Finite Group Databasebeta

← #13166075

Subgroups — #13166075

Order 1728.

346 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 44 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×#64259C212yesmaximal
2A4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
3#10593180—32nomaximal
4#10591188—31nomaximal
5C32×C4×A4C2211yescharacteristic
6C3×C3⋊SL(2,ℤ/4)—22no—
7#88812—41nomaximal
8#64259D312yes—
9C3×A4⋊Q8—32no—
10C3×A4⋊Q8—62no—
11C22×C3⋊(C3⋊C8)D311yescharacteristic
12C3×#924—32no—
13C2×#3718—31no—
14A4×C3⋊C8—32no—
15A4×C3⋊C8—61no—
16#63576—31no—
17#63565—32no—
18C2×C32×A4D411yescharacteristic
19C3×#634—42no—
20C3×C3⋊(C3⋊C8)—41no—
21#5398—91no—
22#5051—92no—
23C22×C32×C4C2×D311yescharacteristic
24C2×C3×Dic6—32no—
25C3×C4×A4—22no—
26C3×C4×A4—21no—
27C3×C4×A4—41no—
28C3×C4×A4C2×D312yes—
29C3⋊SL(2,ℤ/4)—22no—
30C3×SL(2,ℤ/4)—64no—
31C3×SL(2,ℤ/4)—62no—
32C2×C3⋊(C3⋊C8)—31no—
33C2×C3⋊(C3⋊C8)—31no—
34C3×#217—62no—
35C3×C4⋊Dic3—32no—
36C3×Dic3⋊C4—62no—
37#3719—122no—
38#3718—61no—
39#3718—61no—
40C32×A4Q1611yescharacteristic, complemented
41C33×C4—41no—
42C3×C3⋊Dic3—82no—
43A4⋊Q8—32no—
44A4⋊Q8—62no—
45C3×C22⋊Q8—92no—
46C2×#216—92no—
47#924—32no—
48C22×C3⋊C8—32no—
49C22×C3⋊C8—61no—
50C8×A4—91no—
51#835—92no—
52#810—92no—
53#807—92no—
54C23×C32D1211yescharacteristic
55C2×C3×A4—22no—
56C2×C3×A4—21no—
57C2×C3×A4—41no—
58C2×C3×A4#6712yes—
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61#634—42no—
62C2×C3×Dic3—62no—
63C2×C3×Dic3—62no—
64C3×Dic6—62no—
65C3×Dic6—62no—
66C3×Dic6—122no—
67C3×Dic6—242no—
68C3⋊(C3⋊C8)—31no—
69C3⋊(C3⋊C8)S411yescharacteristic
70C3×C3⋊C8—122no—
71C3×C3⋊C8—241no—
72#467—271noSylow
73C2×C33—41no—
74C2×C3×Q8—92no—
75C22×C3×C4—21no—
76C22×C3×C4D3212yes—
77C2×Dic6—32no—
78C4×A4—22no—
79C4×A4—41no—
80C4×A4D3211yescharacteristic
81SL(2,ℤ/4)—64no—
82SL(2,ℤ/4)—62no—
83C3×C4⋊C4—92no—
84C3×C4⋊C4—182no—
85C3×C22⋊C4—182no—
86#217—62no—
87#216—182no—
88#216—182no—
89C4⋊Dic3—32no—
90Dic3⋊C4—62no—
91C2×C3⋊C8—92no—
92C2×C3⋊C8—92no—
93C2×C3⋊C8—181no—
94C2×C3⋊C8—181no—
95Dic12—361no—
96C22×C32—31no—
97C22×C32—31no—
98C22×C32Dic1211yescharacteristic, complemented
99C3×A4—22no—
100C3×A4—21no—
101C3×A4—41no—
102C3×A4#21612yescomplemented
103C32×C4—31no—
104C32×C4—42no—
105C32×C4—81no—
106C32×C4—82no—
107C32×C4—161no—
108C32×C4C2×S411yescharacteristic
109C3⋊Dic3—82no—
110C3×Dic3—122no—
111C3×Dic3—122no—
112C3×Dic3—242no—
113C3×Dic3—244no—
114C2×Q16—271no—
