Finite Group Databasebeta

← #11920972

Subgroups — #11920972

Order 1152.

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

normal subgroups only

This group has 240 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
1#10593625C211yesmaximal, characteristic
2#10593620C211yesmaximal, characteristic
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic
4#73016—33nomaximal
5#70947—31nomaximal
6C3×C4×SL(2,𝔽3)C2211yescharacteristic
7C3⋊#859—21no—
8C3⋊#859—21no—
9#63643—41nomaximal
10#5391—33no—
11#5386—33no—
12C3×C8×Q8D311yescharacteristic
13#5021—31no—
14#5020—31no—
15#4681—31no—
16C8×SL(2,𝔽3)D313yes—
17#4482—31no—
18#4481—31no—
19#4458—31no—
20C2×#634—41no—
21C2×C3×SL(2,𝔽3)D411yescharacteristic
22C2×C32×C8—41no—
23C4⋊(C3⋊Dic3)—41no—
24#1684—91noSylow
25C3×C4×Q8C2×D311yescharacteristic
26#869—31no—
27#869—31no—
28C4×SL(2,𝔽3)C2×D313yes—
29#859—63no—
30#859—63no—
31C3×C4⋊C8—31no—
32C3×C4×C8—31no—
33#835—61no—
34#835—61no—
35#816—61no—
36#816—61no—
37#816—123no—
38#808—61no—
39#807—61no—
40C4⋊(C3⋊C8)—31no—
41C4⋊(C3⋊C8)—31no—
42C2×C32×C4—41no—
43C2×C3⋊Dic3—81no—
44C2×C3⋊Dic3—81no—
45#634—42no—
46#634—81no—
47C3×SL(2,𝔽3)Q1611yescharacteristic
48C32×C8—81no—
49#487—91no—
50#477—91no—
51#475—91no—
52C8×Q8C3⋊D311yescharacteristic
53#335—91no—
54#333—91no—
55#332—91no—
56C2×C3×Q8D1211yescharacteristic
57C2×Dic6—61no—
58C2×Dic6—61no—
59C2×Dic6—123no—
60C2×SL(2,𝔽3)D1213yes—
61C2×C3×C8—31no—
62C2×C3×C8—43no—
63C2×C3×C8S411yescharacteristic
64C3×C4⋊C4—31no—
65C3×C42—31no—
66C4⋊Dic3—61no—
67C4⋊Dic3—61no—
68C4⋊Dic3—61no—
69C4⋊Dic3—61no—
70C4⋊Dic3—123no—
71C2×C3⋊C8—61no—
72C2×C3⋊C8—61no—
73C22×C32—41no—
74C32×C4—41no—
75C32×C4—41no—
76C3⋊Dic3—82no—
77C3⋊Dic3—161no—
78C4⋊Q8—91no—
79C4⋊Q8—91no—
80C4×Q8C2×C3⋊D311yescharacteristic
81#107—181no—
82#106—181no—
83C4⋊C8—31no—
84C4⋊C8—91no—
85C4⋊C8—91no—
86#103—181no—
87#103—181no—
88#103—181no—
89#103—181no—
90C4×C8—31no—
91C3×Q8—31no—
92C3×Q8Dic1211yescharacteristic
93C2×C3×C4—31no—
94C2×C3×C4—31no—
95C2×C3×C4—43no—
96C2×C3×C4C2×S411yescharacteristic
97C2×Dic3—121no—
98C2×Dic3—121no—
99C2×Dic3—243no—
100C2×Dic3—243no—
101Dic6—126no—
102Dic6—121no—
103Dic6—121no—
104Dic6—243no—
105Dic6—241no—
106SL(2,𝔽3)Dic1213yes—
107C3×C8—21no—
108C3×C8—61no—
109C3×C8—83no—
110C3⋊C8—121no—
111C3⋊C8—121no—
112C2×C32—41no—
113C2×C32—41no—
114C2×C32—41no—
115C2×Q8—181no—
116C2×Q8—181no—
117C2×Q8C4⋊(C3⋊D3)11yescharacteristic
118C2×C8—31no—
119C2×C8—181no—
120C2×C8—181no—
121C2×C8C3⋊S411yescharacteristic
122C4⋊C4—31no—
123C4⋊C4—181no—
124C4⋊C4—181no—
125C4⋊C4—181no—
126C4⋊C4—181no—
127C42—31no—
128C22×C3—43no—
129C22×C3C4⋊S411yescharacteristic
130C3×C4—31no—
131C3×C4—31no—
132C3×C4—43no—
133C3×C4—43no—
134C3×C4—61no—
135C3×C4#98311yescharacteristic
136C3×C4#98511yescharacteristic
137Dic3—246no—
138Dic3—241no—
139Dic3—241no—
140Dic3—483no—
141C32—41noSylow
142Q8—31no—
143Q8—361no—
144Q8—361no—
145Q8—721no—
146Q8#374611yescharacteristic
147C2×C4—31no—
148C2×C4—31no—
149C2×C4—361no—
150C2×C4—361no—
151C2×C4C2×C3⋊S411yescharacteristic
152C8—21no—
153C8—61no—
154C8—361no—
155C8—361no—
156C2×C3—43no—
157C2×C3—43no—
158C2×C3—43no—
159C2×C3#539511yescharacteristic
160C2×C3#539811yescharacteristic
161C2×C3#540611yescharacteristic
162C22C4⋊(C3⋊S4)11yescharacteristic
163C4—31no—
164C4—31no—
165C4—61no—
166C4—721no—
167C4—721no—
168C4#6426411yescharacteristic
169C4#6426611yescharacteristic
170C3—43no—
171C3#7301611yescharacteristic, complemented
172C2#1059362911yescharacteristic
173C2#1059363211yescharacteristic
174C2#1059364011yescharacteristic
subgroupG/H
centerC22C4⋊(C3⋊S4)
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×C8×Q8D3
socleC22×C3C4⋊S4
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC2×C8C3⋊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 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
171. C3#73016semidirect

Maximal subgroups

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

subgroupindex
#105936252
#105936202
C3×C8×SL(2,𝔽3)2
#730163
#709473
#636434