Finite Group Databasebeta

← C4×A4×SL(2,𝔽3) #11922554

Subgroups — C4×A4×SL(2,𝔽3) #11922554

Order 1152.

327 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C2×A4×SL(2,𝔽3)C211yesmaximal, characteristic
2C4×C22⋊SL(2,𝔽3)C312yesmaximal, complemented
3C22×C4×SL(2,𝔽3)C311yesmaximal, characteristic, complemented
4C4×Q8×A4C311yesmaximal, characteristic, complemented
5C2×C3×C4×A4—41nomaximal
6A4×SL(2,𝔽3)C411yescharacteristic, direct product factor
7C3×C4×SL(2,𝔽3)—41nomaximal
8C2×C22⋊SL(2,𝔽3)C2×C312yes—
9C2×Q8×A4C2×C311yescharacteristic
10C23×SL(2,𝔽3)C2×C311yescharacteristic
11C2×C4×SL(2,𝔽3)—31no—
12C2×C4×SL(2,𝔽3)—31no—
13A4×C4⋊C4—31no—
14C42×A4—31no—
15C22×C3×A4—41no—
16C2×C3×SL(2,𝔽3)—41no—
17C3×C4×A4—42no—
18C22×C4×Q8C3211yescharacteristic, complemented, Sylow
19C23×C3×C4—41no—
20C22⋊SL(2,𝔽3)C3×C412yescomplemented
21Q8×A4—31no—
22Q8×A4C3×C411yescharacteristic, complemented
23C22×SL(2,𝔽3)—31no—
24C22×SL(2,𝔽3)—31no—
25C22×SL(2,𝔽3)C3×C411yescharacteristic, complemented
26C2×C4×A4—31no—
27C2×C4×A4—31no—
28C2×C4×A4—42no—
29C2×C4×A4A411yescharacteristic
30C3×C4×Q8—41no—
31C4×SL(2,𝔽3)—31no—
32C4×SL(2,𝔽3)—42no—
33C4×SL(2,𝔽3)—122no—
34C4×SL(2,𝔽3)A411yescharacteristic, direct product factor
35C2×C3×A4—41no—
36C2×C3×A4—41no—
37C2×C3×A4—41no—
38C2×C32×C4—161no—
39C3×SL(2,𝔽3)—41no—
40C23×Q8C2×C3211yescharacteristic
41C2×C4×Q8—31no—
42C2×C4×Q8—31no—
43C2×C4×Q8—91no—
44C2×C4×Q8—91no—
45C22×C4⋊C4—31no—
46C22×C42—31no—
47C24×C3—41no—
48C22×A4—42no—
49C22×A4C2×A411yescharacteristic
50C2×C3×Q8—41no—
51C22×C3×C4—42no—
52C22×C3×C4—121no—
53C22×C3×C4—121no—
54C22×C3×C4—122no—
55C2×SL(2,𝔽3)—31no—
56C2×SL(2,𝔽3)—31no—
57C2×SL(2,𝔽3)—42no—
58C2×SL(2,𝔽3)—122no—
59C2×SL(2,𝔽3)C2×A411yescharacteristic
60C4×A4—31no—
61C4×A4—31no—
62C4×A4—44no—
63C4×A4—61no—
64C4×A4SL(2,𝔽3)12yesdirect product factor
65C3×C4⋊C4—121no—
66C3×C42—121no—
67C22×C32—161no—
68C3×A4—41no—
69C32×C4—162no—
70C22×Q8—31no—
71C22×Q8—31no—
72C22×Q8—31no—
73C22×Q8—91no—
74C22×Q8—91no—
75C22×Q8C32×C411yescharacteristic, complemented
76C23×C4—31no—
77C23×C4—31no—
78C23×C4C3×A411yescharacteristic
79C4×Q8—32no—
80C4×Q8—31no—
81C4×Q8—91no—
82C4×Q8—92no—
83C4×Q8—91no—
84C4×Q8—92no—
85C4×Q8C3×A411yescharacteristic, complemented
86C2×C4⋊C4—91no—
87C2×C4⋊C4—91no—
88C2×C4⋊C4—92no—
89C2×C42—91no—
90C2×C42—91no—
91C2×C42—91no—
92C2×C42—91no—
93C23×C3—41no—
94C23×C3—41no—
95C23×C3—41no—
96C23×C3—121no—
97C23×C3—121no—
98C23×C3—121no—
99C23×C3—121no—
100C2×A4—42no—
101C2×A4—42no—
102C2×A4—42no—
103C2×A4C4×A411yescharacteristic
104C2×A4C2×SL(2,𝔽3)11yescharacteristic
105C2×A4#41⋊C311yescharacteristic
106C3×Q8—41no—
107C3×Q8—121no—
108C2×C3×C4—41no—
109C2×C3×C4—41no—
110C2×C3×C4—122no—
111C2×C3×C4—122no—
112C2×C3×C4—121no—
113C2×C3×C4—121no—
114C2×C3×C4—122no—
115C2×C3×C4—122no—
116C2×C3×C4—121no—
117C2×C3×C4—162no—
118SL(2,𝔽3)—42no—
119SL(2,𝔽3)—122no—
120SL(2,𝔽3)C4×A411yescharacteristic, direct product factor
121C2×C32—161no—
122C2×C32—161no—
123C2×C32—161no—
124C24C2×C3×A411yescharacteristic
125C2×Q8—32no—
126C2×Q8—31no—
127C2×Q8—31no—
128C2×Q8—91no—
129C2×Q8—92no—
130C2×Q8—92no—
131C2×Q8—91no—
132C2×Q8—91no—
133C2×Q8—91no—
134C2×Q8—91no—
