Finite Group Databasebeta

← D3×SL(2,𝔽3)⋊C8 #11920559

Subgroups — D3×SL(2,𝔽3)⋊C8 #11920559

Order 1152.

317 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 61 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4C2×SL(2,𝔽3)⋊C8—31nomaximal
5D3×Q8⋊C8—31nomaximal
6C2×D3×SL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×D3×C3⋊C8—41nomaximal
9Dic3×SL(2,𝔽3)C411yescharacteristic
10C4×D3×Q8D311yescharacteristic
11C2×C4×SL(2,𝔽3)—31no—
12D3×C4⋊C8—31no—
13C4×D3×C8—31no—
14SL(2,𝔽3)⋊C8—31no—
15SL(2,𝔽3)⋊C8—61no—
16SL(2,𝔽3)⋊C8D311yescharacteristic, direct product factor
17C3×Q8⋊C8—31no—
18#4535—31no—
19#4484—31no—
20#4456—31no—
21C2×C3×C4×D3—41no—
22C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
23D3×SL(2,𝔽3)C812yescomplemented
24C2×C3⋊(C3⋊C8)—41no—
25C2×C3×C3⋊C8—41no—
26D3×C3⋊C8—44no—
27C2×Q8⋊C8—91noSylow
28C2×D3×Q8Dic311yescharacteristic
29C22×SL(2,𝔽3)—31no—
30C3×C4×Q8C2×D311yescharacteristic
31Q8×Dic3Dic311yescharacteristic
32C22×C3⋊C8—121no—
33C2×D3×C8—61no—
34C2×D3×C8—61no—
35D3×C4⋊C4—31no—
36#887—31no—
37C42×D3—31no—
38C4×Dic6—31no—
39C4×SL(2,𝔽3)—21no—
40C4×SL(2,𝔽3)—31no—
41C4×SL(2,𝔽3)C2×D311yescharacteristic
42C3×C4⋊C8—31no—
43C3×C4×C8—31no—
44Dic3⋊C8—61no—
45C8×Dic3—61no—
46C4⋊(C3⋊C8)—31no—
47C4×C3⋊C8—31no—
48C22×C3×D3—41no—
49C2×C32×C4—41no—
50C2×C3×Dic3—41no—
51C3×C4×D3—42no—
52C3×C4×D3—42no—
53C3×SL(2,𝔽3)C2×C811yescharacteristic, complemented
54C3⋊(C3⋊C8)—42no—
55C3×C3⋊C8—42no—
56C2×C4×Q8—31no—
57C2×C4⋊C8—91no—
58C2×C4×C8—91no—
59Q8⋊C8—31no—
60Q8⋊C8—91no—
61Q8⋊C8—91no—
62Q8⋊C8—91no—
63C2×C3×Q8C2×Dic311yescharacteristic
64C22×C3×C4—121no—
65D3×Q8—32no—
66D3×Q8C3⋊C812yescomplemented
67C2×C4×D3—31no—
68C2×C4×D3—31no—
69C2×C4×D3S411yescharacteristic
70C2×Dic6—31no—
71C2×SL(2,𝔽3)—21no—
72C2×SL(2,𝔽3)—32no—
73C2×SL(2,𝔽3)C4×D311yescharacteristic
74C2×C3×C8—61no—
75C2×C3×C8—61no—
76C3×C4⋊C4—31no—
77C3×C42—31no—
78C4⋊Dic3—31no—
79Dic3⋊C4—61no—
80C4×Dic3—31no—
81C4×Dic3—31no—
82C4×Dic3—31no—
83C2×C3⋊C8—41no—
84C2×C3⋊C8—61no—
85C2×C3⋊C8—61no—
86C2×C3⋊C8—121no—
87C2×C3⋊C8—124no—
88C2×C3⋊C8—241no—
89D3×C8—64no—
90D3×C8—122no—
91C22×C32—41no—
92C2×C3×D3—42no—
93C2×C3×D3—42no—
94C2×C3×D3—42no—
95C32×C4—41no—
96C32×C4—41no—
97C3×Dic3—42no—
98C22×Q8—31no—
99C22×C8—181no—
100C22×C8—181no—
101C4×Q8—31no—
102C4×Q8—91no—
103C4×Q8—91no—
104C4×Q8D3211yescharacteristic
105C2×C4⋊C4—91no—
106C2×C42—91no—
107C4⋊C8—31no—
108C4⋊C8—91no—
109C4⋊C8—181no—
110C4×C8—31no—
111C4×C8—91no—
112C4×C8—181no—
113C23×C3—121no—
114C22×D3SL(2,ℤ/4)11yescharacteristic
115C3×Q8—31no—
116C3×Q8C2×C3⋊C811yescharacteristic, complemented
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—41no—
120C2×C3×C4—81no—
121C2×C3×C4—122no—
122C2×C3×C4—122no—
123C2×C3×C4—121no—
124C2×C3×C4C2×S411yescharacteristic
125C2×Dic3—31no—
126C2×Dic3—31no—
127C2×Dic3SL(2,ℤ/4)11yescharacteristic
