Finite Group Databasebeta

← #13157208

Subgroups — #13157208

Order 1728.

356 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 65 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#11724511C211yesmaximal, characteristic
2C2×#88988C211yesmaximal, characteristic
3C3×M16×C3⋊D3C211yesmaximal, characteristic
4#10590190—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6#88993—21no—
7C2×C3⋊#622—21no—
8#88989C412yes—
9#88988—21no—
10#88988—21no—
11#88988C412yes—
12C33×M16C411yescharacteristic
13C3×#3748C411yescharacteristic
14C3×#3743—21no—
15C3×C8×C3⋊D3—21no—
16#64283—31no—
17C2×#3787—31no—
18M16×C3⋊D3D311yescharacteristic
19C3×D3×M16—62no—
20C22×C3×C3⋊D3Q811yescharacteristic
21C2×C33×C4C2×C411yescharacteristic
22C2×C3×C3⋊Dic3D411yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3⋊#622—22no—
28C3⋊#622—22no—
29C33×C8—21no—
30C3×C3⋊(C3⋊C8)—21no—
31#4558—91nomaximal
32C2×C4×C3⋊D3C2×D311yescharacteristic
33C2×C3×C4×D3—62no—
34#3792—61no—
35C2×#622—61no—
36#3788—32no—
37#3787—32no—
38#3787—61no—
39#3787—61no—
40C32×M16—22no—
41C32×M16—42no—
42C32×M16Dic311yescharacteristic
43#3748Dic311yescharacteristic
44#3743—21no—
45C8×C3⋊D3—21no—
46C3×#208—62no—
47C3×ΓL(1,𝔽25)—64no—
48C3×D3×C8—64no—
49C22×C33C4⋊C411yescharacteristic
50C2×C3×C3⋊D3—22no—
51C2×C3×C3⋊D3C4⋊C411yescharacteristic
52C2×C3×C3⋊D3C4⋊C411yescharacteristic
53C33×C4C4211yescharacteristic
54C33×C4C22⋊C411yescharacteristic
55C3×C3⋊Dic3C22⋊C411yescharacteristic
56C3×C3⋊Dic3C22⋊C411yescharacteristic
57C2×C3×M16—91no—
58C2×#208—91no—
59C22×C3⋊C8—91no—
60D3×M16—62no—
61C22×C3⋊D3Dic611yescharacteristic
62C22×C3×D3—62no—
63C2×C32×C4—22no—
64C2×C32×C4—42no—
65C2×C32×C4C2×Dic311yescharacteristic
66C2×C3⋊Dic3D1211yescharacteristic
67C4×C3⋊D3C4×D311yescharacteristic
68C4×C3⋊D3C4×D311yescharacteristic
69C4×C3⋊D3#6711yescharacteristic
70C4×C3⋊D3#6711yescharacteristic
71C2×C3×Dic3—62no—
72C3×C4×D3—62no—
73C3×C4×D3—62no—
74C3×C4×D3—62no—
75C3×C4×D3—62no—
76#622—62no—
77#622—62no—
78C32×C8—21no—
79C32×C8—24no—
80C32×C8—44no—
81C3⋊(C3⋊C8)—21no—
82C3×C3⋊C8—64no—
83#341—271noSylow
84C2×C33—21no—
85C2×C33#9511yescharacteristic
86C3×C3⋊D3—21no—
87C3×C3⋊D3#10812yes—
88C22×C3×C4—91no—
89C2×C4×D3—62no—
90C3×M16—22no—
91C3×M16—42no—
92C3×M16—91no—
93C3×M16—181no—
94C3×M16#15711yescharacteristic
95C2×C3×C8—181no—
96#208—62no—
97#208—92no—
98#208—181no—
99C2×C3⋊C8—92no—
100C2×C3⋊C8—181no—
101C2×C3⋊C8—181no—
102C2×C3⋊C8—181no—
103ΓL(1,𝔽25)—64no—
104D3×C8—64no—
105C22×C32—22no—
106C22×C32—42no—
107C22×C32C4⋊Dic311yescharacteristic
108C2×C3⋊D3—22no—
109C2×C3⋊D3Dic3⋊C411yescharacteristic
110C2×C3⋊D3Dic3⋊C411yescharacteristic
111C2×C3×D3—62no—
112C2×C3×D3—62no—
113C2×C3×D3—124no—
