Finite Group Databasebeta

← 3−1+2×#237 #13048856

Subgroups — 3−1+2×#237 #13048856

Order 1296.

408 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 161 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×D4×3−1+2C211yesmaximal, characteristic, complemented
2#67×3−1+2C212yesmaximal, complemented
3C2×Dic3×3−1+2C212yesmaximal, complemented
4C4×D3×3−1+2C211yesmaximal, characteristic, complemented
5Dic6×3−1+2C211yesmaximal, characteristic, complemented
6C32×#237C311yesmaximal, characteristic
7#41×3−1+2—31nomaximal
8C9×#237C313yesmaximal, complemented
9C22×C3×3−1+2C2212yes—
10C2×D3×3−1+2C2211yescharacteristic
11C3×C4×3−1+2C2211yescharacteristic
12Dic3×3−1+2C2212yes—
13Dic3×3−1+2C2211yescharacteristic
14C33×D4C2×C311yescharacteristic
15C32×#67C2×C312yes—
16C2×C32×Dic3C2×C312yes—
17C32×C4×D3C2×C311yescharacteristic
18C32×Dic6C2×C311yescharacteristic
19Q8×3−1+2—31no—
20D4×3−1+2—21no—
21D4×3−1+2—23no—
22D4×3−1+2—32no—
23D4×3−1+2D311yescharacteristic, complemented
24C3×D4×C9C2×C313yescomplemented
25C2×C4×3−1+2—32no—
26C2×C4×3−1+2—31no—
27C9×#67C2×C316yescomplemented
28C2×C9×Dic3C2×C316yescomplemented
29C4×D3×C9C2×C313yescomplemented
30C9×Dic6C2×C313yescomplemented
31C2×C3×3−1+2—22no—
32C2×C3×3−1+2C2311yescharacteristic
33D3×3−1+2—21no—
34C32×#41—31no—
35C3×#237—31no—
36C3×#237C3211yescharacteristic
37C9×#41—33no—
38C22×C33C22×C312yes—
39C2×C32×D3C22×C311yescharacteristic
40C33×C4C22×C311yescharacteristic
41C32×Dic3C22×C312yes—
42C32×Dic3C22×C311yescharacteristic
43C22×3−1+2—22no—
44C22×3−1+2—26no—
45C22×3−1+2—31no—
46C22×3−1+2C2×D312yes—
47C22×C3×C9C22×C316yes—
48C2×D3×C9C22×C313yes—
49C4×3−1+2—21no—
50C4×3−1+2—23no—
51C4×3−1+2—31no—
52C4×3−1+2—32no—
53C4×3−1+2C2×D311yescharacteristic
54C3×C4×C9C22×C313yes—
55C9×Dic3C22×C316yes—
56C9×Dic3C22×C313yes—
57C3×3−1+2#4111yescharacteristic, complemented, Sylow
58C32×Q8—31no—
59C32×D4—21no—
60C32×D4—31no—
61C32×D4—32no—
62C32×D4—61no—
63C32×D4C3×D311yescharacteristic
64C32×D4C2×C3211yescharacteristic
65C2×C32×C4—32no—
66C2×C32×C4—31no—
67C3×#67—32no—
68C3×#67C2×C3212yes—
69C2×C3×Dic3—32no—
70C2×C3×Dic3C2×C3212yes—
71C3×C4×D3—31no—
72C3×C4×D3C2×C3211yescharacteristic
73C3×Dic6—31no—
74C3×Dic6C2×C3211yescharacteristic
75Q8×C9—33no—
76D4×C9—23no—
77D4×C9—36no—
78D4×C9C3×D313yescomplemented
79C2×C4×C9—36no—
80C2×C4×C9—33no—
81C2×C33—22no—
82C2×C33C23×C311yescharacteristic
83C32×D3—21no—
84C2×3−1+2—22no—
85C2×3−1+2—21no—
86C2×3−1+2—23no—
87C2×3−1+2—46no—
88C2×3−1+2—42no—
89C2×3−1+2—61no—
90C2×3−1+2C22×D311yescharacteristic
91C2×C3×C9—26no—
92C2×C3×C9C23×C313yes—
93D3×C9—23no—
94C3×#41—31no—
95C3×#41—91no—
96#2373−1+211yescharacteristic, direct product factor
97C22×C32—22no—
98C22×C32—32no—
99C22×C32—31no—
100C22×C32—62no—
101C22×C32C2×C3×D312yes—
102C22×C32C22×C3212yes—
103C2×C3×D3—31no—
104C2×C3×D3C22×C3211yescharacteristic
105C32×C4—21no—
106C32×C4—31no—
107C32×C4—31no—
108C32×C4—32no—
109C32×C4—61no—
110C32×C4C2×C3×D311yescharacteristic
111C32×C4C22×C3211yescharacteristic
112C3×Dic3—32no—
113C3×Dic3—31no—
114C3×Dic3C22×C3212yes—
115C3×Dic3C22×C3211yescharacteristic
116C22×C9—26no—
117C22×C9—33no—
118C22×C9C2×C3×D316yes—
119C4×C9—23no—
120C4×C9—33no—
121C4×C9—36no—
122C4×C9C2×C3×D313yes—
123C33C3×#4111yescharacteristic
1243−1+2—21no—
1253−1+2—23no—
1263−1+2#23711yescharacteristic, direct product factor
127C3×C9C3×#4113yescomplemented
128C3×Q8—31no—
129C3×Q8—91no—
130C3×D4—21no—
131C3×D4—32no—
132C3×D4—31no—
133C3×D4—61no—
134C3×D4—92no—
135C3×D4C2×3−1+211yescharacteristic, complemented
136C3×D4C32×D311yescharacteristic
137C2×C3×C4—32no—
138C2×C3×C4—31no—
139C2×C3×C4—91no—
140C2×C3×C4—92no—
141#67C2×3−1+212yescomplemented
142C2×Dic3C2×3−1+212yescomplemented
143C4×D3C2×3−1+211yescharacteristic, complemented
144Dic6C2×3−1+211yescharacteristic, complemented
145C2×C32—22no—
146C2×C32—22no—
147C2×C32—21no—
148C2×C32—31no—
149C2×C32—42no—
150C2×C32—61no—
151C2×C32—62no—
152C2×C32—61no—
153C2×C32—122no—
154C2×C32C22×C3×D311yescharacteristic
155C2×C32C23×C3211yescharacteristic
156C3×D3—21no—
157C3×D3—61no—
158C2×C9—26no—
159C2×C9—23no—
160C2×C9—46no—
161C2×C9—63no—
162C2×C9C22×C3×D313yes—
163#41—31noSylow
164C22×C3—22no—
165C22×C3—31no—
166C22×C3—32no—
167C22×C3—62no—
168C22×C3—91no—
169C22×C3C22×3−1+212yes—
170C22×C3C2×C32×D312yes—
171C2×D3C22×3−1+211yescharacteristic
172C3×C4—21no—
173C3×C4—31no—
174C3×C4—32no—
175C3×C4—31no—
176C3×C4—61no—
177C3×C4—91no—
178C3×C4—92no—
179C3×C4C22×3−1+211yescharacteristic
180C3×C4C2×C32×D311yescharacteristic
181Dic3C22×3−1+212yes—
182Dic3C22×3−1+211yescharacteristic
183C32—21no—
184C32—31no—
185C32—61no—
186C32C3×#23711yescharacteristic
187C32C32×#4111yescharacteristic
188C9—23no—
189C9C3×#23713yescomplemented
190Q8—31no—
191D4—32no—
192D4D3×3−1+211yescharacteristic, complemented
193C2×C4—32no—
194C2×C4—31no—
195C2×C3—22no—
196C2×C3—22no—
197C2×C3—21no—
198C2×C3—31no—
199C2×C3—42no—
200C2×C3—62no—
201C2×C3—61no—
202C2×C3—61no—
203C2×C3—122no—
204C2×C3—181no—
205C2×C3C23×3−1+211yescharacteristic
206C2×C3C22×C32×D311yescharacteristic
207D3—21no—
208C22—31no—
209C22C2×D3×3−1+212yes—
210C4—31no—
211C4—32no—
212C4C2×D3×3−1+211yescharacteristic
213C3—21no—
214C3—31no—
215C3—61no—
216C3#41×3−1+211yescharacteristic, complemented
217C3C32×#23711yescharacteristic
218C2—22no—
219C2—61no—
220C2C22×D3×3−1+211yescharacteristic
subgroupG/H
center, FrattiniC2×C3C22×C32×D3
commutator, socle, cosocleC2×C32C23×C32
FittingC3×D4×3−1+2C2
prime radicalC32×#237C3
hypercenterD4×3−1+2D3
hypocenterC3#41×3−1+2
LCSG ▷ C2×C32 ▷ C3
UCSC1 ◁ C2×C3 ◁ D4×3−1+2