128C4×D3—32no—
129C4×D3—32no—
130C4×D3—62no—
131C4×D3#22612yes—
132C4×D3GL(2,𝔽3)12yes—
133Dic6—32no—
134Dic6—61no—
135SL(2,𝔽3)—21no—
136SL(2,𝔽3)D3×C811yescharacteristic, complemented
137C3×C8—62no—
138C3×C8—121no—
139C3⋊C8—42no—
140C3⋊C8—62no—
141C3⋊C8—122no—
142C3⋊C8—121no—
143C3⋊C8—242no—
144C2×C32—41no—
145C2×C32—41no—
146C2×C32—41no—
147C3×D3—44no—
148C2×Q8—32no—
149C2×Q8—91no—
150C2×Q8—92no—
151C2×Q8D3×Dic311yescharacteristic
152C22×C4—31no—
153C22×C4—91no—
154C22×C4—91no—
155C2×C8—61no—
156C2×C8—61no—
157C2×C8—184no—
158C2×C8—181no—
159C2×C8—181no—
160C2×C8—362no—
161C4⋊C4—31no—
162C4⋊C4—91no—
163C4⋊C4—181no—
164C42—31no—
165C42—91no—
166C42—91no—
167C42—91no—
168C22×C3—41no—
169C22×C3—81no—
170C22×C3—122no—
171C22×C3—122no—
172C22×C3—122no—
173C22×C3C2×SL(2,ℤ/4)11yescharacteristic
174C2×D3A4⋊C812yes—
175C2×D3#85912yes—
176C2×D3GU(2,𝔽3)12yes—
177C3×C4—31no—
178C3×C4—31no—
179C3×C4—41no—
180C3×C4—41no—
181C3×C4—61no—
182C3×C4—81no—
183C3×C4—81no—
184C3×C4—122no—
185C3×C4C2×#22611yescharacteristic
186C3×C4C2×GL(2,𝔽3)11yescharacteristic
187Dic3—32no—
188Dic3—61no—
189Dic3#85912yes—
190C32—41noSylow
191C23—31no—
192Q8—31no—
193Q8—92no—
194Q8—181no—
195Q8D3×C3⋊C811yescharacteristic, complemented
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—32no—
199C2×C4—32no—
200C2×C4—31no—
201C2×C4—92no—
202C2×C4—92no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—182no—
206C2×C4D3×S411yescharacteristic
207C8—62no—
208C8—121no—
209C8—182no—
210C8—361no—
211C2×C3—41no—
212C2×C3—41no—
213C2×C3—41no—
214C2×C3—81no—
215C2×C3—81no—
216C2×C3—81no—
217C2×C3—124no—
218C2×C3C2×A4⋊C811yescharacteristic
219C2×C3C2×#85911yescharacteristic
220C2×C3C2×GU(2,𝔽3)11yescharacteristic
221D3SL(2,𝔽3)⋊C814yesdirect product factor
222C22—32no—
223C22—32no—
224C22—32no—
225C22D3×SL(2,ℤ/4)11yescharacteristic
226C4—31no—
227C4—31no—
228C4—32no—
229C4—61no—
230C4—92no—
231C4—181no—
232C4D3×#22611yescharacteristic
233C4D3×GL(2,𝔽3)11yescharacteristic
234C3—41no—
235C3—81no—
236C3C2×SL(2,𝔽3)⋊C811yescharacteristic, complemented
237C2—34no—
238C2D3×A4⋊C811yescharacteristic
239C2D3×#85911yescharacteristic
240C2D3×GU(2,𝔽3)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C8
FittingC3×C4×Q8C2×D3
socleC22×C3C2×SL(2,ℤ/4)
prime radicalC2×D3×SL(2,𝔽3)C4
cosocleC3×C4×SL(2,𝔽3)C22
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
3. C3×SL(2,𝔽3)⋊C8C2semidirect
16. SL(2,𝔽3)⋊C8D3direct (N × H)
23. D3×SL(2,𝔽3)C8semidirect
53. C3×SL(2,𝔽3)C2×C8semidirect
66. D3×Q8C3⋊C8semidirect
116. C3×Q8C2×C3⋊C8semidirect
136. SL(2,𝔽3)D3×C8semidirect
195. Q8D3×C3⋊C8semidirect
221. D3SL(2,𝔽3)⋊C8direct (N × H)
236. C3C2×SL(2,𝔽3)⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×D3×SL(2,𝔽3)2
C3⋊(SL(2,𝔽3)⋊C8)2
C3×SL(2,𝔽3)⋊C82
C2×SL(2,𝔽3)⋊C83
D3×Q8⋊C83
C2×D3×C3⋊C84