Finite Group Databasebeta

← C22×D3×AGL(1,𝔽9) #13181819

Subgroups — C22×D3×AGL(1,𝔽9) #13181819

Order 1728.

842 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 228 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×D3×#157C211yesmaximal, characteristic
2C22×C3⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
3C22×C3×AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
4C2×D3×AGL(1,𝔽9)C2112yesmaximal, direct product factor
5C23×AGL(1,𝔽9)—31nomaximal
6C22×D3×C3⋊D3C411yescharacteristic
7C22×C3⋊#157C411yescharacteristic
8C22×C3×#157C2211yescharacteristic
9C2×D3×#157C416yes—
10C2×D3×#157C2216yes—
11C2×C3⋊AGL(1,𝔽9)C2216yescomplemented
12C2×C3×AGL(1,𝔽9)C2216yescomplemented
13D3×AGL(1,𝔽9)C22116yesdirect product factor
14C23×#157—31no—
15C22×AGL(1,𝔽9)—31no—
16C22×AGL(1,𝔽9)—312no—
17C22×AGL(1,𝔽9)D311yescharacteristic, direct product factor
18C22×C3⋊(C3⋊D3)C811yescharacteristic, complemented
19C22×C3×C3⋊D3C2×C411yescharacteristic
20C22×C32×D3C811yescharacteristic, complemented
21C2×D3×C3⋊D3C816yescomplemented
22C2×D3×C3⋊D3C2×C416yes—
23C2×C3⋊#157C2×C416yes—
24C2×C3×#157C2×C413yes—
25C2×C3×#157C2313yes—
26D3×#157C2×C4112yes—
27D3×#157C2314yes—
28C3⋊AGL(1,𝔽9)C2314yescomplemented
29C3×AGL(1,𝔽9)C2314yescomplemented
30C22×D3×C8—91nomaximal
31C23×C3⋊D3—31no—
32C22×D32—121no—
33C22×#157—31no—
34C22×#157—36no—
35C22×#157—36no—
36C22×#157C2×D311yescharacteristic
37C2×AGL(1,𝔽9)—36no—
38C2×AGL(1,𝔽9)—316no—
39C2×AGL(1,𝔽9)C2×D316yesdirect product factor
40C22×C33C2×C811yescharacteristic, complemented
41C2×C3⋊(C3⋊D3)C2×C816yescomplemented
42C2×C3×C3⋊D3C2×C813yescomplemented
43C2×C3×C3⋊D3C22×C413yes—
44C2×C32×D3C2×C816yescomplemented
45D3×C3⋊D3C2×C8112yescomplemented
46D3×C3⋊D3C22×C414yes—
47C3⋊#157C22×C414yes—
48C3×#157C22×C413yes—
49C3×#157C2411yescharacteristic
50C22×C4×D3—91no—
51C22×C3×C8—91no—
52C22×C3⋊C8—91no—
53C2×D3×C8—912no—
54C23×C32—31no—
55C22×C3⋊D3—31no—
56C22×C3⋊D3—36no—
57C22×C3⋊D3—36no—
58C22×C3⋊D3—121no—
59C22×C3⋊D3—241no—
60C22×C3⋊D3C4×D311yescharacteristic
61C22×C3×D3—41no—
62C22×C3×D3—121no—
63C2×D32—126no—
64C2×D32—126no—
65C2×#157—36no—
66C2×#157—312no—
67C2×#157—34no—
68C2×#157C4×D313yes—
69C2×#157C22×D313yes—
70AGL(1,𝔽9)—34no—
71AGL(1,𝔽9)C22×D314yesdirect product factor
72C23×C8—271noSylow
73C2×C33C22×C813yescomplemented
74C3⋊(C3⋊D3)C22×C814yescomplemented
75C3×C3⋊D3C22×C813yescomplemented
76C3×C3⋊D3C23×C411yescharacteristic
77C32×D3C22×C814yescomplemented
78C23×D3—91no—
79C23×D3—361no—
80C22×C3×C4—91no—
81C22×Dic3—91no—
82C2×C4×D3—96no—
83C2×C4×D3—96no—
84C2×C3×C8—96no—
85C2×C3⋊C8—96no—
86D3×C8—916no—
87C22×C32—36no—
88C22×C32—41no—
89C22×C32—81no—
90C22×C32D3×C811yescharacteristic, complemented
91C2×C3⋊D3—36no—
92C2×C3⋊D3—312no—
93C2×C3⋊D3—34no—
94C2×C3⋊D3—126no—
95C2×C3⋊D3—246no—
96C2×C3⋊D3D3×C813yescomplemented
97C2×C3⋊D3C2×C4×D313yes—
98C2×C3×D3—46no—
99C2×C3×D3—123no—
100C2×C3×D3—123no—
101D32—1212no—
102D32—124no—
103#157—34no—
104#157C2×C4×D313yes—
105#157C23×D311yescharacteristic
106C23×C4—271no—
107C22×C8—91no—
108C22×C8—2712no—
109C22×C8—271no—
110C33C23×C811yescharacteristic, complemented, Sylow
111C23×C3—91no—
112C23×C3—121no—
113C22×D3—91no—
114C22×D3—96no—
115C22×D3—96no—
116C22×D3—121no—
117C22×D3—366no—
118C22×D3—366no—
119C22×D3—361no—
120C22×D3—721no—
