Finite Group Databasebeta

← #11920564

Subgroups — #11920564

Order 1152.

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

normal subgroups only

This group has 277 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
1SL(2,𝔽3)×D12C211yesmaximal, characteristic
2#10593620C211yesmaximal, characteristic, complemented
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#73087—31nomaximal
5#68251—31nomaximal
6C2×D3×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×#3716—41nomaximal
9C3⋊#859—21no—
10Q8×D12D311yescharacteristic
11D4×SL(2,𝔽3)—31no—
12#5386—31no—
13#5386—61no—
14#5020—31no—
15#4841—31no—
16#4680—31no—
17SL(2,𝔽3)⋊C8D311yescharacteristic, complemented
18C3×Q8⋊C8—31no—
19#4486—31no—
20#4459—31no—
21C2×#634—41no—
22C2×C3×D12—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24D3×SL(2,𝔽3)—41no—
25C2×C3×C3⋊C8—41no—
26#3716—44no—
27#1651—91noSylow
28C2×D3×Q8—21no—
29C22×SL(2,𝔽3)—61no—
30C3×C4×Q8C2×D311yescharacteristic
31C2×#214—121no—
32C2×#204—61no—
33#897—61no—
34C4×D12—31no—
35#869—31no—
36#869—31no—
37C4×SL(2,𝔽3)—21no—
38C4×SL(2,𝔽3)C2×D311yescharacteristic
39#859—62no—
40#859—61no—
41C3×C4⋊C8—31no—
42C3×C4×C8—31no—
43#835—61no—
44#817—62no—
45#816—61no—
46#808—61no—
47C4⋊(C3⋊C8)—31no—
48C22×C3×D3—81no—
49C2×C32×C4—41no—
50C2×C3⋊Dic3—81no—
51#634—42no—
52#634—81no—
53C3×D12—42no—
54C3×D12—81no—
55C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
56C3×C3⋊C8—42no—
57D4×Q8—31no—
58#498—91no—
59#475—91no—
60#461—91no—
61#332—91no—
62Q8⋊C8—31no—
63D4⋊C8—91no—
64C2×C3×Q8#6711yescharacteristic
65C2×C3×D4—121no—
66D3×Q8—41no—
67D3×Q8—121no—
68C2×D12S411yescharacteristic
69C2×C4×D3—61no—
70C2×Dic6—61no—
71C2×Dic6—61no—
72C2×Dic6—121no—
73C2×Dic6—241no—
74C2×SL(2,𝔽3)—21no—
75C2×SL(2,𝔽3)—121no—
76C2×SL(2,𝔽3)D1211yescharacteristic
77C2×C3×C8—61no—
78C2×C3×C8—61no—
79C3×C4⋊C4—31no—
80C3×C42—31no—
81#214—124no—
82(C2×D3)⋊C4—61no—
83C4⋊Dic3—32no—
84C4⋊Dic3—31no—
85C4⋊Dic3—61no—
86C4⋊Dic3—61no—
87C2×C3⋊C8—41no—
88C2×C3⋊C8—61no—
89#204—122no—
90C22×C32—41no—
91C2×C3×D3—82no—
92C2×C3×D3—161no—
93C2×C3×D3—161no—
94C32×C4—41no—
95C32×C4—41no—
96C3⋊Dic3—82no—
97C22×Q8—61no—
98C2×SD16—181no—
99C4⋊Q8—91no—
100C4⋊Q8—91no—
101C22⋊Q8—181no—
102C4×Q8D3211yescharacteristic
103C4×D4—91no—
104#106—181no—
105#106—182no—
106C4⋊C8—31no—
107C4⋊C8—91no—
108#103—181no—
109#103—181no—
110C4×C8—31no—
111C23×C3—241no—
112C22×D3—21no—
113C3×Q8—31no—
114C3×Q8#21411yescharacteristic, complemented
115C3×D4—122no—
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—61no—
123C2×Dic3—121no—
124C2×Dic3—241no—
125C2×Dic3—242no—
126D12—21no—
