Finite Group Databasebeta

← C3×C4⋊GU(2,𝔽3) #11920848

Subgroups — C3×C4⋊GU(2,𝔽3) #11920848

Order 1152.

311 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 64 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×C4×#41⋊C3C211yesmaximal, characteristic
2C2×C3×GU(2,𝔽3)C212yesmaximal
3C4⋊GU(2,𝔽3)C311yesmaximal, characteristic, direct product factor
4C3×#1806—31nomaximal
5C2×C3×#41⋊C3C2211yescharacteristic
6C3×C4×SL(2,𝔽3)C411yescharacteristic, complemented
7C3×C4×SL(2,𝔽3)C411yescharacteristic, complemented
8C3×GU(2,𝔽3)—22no—
9C3×C4⋊(C3⋊C8)—41nomaximal
10C3×C4×#41D311yescharacteristic
11C4×#41⋊C3—21no—
12C4×#41⋊C3C2×C311yescharacteristic
13C2×GU(2,𝔽3)C2×C312yes—
14C3×#423—31no—
15C2×C3×#104—32no—
16C3×C4×C4⋊C4—31no—
17C3×#339—32no—
18C3×#41⋊C3D411yescharacteristic
19C3×#41⋊C3Q811yescharacteristic
20C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
21C32×C42—41no—
22C2×C3×C3⋊C8—42no—
23#1806—31noSylow
24C2×C3×#41C2×D311yescharacteristic
25C2×#41⋊C3—21no—
26C2×#41⋊C3C22×C311yescharacteristic
27C2×C3×M16—32no—
28C3×C4×Q8Dic311yescharacteristic
29C3×C4×Q8Dic311yescharacteristic
30C3×C4×D4—31no—
31C3×C4×D4—31no—
32C3×#117—31no—
33C2×C3×C4⋊C4—61no—
34C2×C3×C42—31no—
35C2×C3×C42—32no—
36C4×SL(2,𝔽3)—21no—
37C4×SL(2,𝔽3)—21no—
38C4×SL(2,𝔽3)C3×C411yescharacteristic, complemented
39C4×SL(2,𝔽3)C3×C411yescharacteristic, complemented
40GU(2,𝔽3)—22no—
41C3×C4⋊C8—61no—
42C3×C4⋊C8—61no—
43C3×#104—62no—
44C3×#104—62no—
45C3×#95—61no—
46C4⋊(C3⋊C8)—41no—
47C2×C32×C4—41no—
48C2×C32×C4—41no—
49C2×C32×C4—41no—
50C3×SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
51C3×C3⋊C8—82no—
52C4×#41C3×D311yescharacteristic
53#423—31no—
54C2×#104—32no—
55C4×C4⋊C4—31no—
56#339—32no—
57C3×#41—31no—
58C3×#41—31no—
59C3×#41Dic611yescharacteristic
60C3×#41D1211yescharacteristic
61C2×C3×Q8C2×Dic311yescharacteristic
62C2×C3×D4—31no—
63C22×C3×C4—31no—
64C22×C3×C4—31no—
65C22×C3×C4—31no—
66C22×C3×C4—62no—
67#41⋊C3—21no—
68#41⋊C3—21no—
69#41⋊C3C3×D411yescharacteristic
70#41⋊C3C3×Q811yescharacteristic
71C2×SL(2,𝔽3)—21no—
72C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
73C3×M16—62no—
74C3×M16—62no—
75C2×C3×C8—62no—
76C3×C4⋊C4—31no—
77C3×C4⋊C4—31no—
78C3×C4⋊C4—64no—
79C3×C22⋊C4—61no—
80C3×C42—31no—
81C3×C42—31no—
82C3×C42—31no—
83C3×C42—41no—
84C3×C42—62no—
85C3×C42—62no—
86C3×C42—81no—
87C3×C42S411yescharacteristic
88C2×C3⋊C8—42no—
89C22×C32—41no—
90C32×C4—42no—
91C32×C4—41no—
92C32×C4—41no—
93C32×C4—81no—
94C2×#41C2×C3×D311yescharacteristic
95C2×M16—32no—
96C4×Q8C3×Dic311yescharacteristic
97C4×Q8C3×Dic311yescharacteristic
98C4×D4—31no—
99C4×D4—31no—
100#117—31no—
101C2×C4⋊C4—61no—
102C2×C42—31no—
103C2×C42—32no—
104C4⋊C8—61no—
105C4⋊C8—61no—
106#104—62no—
107#104—62no—
108#95—61no—
109C23×C3—31no—
110C3×Q8—31no—
111C3×Q8C4⋊Dic311yescharacteristic
112C3×D4—31no—
113C3×D4—31no—
114C3×D4—61no—
115C2×C3×C4—31no—
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—31no—
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—41no—
123C2×C3×C4—41no—
124C2×C3×C4—41no—
125C2×C3×C4—62no—
126C2×C3×C4—64no—
