Finite Group Databasebeta

← #13166324

Subgroups — #13166324

Order 1728.

568 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C2×#88988C211yesmaximal, characteristic
2C2×C3×#3787C211yesmaximal, characteristic
3C4×Dic3×C3⋊D3C211yesmaximal, characteristic
4#11722724C214yesmaximal, complemented
5#10593723—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×Dic3×C3⋊D3C412yes—
8C2×C3⋊#622C2212yes—
9#88988C414yescomplemented
10C2×C3×#622C2212yes—
11C3×#3787C414yescomplemented
12C4×C3⋊(C3⋊Dic3)C411yescharacteristic
13C32×C4×Dic3C411yescharacteristic
14Dic3×C3⋊Dic3C2212yes—
15C2×#3787—31no—
16C2×#3787D311yescharacteristic
17C42×C3⋊D3—31no—
18C4×D3×Dic3—62no—
19#63777—34no—
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C2×C33×C4C2×C411yescharacteristic
22C2×C3⋊(C3⋊Dic3)C2×C412yes—
23C2×C3×C3⋊Dic3C2311yescharacteristic
24C3×C4×C3⋊D3C2×C412yes—
25C3×C4×C3⋊D3C2×C412yes—
26C2×C32×Dic3C2×C412yes—
27Dic3×C3⋊D3—24no—
28C3⋊#622C2×C414yescomplemented
29C3×#622C2×C414yescomplemented
30C2×#815—91nomaximal
31C2×C4×C3⋊D3—32no—
32C2×C4×C3⋊D3C2×D311yescharacteristic
33C2×C3×C4×D3—62no—
34C2×D3×Dic3—64no—
35C2×#622—32no—
36C2×#622C2×D312yes—
37#3787—34no—
38#3787Dic314yescomplemented
39C32×C42—31no—
40C4×C3⋊Dic3—31no—
41C4×C3⋊Dic3—32no—
42C4×C3⋊Dic3—62no—
43C4×C3⋊Dic3—122no—
44C3×C4×Dic3—22no—
45Dic32—64no—
46C22×C33C22×C411yescharacteristic
47C2×C3×C3⋊D3C4212yes—
48C2×C3×C3⋊D3M1614yes—
49C33×C4C4212yes—
50C3⋊(C3⋊Dic3)—22no—
51C3×C3⋊Dic3C22×C411yescharacteristic
52C3×C3⋊Dic3C22×C411yescharacteristic
53C32×Dic3—22no—
54C22×C3×C8—91no—
55C2×C4×Dic3—91no—
56C22×C3⋊C8—91no—
57C42×D3—182no—
58#815—94no—
59C22×C3⋊D3C4×D311yescharacteristic
60C22×C3×D3—62no—
61C2×C32×C4—22no—
62C2×C32×C4—32no—
63C2×C32×C4—42no—
64C2×C32×C4C4×D311yescharacteristic
65C2×C3⋊Dic3—32no—
66C2×C3⋊Dic3—64no—
67C2×C3⋊Dic3—124no—
68C2×C3⋊Dic3C22×D311yescharacteristic
69C4×C3⋊D3—64no—
70C4×C3⋊D3C4×D312yes—
71C4×C3⋊D3C2×Dic312yes—
72C2×C3×Dic3—24no—
73C2×C3×Dic3—62no—
74C3×C4×D3—64no—
75C3×C4×D3—64no—
76D3×Dic3—616no—
77#622—34no—
78#622C2×Dic314yescomplemented
79C2×#97—271noSylow
80C2×C33C2×C4211yescharacteristic
81C2×C33C2×M1612yes—
82C3×C3⋊D3#9714yescomplemented
83C22×C3×C4—91no—
84C22×Dic3—92no—
85C2×C4×D3—62no—
86C2×C4×D3—184no—
87C2×C3×C8—92no—
88C2×C3×C8—94no—
89C3×C42—62no—
90C4×Dic3—91no—
91C4×Dic3—92no—
92C4×Dic3—184no—
93C4×Dic3—182no—
94C4×Dic3—362no—
95C4×Dic3#15711yescharacteristic
96C2×C3⋊C8—92no—
97C2×C3⋊C8—94no—
98C22×C32—22no—
99C22×C32—42no—
100C22×C32C2×C4×D311yescharacteristic
101C2×C3⋊D3ΓL(1,𝔽25)14yes—
102C2×C3⋊D3C4×Dic312yes—
103C2×C3×D3—64no—
104C2×C3×D3—68no—
105C32×C4—24no—
106C32×C4—44no—
