Finite Group Databasebeta

← C3⋊(GL(2,𝔽3)⋊A4) #13180325

Subgroups — C3⋊(GL(2,𝔽3)⋊A4) #13180325

Order 1728.

296 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 29 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×Ω+(4,𝔽3)C211yesmaximal, characteristic, complemented
2#10596587C311yesmaximal, characteristic, complemented
3GL(2,𝔽3)⋊A4—33nomaximal
4#5836⋊C3—31nomaximal
5C3⋊D3×SL(2,𝔽3)—41nomaximal
6C3×C3⋊GL(2,𝔽3)—41nomaximal
7C3×#997—21no—
8C3×#994D311yescharacteristic, complemented
9C3×#994C2×C311yescharacteristic, complemented
10#1010⋊C3—31no—
11#64266—31no—
12#64265—31no—
13Ω+(4,𝔽3)—23no—
14Ω+(4,𝔽3)D313yescomplemented
15C3⋊(#131⋊C3)—31no—
16C32×SL(2,𝔽3)—41no—
17C32×SL(2,𝔽3)—41no—
18#5924—33no—
19#5836—31no—
20#577⋊C3—91no—
21Q8×C3⋊D3—41no—
22C3×#41⋊C3—31no—
23C3×#41⋊C3—31no—
24D3×SL(2,𝔽3)—61no—
25D3×SL(2,𝔽3)—123no—
26C3⋊GL(2,𝔽3)A411yescharacteristic, complemented
27C3⋊#226—31no—
28C3×GL(2,𝔽3)—123no—
29C3×#215—121no—
30C2×C3×C3⋊D3—161no—
31C3×H(5,𝔽2)C3×D311yescharacteristic, complemented
32#1010—31no—
33#997—23no—
342−1+4⋊C3—91no—
35#994—21no—
36#994C3×D313yescomplemented
37#994C3⋊D311yescharacteristic, complemented
38#985—93no—
39#984—93no—
40#951—91no—
41#950—91no—
42#948—31no—
43C2×#214—91no—
44#932—91no—
45#131⋊C3—91no—
46C2×C3×A4—241no—
47C32×Q8—41no—
48C32×Q8—41no—
49C4×C3⋊D3—121no—
50#634—121no—
51C3×D12—121no—
52C3×SL(2,𝔽3)—43no—
53C3×SL(2,𝔽3)—43no—
54C3×SL(2,𝔽3)—83no—
55C3×SL(2,𝔽3)—81no—
56C3×SL(2,𝔽3)—81no—
57C3×SL(2,𝔽3)—83no—
58C3×SL(2,𝔽3)S411yescharacteristic
59C3×SL(2,𝔽3)C2×A411yescharacteristic
60C3×C3⋊C8—121no—
61#577—91noSylow
62C2×C33—161no—
63C3×C3⋊D3—162no—
64C3×#41—31no—
65C3×#41—31no—
66C2×C3×D4—91no—
67D3×Q8—61no—
68D3×Q8—123no—
69#237—181no—
70#235—91no—
71C2×Dic6—91no—
72#41⋊C3—31no—
73#41⋊C3—33no—
74#41⋊C3—61no—
75C2×SL(2,𝔽3)—181no—
76GL(2,𝔽3)—33no—
77#226—93no—
78C3×SD16—361no—
79#216—91no—
80#215—31no—
81#214—91no—
82#214—92no—
83#213—91no—
84#208—91no—
85C2×C3⋊C8—91no—
86C2×C3⋊D3—41no—
87C2×C3×D3—241no—
88C2×C3×D3—483no—
89C3×A4—241no—
90C32×C4—121no—
91C32×C4—121no—
92C3⋊Dic3—121no—
932−1+4—91no—
94H(5,𝔽2)C3×C3⋊D311yescharacteristic, complemented
95#137—271no—
96AGL(1,ℤ/8)—271no—
97#135—271no—
98C2×SD16—271no—
99#131—91no—
100C33—161noSylow
101C23×C3—61no—
102C2×A4—243no—
103C3×Q8—43no—
104C3×Q8—41no—
