Finite Group Databasebeta

← #11920566

Subgroups — #11920566

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
1SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#73100—31nomaximal
5#68264—31nomaximal
6C2×C3⋊GL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×#3717—41nomaximal
9Dic3×SL(2,𝔽3)—21no—
10Q8×Dic6D311yescharacteristic, complemented
11Q8×SL(2,𝔽3)—31no—
12C4⋊GL(2,𝔽3)—31no—
13C4⋊GL(2,𝔽3)—61no—
14#5027—31no—
15#4848—31no—
16#4693—31no—
17SL(2,𝔽3)⋊C8D311yescharacteristic, complemented
18C3×Q8⋊C8—31no—
19#4481—31no—
20#4456—31no—
21C2×C4⋊(C3⋊D3)—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3⋊GL(2,𝔽3)—41no—
25C2×C3×C3⋊C8—41no—
26#3717—44no—
27#1623—91noSylow
28C2×GL(2,𝔽3)—62no—
29C2×GL(2,𝔽3)—61no—
30C3×C4×Q8C2×D311yescharacteristic
31Q8×Dic3—21no—
32C2×#215—61no—
33C2×#215—121no—
34C2×#204—62no—
35C2×#204—61no—
36#888—61no—
37#874—31no—
38#869—31no—
39C4×Dic6—31no—
40C4×SL(2,𝔽3)—21no—
41C4×SL(2,𝔽3)—61no—
42C4×SL(2,𝔽3)C2×D311yescharacteristic
43C3×C4⋊C8—31no—
44C3×C4×C8—31no—
45#821—61no—
46#809—61no—
47C4⋊(C3⋊C8)—31no—
48C22×C3⋊D3—81no—
49C2×C32×C4—41no—
50C4⋊(C3⋊D3)—42no—
51C4⋊(C3⋊D3)—81no—
52C2×C3×Dic3—81no—
53C3×Dic6—42no—
54C3×Dic6—81no—
55C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
56C3×C3⋊C8—42no—
57Q82—31no—
58#492—91no—
59#460—91no—
60#460—91no—
61#333—91no—
62Q8⋊C8—31no—
63Q8⋊C8—91no—
64C2×C3×Q8—121no—
65C2×C3×Q8#6711yescharacteristic
66C2×D12—61no—
67C2×D12—61no—
68C2×D12—61no—
69C2×D12—121no—
70C2×D12—241no—
71C2×Dic6—32no—
72C2×Dic6S411yescharacteristic
73C2×SL(2,𝔽3)—21no—
74C2×SL(2,𝔽3)D1211yescharacteristic
75GL(2,𝔽3)—122no—
76GL(2,𝔽3)—121no—
77C2×C3×C8—61no—
78C2×C3×C8—61no—
79C3×C4⋊C4—31no—
80C3×C42—31no—
81#215—124no—
82#215—121no—
83#215—121no—
84C4⋊Dic3—31no—
85C4⋊Dic3—61no—
86Dic3⋊C4—61no—
87C4×Dic3—61no—
88C2×C3⋊C8—41no—
89C2×C3⋊C8—61no—
90#204—124no—
91#204—122no—
92C22×C32—41no—
93C2×C3⋊D3—82no—
94C2×C3⋊D3—161no—
95C2×C3⋊D3—161no—
96C32×C4—41no—
97C32×C4—41no—
98C3×Dic3—82no—
99C2×SD16—181no—
100C2×SD16—182no—
101C2×SD16—181no—
102C4⋊Q8—91no—
103C4⋊Q8—181no—
104#127—91no—
105C4×Q8—61no—
106C4×Q8—91no—
107C4×Q8D3211yescharacteristic
108C4⋊C8—31no—
109C4⋊C8—91no—
110#102—181no—
111#102—181no—
112C4×C8—31no—
113C22×D3—121no—
114C22×D3—241no—
115C22×D3—242no—
116C3×Q8—31no—
117C3×Q8—122no—
118C3×Q8—241no—
119C3×Q8#21511yescharacteristic, complemented
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—41no—
123C2×C3×C4—81no—
124C2×C3×C4—241no—
125C2×C3×C4C2×S411yescharacteristic
126C2×Dic3—21no—
127C2×Dic3—61no—
128D12—121no—
129D12—121no—
130D12—122no—
