Finite Group Databasebeta

← #69712

Subgroups — #69712

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
2C3×#398C211yesmaximal, characteristic, complemented
3#4967C212yesmaximal, complemented
4#4528C211yesmaximal, characteristic, complemented
5#4451C212yesmaximal, complemented
6#1610—31nomaximal, Sylow
7C2×#235—21no—
8C2×C3×C4⋊C4—21no—
9C2×C3×C4⋊C4C2211yescharacteristic
10C2×C3×C4⋊C4C2211yescharacteristic
11#930—21no—
12#930C2212yes—
13#897—22no—
14#895—22no—
15C3×#95—21no—
16C3×#95C2211yescharacteristic
17#830C2212yes—
18#809—22no—
19#807—22no—
20#537—31no—
21#480—32no—
22#398D311yescharacteristic, complemented
23#327—32no—
24#324—31no—
25C22×C3×C4—21no—
26C22×C3×C4—21no—
27C22×C3×C4D411yescharacteristic
28C22×C3×C4D411yescharacteristic
29C22×C3×C4C2311yescharacteristic
30C2×#67—22no—
31#235—42no—
32#235—42no—
33C2×D12—21no—
34C2×D12—21no—
35C2×D12D412yes—
36C2×C4×D3—22no—
37C2×Dic6—21no—
38C3×C4⋊C4—22no—
39C3×C4⋊C4—22no—
40C3×C4⋊C4—42no—
41(C2×D3)⋊C4—22no—
42(C2×D3)⋊C4—22no—
43C4⋊Dic3—21no—
44C4⋊Dic3D412yes—
45C2×C3⋊C8—22no—
46C2×#41—61no—
47C22⋊Q8—62no—
48#121—32no—
49#121—62no—
50#121—61no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4C2×D311yescharacteristic
53C2×C4⋊C4C2×D311yescharacteristic
54#107—62no—
55#102—62no—
56C22⋊C8—32no—
57#95—21no—
58#95C2×D311yescharacteristic
59C23×C3C2×D411yescharacteristic
60C22×D3—22no—
61C2×C3×C4—21no—
62C2×C3×C4—21no—
63C2×C3×C4—22no—
64C2×C3×C4—21no—
65C2×C3×C4—41no—
66C2×C3×C4—41no—
67C2×C3×C4—41no—
68C2×C3×C4—41no—
69C2×C3×C4—41no—
70C2×C3×C4—41no—
71C2×C3×C4—41no—
72C2×C3×C4D812yes—
73C2×C3×C4C2×D412yes—
74#67—42no—
75#67—42no—
76C2×Dic3—22no—
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—21no—
91C22×C4—21no—
92C22×C4—62no—
93C22×C4#6711yescharacteristic
94C22×C4#6711yescharacteristic
95C22×C4C22×D311yescharacteristic
96C2×C8—62no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—32no—
100C4⋊C4—42no—
101C4⋊C4—61no—
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—42no—
115C3×C4—41no—
116C3×C4—81no—
117Dic3—42no—
118C23—62no—
119C23C2×#6711yescharacteristic
120Q8—122no—
121D4—122no—
122D4—122no—
123D4—122no—
124D4—122no—
125D4—241no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—22no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—62no—
138C2×C4—124no—
139C2×C4#21312yes—
140C2×C4D3×D412yes—
141C8—122no—
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—
153C22C2×#21311yescharacteristic
154C22#93711yescharacteristic
155C22#94911yescharacteristic
156C4—22no—
157C4—22no—
158C4—42no—
159C4—41no—
160C4—81no—
161C4—122no—
162C3#161011yescharacteristic, complemented, Sylow
163C2—21no—
164C2—21no—
165C2—242no—
166C2#503611yescharacteristic
167C2#505311yescharacteristic
168C2#509511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#398C2
socleC22×C3#120
prime radical#5510C2
hypercenter#398D3
hypocenterC3#1610
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C3×#398C2semidirect
3. #4967C2semidirect
4. #4528C2semidirect
5. #4451C2semidirect
22. #398D3semidirect
162. C3#1610semidirect

Maximal subgroups

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

subgroupindex
#55102
C3×#3982
#49672
#45282
#44512
#16103