Finite Group Databasebeta

← D3×#3757 #11723613

Subgroups — D3×#3757 #11723613

Order 864.

928 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 224 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
1C23×C32×D3C211yesmaximal, characteristic
2C2×D3×C3⋊Dic3C212yesmaximal, complemented
3#88864C211yesmaximal, characteristic, complemented
4C3×#3757C211yesmaximal, characteristic, complemented
5(C2×D3)⋊(C3⋊Dic3)C212yesmaximal, complemented
6C2×#3757—31nomaximal
7D3×#217—34nomaximal
8C23×C33C2211yescharacteristic
9C22×C32×D3—24no—
10C22×C32×D3C414yescomplemented
11C22×C32×D3C2211yescharacteristic
12C22×C32×D3C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)C2212yescomplemented
14C2×C3×C3⋊Dic3C2212yescomplemented
15D3×C3⋊Dic3—24no—
16C24×C32—31no—
17C23×C3×D3D314yes—
18C22×C3⋊Dic3—32no—
19C2×D3×Dic3—38no—
20#3757—31no—
21#3757—34no—
22#3757—32no—
23#3757—64no—
24#3757D311yescharacteristic, direct product factor
25C3×#217—34no—
26(C2×D3)⋊Dic3—38no—
27C22×C33—22no—
28C22×C33C2×C412yescomplemented
29C22×C33C2311yescharacteristic
30C2×C32×D3—22no—
31C2×C32×D3—28no—
32C2×C32×D3C2×C412yes—
33C2×C32×D3C2×C412yescomplemented
34C2×C32×D3D414yes—
35C3⋊(C3⋊Dic3)—22no—
36C3×C3⋊Dic3—22no—
37C2×#217—94no—
38D3×C22⋊C4—91no—
39C23×C32—24no—
40C23×C32—31no—
41C23×C32—34no—
42C23×C32—31no—
43C23×C32—64no—
44C23×C32C2×D314yes—
45C23×C32C2×D311yescharacteristic
46C22×C3×D3—216no—
47C22×C3×D3Dic3116yescomplemented
48C22×C3×D3C2×D314yes—
49C22×C3×D3C2×D314yes—
50C2×C3⋊Dic3—32no—
51C2×C3⋊Dic3—38no—
52C2×C3⋊Dic3—68no—
53C2×C3⋊Dic3—64no—
54C2×C3⋊Dic3C2×D312yescomplemented
55C2×C3×Dic3—38no—
56D3×Dic3—616no—
57C2×C33—22no—
58C2×C33C22×C411yescharacteristic
59C2×C33C2×D412yes—
60C32×D3—22no—
61C32×D3C22⋊C414yescomplemented
62C24×C3—34no—
63C23×D3C3⋊D311yescharacteristic
64C22×Dic3—98no—
65C2×C4×D3—92no—
66C3×C22⋊C4—91no—
67#217—34no—
68#217—98no—
69#217—94no—
70#217—91no—
71#217—184no—
72(C2×D3)⋊C4—92no—
73C22×C32—22no—
74C22×C32—28no—
75C22×C32—24no—
76C22×C32—28no—
77C22×C32—32no—
78C22×C32—34no—
79C22×C32—32no—
80C22×C32—48no—
81C22×C32—68no—
82C22×C32—62no—
83C22×C32C4×D312yescomplemented
84C22×C32C2×Dic318yescomplemented
85C22×C32C22×D314yes—
86C22×C32C22×D311yescharacteristic
87C2×C3×D3—28no—
88C2×C3×D3—232no—
89C2×C3×D3C2×Dic318yes—
90C2×C3×D3C2×Dic318yescomplemented
91C2×C3×D3#67116yes—
92C3⋊Dic3—22no—
93C3⋊Dic3—62no—
94C3⋊Dic3—68no—
95C3⋊Dic3—128no—
96C3×Dic3—68no—
97C2×C22⋊C4—271noSylow
98C33C2×C22⋊C411yescharacteristic, complemented, Sylow
99C23×C3—24no—
100C23×C3—34no—
101C23×C3—316no—
102C23×C3—34no—
103C23×C3—616no—
104C23×C3D3214yes—
105C23×C3C2×C3⋊D311yescharacteristic
106C22×D3—24no—
107C22×D3C3⋊Dic314yescomplemented
108C22×D3C2×C3⋊D311yescharacteristic
109C22×D3C2×C3⋊D311yescharacteristic
110C2×C3×C4—92no—
111C2×Dic3—38no—
112C2×Dic3—98no—
113C2×Dic3—92no—
114C2×Dic3—188no—
115C2×Dic3—1816no—
116C4×D3—184no—
