Finite Group Databasebeta

← #13167520

Subgroups — #13167520

Order 1728.

412 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 67 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×#88943C211yesmaximal, characteristic
2#11723665C212yesmaximal
3C3×C4×#622C211yesmaximal, characteristic
4#11722797C211yesmaximal, characteristic
5#11722794C212yesmaximal
6#10593445—31nomaximal
7C2×#6296C2212yes—
8C2×C3×#622C2211yescharacteristic
9C2×C3×#622C2211yescharacteristic
10#88943C414yescomplemented
11C3×#3750—22no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13#88815C2212yes—
14#88813—22no—
15C2×#3776—31no—
16#63838—32no—
17#63837—62no—
18C4×#622D311yescharacteristic
19#63743—31no—
20#63740—32no—
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3—22no—
23C2×C3×C3⋊Dic3—22no—
24C2×C3×C3⋊Dic3D411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26#6296—22no—
27#6296C2×C414yes—
28C3×#622—21no—
29C3×#622C2×C412yes—
30C4×Dic12—91nomaximal
31C2×#627—32no—
32C2×#627—62no—
33C2×#622C2×D311yescharacteristic
34C2×#622C2×D311yescharacteristic
35#3776—34no—
36#3750—22no—
37C4×C3⋊Dic3C2×D311yescharacteristic
38C3×Dic3⋊C4—62no—
39C3×Dic3⋊C4—62no—
40C3×Dic3⋊C4—122no—
41C3×C4×Dic3—62no—
42#3724—32no—
43Dic3⋊Dic3—62no—
44Dic32—62no—
45Dic32—62no—
46Dic32—62no—
47C22×C33C2×D411yescharacteristic
48C33×C4—21no—
49C3×C3⋊Dic3—24no—
50C3×C3⋊Dic3—42no—
51C3×C3⋊Dic3Q1612yescomplemented
52C3×C3⋊Dic3C22×C411yescharacteristic
53C3×C3⋊Dic3#4111yescharacteristic
54C2×Dic12—91no—
55#887—182no—
56C4×Dic6—92no—
57C3×C4×C8—91no—
58#818—91no—
59#816—92no—
60C2×C32×C4—22no—
61C2×C32×C4—42no—
62C2×C32×C4D1211yescharacteristic
63C2×C3⋊Dic3—22no—
64C2×C3⋊Dic3—22no—
65C2×C3⋊Dic3D1211yescharacteristic
66C2×C3⋊Dic3C22×D311yescharacteristic
67C2×C3×Dic3—62no—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—62no—
71C2×C3×Dic3—62no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—122no—
74C2×C3×Dic3—122no—
75#627—34no—
76#627—64no—
77#627—64no—
78#627—62no—
79#622—21no—
80#622C4×D312yes—
81C4×Q16—271noSylow
82C2×C33C4×D411yescharacteristic
83C2×C33C2×Q1611yescharacteristic
84C2×C33#13511yescharacteristic
85C2×Dic6—92no—
86C2×Dic6—182no—
87C2×C3×C8—91no—
88C2×C3×C8—91no—
89C3×C4⋊C4—182no—
90C3×C42—91no—
91C4⋊Dic3—92no—
92Dic3⋊C4—62no—
93Dic3⋊C4—62no—
94Dic3⋊C4—122no—
95Dic3⋊C4—182no—
96C4×Dic3—62no—
97C4×Dic3—182no—
98C4×Dic3—182no—
99C4×Dic3—182no—
100Dic12—94no—
101C22×C32—22no—
102C22×C32—42no—
103C22×C32C2×D1211yescharacteristic
104C32×C4—21no—
105C32×C4—42no—
106C32×C4—82no—
107C3⋊Dic3—24no—
108C3⋊Dic3—42no—
109C3⋊Dic3Dic1212yescomplemented
110C3⋊Dic3C2×C4×D311yescharacteristic
111C3⋊Dic3#23511yescharacteristic
112C3×Dic3—64no—
113C3×Dic3—62no—
114C3×Dic3—62no—
115C3×Dic3—64no—
116C3×Dic3—64no—
117C3×Dic3—124no—
118C3×Dic3—122no—
119C3×Dic3—122no—
120C3×Dic3—242no—
121C2×Q16—271no—
122C4×Q8—272no—
123#107—271no—
124#103—272no—
125C4×C8—91no—
126C33C4×Q1611yescharacteristic, complemented, Sylow
127C2×C3×C4—22no—
128C2×C3×C4—42no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—182no—
132C2×C3×C4—182no—
133C2×C3×C4GO+(4,𝔽2)11yescharacteristic
134C2×Dic3—62no—
135C2×Dic3—62no—
136C2×Dic3—62no—
137C2×Dic3—62no—
138C2×Dic3—62no—
139C2×Dic3—62no—
140C2×Dic3—122no—
141C2×Dic3—122no—
142Dic6—94no—
143Dic6—184no—
144Dic6—182no—
145Dic6—184no—
146C3×C8—92no—
147C3×C8—181no—
148C2×C32—22no—
149C2×C32—22no—
150C2×C32—22no—
151C2×C32—42no—
152C2×C32—42no—
153C2×C32—42no—
154C2×C32C4×D1211yescharacteristic
155C2×C32#90411yescharacteristic
156C2×C32C2×Dic1211yescharacteristic
157C2×Q8—272no—
158Q16—274no—
159C2×C8—91no—
160C2×C8—91no—
161C4⋊C4—182no—
162C4⋊C4—272no—
163C42—91no—
164C42—542no—
165C22×C3—22no—
166C22×C3—42no—
167C22×C3C2×GO+(4,𝔽2)11yescharacteristic
168C3×C4—21no—
169C3×C4—42no—
170C3×C4—82no—
171C3×C4—92no—
172C3×C4—91no—
173C3×C4—91no—
174C3×C4—184no—
175C3×C4—362no—
176Dic3—62no—
177Dic3—62no—
178Dic3—64no—
179Dic3—64no—
180Dic3—64no—
181Dic3—124no—
182Dic3—122no—
183Dic3—122no—
184Dic3—242no—
185C32—22no—
186C32—42no—
187C32C4×Dic1211yescharacteristic, complemented
188Q8—274no—
189Q8—542no—
190C2×C4—91no—
191C2×C4—91no—
192C2×C4—182no—
193C2×C4—182no—
194C2×C4#157⋊D311yescharacteristic
195C8—92no—
196C8—181no—
197C2×C3—22no—
198C2×C3—22no—
199C2×C3—22no—
200C2×C3—42no—
201C2×C3—42no—
202C2×C3—42no—
203C2×C3#6422811yescharacteristic
204C2×C3C4×GO+(4,𝔽2)11yescharacteristic
205C2×C3#6423711yescharacteristic
206C22C2×#157⋊D311yescharacteristic
207C4—21no—
208C4—92no—
209C4—91no—
210C4—91no—
211C4—184no—
212C4—362no—
213C3—22no—
214C3—42no—
215C3#1059344511yescharacteristic, complemented
216C2#1172434111yescharacteristic
217C2C4×#157⋊D311yescharacteristic
218C2#1172435111yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4#157⋊D3
hypocenterC33C4×Q16
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
10. #88943C4semidirect
51. C3×C3⋊Dic3Q16semidirect
109. C3⋊Dic3Dic12semidirect
126. C33C4×Q16semidirect
187. C32C4×Dic12semidirect
215. C3#10593445semidirect

Maximal subgroups

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

subgroupindex
C2×#889432
#117236652
C3×C4×#6222
#117227972
#117227942
#105934453
C4×Dic129