Finite Group Databasebeta

← C22×(C3⋊GL(2,𝔽3))⋊C3 #13164415

Subgroups — C22×(C3⋊GL(2,𝔽3))⋊C3 #13164415

Order 1728.

830 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×(C3⋊GL(2,𝔽3))⋊C3C216yesmaximal, direct product factor
3C22×C3⋊GL(2,𝔽3)C311yesmaximal, characteristic, complemented
4C22×C3×#215—31nomaximal
5C23×(C3⋊D3)⋊C3—41nomaximal
6C2×Q8⋊H(3,𝔽3)C2213yescomplemented
7(C3⋊GL(2,𝔽3))⋊C3C2214yesdirect product factor
8C22×C32×Q8D311yescharacteristic, complemented
9C22×C3×D12—31no—
10C22×C3×SL(2,𝔽3)—21no—
11C22×C3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
12C2×C3⋊GL(2,𝔽3)C2×C316yescomplemented
13C2×C3×#215—36no—
14C2×C3×#215—36no—
15C22×C3×C3⋊C8—31no—
16C23×H(3,𝔽3)—41no—
17C22×(C3⋊D3)⋊C3—46no—
18C22×(C3⋊D3)⋊C3—48no—
19Q8⋊H(3,𝔽3)C2311yescharacteristic, complemented
20C22×GL(2,𝔽3)—91no—
21C22×C3×SD16—91no—
22C22×#215—31no—
23C23×C3⋊D3—41no—
24C23×C3×D3—61no—
25C2×C32×Q8—33no—
26C2×C32×Q8C2×D313yescomplemented
27C22×C32×C4—31no—
28C2×C3×D12—36no—
29C2×C3×D12—63no—
30C2×C3×SL(2,𝔽3)—23no—
31C2×C3×SL(2,𝔽3)C22×C313yescomplemented
32C3⋊GL(2,𝔽3)C22×C314yescomplemented
33C3×#215—312no—
34C3×#215—34no—
35C2×C3×C3⋊C8—36no—
36C22×H(3,𝔽3)—44no—
37C22×H(3,𝔽3)—43no—
38C2×(C3⋊D3)⋊C3—44no—
39C2×(C3⋊D3)⋊C3—424no—
40C22×C3×Q8—31no—
41C22×C3×Q8C3×D311yescharacteristic
42C22×C3×D4—91no—
43C22×D12—31no—
44C22×SL(2,𝔽3)—31no—
45C22×SL(2,𝔽3)—61no—
46C2×GL(2,𝔽3)—96no—
47C2×C3×SD16—96no—
48C2×C3×SD16—96no—
49C22×C3×C8—91no—
50C2×#215—36no—
51C2×#215—36no—
52C22×C3⋊C8—31no—
53C23×C32—41no—
54C23×C32—81no—
55C23×C32S411yescharacteristic
56C22×C3⋊D3—46no—
57C22×C3⋊D3—48no—
58C22×C3×D3—66no—
59C22×C3×D3—124no—
60C32×Q8—33no—
61C32×Q8—61no—
62C32×Q8C22×D311yescharacteristic, complemented
63C2×C32×C4—33no—
64C2×C32×C4—33no—
65C3×D12—34no—
66C3×D12—66no—
67C3×SL(2,𝔽3)—21no—
68C3×SL(2,𝔽3)C23×C311yescharacteristic, complemented
69C3×C3⋊C8—34no—
70C22×SD16—91noSylow
71C2×H(3,𝔽3)—46no—
72C2×H(3,𝔽3)—41no—
73(C3⋊D3)⋊C3—48no—
74C24×C3—181no—
75C23×D3—61no—
76C23×D3—361no—
77C2×C3×Q8—33no—
78C2×C3×Q8—33no—
79C2×C3×Q8—33no—
80C2×C3×Q8—63no—
81C2×C3×Q8C2×C3×D313yes—
82C2×C3×D4—96no—
83C2×C3×D4—183no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—61no—
87C2×D12—36no—
88C2×D12—63no—
89C2×SL(2,𝔽3)—33no—
90C2×SL(2,𝔽3)—63no—
91GL(2,𝔽3)—94no—
92C3×SD16—912no—
93C3×SD16—94no—
94C2×C3×C8—96no—
95#215—312no—
96#215—34no—
97C2×C3⋊C8—36no—
98C22×C32—43no—
99C22×C32—44no—
100C22×C32—83no—
101C22×C32—84no—
102C22×C32GL(2,𝔽3)14yescomplemented
103C22×C32C2×S413yes—
104C2×C3⋊D3—44no—
105C2×C3⋊D3—424no—
106C2×C3×D3—64no—
107C2×C3×D3—1212no—
108C32×C4—31no—
109C32×C4—33no—
110C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
111C22×D4—91no—
112C2×SD16—96no—
113C2×SD16—96no—
114C22×C8—91no—
115H(3,𝔽3)—41noSylow
116C23×C3—31no—
117C23×C3—121no—
118C23×C3—186no—
119C23×C3—241no—
120C23×C3—364no—
121C23×C3C3×S411yescharacteristic
122C22×D3—66no—