127C2×C3×C4—81no—
128C2×C3×C4—81no—
129C2×C3×C4—81no—
130C2×C3×C4—121no—
131C2×C3×C4—122no—
132C2×C3×C4—122no—
133C2×C3×C4SL(2,ℤ/4)11yescharacteristic
134C2×C3×C4SL(2,ℤ/4)11yescharacteristic
135C2×C3×C4C2×S411yescharacteristic
136SL(2,𝔽3)—21no—
137SL(2,𝔽3)C3×C4⋊C411yescharacteristic, complemented
138C3×C8—122no—
139C3⋊C8—82no—
140C2×C32—41no—
141C2×C32—41no—
142C2×C32—41no—
143#41—31no—
144#41—31no—
145#41C3×Dic611yescharacteristic
146#41C3×D1211yescharacteristic
147C2×Q8C2×C3×Dic311yescharacteristic
148C2×D4—31no—
149C22×C4—31no—
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—62no—
153M16—62no—
154M16—62no—
155C2×C8—62no—
156C4⋊C4—31no—
157C4⋊C4—31no—
158C4⋊C4—64no—
159C22⋊C4—61no—
160C42—31no—
161C42—31no—
162C42—31no—
163C42—62no—
164C42—62no—
165C42C3×S411yescharacteristic
166C22×C3—31no—
167C22×C3—31no—
168C22×C3—41no—
169C22×C3—61no—
170C22×C3—61no—
171C22×C3—81no—
172C22×C3C2×SL(2,ℤ/4)11yescharacteristic
173C3×C4—21no—
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—42no—
177C3×C4—41no—
178C3×C4—41no—
179C3×C4—61no—
180C3×C4—61no—
181C3×C4—81no—
182C3×C4—82no—
183C3×C4—82no—
184C3×C4—81no—
185C3×C4—81no—
186C3×C4—124no—
187C3×C4GU(2,𝔽3)12yescomplemented
188C3×C4A4⋊Q811yescharacteristic
189C3×C4C4⋊S411yescharacteristic
190C32—41noSylow
191C23—31no—
192Q8—31no—
193Q8C3×C4⋊Dic311yescharacteristic
194D4—31no—
195D4—31no—
196D4—61no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—31no—
204C2×C4—62no—
205C2×C4—64no—
206C2×C4—121no—
207C2×C4—122no—
208C2×C4—122no—
209C2×C4C3×SL(2,ℤ/4)11yescharacteristic
210C2×C4C3×SL(2,ℤ/4)11yescharacteristic
211C2×C4C2×C3×S411yescharacteristic
212C8—122no—
213C2×C3—41no—
214C2×C3—41no—
215C2×C3—41no—
216C2×C3—61no—
217C2×C3—61no—
218C2×C3—81no—
219C2×C3—81no—
220C2×C3—81no—
221C2×C3C4⋊SL(2,ℤ/4)11yescharacteristic
222C2×C3C2×GU(2,𝔽3)11yescharacteristic
223C2×C3#542311yescharacteristic
224C22—31no—
225C22—31no—
226C22—61no—
227C22—61no—
228C22C2×C3×SL(2,ℤ/4)11yescharacteristic
229C4—21no—
230C4—31no—
231C4—31no—
232C4—61no—
233C4—61no—
234C4—124no—
235C4C3×GU(2,𝔽3)12yescomplemented
236C4C3×A4⋊Q811yescharacteristic
237C4C3×C4⋊S411yescharacteristic
238C3—41no—
239C3—81no—
240C3C4⋊GU(2,𝔽3)11yescharacteristic, direct product factor
241C2—61no—
242C2—61no—
243C2C3×C4⋊SL(2,ℤ/4)11yescharacteristic
244C2C2×C3×GU(2,𝔽3)11yescharacteristic
245C2C3×#542311yescharacteristic
subgroupG/H
centerC2×C3×C4C2×S4
commutatorC2×SL(2,𝔽3)C2×C3×C4
FrattiniC2×C4C2×C3×S4
FittingC3×C4×#41D3
socleC22×C3C2×SL(2,ℤ/4)
prime radicalC2×C3×#41⋊C3C22
cosocleC2×#41⋊C3C22×C3
hypercenterC3×C42S4
hypocenterSL(2,𝔽3)C3×C4⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
3. C4⋊GU(2,𝔽3)C3direct (N × H)
6. C3×C4×SL(2,𝔽3)C4semidirect
7. C3×C4×SL(2,𝔽3)C4semidirect
38. C4×SL(2,𝔽3)C3×C4semidirect
39. C4×SL(2,𝔽3)C3×C4semidirect
50. C3×SL(2,𝔽3)C4⋊C4semidirect
137. SL(2,𝔽3)C3×C4⋊C4semidirect
187. C3×C4GU(2,𝔽3)semidirect
235. C4C3×GU(2,𝔽3)semidirect
240. C3C4⋊GU(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
C3×C4×#41⋊C32
C2×C3×GU(2,𝔽3)2
C4⋊GU(2,𝔽3)3
C3×#18063
C3×C4⋊(C3⋊C8)4