Finite Group Databasebeta

← #11920555

Subgroups — #11920555

Order 1152.

261 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 37 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#10593620C211yesmaximal, characteristic, complemented
2C3×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73021—31nomaximal
5#71020—31nomaximal
6C2×C3×GL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×#3716—41nomaximal
9C3⋊#859—21no—
10C4⋊GL(2,𝔽3)D311yescharacteristic
11#5386—31no—
12#5386—61no—
13C3×#460—31no—
14#5076—31no—
15Q8×C3⋊C8D311yescharacteristic, complemented
16#5020—31no—
17C8×SL(2,𝔽3)—31no—
18#4544—31no—
19#4486—31no—
20#4480—31no—
21C2×#634—41no—
22C2×C3×D12—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3×GL(2,𝔽3)—41no—
25C2×C3×C3⋊C8—41no—
26#3716—44no—
27#1666—91noSylow
28C2×GL(2,𝔽3)—21no—
29C2×C3×SD16—61no—
30C3×#127—31no—
31C3×C4×Q8C2×D311yescharacteristic
32C2×#214—61no—
33C2×#214—61no—
34C2×#204—121no—
35#869—31no—
36C4×SL(2,𝔽3)—21no—
37C4×SL(2,𝔽3)C2×D311yescharacteristic
38#859—62no—
39#859—61no—
40C3×C4⋊C8—31no—
41C3×#102—61no—
42#835—61no—
43#808—61no—
44C4⋊(C3⋊C8)—31no—
45C4⋊(C3⋊C8)—31no—
46C4×C3⋊C8—31no—
47C22×C3×D3—81no—
48C2×C32×C4—41no—
49C2×C3⋊Dic3—81no—
50#634—42no—
51#634—81no—
52C3×D12—42no—
53C3×D12—81no—
54C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
55C3×C3⋊C8—42no—
56#492—91no—
57#475—91no—
58#460—31no—
59C8×Q8—31no—
60#334—91no—
61#332—91no—
62#331—91no—
63C2×C3×Q8D1211yescharacteristic
64C2×C3×D4—61no—
65C2×C3×D4—61no—
66C2×C3×D4—61no—
67C2×D12—41no—
68C2×Dic6—61no—
69C2×Dic6—121no—
70C2×Dic6—241no—
71C2×SL(2,𝔽3)—21no—
72C2×SL(2,𝔽3)#6711yescharacteristic
73GL(2,𝔽3)—41no—
74C3×SD16—121no—
75C3×SD16—121no—
76C2×C3×C8—61no—
77C2×C3×C8—121no—
78C3×C4⋊C4—31no—
79C3×C42—31no—
80#214—122no—
81#214—122no—
82C4⋊Dic3—61no—
83C4⋊Dic3—61no—
84C2×C3⋊C8—31no—
85C2×C3⋊C8—61no—
86C2×C3⋊C8S411yescharacteristic
87#204—124no—
88C22×C32—41no—
89C2×C3×D3—82no—
90C2×C3×D3—161no—
91C2×C3×D3—161no—
92C32×C4—41no—
93C32×C4—41no—
94C3⋊Dic3—82no—
95C2×SD16—61no—
96C2×SD16—181no—
97C2×SD16—181no—
98C4⋊Q8—91no—
99#127—31no—
100C4×Q8D3211yescharacteristic
101#106—181no—
102C4⋊C8—31no—
103C4⋊C8—91no—
104C4⋊C8—91no—
105#103—181no—
106#102—61no—
107C4×C8—91no—
108C23×C3—121no—
109C22×D3—81no—
110C3×Q8—31no—
111C3×Q8#20411yescharacteristic, complemented
112C3×D4—121no—
113C3×D4—121no—
114C3×D4—121no—
115C3×D4—121no—
116C3×D4—241no—
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—41no—
120C2×C3×C4—81no—
121C2×C3×C4C2×S411yescharacteristic
122C2×Dic3—121no—
123C2×Dic3—241no—
124C2×Dic3—242no—
125D12—42no—