105C3×Q8—81no—
106C3×Q8C3×S411yescharacteristic
107C3×Q8D3×A411yescharacteristic
108C3×D4—91no—
109C3×D4—91no—
110C3×D4—361no—
111C2×C3×C4—91no—
112#67—181no—
113C2×Dic3—181no—
114D12—31no—
115C4×D3—181no—
116C4×D3—363no—
117Dic6—91no—
118Dic6—181no—
119Dic6—363no—
120SL(2,𝔽3)—21no—
121SL(2,𝔽3)—83no—
122SL(2,𝔽3)—83no—
123SL(2,𝔽3)C3⋊S411yescharacteristic
124SL(2,𝔽3)D3×A413yes—
125C3×C8—361no—
126C3⋊C8—31no—
127C3⋊C8—91no—
128C2×C32—41no—
129C2×C32—41no—
130C2×C32—163no—
131C2×C32—323no—
132C2×C32—321no—
133C3⋊D3—42no—
134C3×D3—481no—
135C3×D3—486no—
136#41—31no—
137#41—31no—
138#41—271no—
139#41—541no—
140C2×Q8—181no—
141C2×Q8—271no—
142C2×D4—91no—
143Q16—271no—
144SD16—91no—
145SD16—271no—
146SD16—272no—
147D8—271no—
148M16—271no—
149C2×C8—271no—
150C22×C3—91no—
151C22×C3—241no—
152C22×C3—721no—
153C2×D3—61no—
154C2×D3—123no—
155A4—243no—
156C3×C4—31no—
157C3×C4—31no—
158C3×C4—121no—
159C3×C4—123no—
160C3×C4—241no—
161Dic3—181no—
162Dic3—363no—
163C32—41no—
164C32—41no—
165C32—163no—
166C32—323no—
167C32—321no—
168C23—61no—
169Q8—271no—
170Q8—541no—
171Q8C3×C3⋊S411yescharacteristic
172Q8A4×C3⋊D311yescharacteristic
173D4—91no—
174D4—91no—
175D4—91no—
176D4—541no—
177C2×C4—91no—
178C2×C4—541no—
179C2×C4—541no—
180C8—91no—
181C8—271no—
182C2×C3—43no—
183C2×C3—41no—
184C2×C3—81no—
185C2×C3—181no—
186C2×C3—323no—
187C2×C3—1441no—
188C2×C3A4×S411yescharacteristic
189D3—126no—
190D3—121no—
191C22—91no—
192C22—181no—
193C22—241no—
194C4—31no—
195C4—31no—
196C4—541no—
197C3—43no—
198C3—41no—
199C3—81no—
200C3—323no—
201C3GL(2,𝔽3)⋊A411yescharacteristic, complemented
202C2—181no—
203C2—361no—
204C2A4×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2A4×C3⋊S4
commutator, cosocle, hypocenterC3×#994C2×C3
FittingC3×H(5,𝔽2)C3×D3
socleC2×C3A4×S4
LCSG ▷ C3×#994
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C3×Ω+(4,𝔽3)C2semidirect
2. #10596587C3semidirect
8. C3×#994D3semidirect
9. C3×#994C2×C3semidirect
14. Ω+(4,𝔽3)D3semidirect
26. C3⋊GL(2,𝔽3)A4semidirect
31. C3×H(5,𝔽2)C3×D3semidirect
36. #994C3×D3semidirect
37. #994C3⋊D3semidirect
94. H(5,𝔽2)C3×C3⋊D3semidirect
201. C3GL(2,𝔽3)⋊A4semidirect

Maximal subgroups

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

subgroupindex
C3×Ω+(4,𝔽3)2
#105965873
GL(2,𝔽3)⋊A43
#5836⋊C33
C3⋊D3×SL(2,𝔽3)4
C3×C3⋊GL(2,𝔽3)4