Finite Group Databasebeta

← #11921044

Subgroups — #11921044

Order 1152.

376 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 49 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
1C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
4#73088—33nomaximal
5#71914—31nomaximal
6C22×C3×SL(2,𝔽3)—21no—
7C2×C3⋊GL(2,𝔽3)—21no—
8C2×#3753—41nomaximal
9C3×C4×SL(2,𝔽3)C2211yescharacteristic
10C3×D4×Q8D311yescharacteristic, complemented
11D4×SL(2,𝔽3)D313yescomplemented
12C4⋊GL(2,𝔽3)—33no—
13#5088—31no—
14#5027—31no—
15#5015—31no—
16SL(2,𝔽3)⋊C8—33no—
17#4547—31no—
18#4535—31no—
19D4⋊(C3⋊C8)—31no—
20C2×C32×D4—41no—
21C2×C4⋊(C3⋊D3)—41no—
22C2×C3×SL(2,𝔽3)—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3⋊GL(2,𝔽3)—41no—
25#3753—44no—
26C2×C3⋊(C3⋊C8)—41no—
27#1627—91noSylow
28C22×C3×Q8—21no—
29C22×SL(2,𝔽3)—23no—
30C2×GL(2,𝔽3)—63no—
31C3×C4⋊Q8—31no—
32C3×C22⋊Q8—61no—
33C3×C4×Q8C2×D311yescharacteristic
34C3×C4×D4—31no—
35C2×#215—61no—
36C2×#213—61no—
37C2×#213—123no—
38#874—31no—
39C4×SL(2,𝔽3)C2×D313yes—
40#809—61no—
41#809—61no—
42#809—62no—
43C4⋊(C3⋊C8)—31no—
44C4⋊(C3⋊C8)—31no—
45C4×C3⋊C8—31no—
46C23×C32—81no—
47C22×C3⋊D3—81no—
48C32×D4—42no—
49C32×D4—81no—
50C2×C32×C4—41no—
51C4⋊(C3⋊D3)—42no—
52C4⋊(C3⋊D3)—81no—
53C3×SL(2,𝔽3)D811yescharacteristic, complemented
54C3⋊(C3⋊C8)—42no—
55D4×Q8C3⋊D311yescharacteristic, complemented
56#486—91no—
57#460—91no—
58#459—91no—
59#332—91no—
60Q8⋊C8—91no—
61D4⋊C8—91no—
62C2×C3×Q8—41no—
63C2×C3×Q8—61no—
64C2×C3×Q8—121no—
65C2×C3×Q8#6711yescharacteristic
66C2×C3×D4—43no—
67C2×C3×D4S411yescharacteristic
68C22×C3×C4—61no—
69C2×D12—61no—
70C2×D12—61no—
71C2×D12—61no—
72C2×D12—123no—
73C2×SL(2,𝔽3)—43no—
74C2×SL(2,𝔽3)#6713yes—
75GL(2,𝔽3)—123no—
76C3×C4⋊C4—31no—
77C3×C4⋊C4—32no—
78C3×C4⋊C4—31no—
79C3×C22⋊C4—61no—
80C3×C42—31no—
81#215—121no—
82#215—121no—
83#213—1212no—
84#213—122no—
85C2×C3⋊C8—61no—
86C2×C3⋊C8—61no—
87C2×C3⋊C8—61no—
88C2×C3⋊C8—123no—
89C22×C32—41no—
90C22×C32—82no—
91C22×C32—161no—
92C22×C32—161no—
93C2×C3⋊D3—82no—
94C2×C3⋊D3—161no—
95C2×C3⋊D3—161no—
96C32×C4—41no—
97C32×C4—41no—
98C22×Q8—21no—
99C2×SD16—181no—
100C2×D8—181no—
101C4⋊Q8—31no—
102#127—91no—
103C22⋊Q8—61no—
104C4×Q8C2×C3⋊D311yescharacteristic
105C4×D4—31no—
106C4⋊C8—91no—
107C4⋊C8—91no—
108#102—181no—
109#102—181no—
110#102—182no—
111C4×C8—91no—
112C23×C3—21no—
113C23×C3—83no—
114C22×D3—121no—
115C22×D3—243no—
116C3×Q8—31no—
117C3×Q8—121no—
118C3×Q8—121no—
119C3×Q8#21311yescharacteristic, complemented
120C3×D4—21no—
121C3×D4—46no—
122C3×D4—83no—
123C3×D4GL(2,𝔽3)12yescomplemented
124C2×C3×C4—31no—
125C2×C3×C4—31no—
126C2×C3×C4—43no—
127C2×C3×C4—61no—
128C2×C3×C4—121no—
129C2×C3×C4—121no—
130C2×C3×C4C2×S411yescharacteristic
131D12—121no—
132D12—121no—
133D12—126no—
