Finite Group Databasebeta

← SL(2,𝔽3)×#214 #11921302

Subgroups — SL(2,𝔽3)×#214 #11921302

Order 1152.

305 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 55 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
2SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4Q8×#214C311yesmaximal, characteristic, complemented
5SD16×SL(2,𝔽3)—31nomaximal
6C22×C3×SL(2,𝔽3)—21no—
7C2×C3×#214—41nomaximal
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9Dic3×SL(2,𝔽3)—21no—
10C3×D4×Q8C2×C311yescharacteristic
11Q8×Dic6C2×C311yescharacteristic, complemented
12Q8×SL(2,𝔽3)—31no—
13D4×SL(2,𝔽3)—21no—
14D4×SL(2,𝔽3)D311yescharacteristic
15#5094—31no—
16#5080—31no—
17Q8×C3⋊C8C2×C311yescharacteristic, complemented
18C4×#214—31no—
19#5008—31no—
20C8×SL(2,𝔽3)—31no—
21C2×C32×D4—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)—41no—
24C2×C3×SL(2,𝔽3)D411yescharacteristic
25C3×#214—44no—
26C2×C3×C3⋊C8—41no—
27Q8×SD16—31noSylow
28C22×C3×Q8—21no—
29C22×SL(2,𝔽3)—22no—
30C22×SL(2,𝔽3)—21no—
31C2×C3×SD16—121no—
32C3×C4⋊Q8—31no—
33C3×C22⋊Q8—61no—
34C3×C4×Q8C22×C311yescharacteristic
35C3×C4×D4—31no—
36Q8×Dic3—21no—
37C2×#214A411yescharacteristic
38#888—61no—
39#869—31no—
40C4×Dic6—31no—
41C4×SL(2,𝔽3)—21no—
42C4×SL(2,𝔽3)—61no—
43C4×SL(2,𝔽3)C2×D311yescharacteristic
44#832—31no—
45#810—31no—
46#808—32no—
47#808—31no—
48C4⋊(C3⋊C8)—31no—
49C4×C3⋊C8—31no—
50C23×C32—81no—
51C32×D4—42no—
52C32×D4—81no—
53C2×C32×C4—41no—
54C2×C3×Dic3—81no—
55C3×Dic6—42no—
56C3×Dic6—81no—
57C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
58C3×C3⋊C8—42no—
59Q82—31no—
60D4×Q8C3×D311yescharacteristic
61#498—91no—
62#476—91no—
63#475—91no—
64C8×Q8—31no—
65C4×SD16—91no—
66C2×C3×Q8—41no—
67C2×C3×Q8—61no—
68C2×C3×Q8—121no—
69C2×C3×Q8—121no—
70C2×C3×Q8C3×D411yescharacteristic
71C2×C3×D4—41no—
72C2×C3×D4—81no—
73C2×C3×D4C2×A411yescharacteristic
74C22×C3×C4—61no—
75C2×Dic6—61no—
76C2×Dic6C2×A411yescharacteristic
77C2×SL(2,𝔽3)—21no—
78C2×SL(2,𝔽3)—42no—
79C2×SL(2,𝔽3)—41no—
80C2×SL(2,𝔽3)#6711yescharacteristic
81C3×SD16—124no—
82C2×C3×C8—121no—
83C3×C4⋊C4—31no—
84C3×C4⋊C4—31no—
85C3×C4⋊C4—61no—
86C3×C22⋊C4—61no—
87C3×C42—31no—
88#214SL(2,𝔽3)14yesdirect product factor
89C4⋊Dic3—31no—
90C4⋊Dic3—61no—
91Dic3⋊C4—61no—
92C4×Dic3—61no—
93C2×C3⋊C8—31no—
94C2×C3⋊C8C2×A411yescharacteristic
95C22×C32—41no—
96C22×C32—82no—
97C22×C32—161no—
98C22×C32—161no—
99C32×C4—41no—
100C32×C4—41no—
101C3×Dic3—82no—
102C22×Q8—21no—
103C2×SD16—31no—
104C4⋊Q8—31no—
105C4⋊Q8—91no—
106C4⋊Q8—181no—
107C22⋊Q8—61no—
108C4×Q8—61no—
109C4×Q8—91no—
110C4×Q8C2×C3×D311yescharacteristic
111C4×D4—31no—
112#106—91no—
113#106—92no—
114C4⋊C8—91no—
115#103—91no—
116#102—91no—
117C4×C8—91no—
118C23×C3—21no—
119C23×C3—81no—
120C23×C3—82no—
121C3×Q8—31no—
122C3×Q8—122no—
123C3×Q8—121no—
124C3×Q8—121no—
125C3×Q8—241no—
126C3×Q8C3×SD1611yescharacteristic, complemented
127C3×D4—21no—
128C3×D4—42no—
129C3×D4—81no—
130C3×D4—82no—
131C3×D4—82no—
132C3×D4C2×SL(2,𝔽3)12yes—
133C2×C3×C4—31no—
134C2×C3×C4—31no—
