Finite Group Databasebeta

← #11920557

Subgroups — #11920557

Order 1152.

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

normal subgroups only

This group has 297 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
1C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
2C3×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73022—31nomaximal
5#71021—31nomaximal
6C2×C3⋊GL(2,𝔽3)—21no—
7C2×C3×GL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×#3714—41nomaximal
10C4⋊GL(2,𝔽3)—31no—
11C4⋊GL(2,𝔽3)—61no—
12C4⋊GL(2,𝔽3)D311yescharacteristic, complemented
13C3×#460—31no—
14#5073—31no—
15#5027—31no—
16Q8×C3⋊C8D311yescharacteristic, complemented
17C8×SL(2,𝔽3)—31no—
18#4544—31no—
19#4485—31no—
20#4479—31no—
21C2×C4⋊(C3⋊D3)—41no—
22C2×C3×D12—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3⋊GL(2,𝔽3)—41no—
25C3×GL(2,𝔽3)—41no—
26C2×C3×C3⋊C8—41no—
27#3714—44no—
28#1668—91noSylow
29C2×GL(2,𝔽3)—21no—
30C2×GL(2,𝔽3)—62no—
31C2×GL(2,𝔽3)—61no—
32C2×C3×SD16—61no—
33C3×#127—31no—
34C3×C4×Q8C2×D311yescharacteristic
35C2×#215—61no—
36C2×#213—61no—
37C2×#213—61no—
38C2×D24—121no—
39#874—31no—
40C4×SL(2,𝔽3)—21no—
41C4×SL(2,𝔽3)C2×D311yescharacteristic
42C3×C4⋊C8—31no—
43C3×#102—61no—
44#809—61no—
45C4⋊(C3⋊C8)—31no—
46C4⋊(C3⋊C8)—31no—
47C4×C3⋊C8—31no—
48C22×C3⋊D3—81no—
49C22×C3×D3—81no—
50C2×C32×C4—41no—
51C4⋊(C3⋊D3)—42no—
52C4⋊(C3⋊D3)—81no—
53C3×D12—42no—
54C3×D12—81no—
55C3×SL(2,𝔽3)D811yescharacteristic, complemented
56C3×C3⋊C8—42no—
57#493—91no—
58#460—31no—
59#460—91no—
60C8×Q8—31no—
61#335—91no—
62#331—91no—
63#331—91no—
64C2×C3×Q8D1211yescharacteristic
65C2×C3×D4—61no—
66C2×C3×D4—61no—
67C2×C3×D4—61no—
68C2×D12—41no—
69C2×D12—61no—
70C2×D12—61no—
71C2×D12—61no—
72C2×D12—121no—
73C2×D12—241no—
74C2×SL(2,𝔽3)—21no—
75C2×SL(2,𝔽3)#6711yescharacteristic
76GL(2,𝔽3)—41no—
77GL(2,𝔽3)—122no—
78GL(2,𝔽3)—121no—
79C3×SD16—121no—
80C3×SD16—121no—
81C2×C3×C8—61no—
82C2×C3×C8—121no—
83C3×C4⋊C4—31no—
84C3×C42—31no—
85#215—121no—
86#215—121no—
87#213—122no—
88#213—122no—
89C2×C3⋊C8—31no—
90C2×C3⋊C8—61no—
91C2×C3⋊C8S411yescharacteristic
92D24—124no—
93C22×C32—41no—
94C2×C3⋊D3—82no—
95C2×C3⋊D3—161no—
96C2×C3⋊D3—161no—
97C2×C3×D3—82no—
98C2×C3×D3—161no—
99C2×C3×D3—161no—
100C32×C4—41no—
101C32×C4—41no—
102C2×SD16—61no—
103C2×SD16—181no—
104C2×D8—181no—
105C2×D8—181no—
106#127—31no—
107#127—91no—
108C4×Q8D3211yescharacteristic
109C4⋊C8—31no—
110C4⋊C8—91no—
111C4⋊C8—91no—
112#102—61no—
113#102—181no—
114C4×C8—91no—
115C23×C3—121no—
116C22×D3—81no—
117C22×D3—121no—
118C22×D3—241no—
119C22×D3—242no—
120C3×Q8—31no—
121C3×Q8D2411yescharacteristic, complemented
122C3×D4—121no—
123C3×D4—121no—
124C3×D4—121no—
125C3×D4—121no—
126C3×D4—241no—
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—41no—
130C2×C3×C4—81no—
131C2×C3×C4C2×S411yescharacteristic
132D12—42no—
133D12—81no—
134D12—121no—
135D12—121no—
136D12—122no—
137D12—121no—
138D12—121no—