127D12GL(2,𝔽3)12yes—
128C4×D3—121no—
129C4×D3—121no—
130Dic6—121no—
131Dic6—121no—
132Dic6—122no—
133Dic6—121no—
134Dic6—121no—
135Dic6—241no—
136Dic6—242no—
137Dic6—242no—
138SL(2,𝔽3)—21no—
139SL(2,𝔽3)#20411yescharacteristic, complemented
140C3×C8—62no—
141C3×C8—121no—
142C3⋊C8—42no—
143C3⋊C8—121no—
144C2×C32—41no—
145C2×C32—41no—
146C2×C32—41no—
147C3×D3—162no—
148C2×Q8—121no—
149C2×Q8—181no—
150C2×Q8—181no—
151C2×Q8—361no—
152C2×Q8#62611yescharacteristic
153C2×D4—31no—
154C22×C4—181no—
155SD16—362no—
156C2×C8—61no—
157C2×C8—61no—
158C2×C8—181no—
159C4⋊C4—31no—
160C4⋊C4—92no—
161C4⋊C4—91no—
162C4⋊C4—181no—
163C4⋊C4—181no—
164C22⋊C4—181no—
165C42—31no—
166C22×C3—41no—
167C22×C3—81no—
168C22×C3—242no—
169C22×C3—481no—
170C22×C3—481no—
171C22×C3GL(2,ℤ/4)11yescharacteristic
172C2×D3—22no—
173C2×D3—41no—
174C2×D3—41no—
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—41no—
178C3×C4—41no—
179C3×C4—61no—
180C3×C4—81no—
181C3×C4—81no—
182C3×C4C2×GL(2,𝔽3)11yescharacteristic
183C3×C4#98311yescharacteristic
184Dic3—121no—
185Dic3—242no—
186Dic3—244no—
187Dic3—241no—
188C32—41noSylow
189C23—61no—
190Q8—31no—
191Q8—361no—
192Q8—361no—
193Q8—361no—
194Q8—361no—
195Q8#371611yescharacteristic, complemented
196D4—32no—
197D4—61no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—181no—
201C2×C4—361no—
202C2×C4—361no—
203C2×C4—361no—
204C2×C4D3×S411yescharacteristic
205C8—62no—
206C8—121no—
207C8—361no—
208C2×C3—41no—
209C2×C3—41no—
210C2×C3—41no—
211C2×C3—81no—
212C2×C3—81no—
213C2×C3—81no—
214C2×C3—482no—
215C2×C3#541411yescharacteristic
216C2×C3#542111yescharacteristic
217C2×C3#543011yescharacteristic
218D3—42no—
219C22—62no—
220C22—121no—
221C22—121no—
222C22#6421011yescharacteristic
223C4—31no—
224C4—31no—
225C4—61no—
226C4—361no—
227C4—721no—
228C4#6420211yescharacteristic
229C4D3×GL(2,𝔽3)11yescharacteristic
230C3—41no—
231C3—81no—
232C3#7308711yescharacteristic, complemented
233C2—122no—
234C2#1059318611yescharacteristic
235C2#1059321811yescharacteristic
236C2#1059323211yescharacteristic
subgroupG/H
centerC22#64210
commutator, cosocleC3×C4×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D3×S4
FittingC3×C4×Q8C2×D3
socleC22×C3GL(2,ℤ/4)
prime radicalSL(2,𝔽3)×D12C2
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
2. #10593620C2semidirect
3. C3×SL(2,𝔽3)⋊C8C2semidirect
17. SL(2,𝔽3)⋊C8D3semidirect
55. C3×SL(2,𝔽3)SD16semidirect
114. C3×Q8#214semidirect
139. SL(2,𝔽3)#204semidirect
195. Q8#3716semidirect
232. C3#73087semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D122
#105936202
C3×SL(2,𝔽3)⋊C82
#730873
#682513
C2×#37164