135C2×Q8—92no—
136C2×Q8C2×C3×A411yescharacteristic
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—32no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—61no—
143C22×C4—91no—
144C22×C4—91no—
145C22×C4—91no—
146C22×C4—91no—
147C22×C4—91no—
148C22×C4—91no—
149C22×C4—181no—
150C22×C4C3×SL(2,𝔽3)12yescomplemented
151C4⋊C4—31no—
152C4⋊C4—91no—
153C4⋊C4—92no—
154C4⋊C4—92no—
155C42—31no—
156C42—91no—
157C42—91no—
158C42—92no—
159C42—91no—
160C22×C3—41no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3—121no—
164C22×C3—121no—
165C22×C3—121no—
166C22×C3—121no—
167C22×C3—121no—
168C22×C3—121no—
169C22×C3—121no—
170C22×C3—121no—
171C22×C3—121no—
172C22×C3—122no—
173C22×C3—162no—
174A4—42no—
175A4C4×SL(2,𝔽3)11yescharacteristic, direct product factor
176C3×C4—42no—
177C3×C4—42no—
178C3×C4—122no—
179C3×C4—121no—
180C3×C4—121no—
181C3×C4—164no—
182C3×C4—241no—
183C32—161noSylow
184C23—31no—
185C23—31no—
186C23—31no—
187C23—31no—
188C23C3×C4×A411yescharacteristic
189C23C2×C3×SL(2,𝔽3)11yescharacteristic
190C23C3×#41⋊C311yescharacteristic
191Q8—31no—
192Q8—32no—
193Q8—91no—
194Q8—91no—
195Q8—92no—
196Q8—92no—
197Q8C3×C4×A411yescharacteristic, complemented
198C2×C4—32no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—32no—
202C2×C4—32no—
203C2×C4—32no—
204C2×C4—31no—
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—92no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—181no—
215C2×C4—181no—
216C2×C4—181no—
217C2×C4—181no—
218C2×C4A4211yescharacteristic
219C2×C3—41no—
220C2×C3—41no—
221C2×C3—41no—
222C2×C3—41no—
223C2×C3—41no—
224C2×C3—41no—
225C2×C3—121no—
226C2×C3—121no—
227C2×C3—121no—
228C2×C3—121no—
229C2×C3—162no—
230C2×C3—162no—
231C2×C3—162no—
232C22—31no—
233C22—31no—
234C22—31no—
235C22—32no—
236C22—31no—
237C22—31no—
238C22—31no—
239C22—31no—
240C22—31no—
241C22—31no—
242C22C3×C4×SL(2,𝔽3)11yescharacteristic, complemented
243C22C2×A4211yescharacteristic
244C4—31no—
245C4—31no—
246C4—32no—
247C4—61no—
248C4—91no—
249C4—91no—
250C4—181no—
251C4A4×SL(2,𝔽3)12yesdirect product factor
252C3—41no—
253C3—41no—
254C3—162no—
255C2—31no—
256C2—31no—
257C2—31no—
258C2—31no—
259C2C4×A4211yescharacteristic
260C2C2×A4×SL(2,𝔽3)11yescharacteristic
261C2A4×#41⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2×C4A42
commutator, hypocenterC22×Q8C32×C4
FrattiniC22C2×A42
FittingC22×C4×Q8C32
socleC24C2×C3×A4
prime radicalC2×A4×SL(2,𝔽3)C2
cosocleC23×Q8C2×C32
LCSG ▷ C22×Q8
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊SL(2,𝔽3)C3semidirect
3. C22×C4×SL(2,𝔽3)C3semidirect
4. C4×Q8×A4C3semidirect
6. A4×SL(2,𝔽3)C4direct (N × H)
18. C22×C4×Q8C32semidirect
20. C22⋊SL(2,𝔽3)C3×C4semidirect
22. Q8×A4C3×C4semidirect
25. C22×SL(2,𝔽3)C3×C4semidirect
34. C4×SL(2,𝔽3)A4direct (N × H)
64. C4×A4SL(2,𝔽3)direct (N × H)
75. C22×Q8C32×C4semidirect
85. C4×Q8C3×A4semidirect
120. SL(2,𝔽3)C4×A4direct (N × H)
150. C22×C4C3×SL(2,𝔽3)semidirect
175. A4C4×SL(2,𝔽3)direct (N × H)
197. Q8C3×C4×A4semidirect
242. C22C3×C4×SL(2,𝔽3)semidirect
251. C4A4×SL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×SL(2,𝔽3)2
C4×C22⋊SL(2,𝔽3)3
C22×C4×SL(2,𝔽3)3
C4×Q8×A43
C2×C3×C4×A44
C3×C4×SL(2,𝔽3)4