117C2×C32—22no—
118C2×C32—24no—
119C2×C32—28no—
120C2×C32—28no—
121C2×C32—34no—
122C2×C32—48no—
123C2×C32—62no—
124C2×C32C2×C4×D311yescharacteristic
125C2×C32D3×D412yes—
126C2×C32C22×Dic314yes—
127C2×C32C2×#6718yes—
128C3×D3—28no—
129C3×D3#217116yescomplemented
130C24—31no—
131C22×C4—272no—
132C22⋊C4—91no—
133C22⋊C4—272no—
134C22⋊C4—271no—
135C22×C3—28no—
136C22×C3—22no—
137C22×C3—28no—
138C22×C3—24no—
139C22×C3—38no—
140C22×C3—316no—
141C22×C3—38no—
142C22×C3—48no—
143C22×C3—632no—
144C22×C3—68no—
145C22×C3D3×Dic318yescomplemented
146C22×C3C2×C3⋊Dic312yescomplemented
147C22×C3C2×D3214yes—
148C22×C3C22×C3⋊D311yescharacteristic
149C2×D3—22no—
150C2×D3—28no—
151C2×D3C2×C3⋊Dic312yes—
152C2×D3C2×C3⋊Dic312yescomplemented
153C2×D3#63814yes—
154C3×C4—182no—
155Dic3—68no—
156Dic3—188no—
157Dic3—182no—
158Dic3—368no—
159C32—24no—
160C32D3×C22⋊C411yescharacteristic, complemented
161C32C2×#21714yescomplemented
162C23—31no—
163C23—34no—
164C23—31no—
165C23—64no—
166C23D3×C3⋊D311yescharacteristic
167C2×C4—92no—
168C2×C4—272no—
169C2×C4—544no—
170C2×C3—22no—
171C2×C3—28no—
172C2×C3—24no—
173C2×C3—28no—
174C2×C3—316no—
175C2×C3—48no—
176C2×C3—68no—
177C2×C3C2×D3×Dic314yes—
178C2×C3D3×#6718yes—
179C2×C3C22×C3⋊Dic311yescharacteristic
180C2×C3C2×#63812yes—
181D3—22no—
182D3#375714yesdirect product factor
183C22—22no—
184C22—32no—
185C22—34no—
186C22—32no—
187C22—68no—
188C22—62no—
189C22D3×C3⋊Dic312yescomplemented
190C22C2×D3×C3⋊D311yescharacteristic
191C4—182no—
192C4—542no—
193C3—24no—
194C3D3×#21714yescomplemented
195C3C2×#375711yescharacteristic, complemented
196C2—22no—
197C2—34no—
198C2—62no—
199C2C2×D3×C3⋊Dic311yescharacteristic
200C2D3×#63812yes—
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutatorC2×C33C22×C4
FittingC23×C33C22
socle, cosocleC22×C33C23
prime radicalC23×C32×D3C2
hypercenterC23D3×C3⋊D3
hypocenterC33C2×C22⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×D3×C3⋊Dic3C2semidirect
3. #88864C2semidirect
4. C3×#3757C2semidirect
5. (C2×D3)⋊(C3⋊Dic3)C2semidirect
10. C22×C32×D3C4semidirect
13. C2×C3⋊(C3⋊Dic3)C22semidirect
14. C2×C3×C3⋊Dic3C22semidirect
24. #3757D3direct (N × H)
28. C22×C33C2×C4semidirect
33. C2×C32×D3C2×C4semidirect
47. C22×C3×D3Dic3semidirect
54. C2×C3⋊Dic3C2×D3semidirect
61. C32×D3C22⋊C4semidirect
83. C22×C32C4×D3semidirect
84. C22×C32C2×Dic3semidirect
90. C2×C3×D3C2×Dic3semidirect
98. C33C2×C22⋊C4semidirect
107. C22×D3C3⋊Dic3semidirect
129. C3×D3#217semidirect
145. C22×C3D3×Dic3semidirect
146. C22×C3C2×C3⋊Dic3semidirect
152. C2×D3C2×C3⋊Dic3semidirect
160. C32D3×C22⋊C4semidirect
161. C32C2×#217semidirect
182. D3#3757direct (N × H)
189. C22D3×C3⋊Dic3semidirect
194. C3D3×#217semidirect
195. C3C2×#3757semidirect

Maximal subgroups

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

subgroupindex
C23×C32×D32
C2×D3×C3⋊Dic32
#888642
C3×#37572
(C2×D3)⋊(C3⋊Dic3)2
C2×#37573
D3×#2173