139D12—241no—
140D12—242no—
141D12—242no—
142D12—241no—
143SL(2,𝔽3)—21no—
144SL(2,𝔽3)#21311yescharacteristic, complemented
145C3×C8—122no—
146C3×C8—121no—
147C3⋊C8—61no—
148C3⋊C8—121no—
149C3⋊C8GL(2,𝔽3)12yescomplemented
150C2×C32—41no—
151C2×C32—41no—
152C2×C32—41no—
153C3⋊D3—162no—
154C3×D3—162no—
155C2×Q8#62611yescharacteristic
156C2×D4—61no—
157C2×D4—61no—
158C2×D4—61no—
159C2×D4—181no—
160C2×D4—181no—
161C2×D4—181no—
162SD16—121no—
163SD16—121no—
164SD16—361no—
165SD16—361no—
166D8—362no—
167D8—362no—
168C2×C8—31no—
169C2×C8—61no—
170C2×C8—91no—
171C2×C8—181no—
172C4⋊C4—31no—
173C42—31no—
174C22×C3—41no—
175C22×C3—81no—
176C22×C3—241no—
177C22×C3—241no—
178C22×C3—241no—
179C22×C3C4⋊S411yescharacteristic
180C2×D3—82no—
181C2×D3—161no—
182C2×D3—161no—
183C2×D3—242no—
184C2×D3—244no—
185C2×D3—241no—
186C2×D3—241no—
187C2×D3—241no—
188C2×D3—481no—
189C2×D3—481no—
190C2×D3—482no—
191C2×D3—482no—
192C3×C4—31no—
193C3×C4—31no—
194C3×C4—41no—
195C3×C4—41no—
196C3×C4—61no—
197C3×C4—81no—
198C3×C4—81no—
199C3×C4C2×GL(2,𝔽3)11yescharacteristic
200C3×C4#98611yescharacteristic
201C32—41noSylow
202C23—121no—
203C23—361no—
204Q8—31no—
205Q8#371411yescharacteristic, complemented
206D4—121no—
207D4—121no—
208D4—121no—
209D4—121no—
210D4—241no—
211D4—361no—
212D4—361no—
213D4—361no—
214D4—361no—
215D4—721no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4D3×S411yescharacteristic
219C8—32no—
220C8—121no—
221C8—181no—
222C8—361no—
223C2×C3—41no—
224C2×C3—41no—
225C2×C3—41no—
226C2×C3—81no—
227C2×C3—81no—
228C2×C3—81no—
229C2×C3—481no—
230C2×C3C4⋊GL(2,𝔽3)11yescharacteristic
231C2×C3#540211yescharacteristic
232C2×C3#540711yescharacteristic
233D3—162no—
234D3—482no—
235D3—484no—
236D3—481no—
237C22—241no—
238C22—241no—
239C22—241no—
240C22—721no—
241C22—721no—
242C22—721no—
243C22#6420711yescharacteristic
244C4—31no—
245C4—31no—
246C4—61no—
247C4#6419911yescharacteristic
248C4D3×GL(2,𝔽3)11yescharacteristic
249C3—41no—
250C3—81no—
251C3#7302211yescharacteristic, complemented
252C2—481no—
253C2—1441no—
254C2#1059317911yescharacteristic
255C2#1059321511yescharacteristic
256C2#1059322511yescharacteristic
subgroupG/H
centerC22#64207
commutator, cosocleC3×C4×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D3×S4
FittingC3×C4×Q8C2×D3
socleC22×C3C4⋊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

Complements

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

NH ≅ G/Ntype
1. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
2. C3×C4⋊GL(2,𝔽3)C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
12. C4⋊GL(2,𝔽3)D3semidirect
16. Q8×C3⋊C8D3semidirect
55. C3×SL(2,𝔽3)D8semidirect
121. C3×Q8D24semidirect
144. SL(2,𝔽3)#213semidirect
149. C3⋊C8GL(2,𝔽3)semidirect
205. Q8#3714semidirect
251. C3#73022semidirect

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
C3×C4⋊GL(2,𝔽3)2
C3⋊C8×SL(2,𝔽3)2
#730223
#710213
C2×#37144