Finite Group Databasebeta

← #13145713

Subgroups — #13145713

Order 1728.

374 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 87 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×C3×#3743C211yesmaximal, characteristic
2C3×#63639C212yesmaximal, complemented
3#10589110—31nomaximal
4#10589109—32nomaximal
5C2×C3×C4×C3⋊D3C411yescharacteristic
6C2×C33×C8C2211yescharacteristic
7C2×C3×C3⋊(C3⋊C8)C411yescharacteristic
8C3×#3743—22no—
9C3×C3⋊(C3⋊C16)—22no—
10C2×#3743D311yescharacteristic
11C2×C3×ΓL(1,𝔽25)—32no—
12C2×C3×ΓL(1,𝔽25)—61no—
13#63639D312yescomplemented
14C3×#812—32no—
15C3×#812—32no—
16C3×#812—62no—
17C22×C3×C3⋊D3C811yescharacteristic
18C2×C33×C4C2×C411yescharacteristic
19C2×C3×C3⋊Dic3C811yescharacteristic
20C3×C4×C3⋊D3—21no—
21C3×C4×C3⋊D3—21no—
22C33×C8D412yes—
23C3×C3⋊(C3⋊C8)—21no—
24#4553—91no—
25#4514—92no—
26C2×C4×C3⋊D3Dic311yescharacteristic
27C2×C3×C4×D3—32no—
28C2×C3×C4×D3—61no—
29C2×C32×C8—21no—
30C2×C32×C8—22no—
31C2×C32×C8—41no—
32C2×C32×C8C2×D312yes—
33C2×C32×C8C2×D311yescharacteristic
34C2×C3⋊(C3⋊C8)Dic311yescharacteristic
35#3743—22no—
36C2×C3×C3⋊C8—32no—
37C2×C3×C3⋊C8—61no—
38C3×ΓL(1,𝔽25)—64no—
39C3×ΓL(1,𝔽25)—62no—
40C3×ΓL(1,𝔽25)—62no—
41C3⋊(C3⋊C16)—22no—
42C3×C3⋊C16—64no—
43C3×C3⋊C16—62no—
44C3×C3⋊C16—62no—
45C22×C33C2×C811yescharacteristic
46C2×C3×C3⋊D3—21no—
47C2×C3×C3⋊D3—41no—
48C33×C4C22⋊C411yescharacteristic
49C33×C4M1611yescharacteristic
50C3×C3⋊Dic3—21no—
51C2×C3×M16—91no—
52C2×ΓL(1,𝔽25)—32no—
53C2×ΓL(1,𝔽25)—61no—
54C3×M32—92no—
55#812—32no—
56#812—32no—
57#812—62no—
58C22×C3⋊D3C3⋊C811yescharacteristic
59C22×C3×D3—32no—
60C22×C3×D3—61no—
61C2×C32×C4—22no—
62C2×C32×C4—21no—
63C2×C32×C4—41no—
64C2×C32×C4C4×D312yes—
65C2×C32×C4C2×Dic311yescharacteristic
66C2×C3⋊Dic3C3⋊C811yescharacteristic
67C4×C3⋊D3—21no—
68C4×C3⋊D3—21no—
69C2×C3×Dic3—32no—
70C2×C3×Dic3—61no—
71C3×C4×D3—62no—
72C3×C4×D3—62no—
73C3×C4×D3—62no—
74C3×C4×D3—62no—
75C32×C8—22no—
76C32×C8—22no—
77C32×C8—22no—
78C32×C8—42no—
79C32×C8D1212yes—
80C32×C8#6712yes—
81C32×C8#6712yes—
82C3⋊(C3⋊C8)—21no—
83C3×C3⋊C8—62no—
84C3×C3⋊C8—62no—
85#349—271noSylow
86C2×C33—21no—
87C2×C33C22⋊C811yescharacteristic
88C3×C3⋊D3—41no—
89C22×C3×C4—91no—
90C2×C4×D3—32no—
91C2×C4×D3—61no—
92C3×M16—182no—
93C2×C3×C8—22no—
94C2×C3×C8—21no—
95C2×C3×C8—41no—
96C2×C3×C8—91no—
97C2×C3×C8D3212yes—
98C2×C3×C8D3211yescharacteristic
99C2×C3⋊C8—32no—
100C2×C3⋊C8—61no—
101ΓL(1,𝔽25)—64no—
102ΓL(1,𝔽25)—62no—
103ΓL(1,𝔽25)—62no—
104C3×C16—182no—
105C3⋊C16—64no—
106C3⋊C16—62no—
107C3⋊C16—62no—
108C22×C32—22no—
109C22×C32—21no—
110C22×C32—41no—
111C22×C32D3×C812yes—
112C22×C32C2×C3⋊C811yescharacteristic
113C2×C3⋊D3—21no—
114C2×C3⋊D3—41no—
115C2×C3×D3—62no—
116C2×C3×D3—62no—
117C2×C3×D3—122no—
118C2×C3×D3—122no—
119C32×C4—22no—
120C32×C4—21no—
121C32×C4—22no—
122C32×C4—21no—