123C22×D3—124no—
124C22×D3—366no—
125C22×D3—368no—
126C3×Q8—33no—
127C3×Q8—33no—
128C3×Q8—31no—
129C3×Q8—63no—
130C3×Q8—61no—
131C3×Q8—63no—
132C3×Q8C22×C3×D311yescharacteristic
133C3×D4—94no—
134C3×D4—186no—
135C2×C3×C4—33no—
136C2×C3×C4—33no—
137C2×C3×C4—33no—
138C2×C3×C4—33no—
139C2×C3×C4—63no—
140C2×C3×C4—63no—
141D12—34no—
142D12—66no—
143SL(2,𝔽3)—31no—
144SL(2,𝔽3)—61no—
145C3×C8—94no—
146C3⋊C8—34no—
147C2×C32—41no—
148C2×C32—46no—
149C2×C32—81no—
150C2×C32—86no—
151C2×C32C2×GL(2,𝔽3)16yescomplemented
152C2×C32C22×S411yescharacteristic
153C3⋊D3—48no—
154C3×D3—124no—
155C24—181no—
156C2×Q8—33no—
157C2×Q8C2×(C3⋊D3)⋊C313yescomplemented
158C2×D4—96no—
159C2×D4—183no—
160C22×C4—31no—
161SD16—912no—
162SD16—94no—
163C2×C8—96no—
164C22×C3—34no—
165C22×C3—33no—
166C22×C3—124no—
167C22×C3—123no—
168C22×C3—184no—
169C22×C3—244no—
170C22×C3—243no—
171C22×C3—3612no—
172C22×C3C3×GL(2,𝔽3)14yes—
173C22×C3C2×C3×S413yes—
174C2×D3—64no—
175C2×D3—1212no—
176C2×D3—364no—
177C2×D3—3624no—
178C3×C4—31no—
179C3×C4—33no—
180C3×C4—31no—
181C3×C4—33no—
182C3×C4—61no—
183C3×C4—63no—
184C32—41no—
185C32—81no—
186C32C22×GL(2,𝔽3)11yescharacteristic, complemented
187C23—186no—
188C23—364no—
189C23(C3⋊S4)⋊C311yescharacteristic
190Q8—33no—
191Q8—61no—
192Q8C22×(C3⋊D3)⋊C311yescharacteristic, complemented
193D4—94no—
194D4—186no—
195C2×C4—33no—
196C2×C4—33no—
197C8—94no—
198C2×C3—31no—
199C2×C3—36no—
200C2×C3—121no—
201C2×C3—126no—
202C2×C3—241no—
203C2×C3—246no—
204C2×C3—364no—
205C2×C3C2×C3×GL(2,𝔽3)16yes—
206C2×C3C22×C3×S411yescharacteristic
207D3—124no—
208D3—368no—
209C22—184no—
210C22—3612no—
211C22(C3⋊GL(2,𝔽3))⋊C314yesdirect product factor
212C22C2×(C3⋊S4)⋊C313yes—
213C4—31no—
214C4—33no—
215C3—31no—
216C3—121no—
217C3—241no—
218C3C22×C3×GL(2,𝔽3)11yescharacteristic
219C2—364no—
220C2C2×(C3⋊GL(2,𝔽3))⋊C316yesdirect product factor
221C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC23(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×Q8D3
socleC23×C3C3×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×Q8⋊H(3,𝔽3)C2semidirect
2. C2×(C3⋊GL(2,𝔽3))⋊C3C2direct (N × H)
3. C22×C3⋊GL(2,𝔽3)C3semidirect
6. C2×Q8⋊H(3,𝔽3)C22semidirect
7. (C3⋊GL(2,𝔽3))⋊C3C22direct (N × H)
8. C22×C32×Q8D3semidirect
11. C22×C3×SL(2,𝔽3)C2×C3semidirect
12. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
19. Q8⋊H(3,𝔽3)C23semidirect
26. C2×C32×Q8C2×D3semidirect
31. C2×C3×SL(2,𝔽3)C22×C3semidirect
32. C3⋊GL(2,𝔽3)C22×C3semidirect
62. C32×Q8C22×D3semidirect
68. C3×SL(2,𝔽3)C23×C3semidirect
102. C22×C32GL(2,𝔽3)semidirect
110. C22×Q8(C3⋊D3)⋊C3semidirect
151. C2×C32C2×GL(2,𝔽3)semidirect
157. C2×Q8C2×(C3⋊D3)⋊C3semidirect
186. C32C22×GL(2,𝔽3)semidirect
192. Q8C22×(C3⋊D3)⋊C3semidirect
211. C22(C3⋊GL(2,𝔽3))⋊C3direct (N × H)
220. C2C2×(C3⋊GL(2,𝔽3))⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
C2×(C3⋊GL(2,𝔽3))⋊C32
C22×C3⋊GL(2,𝔽3)3
C22×C3×#2153
C23×(C3⋊D3)⋊C34