Finite Group Databasebeta

← #13049508

Subgroups — #13049508

Order 1296.

780 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 114 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
1#10620155C211yesmaximal, characteristic
2C3⋊D3×C3⋊Dic3C211yesmaximal, characteristic
3#10620148C212yesmaximal, complemented
4C3×#6293C212yesmaximal, complemented
5C32×#624C211yesmaximal, characteristic
6D3×#634—32nomaximal
7#88957—34nomaximal
8C2×C32×C3⋊D3C2211yescharacteristic
9C3⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
10C3×C3⋊(C3⋊Dic3)C2212yes—
11C32×C3⋊Dic3C2211yescharacteristic
12C33×Dic3C2212yes—
13#6308—32no—
14C3×#634—32no—
15C32×C4×D3—32no—
16#6294—34no—
17#6293—32no—
18#6293—38no—
19#6293—68no—
20#6293D312yescomplemented
21Dic3×C3⋊D3—34no—
22D3×C3⋊Dic3—32no—
23D3×C3⋊Dic3—32no—
24D3×C3⋊Dic3—61no—
25C3×#627—38no—
26C3×#624D314yes—
27C2×C34C2311yescharacteristic
28C32×C3⋊D3Q812yescomplemented
29C2×#634—91no—
30#3797—91no—
31D3×Dic6—98no—
32C2×C3×C3⋊D3C2×D314yes—
33C2×C32×D3—32no—
34C2×C32×D3—61no—
35C33×C4—32no—
36C3⋊(C3⋊Dic3)—32no—
37C3⋊(C3⋊Dic3)—34no—
38C3⋊(C3⋊Dic3)—61no—
39C3⋊(C3⋊Dic3)—68no—
40C3⋊(C3⋊Dic3)—124no—
41C3⋊(C3⋊Dic3)C2×D312yes—
42C3×C3⋊Dic3—21no—
43C3×C3⋊Dic3—32no—
44C3×C3⋊Dic3—38no—
45C3×C3⋊Dic3—68no—
46C3×C3⋊Dic3C2×D312yes—
47C32×Dic3—28no—
48C32×Dic3—34no—
49C32×Dic3C2×D318yes—
50C32×Dic3C2×D312yes—
51C34C2×Q811yescharacteristic, complemented, Sylow
52C2×C32×C4—91no—
53C2×C3⋊Dic3—91no—
54C2×C3⋊Dic3—91no—
55C4×C3⋊D3—91no—
56#634—32no—
57#634—92no—
58#634—98no—
59#634—188no—
60C3×C4×D3—38no—
61C3×Dic6—98no—
62C3×Dic6—92no—
63#627—38no—
64#627—98no—
65#627—92no—
66#627—188no—
67#624—91no—
68#624C3⋊D311yescharacteristic
69D3×Dic3—98no—
70D3×Dic3—98no—
71D3×Dic3—184no—
72C2×C33—28no—
73C2×C33—21no—
74C2×C33—44no—
75C2×C33C22×D312yes—
76C2×C33C22×D314yes—
77C3×C3⋊D3Dic618yescomplemented
78C32×D3—34no—
79C32×D3—62no—
80D3×Q8—272no—
81C2×Dic6—274no—
82C22×C32—91no—
83C2×C3⋊D3C2×C3⋊D311yescharacteristic
84C2×C3×D3—38no—
85C2×C3×D3—64no—
86C32×C4—32no—
87C32×C4—38no—
88C32×C4—68no—
89C32×C4—91no—
90C3⋊Dic3—32no—
91C3⋊Dic3—38no—
92C3⋊Dic3—68no—
93C3⋊Dic3—91no—
94C3⋊Dic3—98no—
95C3⋊Dic3—91no—
96C3⋊Dic3—188no—
97C3⋊Dic3—188no—
98C3⋊Dic3—184no—
99C3⋊Dic3—364no—
100C3⋊Dic3—364no—
101C3⋊Dic3—3612no—
102C3⋊Dic3D3211yescharacteristic
103C3×Dic3—28no—
104C3×Dic3—38no—
105C3×Dic3—64no—
106C3×Dic3—98no—
107C3×Dic3—92no—
108C3×Dic3—188no—
109C3×Dic3D3218yes—
110C3×Dic3C2×C3⋊D312yes—
111C33—28no—
112C33—21no—
113C33—44no—
114C33C2×Dic614yescomplemented
115C33D3×Q812yescomplemented
116C3×Q8—272no—
117C2×C3×C4—94no—
118C2×Dic3—274no—
119C2×Dic3—274no—
120C4×D3—32no—
121C4×D3—272no—
122C4×D3—272no—
123C4×D3—541no—
124Dic6—98no—
125Dic6—92no—
126Dic6—272no—
127Dic6—278no—
128Dic6—272no—
129Dic6—548no—
130C2×C32—28no—
131C2×C32—24no—
132C2×C32—28no—
133C2×C32—44no—
134C2×C32—412no—
135C2×C32—44no—
136C2×C32—92no—
137C2×C32C2×D3218yes—
138C2×C32C2×D3211yescharacteristic
139C2×C32C22×C3⋊D311yescharacteristic
140C3⋊D3#63412yescomplemented
141C3×D3—316no—
142C3×D3—68no—
143C2×Q8—811noSylow
144C22×C3—94no—
145C2×D3—32no—
146C2×D3—61no—
147C3×C4—38no—
148C3×C4—32no—
149C3×C4—68no—
150C3×C4—92no—
151C3×C4—181no—
152C3×C4—272no—
153Dic3—34no—
154Dic3—98no—
155Dic3—92no—
156Dic3—188no—
157Dic3—274no—
158Dic3—272no—
159Dic3—548no—
160Dic3—541no—
161Dic3—1084no—
162Dic3D3×C3⋊D312yes—
163C32—28no—
164C32—24no—
165C32—28no—
166C32—44no—
167C32—412no—
168C32—44no—
169C32D3×Dic618yescomplemented
170C32#379711yescharacteristic, complemented
171C32C2×#63411yescharacteristic, complemented
172Q8—272no—
173Q8—812no—
174C2×C4—91no—
175C2×C4—811no—
176C2×C4—811no—
177C2×C3—28no—
178C2×C3—21no—
179C2×C3—44no—
180C2×C3—98no—
181C2×C3D3314yes—
182C2×C3C2×D3×C3⋊D312yes—
183D3—34no—
184D3—62no—
185C22—91no—
186C4—32no—
187C4—91no—
188C4—272no—
189C4—811no—
190C3—21no—
191C3—28no—
192C3—44no—
193C3#8895714yescomplemented
194C3D3×#63412yescomplemented
195C2—92no—
196C2D32×C3⋊D311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D32×C3⋊D3
commutator, Fitting, socle, cosocleC2×C34C23
prime radicalC2×C32×C3⋊D3C22
hypocenterC34C2×Q8
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
3. #10620148C2semidirect
4. C3×#6293C2semidirect
20. #6293D3semidirect
28. C32×C3⋊D3Q8semidirect
51. C34C2×Q8semidirect
77. C3×C3⋊D3Dic6semidirect
114. C33C2×Dic6semidirect
115. C33D3×Q8semidirect
140. C3⋊D3#634semidirect
169. C32D3×Dic6semidirect
170. C32#3797semidirect
171. C32C2×#634semidirect
193. C3#88957semidirect
194. C3D3×#634semidirect

Maximal subgroups

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

subgroupindex
#106201552
C3⋊D3×C3⋊Dic32
#106201482
C3×#62932
C32×#6242
D3×#6343
#889573