121C22×D3AGL(1,𝔽9)11yescharacteristic, direct product factor
122C2×C3×C4—93no—
123C2×C3×C4—93no—
124C2×Dic3—96no—
125C4×D3—912no—
126C4×D3—94no—
127C3×C8—94no—
128C3⋊C8—94no—
129C2×C32—34no—
130C2×C32—43no—
131C2×C32—83no—
132C2×C32C2×D3×C813yescomplemented
133C3⋊D3—34no—
134C3⋊D3—124no—
135C3⋊D3—244no—
136C3⋊D3C2×D3×C813yescomplemented
137C3⋊D3C22×C4×D311yescharacteristic
138C3×D3—44no—
139C3×D3—123no—
140C3×D3—121no—
141C24—271no—
142C22×C4—91no—
143C22×C4—276no—
144C22×C4—276no—
145C22×C4—271no—
146C2×C8—96no—
147C2×C8—2716no—
148C2×C8—276no—
149C22×C3—41no—
150C22×C3—81no—
151C22×C3—93no—
152C22×C3—93no—
153C22×C3—126no—
154C22×C3C2×AGL(1,𝔽9)11yescharacteristic, complemented
155C2×D3—96no—
156C2×D3—94no—
157C2×D3—912no—
158C2×D3—123no—
159C2×D3—123no—
160C2×D3—364no—
161C2×D3—3612no—
162C2×D3—366no—
163C2×D3—726no—
164C2×D3C2×AGL(1,𝔽9)16yesdirect product factor
165C3×C4—91no—
166C3×C4—93no—
167Dic3—94no—
168C32—41no—
169C32—81no—
170C32C22×D3×C811yescharacteristic, complemented
171C23—31no—
172C23—91no—
173C23—276no—
174C23—276no—
175C23—271no—
176C2×C4—93no—
177C2×C4—93no—
178C2×C4—274no—
179C2×C4—2712no—
180C2×C4—276no—
181C8—94no—
182C8—274no—
183C2×C3—43no—
184C2×C3—83no—
185C2×C3—91no—
186C2×C3—93no—
187C2×C3—124no—
188C2×C3C22×AGL(1,𝔽9)13yescomplemented
189D3—94no—
190D3—123no—
191D3—121no—
192D3—364no—
193D3—724no—
194D3C22×AGL(1,𝔽9)14yesdirect product factor
195C22—36no—
196C22—93no—
197C22—93no—
198C22—274no—
199C22—2712no—
200C22—276no—
201C22D3×AGL(1,𝔽9)11yescharacteristic, direct product factor
202C4—91no—
203C4—93no—
204C4—274no—
205C3—41no—
206C3—81no—
207C3C23×AGL(1,𝔽9)11yescharacteristic, complemented
208C2—34no—
209C2—91no—
210C2—93no—
211C2—274no—
212C2C2×D3×AGL(1,𝔽9)13yesdirect product factor
subgroupG/H
center, hypercenterC22D3×AGL(1,𝔽9)
commutator, hypocenterC33C23×C8
FrattiniC1G
Fitting, socleC22×C33C2×C8
prime radicalC22×D3×C3⋊D3C4
cosocleC3×#157C24
LCSG ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊AGL(1,𝔽9)C2semidirect
3. C22×C3×AGL(1,𝔽9)C2semidirect
4. C2×D3×AGL(1,𝔽9)C2direct (N × H)
11. C2×C3⋊AGL(1,𝔽9)C22semidirect
12. C2×C3×AGL(1,𝔽9)C22semidirect
13. D3×AGL(1,𝔽9)C22direct (N × H)
17. C22×AGL(1,𝔽9)D3direct (N × H)
18. C22×C3⋊(C3⋊D3)C8semidirect
20. C22×C32×D3C8semidirect
21. C2×D3×C3⋊D3C8semidirect
28. C3⋊AGL(1,𝔽9)C23semidirect
29. C3×AGL(1,𝔽9)C23semidirect
39. C2×AGL(1,𝔽9)C2×D3direct (N × H)
40. C22×C33C2×C8semidirect
41. C2×C3⋊(C3⋊D3)C2×C8semidirect
42. C2×C3×C3⋊D3C2×C8semidirect
44. C2×C32×D3C2×C8semidirect
45. D3×C3⋊D3C2×C8semidirect
71. AGL(1,𝔽9)C22×D3direct (N × H)
73. C2×C33C22×C8semidirect
74. C3⋊(C3⋊D3)C22×C8semidirect
75. C3×C3⋊D3C22×C8semidirect
77. C32×D3C22×C8semidirect
90. C22×C32D3×C8semidirect
96. C2×C3⋊D3D3×C8semidirect
110. C33C23×C8semidirect
121. C22×D3AGL(1,𝔽9)direct (N × H)
132. C2×C32C2×D3×C8semidirect
136. C3⋊D3C2×D3×C8semidirect
154. C22×C3C2×AGL(1,𝔽9)semidirect
164. C2×D3C2×AGL(1,𝔽9)direct (N × H)
170. C32C22×D3×C8semidirect
188. C2×C3C22×AGL(1,𝔽9)semidirect
194. D3C22×AGL(1,𝔽9)direct (N × H)
201. C22D3×AGL(1,𝔽9)direct (N × H)
207. C3C23×AGL(1,𝔽9)semidirect
212. C2C2×D3×AGL(1,𝔽9)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×#1572
C22×C3⋊AGL(1,𝔽9)2
C22×C3×AGL(1,𝔽9)2
C2×D3×AGL(1,𝔽9)2
C23×AGL(1,𝔽9)3
C22×D3×C89