131D12—122no—
132D12—121no—
133D12—121no—
134D12—241no—
135D12—242no—
136D12—242no—
137Dic6—21no—
138Dic6—122no—
139Dic6GL(2,𝔽3)12yescomplemented
140SL(2,𝔽3)—21no—
141SL(2,𝔽3)#20411yescharacteristic, complemented
142C3×C8—62no—
143C3×C8—121no—
144C3⋊C8—42no—
145C3⋊C8—121no—
146C2×C32—41no—
147C2×C32—41no—
148C2×C32—41no—
149C3⋊D3—162no—
150C2×Q8—31no—
151C2×Q8—92no—
152C2×Q8#62611yescharacteristic
153C2×D4—181no—
154C2×D4—181no—
155C2×D4—181no—
156SD16—364no—
157SD16—362no—
158SD16—361no—
159SD16—361no—
160C2×C8—61no—
161C2×C8—61no—
162C2×C8—181no—
163C4⋊C4—31no—
164C4⋊C4—91no—
165C4⋊C4—181no—
166C4⋊C4—181no—
167C42—31no—
168C42—181no—
169C22×C3—41no—
170C22×C3—81no—
171C22×C3GL(2,ℤ/4)11yescharacteristic
172C2×D3—242no—
173C2×D3—244no—
174C2×D3—241no—
175C2×D3—241no—
176C2×D3—241no—
177C2×D3—481no—
178C2×D3—481no—
179C2×D3—482no—
180C2×D3—482no—
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—41no—
184C3×C4—41no—
185C3×C4—61no—
186C3×C4—81no—
187C3×C4—81no—
188C3×C4—242no—
189C3×C4C2×GL(2,𝔽3)11yescharacteristic
190C3×C4#98311yescharacteristic
191Dic3—22no—
192Dic3—121no—
193C32—41noSylow
194C23—361no—
195Q8—31no—
196Q8—32no—
197Q8—61no—
198Q8—362no—
199Q8#371711yescharacteristic, complemented
200D4—361no—
201D4—361no—
202D4—362no—
203D4—361no—
204D4—361no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—61no—
208C2×C4—181no—
209C2×C4D3×S411yescharacteristic
210C8—62no—
211C8—121no—
212C8—361no—
213C2×C3—41no—
214C2×C3—41no—
215C2×C3—41no—
216C2×C3—81no—
217C2×C3—81no—
218C2×C3—81no—
219C2×C3#541711yescharacteristic
220C2×C3#542111yescharacteristic
221C2×C3#542911yescharacteristic
222D3—482no—
223D3—484no—
224D3—481no—
225C22—721no—
226C22—721no—
227C22—721no—
228C22#6421011yescharacteristic
229C4—31no—
230C4—31no—
231C4—62no—
232C4—61no—
233C4—361no—
234C4#6420211yescharacteristic
235C4D3×GL(2,𝔽3)11yescharacteristic
236C3—41no—
237C3—81no—
238C3#7310011yescharacteristic, complemented
239C2—1441no—
240C2#1059318811yescharacteristic
241C2#1059321811yescharacteristic
242C2#1059323411yescharacteristic
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 radicalC4⋊(C3⋊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. SL(2,𝔽3)×Dic6C2semidirect
3. C3×SL(2,𝔽3)⋊C8C2semidirect
10. Q8×Dic6D3semidirect
17. SL(2,𝔽3)⋊C8D3semidirect
55. C3×SL(2,𝔽3)SD16semidirect
119. C3×Q8#215semidirect
139. Dic6GL(2,𝔽3)semidirect
141. SL(2,𝔽3)#204semidirect
199. Q8#3717semidirect
238. C3#73100semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×Dic62
C4⋊(C3⋊GL(2,𝔽3))2
C3×SL(2,𝔽3)⋊C82
#731003
#682643
C2×#37174