Finite Group Databasebeta

← #13049708

Subgroups — #13049708

Order 1296.

868 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 107 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×C4×C3⋊(C3⋊D3)C212yesmaximal, complemented
2C32×C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C3⋊#6296C211yesmaximal, characteristic, complemented
4#10620157C212yesmaximal, complemented
5#10620156C211yesmaximal, characteristic, complemented
6#89042—34nomaximal
7#89020—32nomaximal
8C2×C3×C3⋊(C3⋊D3)C2212yes—
9C2×C32×C3⋊D3C2211yescharacteristic
10C34×C4C2211yescharacteristic
11C3×C3⋊(C3⋊Dic3)C2212yes—
12C32×C3⋊Dic3C2211yescharacteristic
13C4×C3⋊(C3⋊D3)D312yescomplemented
14C3×C4×C3⋊D3—32no—
15C3×C4×C3⋊D3—38no—
16C3×C4×C3⋊D3—68no—
17C3×C4×C3⋊D3D314yescomplemented
18C32×C4×D3—32no—
19C32×C4×D3—61no—
20#6296—34no—
21#6295—38no—
22#6295—34no—
23#6293—32no—
24#6292—32no—
25#6291—32no—
26#6290—32no—
27C2×C34C2311yescharacteristic
28C3×C3⋊(C3⋊D3)—22no—
29C32×C3⋊D3—21no—
30#3828—91no—
31#3798—98no—
32#3798—91no—
33C2×C3⋊(C3⋊D3)C2×D312yes—
34C2×C3×C3⋊D3—32no—
35C2×C3×C3⋊D3—38no—
36C2×C3×C3⋊D3—68no—
37C2×C3×C3⋊D3C2×D314yes—
38C2×C32×D3—32no—
39C2×C32×D3—61no—
40C33×C4—21no—
41C33×C4—28no—
42C33×C4—44no—
43C33×C4C2×D312yes—
44C33×C4C2×D314yes—
45C3⋊(C3⋊Dic3)C2×D312yes—
46C3×C3⋊Dic3—32no—
47C3×C3⋊Dic3—38no—
48C3×C3⋊Dic3—68no—
49C3×C3⋊Dic3C2×D314yes—
50C32×Dic3—32no—
51C32×Dic3—61no—
52C34#4111yescharacteristic, complemented, Sylow
53C2×C32×C4—91no—
54#638—92no—
55C4⋊(C3⋊D3)—91no—
56C4×C3⋊D3—32no—
57C4×C3⋊D3—38no—
58C4×C3⋊D3—68no—
59C4×C3⋊D3C3⋊D311yescharacteristic, complemented
60#634—91no—
61C3×C4×D3—38no—
62C3×C4×D3—64no—
63C3×C4×D3—98no—
64C3×C4×D3—92no—
65C3×C4×D3—188no—
66#627—98no—
67#627—91no—
68#626—98no—
69#626—98no—
70#626—92no—
71#625—98no—
72#625—91no—
73C2×C33—21no—
74C2×C33—28no—
75C2×C33—44no—
76C2×C33C22×D312yes—
77C2×C33C22×D314yes—
78C3⋊(C3⋊D3)—22no—
79C3×C3⋊D3—24no—
80C3×C3⋊D3—616no—
81C3×C3⋊D3—68no—
82C3×C3⋊D3—62no—
83C32×D3—62no—
84C32×D3—62no—
85#235—272no—
86#235—274no—
87C22×C32—91no—
88C2×C3⋊D3—32no—
89C2×C3⋊D3—38no—
90C2×C3⋊D3—68no—
91C2×C3⋊D3C2×C3⋊D311yescharacteristic
92C2×C3×D3—38no—
93C2×C3×D3—64no—
94C2×C3×D3—98no—
95C2×C3×D3—92no—
96C2×C3×D3—188no—
97C32×C4—28no—
98C32×C4—28no—
99C32×C4—24no—
100C32×C4—44no—
101C32×C4—412no—
102C32×C4—44no—
103C32×C4—91no—
104C32×C4D3218yes—
105C32×C4D3211yescharacteristic
106C32×C4C2×C3⋊D311yescharacteristic
107C3⋊Dic3—32no—
108C3⋊Dic3—38no—
109C3⋊Dic3—68no—
110C3⋊Dic3C2×C3⋊D311yescharacteristic
