Finite Group Databasebeta

← #11921049

Subgroups — #11921049

Order 1152.

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

normal subgroups only

This group has 242 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
2#10593625C211yesmaximal, characteristic
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic
4#73093—33nomaximal
5#71936—31nomaximal
6C3×C4×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3⋊#859—21no—
9#63634—41nomaximal
10C3×#557D311yescharacteristic
11#557⋊C3D313yes—
12#5391—33no—
13#5093—31no—
14#5021—31no—
15#5021—31no—
16SL(2,𝔽3)⋊C8—33no—
17#4547—31no—
18#4535—31no—
19#4535—31no—
20C2×C3×SL(2,𝔽3)D411yescharacteristic
21C32×C4⋊C4—41no—
22C4⋊(C3⋊Dic3)—41no—
23C2×C3⋊(C3⋊C8)—41no—
24#1636—91noSylow
25C3×C4⋊Q8—31no—
26C3×#125—61no—
27C3×C4×Q8—21no—
28C3×C4×Q8—31no—
29C3×C4×Q8C2×D311yescharacteristic
30#869—31no—
31C4×SL(2,𝔽3)—23no—
32C4×SL(2,𝔽3)C2×D313yes—
33#859—63no—
34#835—61no—
35#835—61no—
36#810—62no—
37#807—61no—
38#807—61no—
39#807—123no—
40C4⋊(C3⋊C8)—31no—
41C4⋊(C3⋊C8)—31no—
42C4×C3⋊C8—31no—
43C2×C32×C4—41no—
44C2×C32×C4—81no—
45C2×C3⋊Dic3—81no—
46C3×SL(2,𝔽3)Q1611yescharacteristic
47C3⋊(C3⋊C8)—42no—
48#557C3⋊D311yescharacteristic
49#487—91no—
50#477—91no—
51#477—91no—
52#332—91no—
53Q8⋊C8—91no—
54Q8⋊C8—91no—
55C2×C3×Q8—31no—
56C2×C3×Q8#6711yescharacteristic
57C2×Dic6—61no—
58C2×SL(2,𝔽3)#6713yes—
59C3×C4⋊C4—31no—
60C3×C4⋊C4—32no—
61C3×C4⋊C4—43no—
62C3×C4⋊C4—61no—
63C3×C4⋊C4—61no—
64C3×C4⋊C4S411yescharacteristic
65C3×C42—31no—
66C3×C42—61no—
67C4⋊Dic3—61no—
68C4⋊Dic3—61no—
69C4⋊Dic3—123no—
70C2×C3⋊C8—61no—
71C2×C3⋊C8—61no—
72C2×C3⋊C8—61no—
73C2×C3⋊C8—123no—
74C22×C32—41no—
75C32×C4—41no—
76C32×C4—41no—
77C32×C4—161no—
78C3⋊Dic3—161no—
79C4⋊Q8—31no—
80C4⋊Q8—91no—
81#125—61no—
82C4×Q8—21no—
83C4×Q8—31no—
84C4×Q8C2×C3⋊D311yescharacteristic
85#107—181no—
86#107—181no—
87C4⋊C8—91no—
88C4⋊C8—91no—
89#103—182no—
90#103—181no—
91#103—181no—
92C4×C8—91no—
93C3×Q8—31no—
94C3×Q8—61no—
95C3×Q8—61no—
96C3×Q8#21611yescharacteristic
97C2×C3×C4—21no—
98C2×C3×C4—31no—
99C2×C3×C4—31no—
100C2×C3×C4—43no—
101C2×C3×C4—61no—
102C2×C3×C4—83no—
103C2×C3×C4C2×S411yescharacteristic
104C2×Dic3—121no—
105C2×Dic3—243no—
106Dic6—122no—
107SL(2,𝔽3)#21613yes—
108C3⋊C8—121no—
109C3⋊C8—126no—
110C3⋊C8—121no—
111C3⋊C8—121no—
112C2×C32—41no—
113C2×C32—41no—
114C2×C32—41no—
115C2×Q8—31no—
116C2×Q8—181no—
117C2×Q8#63811yescharacteristic
118C2×C8—181no—
119C2×C8—181no—
120C2×C8—181no—
121C4⋊C4—31no—
122C4⋊C4—32no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—181no—
126C4⋊C4—181no—
127C4⋊C4C3⋊S411yescharacteristic
128C42—31no—
129C42—61no—
130C22×C3—43no—
131C22×C3GL(2,ℤ/4)11yescharacteristic
132C3×C4—31no—
133C3×C4—31no—
134C3×C4—41no—
135C3×C4—43no—
136C3×C4—43no—
137C3×C4—61no—
138C3×C4—121no—
139C3×C4—163no—
140C3×C4#98411yescharacteristic
141C3×C4#98511yescharacteristic
142Dic3—241no—
143Dic3—483no—
144C32—41noSylow
145Q8—31no—
146Q8—61no—
147Q8—61no—
148Q8—362no—
149Q8#375611yescharacteristic
150C2×C4—21no—
151C2×C4—31no—
152C2×C4—31no—
153C2×C4—61no—
154C2×C4—361no—
155C2×C4C2×C3⋊S411yescharacteristic
156C8—361no—
157C8—361no—
158C8—361no—
159C2×C3—43no—
160C2×C3—43no—
161C2×C3—43no—
162C2×C3#541611yescharacteristic
163C2×C3#542511yescharacteristic
164C2×C3#543111yescharacteristic
165C22#6426911yescharacteristic
166C4—31no—
167C4—31no—
168C4—41no—
169C4—61no—
170C4—121no—
171C4—721no—
172C4#6426511yescharacteristic
173C4#6426611yescharacteristic
174C3—43no—
175C3#7309311yescharacteristic, complemented
176C2#1059365011yescharacteristic
177C2#1059365911yescharacteristic
178C2#1059366511yescharacteristic
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)Q16
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
175. C3#73093semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#105936252
C3⋊(SL(2,𝔽3)⋊C8)2
#730933
#719363
#636344