Finite Group Databasebeta

← #11921050

Subgroups — #11921050

Order 1152.

252 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C3×#557⋊C3C211yesmaximal, characteristic
2C4⋊(C3⋊#226)C211yesmaximal, characteristic
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic
4#73094—33nomaximal
5#71938—31nomaximal
6C2×C3⋊#226—21no—
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9#63635—41nomaximal
10C3×#557D311yescharacteristic
11#557⋊C3D313yes—
12C4⋊#226—33no—
13#5080—31no—
14#5031—31no—
15#5020—31no—
16SL(2,𝔽3)⋊C8—33no—
17#4547—31no—
18#4535—31no—
19#4535—31no—
20C2×C3×SL(2,𝔽3)D411yescharacteristic
21C3⋊#226—41no—
22C32×C4⋊C4—41no—
23C4⋊(C3⋊Dic3)—41no—
24C2×C3⋊(C3⋊C8)—41no—
25#1653—91noSylow
26C2×#226—63no—
27C3×C4⋊Q8—31no—
28C3×#125—61no—
29C3×C4×Q8—21no—
30C3×C4×Q8—31no—
31C3×C4×Q8C2×D311yescharacteristic
32C2×#216—61no—
33#869—31no—
34C4×SL(2,𝔽3)—23no—
35C4×SL(2,𝔽3)C2×D313yes—
36#835—61no—
37#810—61no—
38#808—62no—
39#808—61no—
40#808—123no—
41C4⋊(C3⋊C8)—31no—
42C4⋊(C3⋊C8)—31no—
43C4×C3⋊C8—31no—
44C2×C32×C4—41no—
45C2×C32×C4—81no—
46C2×C3⋊Dic3—81no—
47C3×SL(2,𝔽3)SD1611yescharacteristic
48C3⋊(C3⋊C8)—42no—
49#557C3⋊D311yescharacteristic
50#498—91no—
51#475—91no—
52#462—91no—
53#332—91no—
54Q8⋊C8—91no—
55Q8⋊C8—91no—
56C2×C3×Q8—31no—
57C2×C3×Q8#6711yescharacteristic
58C2×Dic6—61no—
59C2×SL(2,𝔽3)#6713yes—
60#226—123no—
61C3×C4⋊C4—31no—
62C3×C4⋊C4—32no—
63C3×C4⋊C4—43no—
64C3×C4⋊C4—61no—
65C3×C4⋊C4—61no—
66C3×C4⋊C4S411yescharacteristic
67C3×C42—31no—
68C3×C42—61no—
69#216—121no—
70#216—121no—
71C4⋊Dic3—61no—
72C4⋊Dic3—61no—
73C4⋊Dic3—123no—
74C2×C3⋊C8—61no—
75C2×C3⋊C8—61no—
76C2×C3⋊C8—61no—
77C2×C3⋊C8—123no—
78C22×C32—41no—
79C32×C4—41no—
80C32×C4—41no—
81C32×C4—161no—
82C3⋊Dic3—161no—
83C2×Q16—181no—
84C4⋊Q8—31no—
85C4⋊Q8—91no—
86#125—61no—
87C4×Q8—21no—
88C4×Q8—31no—
89C4×Q8C2×C3⋊D311yescharacteristic
90#106—182no—
91#106—181no—
92C4⋊C8—91no—
93C4⋊C8—91no—
94#103—181no—
95#103—181no—
96C4×C8—91no—
97C3×Q8—31no—
98C3×Q8—61no—
99C3×Q8—61no—
100C3×Q8#21511yescharacteristic
101C2×C3×C4—21no—
102C2×C3×C4—31no—
103C2×C3×C4—31no—
104C2×C3×C4—43no—
105C2×C3×C4—61no—
106C2×C3×C4—83no—
107C2×C3×C4C2×S411yescharacteristic
108C2×Dic3—121no—
109C2×Dic3—243no—
110Dic6—121no—
111Dic6—121no—
112SL(2,𝔽3)#21513yes—
113C3⋊C8—62no—
114C3⋊C8—121no—
115C3⋊C8—126no—
116C3⋊C8—121no—
117C2×C32—41no—
118C2×C32—41no—
119C2×C32—41no—
120C2×Q8—31no—
121C2×Q8—181no—
122C2×Q8#63811yescharacteristic
123Q16—361no—
124Q16—361no—
125C2×C8—181no—
126C2×C8—181no—
127C2×C8—181no—
128C4⋊C4—31no—
129C4⋊C4—32no—
130C4⋊C4—61no—
131C4⋊C4—61no—
132C4⋊C4—181no—
133C4⋊C4—181no—
134C4⋊C4C3⋊S411yescharacteristic
135C42—31no—
136C42—61no—
137C22×C3—43no—
138C22×C3GL(2,ℤ/4)11yescharacteristic
139C3×C4—31no—
140C3×C4—31no—
141C3×C4—41no—
142C3×C4—43no—
143C3×C4—43no—
144C3×C4—61no—
145C3×C4—121no—
146C3×C4—163no—
147C3×C4#98411yescharacteristic
148C3×C4#98511yescharacteristic
149Dic3—241no—
150Dic3—483no—
151C32—41noSylow
152Q8—31no—
153Q8—61no—
154Q8—61no—
155Q8—361no—
156Q8—361no—
157Q8#375511yescharacteristic
158C2×C4—21no—
159C2×C4—31no—
160C2×C4—31no—
161C2×C4—61no—
162C2×C4—361no—
163C2×C4C2×C3⋊S411yescharacteristic
164C8—182no—
165C8—361no—
166C8—361no—
167C2×C3—43no—
168C2×C3—43no—
169C2×C3—43no—
170C2×C3#541711yescharacteristic
171C2×C3#542511yescharacteristic
172C2×C3#543011yescharacteristic
173C22#6426911yescharacteristic
174C4—31no—
175C4—31no—
176C4—41no—
177C4—61no—
178C4—121no—
179C4—721no—
180C4#6426511yescharacteristic
181C4#6426611yescharacteristic
182C3—43no—
183C3#7309411yescharacteristic, complemented
184C2#1059365111yescharacteristic
185C2#1059365911yescharacteristic
186C2#1059366411yescharacteristic
subgroupG/H
centerC22#64269
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×#557D3
socleC22×C3GL(2,ℤ/4)
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)SD16
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
183. C3#73094semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C4⋊(C3⋊#226)2
C3⋊(SL(2,𝔽3)⋊C8)2
#730943
#719383
#636354