111C3×Dic3—38no—
112C3×Dic3—64no—
113C3×Dic3—98no—
114C3×Dic3—92no—
115C3×Dic3—188no—
116C33—21no—
117C33—28no—
118C33—44no—
119C33#23512yescomplemented
120C33#23514yescomplemented
121C2×C3×C4—94no—
122C2×C3×C4—272no—
123#67—272no—
124#67—272no—
125#67—278no—
126D12—274no—
127D12—272no—
128C4×D3—32no—
129C4×D3—61no—
130C4×D3—92no—
131C4×D3—98no—
132C4×D3—188no—
133Dic6—272no—
134Dic6—274no—
135C2×C32—28no—
136C2×C32—28no—
137C2×C32—24no—
138C2×C32—44no—
139C2×C32—412no—
140C2×C32—44no—
141C2×C32—181no—
142C2×C32C2×D3218yes—
143C2×C32C2×D3211yescharacteristic
144C2×C32C22×C3⋊D311yescharacteristic
145C3⋊D3—21no—
146C3⋊D3—616no—
147C3⋊D3—68no—
148C3⋊D3—62no—
149C3×D3—68no—
150C3×D3—68no—
151C3×D3—1816no—
152C3×D3—188no—
153C3×D3—182no—
154#41—811noSylow
155C22×C3—94no—
156C22×C3—272no—
157C2×D3—32no—
158C2×D3—61no—
159C2×D3—92no—
160C2×D3—98no—
161C2×D3—188no—
162C3×C4—21no—
163C3×C4—28no—
164C3×C4—44no—
165C3×C4—94no—
166C3×C4—272no—
167C3×C4D3×C3⋊D312yes—
168C3×C4#111314yes—
169Dic3—32no—
170Dic3—61no—
171Dic3—92no—
172Dic3—98no—
173Dic3—188no—
174C32—28no—
175C32—28no—
176C32—24no—
177C32—44no—
178C32—412no—
179C32—44no—
180C32#379818yescomplemented
181C32#379811yescharacteristic, complemented
182C32#382811yescharacteristic, complemented
183Q8—811no—
184D4—811no—
185D4—812no—
186C2×C4—91no—
187C2×C4—272no—
188C2×C3—21no—
189C2×C3—28no—
190C2×C3—44no—
191C2×C3—184no—
192C2×C3—542no—
193C2×C3C2×D3×C3⋊D312yes—
194C2×C3C2×#111314yes—
195D3—62no—
196D3—62no—
197D3—1816no—
198D3—182no—
199D3—188no—
200C22—91no—
201C22—272no—
202C4—91no—
203C4—272no—
204C4C3⋊#111311yescharacteristic
205C3—21no—
206C3—28no—
207C3—44no—
208C3#8902012yescomplemented
209C3#8904214yescomplemented
210C2—181no—
211C2—542no—
212C2C2×C3⋊#111311yescharacteristic
subgroupG/H
center, hypercenterC4C3⋊#1113
commutator, socle, cosocleC2×C34C23
FrattiniC2C2×C3⋊#1113
FittingC34×C4C22
hypocenterC34#41
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×C4×C3⋊(C3⋊D3)C2semidirect
2. C32×C4×C3⋊D3C2semidirect
3. C3⋊#6296C2semidirect
4. #10620157C2semidirect
5. #10620156C2semidirect
13. C4×C3⋊(C3⋊D3)D3semidirect
17. C3×C4×C3⋊D3D3semidirect
52. C34#41semidirect
59. C4×C3⋊D3C3⋊D3semidirect
119. C33#235semidirect
120. C33#235semidirect
180. C32#3798semidirect
181. C32#3798semidirect
182. C32#3828semidirect
208. C3#89020semidirect
209. C3#89042semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C3⋊(C3⋊D3)2
C32×C4×C3⋊D32
C3⋊#62962
#106201572
#106201562
#890423
#890203