Finite Group Databasebeta

← #11920980

Subgroups — #11920980

Order 1152.

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

normal subgroups only

This group has 286 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
2C4⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73024—33nomaximal
5#70941—31nomaximal
6C2×C3⋊GL(2,𝔽3)—21no—
7C2×C3⋊#226—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9#63648—41nomaximal
10C4⋊GL(2,𝔽3)—33no—
11C4⋊#226—33no—
12C3×C8×Q8D311yescharacteristic, complemented
13#5031—31no—
14#5027—31no—
15#4694—31no—
16C8×SL(2,𝔽3)D313yescomplemented
17#4486—31no—
18#4485—31no—
19#4458—31no—
20C2×C4⋊(C3⋊D3)—41no—
21C2×C3×SL(2,𝔽3)D411yescharacteristic
22C3⋊GL(2,𝔽3)—41no—
23C3⋊#226—41no—
24C2×C32×C8—41no—
25C4⋊(C3⋊Dic3)—41no—
26#1682—91noSylow
27C2×GL(2,𝔽3)—63no—
28C2×#226—63no—
29C3×C4×Q8C2×D311yescharacteristic
30C2×#216—61no—
31C2×#215—61no—
32#874—31no—
33#869—31no—
34C4×SL(2,𝔽3)C2×D313yes—
35C3×C4⋊C8—31no—
36C3×C4×C8—31no—
37#821—61no—
38#821—61no—
39#821—123no—
40#810—61no—
41#809—61no—
42C4⋊(C3⋊C8)—31no—
43C4⋊(C3⋊C8)—31no—
44C22×C3⋊D3—81no—
45C2×C32×C4—41no—
46C2×C3⋊Dic3—81no—
47C4⋊(C3⋊D3)—42no—
48C4⋊(C3⋊D3)—81no—
49C3×SL(2,𝔽3)D811yescharacteristic
50C32×C8—81no—
51#486—91no—
52#462—91no—
53#460—91no—
54C8×Q8C3⋊D311yescharacteristic, complemented
55#335—91no—
56#332—91no—
57#331—91no—
58C2×C3×Q8D1211yescharacteristic
59C2×D12—61no—
60C2×D12—61no—
61C2×D12—61no—
62C2×D12—123no—
63C2×Dic6—61no—
64C2×SL(2,𝔽3)D1213yes—
65GL(2,𝔽3)—123no—
66#226—123no—
67C2×C3×C8—31no—
68C2×C3×C8—43no—
69C2×C3×C8S411yescharacteristic
70C3×C4⋊C4—31no—
71C3×C42—31no—
72#216—121no—
73#216—121no—
74#215—121no—
75#215—121no—
76C4⋊Dic3—61no—
77C4⋊Dic3—61no—
78C4⋊Dic3—123no—
79C2×C3⋊C8—61no—
80C2×C3⋊C8—61no—
81C22×C32—41no—
82C2×C3⋊D3—82no—
83C2×C3⋊D3—161no—
84C2×C3⋊D3—161no—
85C32×C4—41no—
86C32×C4—41no—
87C3⋊Dic3—161no—
88C2×Q16—181no—
89C2×SD16—181no—
90C4⋊Q8—91no—
91#127—91no—
92C4×Q8C2×C3⋊D311yescharacteristic
93C4⋊C8—31no—
94C4⋊C8—91no—
95C4⋊C8—91no—
96#103—181no—
97#102—181no—
98#102—181no—
99#102—181no—
100C4×C8—31no—
101C22×D3—121no—
102C22×D3—243no—
103C3×Q8—31no—
104C3×Q8D2411yescharacteristic
105C2×C3×C4—31no—
106C2×C3×C4—31no—
107C2×C3×C4—43no—
108C2×C3×C4C2×S411yescharacteristic
109C2×Dic3—121no—
110C2×Dic3—243no—
111D12—121no—
112D12—121no—
113D12—126no—
114D12—121no—
115D12—121no—
116D12—243no—
117D12—241no—
118Dic6—121no—
119Dic6—121no—
120SL(2,𝔽3)D2413yes—
121C3×C8—21no—
122C3×C8—61no—
123C3×C8—83no—
124C3⋊C8—121no—
125C3⋊C8—121no—
126C2×C32—41no—
127C2×C32—41no—
128C2×C32—41no—
129C3⋊D3—162no—
130C2×Q8—181no—
131C2×Q8C4⋊(C3⋊D3)11yescharacteristic
132C2×D4—181no—
133C2×D4—181no—
134C2×D4—181no—
135Q16—361no—
136Q16—361no—
137SD16—361no—
138SD16—361no—
139C2×C8—31no—
140C2×C8—181no—
141C2×C8—181no—
142C2×C8C3⋊S411yescharacteristic
143C4⋊C4—31no—
144C4⋊C4—181no—
145C4⋊C4—181no—
146C42—31no—
147C22×C3—43no—
148C22×C3C4⋊S411yescharacteristic
149C2×D3—246no—
150C2×D3—241no—
151C2×D3—241no—
152C2×D3—241no—
153C2×D3—483no—
154C2×D3—483no—
155C3×C4—31no—
156C3×C4—31no—
157C3×C4—43no—
158C3×C4—43no—
159C3×C4—61no—
160C3×C4#98311yescharacteristic
161C3×C4#98511yescharacteristic
162Dic3—241no—
163Dic3—483no—
164C32—41noSylow
165C23—361no—
166Q8—31no—
167Q8—361no—
168Q8—361no—
169Q8#374511yescharacteristic
170D4—361no—
171D4—361no—
172D4—361no—
173D4—361no—
174D4—721no—
175C2×C4—31no—
176C2×C4—31no—
177C2×C4—361no—
178C2×C4C2×C3⋊S411yescharacteristic
179C8—21no—
180C8—61no—
181C8—361no—
182C8—361no—
183C2×C3—43no—
184C2×C3—43no—
185C2×C3—43no—
186C2×C3#539511yescharacteristic
187C2×C3#540211yescharacteristic
188C2×C3#540611yescharacteristic
189D3—486no—
190D3—481no—
191C22—721no—
192C22—721no—
193C22—721no—
194C22C4⋊(C3⋊S4)11yescharacteristic
195C4—31no—
196C4—31no—
197C4—61no—
198C4—721no—
199C4#6426411yescharacteristic
200C4#6426611yescharacteristic
201C3—43no—
202C3#7302411yescharacteristic, complemented
203C2—1441no—
204C2#1059362911yescharacteristic
205C2#1059363611yescharacteristic
206C2#1059364011yescharacteristic
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)D8
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. C4⋊(C3⋊#226)C2semidirect
3. C3×C8×SL(2,𝔽3)C2semidirect
12. C3×C8×Q8D3semidirect
16. C8×SL(2,𝔽3)D3semidirect
54. C8×Q8C3⋊D3semidirect
202. C3#73024semidirect

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
C4⋊(C3⋊#226)2
C3×C8×SL(2,𝔽3)2
#730243
#709413
#636484