Finite Group Databasebeta

← #69706

Subgroups — #69706

Order 384.

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

normal subgroups only

This group has 282 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×#395C211yesmaximal, characteristic, complemented
3#5227C212yesmaximal, complemented
4#4528C211yesmaximal, characteristic, complemented
5#4451C212yesmaximal, complemented
6#1604—31nomaximal, Sylow
7C22×C3×Q8C2211yescharacteristic
8C2×#235—21no—
9C2×C3×C4⋊C4—21no—
10C2×C3×C4⋊C4C2211yescharacteristic
11C2×#215—22no—
12#930—21no—
13#930C2212yes—
14#897—22no—
15#895—22no—
16C3×#95—21no—
17C3×#95C2211yescharacteristic
18#835—22no—
19#830C2212yes—
20#537—31no—
21#448—32no—
22#395D311yescharacteristic, complemented
23#327—32no—
24#324—31no—
25C2×C3×Q8—22no—
26C2×C3×Q8—42no—
27C22×C3×C4—21no—
28C22×C3×C4—21no—
29C22×C3×C4D411yescharacteristic
30C22×C3×C4D411yescharacteristic
31C22×C3×C4C2311yescharacteristic
32C2×#67—22no—
33#235—42no—
34#235—42no—
35C2×D12—21no—
36C2×D12—21no—
37C2×D12D412yes—
38C2×C4×D3—22no—
39C2×Dic6—21no—
40C3×C4⋊C4—22no—
41C3×C4⋊C4—42no—
42#215—44no—
43(C2×D3)⋊C4—22no—
44(C2×D3)⋊C4—22no—
45C4⋊Dic3—21no—
46C4⋊Dic3D412yes—
47C2×C3⋊C8—22no—
48C2×#41—61no—
49C22×Q8C2×D311yescharacteristic
50C2×SD16—62no—
51C22⋊Q8—62no—
52#121—32no—
53#121—62no—
54#121—61no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4C2×D311yescharacteristic
57#103—62no—
58C22⋊C8—32no—
59#95—21no—
60#95C2×D311yescharacteristic
61C23×C3C2×D411yescharacteristic
62C22×D3—22no—
63C3×Q8—24no—
64C3×Q8—42no—
65C2×C3×C4—21no—
66C2×C3×C4—21no—
67C2×C3×C4—22no—
68C2×C3×C4—22no—
69C2×C3×C4—21no—
70C2×C3×C4—41no—
71C2×C3×C4—41no—
72C2×C3×C4—41no—
73C2×C3×C4—41no—
74C2×C3×C4—41no—
75C2×C3×C4—41no—
76C2×C3×C4SD1612yescomplemented
77C2×C3×C4C2×D412yes—
78#67—42no—
79#67—42no—
80C2×Dic3—22no—
81D12—42no—
82D12—42no—
83D12—81no—
84C4×D3—44no—
85Dic6—42no—
86C3⋊C8—42no—
87#41—122no—
88#41—122no—
89C2×Q8—22no—
90C2×Q8—42no—
91C2×Q8—61no—
92C2×D4—32no—
93C2×D4—61no—
94C2×D4—61no—
95C2×D4—62no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—62no—
99C22×C4#6711yescharacteristic
100C22×C4#6711yescharacteristic
101C22×C4C22×D311yescharacteristic
102SD16—124no—
103C2×C8—62no—
104C4⋊C4—22no—
105C4⋊C4—32no—
106C4⋊C4—42no—
107C4⋊C4—61no—
108C22⋊C4—62no—
109C22⋊C4—62no—
110C22×C3—21no—
111C22×C3—21no—
112C22×C3#12011yescharacteristic
113C22×C3C2×SD1611yescharacteristic
114C22×C3AGL(1,ℤ/8)11yescharacteristic
115C2×D3—42no—
116C2×D3—42no—
117C2×D3—42no—
118C3×C4—22no—
119C3×C4—22no—
120C3×C4—42no—
121C3×C4—41no—
122C3×C4—42no—
123Dic3—42no—
124C23—62no—
125C23C2×#6711yescharacteristic
126Q8—24no—
127Q8—42no—
128Q8—122no—
129D4—122no—
130D4—122no—
131D4—122no—
132D4—122no—
133D4—241no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—62no—
146C2×C4—124no—
147C2×C4#21512yescomplemented
148C2×C4D3×D412yes—
149C8—122no—
150C2×C3—21no—
151C2×C3—21no—
152C2×C3#45011yescharacteristic
153C2×C3#45611yescharacteristic
154C2×C3B(4,𝔽2)11yescharacteristic
155D3—82no—
156C22—21no—
157C22—21no—
158C22—122no—
159C22—122no—
160C22—122no—
161C22#93711yescharacteristic
162C22C2×#21511yescharacteristic
163C22#94911yescharacteristic
164C4—22no—
165C4—22no—
166C4—42no—
167C4—42no—
168C4—41no—
169C4—122no—
170C3#160411yescharacteristic, complemented, Sylow
171C2—21no—
172C2—21no—
173C2—242no—
174C2#503211yescharacteristic
175C2#504611yescharacteristic
176C2#509511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#395C2
socleC22×C3#120
prime radical#5510C2
hypercenter#395D3
hypocenterC3#1604
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C3×#395C2semidirect
3. #5227C2semidirect
4. #4528C2semidirect
5. #4451C2semidirect
22. #395D3semidirect
76. C2×C3×C4SD16semidirect
147. C2×C4#215semidirect
170. C3#1604semidirect

Maximal subgroups

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

subgroupindex
#55102
C3×#3952
#52272
#45282
#44512
#16043