135C2×C3×C4—41no—
136C2×C3×C4—61no—
137C2×C3×C4—81no—
138C2×C3×C4—121no—
139C2×C3×C4—121no—
140C2×C3×C4—241no—
141C2×C3×C4C22×A411yescharacteristic
142C2×Dic3—21no—
143C2×Dic3—61no—
144Dic6—21no—
145Dic6—241no—
146Dic6C2×SL(2,𝔽3)12yescomplemented
147SL(2,𝔽3)—21no—
148SL(2,𝔽3)#21411yescharacteristic, direct product factor
149C3×C8—122no—
150C3⋊C8—61no—
151C3⋊C8C2×SL(2,𝔽3)12yescomplemented
152C2×C32—41no—
153C2×C32—41no—
154C2×C32—41no—
155C2×C32—162no—
156C2×Q8—31no—
157C2×Q8—41no—
158C2×Q8—61no—
159C2×Q8—121no—
160C2×Q8—181no—
161C2×Q8C3×#6711yescharacteristic
162C2×D4D3×A411yescharacteristic
163C22×C4—61no—
164SD16—34no—
165C2×C8—31no—
166C2×C8—91no—
167C4⋊C4—31no—
168C4⋊C4—31no—
169C4⋊C4—61no—
170C4⋊C4—91no—
171C4⋊C4—181no—
172C4⋊C4—181no—
173C22⋊C4—61no—
174C42—31no—
175C42—181no—
176C22×C3—22no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—82no—
181C22×C3—84no—
182C22×C3—81no—
183C22×C3—161no—
184C22×C3—161no—
185C22×C3—162no—
186C22×C3—162no—
187C22×C3D4×A411yescharacteristic
188C3×C4—31no—
189C3×C4—31no—
190C3×C4—41no—
191C3×C4—41no—
192C3×C4—61no—
193C3×C4—81no—
194C3×C4—81no—
195C3×C4—121no—
196C3×C4—242no—
197C3×C4C22×SL(2,𝔽3)11yescharacteristic
198C3×C42−1+4⋊C311yescharacteristic
199Dic3—22no—
200Dic3—121no—
201C32—41noSylow
202C23—21no—
203Q8—31no—
204Q8—32no—
205Q8—61no—
206Q8—121no—
207Q8—121no—
208Q8—721no—
209Q8C3×#21411yescharacteristic, complemented
210D4—21no—
211D4D3×SL(2,𝔽3)12yes—
212C2×C4—31no—
213C2×C4—31no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—121no—
217C2×C4—121no—
218C2×C4—181no—
219C2×C4C2×D3×A411yescharacteristic
220C8—32no—
221C8—181no—
222C2×C3—42no—
223C2×C3—41no—
224C2×C3—41no—
225C2×C3—41no—
226C2×C3—81no—
227C2×C3—81no—
228C2×C3—81no—
229C2×C3—162no—
230C2×C3—164no—
231C2×C3D4×SL(2,𝔽3)11yescharacteristic
232C2×C3A4×SD1611yescharacteristic
233C2×C3#577⋊C311yescharacteristic
234C22—22no—
235C22—41no—
236C22—41no—
237C22A4×#6711yescharacteristic
238C4—31no—
239C4—31no—
240C4—62no—
241C4—61no—
242C4—121no—
243C4—361no—
244C4C2×D3×SL(2,𝔽3)11yescharacteristic
245C4#1007⋊C311yescharacteristic
246C3—41no—
247C3—81no—
248C3SD16×SL(2,𝔽3)11yescharacteristic, complemented
249C2—42no—
250C2A4×#21411yescharacteristic
251C2SL(2,𝔽3)×#6711yescharacteristic
252C2#5837⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×D4×Q8C2×C3
socleC22×C3D4×A4
prime radicalC3×D4×SL(2,𝔽3)C2
hypercenterC2×D4D3×A4
hypocenterC3×Q8C3×SD16
LCSG ▷ C3×C4×Q8 ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×Dic6C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
4. Q8×#214C3semidirect
11. Q8×Dic6C2×C3semidirect
17. Q8×C3⋊C8C2×C3semidirect
57. C3×SL(2,𝔽3)SD16semidirect
88. #214SL(2,𝔽3)direct (N × H)
126. C3×Q8C3×SD16semidirect
146. Dic6C2×SL(2,𝔽3)semidirect
148. SL(2,𝔽3)#214direct (N × H)
151. C3⋊C8C2×SL(2,𝔽3)semidirect
209. Q8C3×#214semidirect
248. C3SD16×SL(2,𝔽3)semidirect

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
SL(2,𝔽3)×Dic62
C3⋊C8×SL(2,𝔽3)2
Q8×#2143
SD16×SL(2,𝔽3)3
C2×C3×#2144