Finite Group Databasebeta

← #13180190

Subgroups — #13180190

Order 1728.

296 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 24 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#11724186C211yesmaximal, characteristic
2A4×#622C211yesmaximal, characteristic, complemented
3A4⋊#622C211yesmaximal, characteristic, complemented
4#10593740—31nomaximal
5C2×C3⋊(C3⋊S4)C411yescharacteristic
6A4×C3⋊Dic3C2211yescharacteristic
7C32×SL(2,ℤ/4)C411yescharacteristic
8#88926—41nomaximal
9C22×#622D311yescharacteristic, complemented
10C2×#3788—31no—
11(C3⋊S4)⋊C4—62no—
12#64090—31no—
13C22⋊#622—31no—
14#63778—31no—
15#63777—31no—
16C4⋊#622—31no—
17C2×C32×A4C2×C411yescharacteristic
18C3⋊(C3⋊S4)—21no—
19#6289—41no—
20C3⋊#622—41no—
21C3×#622—41no—
22#5400—91nomaximal
23C2×C3⋊S4—62no—
24C2×C3⋊S4—122no—
25C2×#638—31no—
26C22×C3⋊Dic3C2×D311yescharacteristic
27C2×C4×C3⋊D3—31no—
28C2×#622—31no—
29C2×#622—31no—
30C2×#622—31no—
31C2×#622—31no—
32#3788—61no—
33#3788—61no—
34Dic3×A4—62no—
35C3×SL(2,ℤ/4)—22no—
36C32×C22⋊C4—31no—
37#3752—31no—
38#3750—31no—
39C4×C3⋊Dic3—31no—
40C2×C3⋊(C3⋊D3)—41no—
41C32×A4M1611yescharacteristic, complemented
42C3×C3⋊Dic3—41no—
43C32×Dic3—41no—
44C4×S4—91no—
45#67⋊C4—182no—
46C8×A4—91no—
47A4⋊C8—91no—
48C23×C32C4×D311yescharacteristic
49C22×C3⋊D3—31no—
50C2×C3×A4—22no—
51C2×C3×A4—42no—
52C3⋊S4—64no—
53C3⋊S4—124no—
54C2×C32×C4—31no—
55C2×C32×C4—31no—
56#638—61no—
57#638—61no—
58C2×C3⋊Dic3—31no—
59C2×C3⋊Dic3—31no—
60C2×C3⋊Dic3—31no—
61C4×C3⋊D3—61no—
62C4×C3⋊D3—61no—
63#624—242no—
64#622—31no—
65#622—61no—
66#622—61no—
67#622S411yescharacteristic, complemented
68#435—271noSylow
69C2×C33—41no—
70C3⋊(C3⋊D3)—81no—
71C2×S4—91no—
72C2×S4—362no—
73C2×#67—182no—
74C22×Dic3—62no—
75C2×C4×D3—182no—
76C4×A4—91no—
77SL(2,ℤ/4)#15711yescharacteristic
78C3×C22⋊C4—62no—
79(C2×D3)⋊C4—182no—
80Dic3⋊C4—182no—
81C4×Dic3—182no—
82ΓL(1,𝔽25)—361no—
83C22×C32—31no—
84C22×C32—31no—
85C22×C32ΓL(1,𝔽25)11yescharacteristic, complemented
86C2×C3⋊D3—61no—
87C2×C3⋊D3—61no—
88C2×C3⋊D3—61no—
89C2×C3⋊D3—242no—
90C2×C3⋊D3—482no—
91C3×A4—22no—
92C3×A4—42no—
93C32×C4—61no—
94C32×C4—61no—
95C3⋊Dic3—31no—
96C3⋊Dic3—61no—
97C3⋊Dic3C2×S411yescharacteristic
98C3×Dic3—82no—
99C3×Dic3—242no—
100C2×M16—271no—
101C22×C8—91no—
102C4×D4—271no—
103C4⋊C8—271no—
104C22⋊C8—271no—
105C22⋊C8—271no—
106#97—271no—
107C33—41noSylow
108C23×C3—22no—
109C22×D3—182no—
110C2×A4—42no—
111C2×A4C2×#15711yescharacteristic
112S4—181no—
113S4—364no—
114C2×C3×C4—62no—
115C2×C3×C4—62no—
116#67—184no—
117#67—184no—
118C2×Dic3—182no—
119C2×Dic3—182no—
120C2×Dic3—182no—
121C4×D3—361no—
122C4×D3—362no—
123C4×D3—362no—
124C3×C8—361no—
125C3⋊C8—361no—
126C2×C32—31no—
127C2×C32—31no—
128C2×C32—82no—
129C2×C32—162no—
130C2×C32C4×S411yescharacteristic
131C3⋊D3—121no—
132C3⋊D3—244no—
133C3⋊D3—484no—
134C2×D4—271no—
135C22×C4—91no—
136C22×C4—271no—
137M16—541no—
138M16—541no—
139C2×C8—271no—
140C2×C8—271no—
141C2×C8—271no—
142C2×C8—271no—
143C4⋊C4—271no—
144C22⋊C4—31no—
145C22⋊C4—271no—
146C42—271no—
147C22×C3—22no—
148C22×C3—62no—
149C22×C3—62no—
150C2×D3—184no—
151C2×D3—362no—
152C2×D3—362no—
153C2×D3—361no—
154C2×D3—1442no—
155A4—42no—
156A4#378811yescharacteristic, complemented
157C3×C4—122no—
158C3×C4—122no—
159C3×C4—361no—
160Dic3—41no—
161Dic3—62no—
162Dic3—182no—
163Dic3—184no—
164C32—82no—
165C32—162no—
166C32#540011yescharacteristic, complemented
167C23—271no—
168C23D3×#15711yescharacteristic
169D4—541no—
170D4—541no—
171C2×C4—31no—
172C2×C4—31no—
173C2×C4—271no—
174C2×C4—271no—
175C2×C4—271no—
176C2×C4—541no—
177C2×C4—541no—
178C8—91no—
179C8—271no—
180C8—541no—
181C8—541no—
182C2×C3—22no—
183C2×C3—41no—
184C2×C3—62no—
185C2×C3—62no—
186C2×C3—162no—
187D3—364no—
188D3—721no—
189D3—1444no—
190C22—31no—
191C22—31no—
192C22—541no—
193C22—541no—
194C22—541no—
195C22#8892611yescharacteristic, complemented
196C4—61no—
197C4—61no—
198C4—91no—
199C4—271no—
200C4—541no—
201C3—22no—
202C3—41no—
203C3—162no—
204C2—31no—
205C2—31no—
206C2—1081no—
207C2S4×#15711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2S4×#157
commutatorC2×C32×A4C2×C4
Fitting, socleC23×C32C4×D3
prime radicalC2×C3⋊(C3⋊S4)C4
cosocleA4×C3⋊Dic3C22
hypocenterC32×A4M16
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. A4×#622C2semidirect
3. A4⋊#622C2semidirect
9. C22×#622D3semidirect
41. C32×A4M16semidirect
67. #622S4semidirect
85. C22×C32ΓL(1,𝔽25)semidirect
156. A4#3788semidirect
166. C32#5400semidirect
195. C22#88926semidirect

Maximal subgroups

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

subgroupindex
#117241862
A4×#6222
A4⋊#6222
#105937403
#889264
#54009