123C32×C4—41no—
124C32×C4—41no—
125C32×C4ΓL(1,𝔽25)12yes—
126C32×C4#20811yescharacteristic
127C32×C4(C2×D3)⋊C412yes—
128C32×C4#21711yescharacteristic
129C3⋊Dic3—21no—
130C3×Dic3—62no—
131C3×Dic3—62no—
132C2×M16—91no—
133M32—92no—
134C33#34911yescharacteristic, complemented, Sylow
135C23×C3—91no—
136C22×D3—32no—
137C22×D3—61no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—41no—
141C2×C3×C4—91no—
142C2×C3×C4—181no—
143C2×C3×C4—181no—
144C2×C3×C4D3×Dic312yes—
145C2×C3×C4#62411yescharacteristic
146C2×Dic3—32no—
147C2×Dic3—61no—
148C4×D3—62no—
149C4×D3—62no—
150C4×D3—62no—
151C4×D3—62no—
152C3×C8—22no—
153C3×C8—22no—
154C3×C8—22no—
155C3×C8—42no—
156C3×C8—181no—
157C3×C8#62512yes—
158C3×C8#62612yes—
159C3×C8#62612yes—
160C3⋊C8—62no—
161C3⋊C8—62no—
162C2×C32—21no—
163C2×C32—22no—
164C2×C32—22no—
165C2×C32—22no—
166C2×C32—21no—
167C2×C32—41no—
168C2×C32—42no—
169C2×C32—42no—
170C2×C32(C2×D3)⋊C812yes—
171C2×C32#83011yescharacteristic
172C3⋊D3—41no—
173C3×D3—122no—
174C3×D3—122no—
175C22×C4—91no—
176M16—182no—
177C2×C8—91no—
178C2×C8#111311yescharacteristic
179C16—182no—
180C22×C3—21no—
181C22×C3—22no—
182C22×C3—41no—
183C22×C3—181no—
184C22×C3—361no—
185C22×C3D3×C3⋊C812yes—
186C22×C3#371011yescharacteristic
187C2×D3—62no—
188C2×D3—62no—
189C2×D3—122no—
190C2×D3—122no—
191C3×C4—21no—
192C3×C4—22no—
193C3×C4—21no—
194C3×C4—22no—
195C3×C4—41no—
196C3×C4—41no—
197C3×C4—181no—
198C3×C4#371112yes—
199C3×C4#371211yescharacteristic
200C3×C4(C2×D3)⋊Dic312yes—
201C3×C4#372211yescharacteristic
202Dic3—62no—
203Dic3—62no—
204C32—22no—
205C32—21no—
206C32—41no—
207C32#451412yescomplemented
208C32#455311yescharacteristic, complemented
209C23—91no—
210C2×C4—91no—
211C2×C4—181no—
212C2×C4—181no—
213C2×C4#629411yescharacteristic
214C8—181no—
215C8#629512yes—
216C2×C3—22no—
217C2×C3—22no—
218C2×C3—21no—
219C2×C3—21no—
220C2×C3—22no—
221C2×C3—42no—
222C2×C3—42no—
223C2×C3—41no—
224C2×C3—361no—
225C2×C3(C2×D3)⋊(C3⋊C8)12yes—
226C2×C3#6355711yescharacteristic
227D3—122no—
228D3—122no—
229C22—181no—
230C22—361no—
231C22#8880811yescharacteristic
232C4—181no—
233C4#8880911yescharacteristic
234C4#8881411yescharacteristic
235C3—21no—
236C3—22no—
237C3—41no—
238C3#1058910912yescomplemented
239C3#1058911011yescharacteristic, complemented
240C2—21no—
241C2—361no—
242C2#1172207211yescharacteristic
subgroupG/H
centerC4#88814
commutatorC22×C33C2×C8
Frattini, hypercenterC2×C8#1113
Fitting, cosocleC2×C33×C8C22
socleC2×C33C22⋊C8
prime radicalC22×C3×C3⋊D3C8
hypocenterC33#349
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C3×#63639C2semidirect
13. #63639D3semidirect
134. C33#349semidirect
207. C32#4514semidirect
208. C32#4553semidirect
238. C3#10589109semidirect
239. C3#10589110semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
C3×#636392
#105891103
#105891093