115C22×C8—91no—
116C22⋊Q8—92no—
117#107—271no—
118#103—272no—
119C33—41noSylow
120C23×C3—21no—
121C23×C3#62612yes—
122C2×A4—22no—
123C2×A4—41no—
124C2×A4#62511yescharacteristic
125C3×Q8—182no—
126C3×Q8—182no—
127C2×C3×C4—32no—
128C2×C3×C4—32no—
129C2×C3×C4—61no—
130C2×C3×C4—61no—
131C2×C3×C4—182no—
132C2×C3×C4—182no—
133C2×Dic3—62no—
134C2×Dic3—62no—
135Dic6—62no—
136Dic6—62no—
137Dic6—122no—
138Dic6—242no—
139C3×C8—361no—
140C3⋊C8—32no—
141C3⋊C8—61no—
142C3⋊C8—92no—
143C3⋊C8—181no—
144C2×C32—31no—
145C2×C32—31no—
146C2×C32—42no—
147C2×C32—81no—
148C2×C32—82no—
149C2×C32—161no—
150C2×C32C4⋊S411yescharacteristic
151C2×Q8—92no—
152C22×C4#111311yescharacteristic
153Q16—541no—
154Q16—541no—
155C2×C8—271no—
156C2×C8—271no—
157C4⋊C4—92no—
158C4⋊C4—182no—
159C22⋊C4—182no—
160C22×C3—21no—
161C22×C3—32no—
162C22×C3—32no—
163C22×C3—61no—
164C22×C3—61no—
165C22×C3#371912yescomplemented
166A4—22no—
167A4—41no—
168A4#371811yescharacteristic, complemented
169C3×C4—21no—
170C3×C4—32no—
171C3×C4—41no—
172C3×C4—61no—
173C3×C4—82no—
174C3×C4—161no—
175C3×C4—362no—
176C3×C4—362no—
177C3×C4D3×S412yes—
178Dic3—122no—
179Dic3—122no—
180Dic3—242no—
181Dic3—244no—
182C32—42no—
183C32—81no—
184C32—82no—
185C32—161no—
186C32#539811yescharacteristic, complemented
187C23#629511yescharacteristic
188Q8—182no—
189Q8—182no—
190C2×C4—31no—
191C2×C4—31no—
192C2×C4—182no—
193C2×C4—182no—
194C8—91no—
195C8—271no—
196C2×C3—21no—
197C2×C3—32no—
198C2×C3—32no—
199C2×C3—41no—
200C2×C3—61no—
201C2×C3—61no—
202C2×C3—82no—
203C2×C3—161no—
204C2×C3#6420712yes—
205C22—31no—
206C22—31no—
207C22#8881211yescharacteristic, complemented
208C4—31no—
209C4—362no—
210C4—362no—
211C4#8910111yescharacteristic
212C3—21no—
213C3—41no—
214C3—82no—
215C3—161no—
216C3#1059318012yescomplemented
217C2—31no—
218C2—31no—
219C2#1172421411yescharacteristic
subgroupG/H
centerC2#11724214
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4#89101
FittingC22×C32×C4C2×D3
socleC23×C32D12
prime radicalC2×C32×A4D4
hypocenterC32×A4Q16
LCSG ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
40. C32×A4Q16semidirect
98. C22×C32Dic12semidirect
102. C3×A4#216semidirect
165. C22×C3#3719semidirect
168. A4#3718semidirect
186. C32#5398semidirect
207. C22#88812semidirect
216. C3#10593180semidirect

Maximal subgroups

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

subgroupindex
C3×#642592
A4×C3⋊(C3⋊C8)2
#105931803
#105911883
#888124