Complements

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

NH ≅ G/Ntype
1. C3×D4×3−1+2C2semidirect
2. #67×3−1+2C2semidirect
3. C2×Dic3×3−1+2C2semidirect
4. C4×D3×3−1+2C2semidirect
5. Dic6×3−1+2C2semidirect
8. C9×#237C3semidirect
23. D4×3−1+2D3semidirect
24. C3×D4×C9C2×C3semidirect
27. C9×#67C2×C3semidirect
28. C2×C9×Dic3C2×C3semidirect
29. C4×D3×C9C2×C3semidirect
30. C9×Dic6C2×C3semidirect
57. C3×3−1+2#41semidirect
78. D4×C9C3×D3semidirect
96. #2373−1+2direct (N × H)
126. 3−1+2#237direct (N × H)
127. C3×C9C3×#41semidirect
135. C3×D4C2×3−1+2semidirect
141. #67C2×3−1+2semidirect
142. C2×Dic3C2×3−1+2semidirect
143. C4×D3C2×3−1+2semidirect
144. Dic6C2×3−1+2semidirect
189. C9C3×#237semidirect
192. D4D3×3−1+2semidirect
216. C3#41×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C3×D4×3−1+22
#67×3−1+22
C2×Dic3×3−1+22
C4×D3×3−1+22
Dic6×3−1+22
C32×#2373
#41×3−1+23
C9×#2373