126D12—81no—
127Dic6—122no—
128Dic6—121no—
129Dic6—121no—
130Dic6—241no—
131Dic6—242no—
132Dic6—242no—
133SL(2,𝔽3)—21no—
134SL(2,𝔽3)#21411yescharacteristic, complemented
135C3×C8—122no—
136C3×C8—121no—
137C3⋊C8—61no—
138C3⋊C8—121no—
139C3⋊C8GL(2,𝔽3)12yescomplemented
140C2×C32—41no—
141C2×C32—41no—
142C2×C32—41no—
143C3×D3—162no—
144C2×Q8—181no—
145C2×Q8#62611yescharacteristic
146C2×D4—61no—
147C2×D4—61no—
148C2×D4—61no—
149SD16—121no—
150SD16—121no—
151SD16—362no—
152SD16—362no—
153C2×C8—31no—
154C2×C8—61no—
155C2×C8—91no—
156C2×C8—181no—
157C4⋊C4—31no—
158C4⋊C4—181no—
159C4⋊C4—181no—
160C42—31no—
161C22×C3—41no—
162C22×C3—81no—
163C22×C3—241no—
164C22×C3—241no—
165C22×C3—241no—
166C22×C3C4⋊S411yescharacteristic
167C2×D3—82no—
168C2×D3—161no—
169C2×D3—161no—
170C3×C4—31no—
171C3×C4—31no—
172C3×C4—41no—
173C3×C4—41no—
174C3×C4—61no—
175C3×C4—81no—
176C3×C4—81no—
177C3×C4C2×GL(2,𝔽3)11yescharacteristic
178C3×C4#98611yescharacteristic
179Dic3—242no—
180Dic3—244no—
181Dic3—241no—
182C32—41noSylow
183C23—121no—
184Q8—31no—
185Q8—361no—
186Q8—361no—
187Q8#371611yescharacteristic, complemented
188D4—121no—
189D4—121no—
190D4—121no—
191D4—121no—
192D4—241no—
193C2×C4—31no—
194C2×C4—31no—
195C2×C4—361no—
196C2×C4D3×S411yescharacteristic
197C8—32no—
198C8—121no—
199C8—181no—
200C8—361no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—41no—
204C2×C3—81no—
205C2×C3—81no—
206C2×C3—81no—
207C2×C3—481no—
208C2×C3C4⋊GL(2,𝔽3)11yescharacteristic
209C2×C3#540111yescharacteristic
210C2×C3#540611yescharacteristic
211D3—162no—
212C22—241no—
213C22—241no—
214C22—241no—
215C22#6420711yescharacteristic
216C4—31no—
217C4—31no—
218C4—61no—
219C4—721no—
220C4#6419911yescharacteristic
221C4D3×GL(2,𝔽3)11yescharacteristic
222C3—41no—
223C3—81no—
224C3#7302111yescharacteristic, complemented
225C2—481no—
226C2#1059317711yescharacteristic
227C2#1059321511yescharacteristic
228C2#1059322311yescharacteristic
subgroupG/H
centerC22#64207
commutator, cosocleC3×C4×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D3×S4
FittingC3×C4×Q8C2×D3
socleC22×C3C4⋊S4
prime radicalC3×C4⋊GL(2,𝔽3)C2
hypocenterC3×SL(2,𝔽3)SD16
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #10593620C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
15. Q8×C3⋊C8D3semidirect
54. C3×SL(2,𝔽3)SD16semidirect
111. C3×Q8#204semidirect
134. SL(2,𝔽3)#214semidirect
139. C3⋊C8GL(2,𝔽3)semidirect
187. Q8#3716semidirect
224. C3#73021semidirect

Maximal subgroups

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

subgroupindex
#105936202
C3×C4⋊GL(2,𝔽3)2
C3⋊C8×SL(2,𝔽3)2
#730213
#710203
C2×#37164