107C32×C4—62no—
108C32×C4C4×Dic312yes—
109C3⋊Dic3—62no—
110C3⋊Dic3—68no—
111C3⋊Dic3—128no—
112C3⋊Dic3C2×C4×D311yescharacteristic
113C3⋊Dic3C22×Dic311yescharacteristic
114C3×Dic3—28no—
115C3×Dic3—62no—
116C3×Dic3—62no—
117C22×C8—91no—
118C22×C8—271no—
119C2×C42—271no—
120#97—274no—
121C33C2×#9711yescharacteristic, complemented, Sylow
122C23×C3—91no—
123C22×D3—62no—
124C2×C3×C4—22no—
125C2×C3×C4—42no—
126C2×C3×C4—64no—
127C2×C3×C4—91no—
128C2×C3×C4—92no—
129C2×C3×C4—92no—
130C2×C3×C4C2×#15711yescharacteristic
131C2×Dic3—62no—
132C2×Dic3—92no—
133C2×Dic3—184no—
134C2×Dic3—184no—
135C2×Dic3—364no—
136C2×Dic3C2×#15712yes—
137C4×D3—64no—
138C4×D3—64no—
139C4×D3—1816no—
140C3×C8—94no—
141C3⋊C8—94no—
142C2×C32—24no—
143C2×C32—22no—
144C2×C32—42no—
145C2×C32—44no—
146C2×C32C2×ΓL(1,𝔽25)12yes—
147C2×C32C2×C4×Dic311yescharacteristic
148C3⋊D3#81514yescomplemented
149C3×D3—68no—
150C22×C4—91no—
151C22×C4—272no—
152C2×C8—92no—
153C2×C8—94no—
154C2×C8—272no—
155C2×C8—274no—
156C42—31no—
157C42—272no—
158C42—271no—
159C22×C3—22no—
160C22×C3—42no—
161C22×C3—94no—
162C22×C3—92no—
163C22×C3C22×#15711yescharacteristic
164C2×D3—64no—
165C2×D3—68no—
166C3×C4—24no—
167C3×C4—44no—
168C3×C4—68no—
169C3×C4—91no—
170C3×C4—91no—
171C3×C4C4×#15712yes—
172Dic3—22no—
173Dic3—62no—
174Dic3—62no—
175Dic3—188no—
176Dic3—182no—
177Dic3—368no—
178C32—22no—
179C32—42no—
180C32C2×#81511yescharacteristic, complemented
181C23—91no—
182C2×C4—32no—
183C2×C4—91no—
184C2×C4—92no—
185C2×C4—92no—
186C2×C4—272no—
187C2×C4—544no—
188C2×C4D3×#15711yescharacteristic
189C8—94no—
190C8—274no—
191C2×C3—24no—
192C2×C3—22no—
193C2×C3—44no—
194C2×C3—42no—
195C2×C3—94no—
196C2×C3#6428312yes—
197C2×C3C2×C4×#15711yescharacteristic
198D3—68no—
199C22—94no—
200C22—92no—
201C22C2×D3×#15711yescharacteristic
202C4—62no—
203C4—91no—
204C4—91no—
205C4—542no—
206C4Dic3×#15712yes—
207C3—22no—
208C3—42no—
209C3#1059372311yescharacteristic, complemented
210C2—94no—
211C2#1172425412yes—
212C2C2×Dic3×#15711yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×#157
commutatorC2×C33C2×C42
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C2×#97
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722724C2semidirect
9. #88988C4semidirect
11. C3×#3787C4semidirect
28. C3⋊#622C2×C4semidirect
29. C3×#622C2×C4semidirect
38. #3787Dic3semidirect
78. #622C2×Dic3semidirect
82. C3×C3⋊D3#97semidirect
121. C33C2×#97semidirect
148. C3⋊D3#815semidirect
180. C32C2×#815semidirect
209. C3#10593723semidirect

Maximal subgroups

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

subgroupindex
C2×#889882
C2×C3×#37872
C4×Dic3×C3⋊D32
#117227242
#105937233
C2×#8159