Finite Group Databasebeta

← D3×A4×H(3,𝔽3) #14519239

Subgroups — D3×A4×H(3,𝔽3) #14519239

Order 1944.

891 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 891 conjugacy classes of subgroups — too many to draw. Only its 117 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×A4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C32×D3×A4C314yesmaximal, complemented
3C22×D3×H(3,𝔽3)C311yesmaximal, characteristic, complemented
4C2×A4×H(3,𝔽3)—31nomaximal
5D3×C22⋊H(3,𝔽3)C318yesmaximal, complemented
6C3×D3×H(3,𝔽3)—41nomaximal
7C33×A4C2×C314yescomplemented
8C22×C3×H(3,𝔽3)C2×C311yescharacteristic, complemented
9C2×D3×H(3,𝔽3)—31no—
10C2×D3×H(3,𝔽3)—31no—
11C3×C22⋊H(3,𝔽3)C2×C318yescomplemented
12A4×H(3,𝔽3)—24no—
13A4×H(3,𝔽3)—21no—
14A4×H(3,𝔽3)—28no—
15A4×H(3,𝔽3)D311yescharacteristic, direct product factor
16C32×H(3,𝔽3)—41noSylow
17C22×C32×D3C3214yescomplemented
18C2×C32×A4—34no—
19C3×D3×A4—312no—
20C3×D3×A4C3211yescharacteristic
21C3×D3×A4C3218yescomplemented
22C23×H(3,𝔽3)—31no—
23C2×C22⋊H(3,𝔽3)—38no—
24C33×D3—44no—
25C2×C3×H(3,𝔽3)—31no—
26C2×C3×H(3,𝔽3)—121no—
27D3×H(3,𝔽3)—31no—
28D3×H(3,𝔽3)—48no—
29D3×H(3,𝔽3)A411yescharacteristic, direct product factor
30C22×C33C2×C3214yescomplemented
31C2×C32×D3—34no—
32C2×C32×D3—34no—
33C32×A4—24no—
34C32×A4—24no—
35C32×A4—28no—
36C32×A4—312no—
37C32×A4—612no—
38C32×A4C3×D314yescomplemented
39C32×A4C2×C3211yescharacteristic
40C32×A4C2×C3218yescomplemented
41C22×H(3,𝔽3)—24no—
42C22×H(3,𝔽3)—91no—
43C22×H(3,𝔽3)—91no—
44C22×H(3,𝔽3)C3×D311yescharacteristic, complemented
45C22⋊H(3,𝔽3)—28no—
46C22⋊H(3,𝔽3)—224no—
47C22⋊H(3,𝔽3)C3×D318yescomplemented
48C34—44no—
49C3×H(3,𝔽3)—41no—
50C3×H(3,𝔽3)—48no—
51C3×H(3,𝔽3)—88no—
52C3×H(3,𝔽3)—84no—
53C3×H(3,𝔽3)—81no—
54C3×H(3,𝔽3)C2×A411yescharacteristic, complemented
55C23×C32—34no—
56C22×C3×D3—34no—
57C22×C3×D3C3311yescharacteristic
58C2×C3×A4—31no—
59C2×C3×A4—38no—
60C2×C3×A4—912no—
61D3×A4—38no—
62D3×A4H(3,𝔽3)13yesdirect product factor
63C2×C33—34no—
64C2×C33—124no—
65C32×D3—34no—
66C32×D3—41no—
67C32×D3—48no—
68C32×D3—1212no—
69C32×D3C3×A414yescomplemented
70C2×H(3,𝔽3)—31no—
71C2×H(3,𝔽3)—31no—
72C2×H(3,𝔽3)—64no—
73C2×H(3,𝔽3)—91no—
74C2×H(3,𝔽3)—128no—
75C22×C32—24no—
76C22×C32—34no—
77C22×C32—64no—
78C22×C32—94no—
79C22×C32—94no—
80C22×C32C32×D314yescomplemented
81C22×C32C2×C3311yescharacteristic
82C2×C3×D3—31no—
83C2×C3×D3—31no—
84C2×C3×D3—94no—
85C2×C3×D3—94no—
86C3×A4—21no—
87C3×A4—23no—
88C3×A4—28no—
89C3×A4—312no—
90C3×A4—38no—
91C3×A4—624no—
92C3×A4—612no—
93C3×A4—68no—
94C3×A4—612no—
95C3×A4C2×H(3,𝔽3)13yescomplemented
96C3×A4C32×D311yescharacteristic
97C3×A4C32×D318yescomplemented
98C33—41no—
99C33—44no—
100C33—48no—
101C33—84no—
102C33—88no—
103C33—84no—
104C33—1212no—
105C33—2412no—
106C33C2×C3×A414yescomplemented
107H(3,𝔽3)—24no—
108H(3,𝔽3)—48no—
109H(3,𝔽3)—824no—
110H(3,𝔽3)—88no—
111H(3,𝔽3)D3×A411yescharacteristic, direct product factor
112C23×C3—31no—
113C23×C3—94no—
114C22×D3C3×H(3,𝔽3)11yescharacteristic, complemented
115C2×A4—33no—
116C2×A4—98no—
117C2×C32—31no—
118C2×C32—34no—
119C2×C32—34no—
120C2×C32—64no—
121C2×C32—94no—
122C2×C32—94no—
123C2×C32—121no—
124C2×C32—128no—
125C2×C32—184no—
126C2×C32—3612no—
127C3×D3—31no—
128C3×D3—34no—
129C3×D3—43no—
130C3×D3—94no—
131C3×D3—128no—
132C3×D3C32×A411yescharacteristic
133C22×C3—21no—
134C22×C3—34no—
135C22×C3—64no—
136C22×C3—91no—
137C22×C3—91no—
138C22×C3—274no—
139C22×C3—274no—
140C22×C3C2×C3×H(3,𝔽3)11yescharacteristic, complemented
141C22×C3C33×D311yescharacteristic
142C2×D3—31no—
143C2×D3—31no—
144A4—23no—
145A4—38no—
146A4—68no—
147A4D3×H(3,𝔽3)13yesdirect product factor
148C32—24no—
149C32—34no—
150C32—41no—
151C32—43no—
152C32—48no—
153C32—64no—
154C32—83no—
155C32—81no—
156C32—88no—
157C32—1212no—
158C32—128no—
159C32—248no—
160C32—2412no—
161C32—2412no—
162C32—2424no—
163C32C3×D3×A414yescomplemented
164C32C2×C32×A411yescharacteristic
165C23—31noSylow
166C2×C3—31no—
167C2×C3—31no—
168C2×C3—31no—
169C2×C3—61no—
170C2×C3—94no—
171C2×C3—94no—
172C2×C3—91no—
173C2×C3—123no—
174C2×C3—184no—
175C2×C3—274no—
176C2×C3—368no—
177D3—31no—
178D3A4×H(3,𝔽3)11yescharacteristic, direct product factor
179C22—91no—
180C22—91no—
181C22C3×D3×H(3,𝔽3)11yescharacteristic, complemented
182C3—21no—
183C3—34no—
184C3—43no—
185C3—64no—
186C3—83no—
187C3—128no—
188C3—248no—
189C3C2×A4×H(3,𝔽3)11yescharacteristic, complemented
190C3C32×D3×A411yescharacteristic
191C2—31no—
192C2—31no—
193C2—91no—
subgroupG/H
center, FrattiniC3C32×D3×A4
commutator, socle, cosocleC22×C32C2×C33
FittingC22×C3×H(3,𝔽3)C2×C3
hypercenterH(3,𝔽3)D3×A4
hypocenterC22×C3C2×C3×H(3,𝔽3)
LCSG ▷ C22×C32 ▷ C22×C3
UCSC1 ◁ C3 ◁ H(3,𝔽3)

