Finite Group Databasebeta

← #11920971

Subgroups — #11920971

Order 1152.

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

normal subgroups only

This group has 250 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
1C4⋊(C3⋊#226)C211yesmaximal, characteristic
2#10593620C211yesmaximal, characteristic
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic
4#73011—33nomaximal
5#70946—31nomaximal
6C2×C3⋊#226—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3⋊#859—21no—
9#63643—41nomaximal
10C4⋊#226—33no—
11#5386—33no—
12C3×C8×Q8D311yescharacteristic
13#5031—31no—
14#5020—31no—
15#4681—31no—
16C8×SL(2,𝔽3)D313yes—
17#4486—31no—
18#4482—31no—
19#4457—31no—
20C2×#634—41no—
21C2×C3×SL(2,𝔽3)D411yescharacteristic
22C3⋊#226—41no—
23C2×C32×C8—41no—
24C4⋊(C3⋊Dic3)—41no—
25#1680—91noSylow
26C2×#226—63no—
27C3×C4×Q8C2×D311yescharacteristic
28C2×#216—61no—
29#869—31no—
30#869—31no—
31C4×SL(2,𝔽3)C2×D313yes—
32#859—63no—
33C3×C4⋊C8—31no—
34C3×C4×C8—31no—
35#835—61no—
36#816—61no—
37#816—61no—
38#816—123no—
39#810—61no—
40#808—61no—
41C4⋊(C3⋊C8)—31no—
42C4⋊(C3⋊C8)—31no—
43C2×C32×C4—41no—
44C2×C3⋊Dic3—81no—
45C2×C3⋊Dic3—81no—
46#634—42no—
47#634—81no—
48C3×SL(2,𝔽3)SD1611yescharacteristic
49C32×C8—81no—
50#487—91no—
51#475—91no—
52#462—91no—
53C8×Q8C3⋊D311yescharacteristic
54#334—91no—
55#332—91no—
56#332—91no—
57C2×C3×Q8D1211yescharacteristic
58C2×Dic6—61no—
59C2×Dic6—61no—
60C2×Dic6—123no—
61C2×SL(2,𝔽3)D1213yes—
62#226—123no—
63C2×C3×C8—31no—
64C2×C3×C8—43no—
65C2×C3×C8S411yescharacteristic
66C3×C4⋊C4—31no—
67C3×C42—31no—
68#216—121no—
69#216—121no—
70C4⋊Dic3—61no—
71C4⋊Dic3—61no—
72C4⋊Dic3—61no—
73C4⋊Dic3—61no—
74C4⋊Dic3—123no—
75C2×C3⋊C8—61no—
76C2×C3⋊C8—61no—
77C22×C32—41no—
78C32×C4—41no—
79C32×C4—41no—
80C3⋊Dic3—82no—
81C3⋊Dic3—161no—
82C2×Q16—181no—
83C4⋊Q8—91no—
84C4⋊Q8—91no—
85C4×Q8C2×C3⋊D311yescharacteristic
86#106—181no—
87C4⋊C8—31no—
88C4⋊C8—91no—
89C4⋊C8—91no—
90#103—181no—
91#103—181no—
92#103—181no—
93#103—181no—
94C4×C8—31no—
95C3×Q8—31no—
96C3×Q8#20411yescharacteristic
97C2×C3×C4—31no—
98C2×C3×C4—31no—
99C2×C3×C4—43no—
100C2×C3×C4C2×S411yescharacteristic
101C2×Dic3—121no—
102C2×Dic3—121no—
103C2×Dic3—243no—
104C2×Dic3—243no—
105Dic6—121no—
106Dic6—121no—
107Dic6—126no—
108Dic6—121no—
109Dic6—121no—
110Dic6—243no—
111SL(2,𝔽3)#20413yes—
112C3×C8—21no—
113C3×C8—61no—
114C3×C8—83no—
115C3⋊C8—121no—
116C3⋊C8—121no—
117C2×C32—41no—
118C2×C32—41no—
119C2×C32—41no—
120C2×Q8—181no—
121C2×Q8—181no—
122C2×Q8C4⋊(C3⋊D3)11yescharacteristic
123Q16—361no—
124Q16—361no—
125C2×C8—31no—
126C2×C8—181no—
127C2×C8—181no—
128C2×C8C3⋊S411yescharacteristic
129C4⋊C4—31no—
130C4⋊C4—181no—
131C4⋊C4—181no—
132C4⋊C4—181no—
133C4⋊C4—181no—
134C42—31no—
135C22×C3—43no—
136C22×C3C4⋊S411yescharacteristic
137C3×C4—31no—
138C3×C4—31no—
139C3×C4—43no—
140C3×C4—43no—
141C3×C4—61no—
142C3×C4#98311yescharacteristic
143C3×C4#98511yescharacteristic
144Dic3—246no—
145Dic3—241no—
146Dic3—241no—
147Dic3—483no—
148C32—41noSylow
149Q8—31no—
150Q8—361no—
151Q8—361no—
152Q8—361no—
153Q8—361no—
154Q8#374411yescharacteristic
155C2×C4—31no—
156C2×C4—31no—
157C2×C4—361no—
158C2×C4—361no—
159C2×C4C2×C3⋊S411yescharacteristic
160C8—21no—
161C8—61no—
162C8—361no—
163C8—361no—
164C2×C3—43no—
165C2×C3—43no—
166C2×C3—43no—
167C2×C3#539511yescharacteristic
168C2×C3#540111yescharacteristic
169C2×C3#540311yescharacteristic
170C22C4⋊(C3⋊S4)11yescharacteristic
171C4—31no—
172C4—31no—
173C4—61no—
174C4—721no—
175C4—721no—
176C4#6426411yescharacteristic
177C4#6426611yescharacteristic
178C3—43no—
179C3#7301111yescharacteristic, complemented
180C2#1059362911yescharacteristic
181C2#1059363511yescharacteristic
182C2#1059363711yescharacteristic
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)SD16
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
179. C3#73011semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊#226)2
#105936202
C3×C8×SL(2,𝔽3)2
#730113
#709463
#636434