Finite Group Databasebeta

← #13049517

Subgroups — #13049517

Order 1296.

868 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 102 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⋊#6296C211yesmaximal, characteristic, complemented
2C3⋊D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3D3×C3⋊(C3⋊Dic3)C212yesmaximal, complemented
4C3×#6290C212yesmaximal, complemented
5C32×#625C211yesmaximal, characteristic, complemented
6#89029—32nomaximal
7#88963—34nomaximal
8C2×C32×C3⋊D3C2211yescharacteristic
9C2×C33×D3C2212yes—
10C3⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
11C3×C3⋊(C3⋊Dic3)C2212yes—
12C32×C3⋊Dic3C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)—32no—
14C3×#638—32no—
15C32×#67—32no—
16#6296—34no—
17#6293—32no—
18#6290D312yescomplemented
19Dic3×C3⋊D3—32no—
20Dic3×C3⋊D3—61no—
21D3×C3⋊Dic3—32no—
22D3×C3⋊Dic3—38no—
23D3×C3⋊Dic3—34no—
24D3×C3⋊Dic3—68no—
25C3×#625—38no—
26C3×#625D314yescomplemented
27C2×C34C2311yescharacteristic
28C32×C3⋊D3—21no—
29C33×D3—22no—
30#3830—91no—
31#3805—98no—
32#3805—91no—
33C22×C33—32no—
34C2×C3×C3⋊D3—21no—
35C2×C3×C3⋊D3C2×D312yes—
36C2×C32×D3—28no—
37C2×C32×D3—34no—
38C2×C32×D3C2×D318yes—
39C2×C32×D3C2×D312yes—
40C3⋊(C3⋊Dic3)—32no—
41C3⋊(C3⋊Dic3)—34no—
42C3⋊(C3⋊Dic3)—68no—
43C3⋊(C3⋊Dic3)—61no—
44C3⋊(C3⋊Dic3)—124no—
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
53C32×D4—91no—
54#638—32no—
55C2×C3⋊Dic3—92no—
56C2×C3⋊Dic3—98no—
57C2×C3⋊Dic3—91no—
58C2×C3⋊Dic3—188no—
59C4×C3⋊D3—91no—
60#634—91no—
61C3×#67—38no—
62C3×#67—98no—
63C3×#67—92no—
64#627—98no—
65#627—91no—
66#625—38no—
67#625C3⋊D311yescharacteristic, complemented
68D3×Dic3—98no—
69D3×Dic3—98no—
70D3×Dic3—92no—
71D3×Dic3—184no—
72D3×Dic3—188no—
73C2×C33—21no—
74C2×C33—28no—
75C2×C33—44no—
76C2×C33—62no—
77C2×C33C22×D312yes—
78C2×C33C22×D314yes—
79C3×C3⋊D3—22no—
80C3×C3⋊D3—22no—
81C32×D3—216no—
82C32×D3—28no—
83C32×D3—22no—
84C32×D3—64no—
85#237—272no—
86#237—274no—
87C22×C32—32no—
88C22×C32—38no—
89C22×C32—68no—
90C22×C32—91no—
91C2×C3⋊D3D3211yescharacteristic
92C2×C3×D3—28no—
93C2×C3×D3—38no—
94C2×C3×D3—64no—
95C2×C3×D3D3218yes—
96C2×C3×D3C2×C3⋊D312yes—
97C32×C4—91no—
98C3⋊Dic3—32no—
99C3⋊Dic3—38no—
100C3⋊Dic3—68no—
101C3⋊Dic3—91no—
102C3⋊Dic3—98no—
103C3⋊Dic3—91no—
104C3⋊Dic3—188no—
105C3⋊Dic3—188no—
106C3⋊Dic3—184no—
107C3⋊Dic3—364no—
108C3⋊Dic3—364no—
109C3⋊Dic3—3612no—
110C3⋊Dic3C2×C3⋊D311yescharacteristic
111C3×Dic3—38no—
112C3×Dic3—64no—
113C3×Dic3—98no—
114C3×Dic3—92no—
115C3×Dic3—188no—
116C33—28no—
117C33—21no—
118C33—44no—
119C33#23712yescomplemented
120C33#23714yescomplemented
121C3×D4—94no—
122C3×D4—272no—
123#67—32no—
124#67—98no—
125#67—92no—
126C2×Dic3—272no—
127C2×Dic3—272no—
128C2×Dic3—278no—
129C2×Dic3—541no—
130C2×Dic3—548no—
131C4×D3—274no—
132C4×D3—272no—
133Dic6—272no—
134Dic6—274no—
135C2×C32—28no—
136C2×C32—24no—
137C2×C32—28no—
138C2×C32—44no—
139C2×C32—412no—
140C2×C32—44no—
141C2×C32—616no—
142C2×C32—68no—
143C2×C32—62no—
144C2×C32—181no—
145C2×C32C2×D3218yes—
146C2×C32C2×D3211yescharacteristic
147C2×C32C22×C3⋊D311yescharacteristic
148C3⋊D3—21no—
149C3×D3—216no—
150C3×D3—28no—
151C3×D3—22no—
152C3×D3—68no—
153C3×D3—68no—
154#41—811noSylow
155C22×C3—38no—
156C22×C3—32no—
157C22×C3—68no—
158C22×C3—92no—
159C22×C3—181no—
160C2×D3—34no—
161C2×D3D3×C3⋊D312yes—
162C3×C4—94no—
163C3×C4—272no—
164Dic3—32no—
165Dic3—61no—
166Dic3—98no—
167Dic3—92no—
168Dic3—188no—
169Dic3—274no—
170Dic3—272no—
171Dic3—541no—
172Dic3—548no—
173Dic3—1084no—
174C32—28no—
175C32—24no—
176C32—28no—
177C32—44no—
178C32—412no—
179C32—44no—
180C32#380518yescomplemented
181C32#380511yescharacteristic, complemented
182C32#383011yescharacteristic, complemented
183Q8—811no—
184D4—91no—
185D4—272no—
186C2×C4—811no—
187C2×C4—812no—
188C2×C3—21no—
189C2×C3—28no—
190C2×C3—44no—
191C2×C3—616no—
192C2×C3—68no—
193C2×C3—62no—
194C2×C3—182no—
195C2×C3—182no—
196C2×C3D3314yes—
197C2×C3C2×D3×C3⋊D312yes—
198D3—22no—
199D3—64no—
200C22—32no—
201C22—91no—
202C4—91no—
203C4—272no—
204C4—811no—
205C3—28no—
206C3—21no—
207C3—44no—
208C3#8896314yescomplemented
209C3#8902912yescomplemented
210C2—62no—
211C2—181no—
212C2D32×C3⋊D311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D32×C3⋊D3
commutator, Fitting, socle, cosocleC2×C34C23
hypocenterC34#41
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C3⋊#6296C2semidirect
2. C3⋊D3×C3⋊Dic3C2semidirect
3. D3×C3⋊(C3⋊Dic3)C2semidirect
4. C3×#6290C2semidirect
5. C32×#625C2semidirect
18. #6290D3semidirect
26. C3×#625D3semidirect
52. C34#41semidirect
67. #625C3⋊D3semidirect
119. C33#237semidirect
120. C33#237semidirect
180. C32#3805semidirect
181. C32#3805semidirect
182. C32#3830semidirect
208. C3#88963semidirect
209. C3#89029semidirect

Maximal subgroups

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

subgroupindex
C3⋊#62962
C3⋊D3×C3⋊Dic32
D3×C3⋊(C3⋊Dic3)2
C3×#62902
C32×#6252
#890293
#889633