114C32×C4—22no—
115C32×C4—22no—
116C32×C4—42no—
117C32×C4—42no—
118C32×C4C4×Dic311yescharacteristic
119C32×C4#21711yescharacteristic
120C3⋊Dic3(C2×D3)⋊C411yescharacteristic
121C3⋊Dic3(C2×D3)⋊C411yescharacteristic
122C3×Dic3—62no—
123C3×Dic3—62no—
124C2×M16—91no—
125C2×M16—271no—
126C22×C8—271no—
127C33#34111yescharacteristic, complemented, Sylow
128C23×C3—91no—
129C22×D3—62no—
130C2×C3×C4—22no—
131C2×C3×C4—42no—
132C2×C3×C4—91no—
133C2×C3×C4—91no—
134C2×C3×C4—91no—
135C2×C3×C4—91no—
136C2×C3×C4—91no—
137C2×C3×C4C2×#15711yescharacteristic
138C2×Dic3—62no—
139C4×D3—62no—
140C4×D3—62no—
141C4×D3—62no—
142C4×D3—62no—
143C3×C8—24no—
144C3×C8—21no—
145C3×C8—44no—
146C3×C8—181no—
147C3⋊C8—64no—
148C3⋊C8—182no—
149C3⋊C8—182no—
150C2×C32—22no—
151C2×C32—21no—
152C2×C32—42no—
153C2×C32—42no—
154C2×C32—82no—
155C2×C32C3⋊#9511yescharacteristic
156C3⋊D3—21no—
157C3⋊D3#82212yes—
158C3×D3—64no—
159C3×D3—122no—
160C22×C4—91no—
161M16—91no—
162M16—181no—
163M16—272no—
164M16—541no—
165M16C3⋊#15711yescharacteristic
166C2×C8—181no—
167C2×C8—272no—
168C2×C8—541no—
169C2×C8—541no—
170C2×C8—541no—
171C22×C3—22no—
172C22×C3—42no—
173C22×C3—91no—
174C22×C3—91no—
175C22×C3—182no—
176C22×C3C4⋊#15711yescharacteristic
177C2×D3—62no—
178C2×D3—62no—
179C2×D3—124no—
180C3×C4—22no—
181C3×C4—22no—
182C3×C4—42no—
183C3×C4—42no—
184C3×C4—91no—
185C3×C4—91no—
186C3×C4C4×#15711yescharacteristic
187C3×C4C22⋊#15711yescharacteristic
188Dic3—62no—
189Dic3—62no—
190C32—22no—
191C32—42no—
192C32#455811yescharacteristic, complemented
193C23—91no—
194C2×C4—91no—
195C2×C4—91no—
196C2×C4—91no—
197C2×C4—91no—
198C2×C4—91no—
199C2×C4C2×C3⋊#15711yescharacteristic
200C8—21no—
201C8—181no—
202C8—542no—
203C8—542no—
204C2×C3—21no—
205C2×C3—22no—
206C2×C3—42no—
207C2×C3—42no—
208C2×C3—82no—
209C2×C3—92no—
210C2×C3—181no—
211C2×C3#6378111yescharacteristic
212D3—64no—
213D3—122no—
214C22—91no—
215C22—91no—
216C22—182no—
217C22C4⋊(C3⋊#157)11yescharacteristic
218C4—91no—
219C4—91no—
220C4C4×C3⋊#15711yescharacteristic
221C4#8899411yescharacteristic
222C3—22no—
223C3—42no—
224C3#1059019011yescharacteristic, complemented
225C2—21no—
226C2—92no—
227C2—181no—
228C2#1172310611yescharacteristic
subgroupG/H
centerC4#88994
commutatorC33×C4C42
FrattiniC2×C4C2×C3⋊#157
FittingC33×M16C4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3Q8
cosocleC2×C3×C4×C3⋊D3C22
hypercenterM16C3⋊#157
hypocenterC33#341
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
127. C33#341semidirect
192. C32#4558semidirect
224. C3#10590190semidirect

Maximal subgroups

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

subgroupindex
#117245112
C2×#889882
C3×M16×C3⋊D32
#105901903
#45589