—32no—
49#121—62no—
50#121—61no—
51C2×C4⋊C4—31no—
52C2×C4⋊C4—61no—
53C2×C4⋊C4C2×D311yescharacteristic
54#107—62no—
55#102—62no—
56C22⋊C8C2×D312yes—
57#95—31no—
58#95—61no—
59C23×C3C2×D411yescharacteristic
60C22×D3—22no—
61C2×C3×C4—21no—
62C2×C3×C4—21no—
63C2×C3×C4—21no—
64C2×C3×C4—41no—
65C2×C3×C4—41no—
66C2×C3×C4D812yes—
67C2×C3×C4C2×D412yes—
68#67—42no—
69#67—42no—
70C2×Dic3—22no—
71C2×Dic3—22no—
72C2×Dic3—41no—
73C2×Dic3—41no—
74C2×Dic3—41no—
75C2×Dic3—41no—
76C2×Dic3—41no—
77D12—42no—
78D12—42no—
79D12—81no—
80C4×D3—44no—
81Dic6—42no—
82C3×C8—42no—
83#41—122no—
84#41—122no—
85C2×Q8—61no—
86C2×D4—32no—
87C2×D4—61no—
88C2×D4—61no—
89C2×D4—62no—
90C22×C4—62no—
91C22×C4—61no—
92C22×C4—61no—
93C22×C4D1211yescharacteristic
94C22×C4D1211yescharacteristic
95C22×C4C22×D311yescharacteristic
96C2×C8—22no—
97C4⋊C4—22no—
98C4⋊C4—32no—
99C4⋊C4—61no—
100C4⋊C4—62no—
101C4⋊C4—122no—
102C22⋊C4—62no—
103C22⋊C4—62no—
104C22×C3—21no—
105C22×C3—21no—
106C22×C3#12011yescharacteristic
107C22×C3C2×D811yescharacteristic
108C22×C3AGL(1,ℤ/8)11yescharacteristic
109C2×D3—42no—
110C2×D3—42no—
111C2×D3—42no—
112C3×C4—22no—
113C3×C4—22no—
114C3×C4—41no—
115Dic3—42no—
116Dic3—42no—
117Dic3—81no—
118C23—62no—
119C23C2×D1211yescharacteristic
120Q8—122no—
121D4—122no—
122D4—122no—
123D4—122no—
124D4—122no—
125D4—241no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—62no—
132C2×C4—62no—
133C2×C4—124no—
134C2×C4—121no—
135C2×C4—121no—
136C2×C4—121no—
137C2×C4—121no—
138C2×C4—121no—
139C2×C4D2412yes—
140C2×C4D3×D412yes—
141C8—42no—
142C2×C3—21no—
143C2×C3—21no—
144C2×C3#44711yescharacteristic
145C2×C3#45611yescharacteristic
146C2×C3#45811yescharacteristic
147D3—82no—
148C22—21no—
149C22—21no—
150C22—122no—
151C22—122no—
152C22—122no—
153C22#88211yescharacteristic
154C22C2×D2411yescharacteristic
155C22#90811yescharacteristic
156C4—22no—
157C4—22no—
158C4—41no—
159C4—122no—
160C4—122no—
161C4—241no—
162C3#161011yescharacteristic, complemented, Sylow
163C2—21no—
164C2—21no—
165C2—242no—
166C2#473211yescharacteristic
167C2#474211yescharacteristic
168C2#475211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3#120
prime radical#5510C2
hypercenter#324D3
hypocenterC3#1610
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. #4981C2semidirect
3. #4736C2semidirect
4. C3×#324C2semidirect
5. #4470C2semidirect
24. #324D3semidirect
162. C3#1610semidirect

Maximal subgroups

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

subgroupindex
#55102
#49812
#47362
C3×#3242
#44702
#16103