Complements

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

NH ≅ G/Ntype
1. C3×A4×H(3,𝔽3)C2semidirect
2. C32×D3×A4C3semidirect
3. C22×D3×H(3,𝔽3)C3semidirect
5. D3×C22⋊H(3,𝔽3)C3semidirect
7. C33×A4C2×C3semidirect
8. C22×C3×H(3,𝔽3)C2×C3semidirect
11. C3×C22⋊H(3,𝔽3)C2×C3semidirect
15. A4×H(3,𝔽3)D3direct (N × H)
17. C22×C32×D3C32semidirect
21. C3×D3×A4C32semidirect
29. D3×H(3,𝔽3)A4direct (N × H)
30. C22×C33C2×C32semidirect
38. C32×A4C3×D3semidirect
40. C32×A4C2×C32semidirect
44. C22×H(3,𝔽3)C3×D3semidirect
47. C22⋊H(3,𝔽3)C3×D3semidirect
54. C3×H(3,𝔽3)C2×A4semidirect
62. D3×A4H(3,𝔽3)direct (N × H)
69. C32×D3C3×A4semidirect
80. C22×C32C32×D3semidirect
95. C3×A4C2×H(3,𝔽3)semidirect
97. C3×A4C32×D3semidirect
106. C33C2×C3×A4semidirect
111. H(3,𝔽3)D3×A4direct (N × H)
114. C22×D3C3×H(3,𝔽3)semidirect
140. C22×C3C2×C3×H(3,𝔽3)semidirect
147. A4D3×H(3,𝔽3)direct (N × H)
163. C32C3×D3×A4semidirect
178. D3A4×H(3,𝔽3)direct (N × H)
181. C22C3×D3×H(3,𝔽3)semidirect
189. C3C2×A4×H(3,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C3×A4×H(3,𝔽3)2
C32×D3×A43
C22×D3×H(3,𝔽3)3
C2×A4×H(3,𝔽3)3
D3×C22⋊H(3,𝔽3)3
C3×D3×H(3,𝔽3)4