Finite Group Databasebeta

← #11920979

Subgroups — #11920979

Order 1152.

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

normal subgroups only

This group has 276 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⋊GL(2,𝔽3))C211yesmaximal, characteristic
2#10593625C211yesmaximal, characteristic, complemented
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73023—33nomaximal
5#70940—31nomaximal
6C2×C3⋊GL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3⋊#859—21no—
9#63648—41nomaximal
10C4⋊GL(2,𝔽3)—33no—
11#5391—33no—
12C3×C8×Q8D311yescharacteristic, complemented
13#5027—31no—
14#5021—31no—
15#4694—31no—
16C8×SL(2,𝔽3)D313yescomplemented
17#4485—31no—
18#4481—31no—
19#4457—31no—
20C2×C4⋊(C3⋊D3)—41no—
21C2×C3×SL(2,𝔽3)D411yescharacteristic
22C3⋊GL(2,𝔽3)—41no—
23C2×C32×C8—41no—
24C4⋊(C3⋊Dic3)—41no—
25#1678—91noSylow
26C2×GL(2,𝔽3)—63no—
27C3×C4×Q8C2×D311yescharacteristic
28C2×#215—61no—
29#874—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#821—61no—
37#821—61no—
38#821—123no—
39#809—61no—
40#807—61no—
41C4⋊(C3⋊C8)—31no—
42C4⋊(C3⋊C8)—31no—
43C22×C3⋊D3—81no—
44C2×C32×C4—41no—
45C2×C3⋊Dic3—81no—
46C4⋊(C3⋊D3)—42no—
47C4⋊(C3⋊D3)—81no—
48C3×SL(2,𝔽3)SD1611yescharacteristic
49C32×C8—81no—
50#486—91no—
51#477—91no—
52#460—91no—
53C8×Q8C3⋊D311yescharacteristic, complemented
54#334—91no—
55#333—91no—
56#331—91no—
57C2×C3×Q8D1211yescharacteristic
58C2×D12—61no—
59C2×D12—61no—
60C2×D12—61no—
61C2×D12—123no—
62C2×Dic6—61no—
63C2×SL(2,𝔽3)D1213yes—
64GL(2,𝔽3)—123no—
65C2×C3×C8—31no—
66C2×C3×C8—43no—
67C2×C3×C8S411yescharacteristic
68C3×C4⋊C4—31no—
69C3×C42—31no—
70#215—121no—
71#215—121no—
72C4⋊Dic3—61no—
73C4⋊Dic3—61no—
74C4⋊Dic3—123no—
75C2×C3⋊C8—61no—
76C2×C3⋊C8—61no—
77C22×C32—41no—
78C2×C3⋊D3—82no—
79C2×C3⋊D3—161no—
80C2×C3⋊D3—161no—
81C32×C4—41no—
82C32×C4—41no—
83C3⋊Dic3—161no—
84C2×SD16—181no—
85C4⋊Q8—91no—
86#127—91no—
87C4×Q8C2×C3⋊D311yescharacteristic
88#107—181no—
89C4⋊C8—31no—
90C4⋊C8—91no—
91C4⋊C8—91no—
92#103—181no—
93#102—181no—
94#102—181no—
95#102—181no—
96C4×C8—31no—
97C22×D3—121no—
98C22×D3—243no—
99C3×Q8—31no—
100C3×Q8#20411yescharacteristic
101C2×C3×C4—31no—
102C2×C3×C4—31no—
103C2×C3×C4—43no—
104C2×C3×C4C2×S411yescharacteristic
105C2×Dic3—121no—
106C2×Dic3—243no—
107D12—121no—
108D12—121no—
109D12—126no—
110D12—121no—
111D12—121no—
112D12—243no—
113D12—241no—
114Dic6—241no—
115SL(2,𝔽3)#20413yes—
116C3×C8—21no—
117C3×C8—61no—
118C3×C8—83no—
119C3⋊C8—121no—
120C3⋊C8—121no—
121C2×C32—41no—
122C2×C32—41no—
123C2×C32—41no—
124C3⋊D3—162no—
125C2×Q8—181no—
126C2×Q8C4⋊(C3⋊D3)11yescharacteristic
127C2×D4—181no—
128C2×D4—181no—
129C2×D4—181no—
130SD16—361no—
131SD16—361no—
132C2×C8—31no—
133C2×C8—181no—
134C2×C8—181no—
135C2×C8C3⋊S411yescharacteristic
136C4⋊C4—31no—
137C4⋊C4—181no—
138C4⋊C4—181no—
139C42—31no—
140C22×C3—43no—
141C22×C3C4⋊S411yescharacteristic
142C2×D3—246no—
143C2×D3—241no—
144C2×D3—241no—
145C2×D3—241no—
146C2×D3—483no—
147C2×D3—483no—
148C3×C4—31no—
149C3×C4—31no—
150C3×C4—43no—
151C3×C4—43no—
152C3×C4—61no—
153C3×C4#98311yescharacteristic
154C3×C4#98511yescharacteristic
155Dic3—241no—
156Dic3—483no—
157C32—41noSylow
158C23—361no—
159Q8—31no—
160Q8—721no—
161Q8#374411yescharacteristic
162D4—361no—
163D4—361no—
164D4—361no—
165D4—361no—
166D4—721no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—361no—
170C2×C4C2×C3⋊S411yescharacteristic
171C8—21no—
172C8—61no—
173C8—361no—
174C8—361no—
175C2×C3—43no—
176C2×C3—43no—
177C2×C3—43no—
178C2×C3#539511yescharacteristic
179C2×C3#540111yescharacteristic
180C2×C3#540711yescharacteristic
181D3—486no—
182D3—481no—
183C22—721no—
184C22—721no—
185C22—721no—
186C22C4⋊(C3⋊S4)11yescharacteristic
187C4—31no—
188C4—31no—
189C4—61no—
190C4—721no—
191C4#6426411yescharacteristic
192C4#6426611yescharacteristic
193C3—43no—
194C3#7302311yescharacteristic, complemented
195C2—1441no—
196C2#1059362911yescharacteristic
197C2#1059363511yescharacteristic
198C2#1059364111yescharacteristic
subgroupG/H
centerC22C4⋊(C3⋊S4)
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×C8×Q8D3
socleC22×C3C4⋊S4
prime radicalC4⋊(C3⋊GL(2,𝔽3))C2
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
2. #10593625C2semidirect
3. C3×C8×SL(2,𝔽3)C2semidirect
12. C3×C8×Q8D3semidirect
16. C8×SL(2,𝔽3)D3semidirect
53. C8×Q8C3⋊D3semidirect
194. C3#73023semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊GL(2,𝔽3))2
#105936252
C3×C8×SL(2,𝔽3)2
#730233
#709403
#636484