134D12—122no—
135D12—121no—
136D12—121no—
137D12—243no—
138SL(2,𝔽3)#21313yescomplemented
139C3⋊C8—121no—
140C3⋊C8—126no—
141C3⋊C8—121no—
142C3⋊C8—121no—
143C2×C32—41no—
144C2×C32—41no—
145C2×C32—41no—
146C2×C32—162no—
147C3⋊D3—162no—
148C2×Q8—41no—
149C2×Q8—61no—
150C2×Q8—121no—
151C2×Q8#63811yescharacteristic
152C2×D4—181no—
153C2×D4—181no—
154C2×D4—181no—
155C2×D4C3⋊S411yescharacteristic
156C22×C4—61no—
157SD16—361no—
158SD16—361no—
159D8—362no—
160C2×C8—181no—
161C2×C8—181no—
162C2×C8—181no—
163C4⋊C4—31no—
164C4⋊C4—32no—
165C4⋊C4—31no—
166C22⋊C4—61no—
167C42—31no—
168C22×C3—22no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—43no—
172C22×C3—86no—
173C22×C3—163no—
174C22×C3—163no—
175C22×C3GL(2,ℤ/4)11yescharacteristic
176C2×D3—246no—
177C2×D3—241no—
178C2×D3—241no—
179C2×D3—241no—
180C2×D3—483no—
181C2×D3—483no—
182C3×C4—31no—
183C3×C4—31no—
184C3×C4—43no—
185C3×C4—43no—
186C3×C4—61no—
187C3×C4—121no—
188C3×C4C2×GL(2,𝔽3)11yescharacteristic
189C3×C4#98311yescharacteristic
190C32—41noSylow
191C23—21no—
192C23—361no—
193Q8—31no—
194Q8—121no—
195Q8—121no—
196Q8#375311yescharacteristic, complemented
197D4—21no—
198D4—361no—
199D4—361no—
200D4—362no—
201D4—361no—
202D4—361no—
203D4C3⋊GL(2,𝔽3)12yescomplemented
204C2×C4—31no—
205C2×C4—31no—
206C2×C4—61no—
207C2×C4—121no—
208C2×C4—121no—
209C2×C4C2×C3⋊S411yescharacteristic
210C8—361no—
211C8—361no—
212C8—361no—
213C2×C3—42no—
214C2×C3—43no—
215C2×C3—43no—
216C2×C3—43no—
217C2×C3—166no—
218C2×C3#541511yescharacteristic
219C2×C3#542111yescharacteristic
220C2×C3#543111yescharacteristic
221D3—486no—
222D3—481no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—721no—
227C22—721no—
228C22—721no—
229C22#6426911yescharacteristic
230C4—31no—
231C4—31no—
232C4—61no—
233C4—121no—
234C4C2×C3⋊GL(2,𝔽3)11yescharacteristic
235C4#6426411yescharacteristic
236C3—43no—
237C3#7308811yescharacteristic, complemented
238C2—42no—
239C2—1441no—
240C2#1059364911yescharacteristic
241C2#1059365511yescharacteristic
242C2#1059366511yescharacteristic
subgroupG/H
centerC22#64269
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×D4×Q8D3
socleC22×C3GL(2,ℤ/4)
hypercenterC2×D4C3⋊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×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×SL(2,𝔽3)C2semidirect
2. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
10. C3×D4×Q8D3semidirect
11. D4×SL(2,𝔽3)D3semidirect
53. C3×SL(2,𝔽3)D8semidirect
55. D4×Q8C3⋊D3semidirect
119. C3×Q8#213semidirect
123. C3×D4GL(2,𝔽3)semidirect
138. SL(2,𝔽3)#213semidirect
196. Q8#3753semidirect
203. D4C3⋊GL(2,𝔽3)semidirect
237. C3#73088semidirect

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
C4⋊(C3⋊GL(2,𝔽3))2
C3⋊(SL(2,𝔽3)⋊C8)2
#730883
#719143
C2×#37534