Finite Group Databasebeta

← #13145712

Subgroups — #13145712

Order 1728.

418 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 107 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×C8×C3⋊D3C211yesmaximal, characteristic
2C2×C3×C3⋊(C3⋊C16)C212yesmaximal, complemented
3#10589108—31nomaximal
4(C2×D3)⋊(C3⋊C16)—32nomaximal
5C2×C3×C4×C3⋊D3C411yescharacteristic
6C2×C33×C8C2211yescharacteristic
7C2×C3×C3⋊(C3⋊C8)C411yescharacteristic
8C3×C8×C3⋊D3—22no—
9C3×C3⋊(C3⋊C16)—22no—
10C2×C8×C3⋊D3D311yescharacteristic
11C2×C3×D3×C8—32no—
12C2×C3×D3×C8—61no—
13C2×C3⋊(C3⋊C16)D312yescomplemented
14C2×C3×C3⋊C16—32no—
15C2×C3×C3⋊C16—32no—
16C2×C3×C3⋊C16—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#4549—91no—
25(C2×D3)⋊C16—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
35C8×C3⋊D3—22no—
36C2×C3×C3⋊C8—32no—
37C2×C3×C3⋊C8—61no—
38C3×D3×C8—64no—
39C3×D3×C8—62no—
40C3×D3×C8—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—21no—
48C2×C3×C3⋊D3C1612yescomplemented
49C33×C4C22⋊C411yescharacteristic
50C33×C4M1611yescharacteristic
51C3×C3⋊Dic3—21no—
52C22×C3×C8—91no—
53C2×D3×C8—32no—
54C2×D3×C8—61no—
55C2×C3×C16—92no—
56C2×C3⋊C16—32no—
57C2×C3⋊C16—32no—
58C2×C3⋊C16—62no—
59C22×C3⋊D3C3⋊C811yescharacteristic
60C22×C3×D3—32no—
61C22×C3×D3—61no—
62C2×C32×C4—22no—
63C2×C32×C4—21no—
64C2×C32×C4—41no—
65C2×C32×C4C4×D312yes—
66C2×C32×C4C2×Dic311yescharacteristic
67C2×C3⋊Dic3C3⋊C811yescharacteristic
68C4×C3⋊D3—21no—
69C4×C3⋊D3—21no—
70C2×C3×Dic3—32no—
71C2×C3×Dic3—61no—
72C3×C4×D3—62no—
73C3×C4×D3—62no—
74C3×C4×D3—62no—
75C3×C4×D3—62no—
76C32×C8—22no—
77C32×C8—22no—
78C32×C8—22no—
79C32×C8—42no—
80C32×C8D1212yes—
81C32×C8#6712yes—
82C32×C8#6712yes—
83C3⋊(C3⋊C8)—21no—
84C3×C3⋊C8—62no—
85C3×C3⋊C8—62no—
86C22⋊C16—271noSylow
87C2×C33C22⋊C811yescharacteristic
88C2×C33C2×C1611yescharacteristic
89C2×C33M3211yescharacteristic
90C3×C3⋊D3—22no—
91C22×C3×C4—91no—
92C2×C4×D3—32no—
93C2×C4×D3—61no—
94C2×C3×C8—22no—
95C2×C3×C8—21no—
96C2×C3×C8—41no—
97C2×C3×C8—91no—
98C2×C3×C8—182no—
99C2×C3×C8D3212yes—
100C2×C3×C8D3211yescharacteristic
101C2×C3⋊C8—32no—
102C2×C3⋊C8—61no—
103D3×C8—64no—
104D3×C8—62no—
105D3×C8—62no—
106C3×C16—182no—
107C3⋊C16—64no—
108C3⋊C16—62no—
109C3⋊C16—62no—
110C22×C32—22no—
111C22×C32—21no—
112C22×C32—41no—
113C22×C32D3×C812yes—
114C22×C32C2×C3⋊C811yescharacteristic
115C2×C3⋊D3—21no—
116C2×C3⋊D3—21no—
117C2×C3⋊D3C3⋊C1612yescomplemented
118C2×C3×D3—34no—
119C2×C3×D3—62no—
120C2×C3×D3—62no—
121C2×C3×D3—62no—
122C2×C3×D3—62no—
123C2×C3×D3—62no—
124C32×C4—22no—
125C32×C4—21no—
126C32×C4—22no—
127C32×C4—21no—
128C32×C4—41no—
129C32×C4—41no—
130C32×C4ΓL(1,𝔽25)12yes—
131C32×C4#20811yescharacteristic
132C32×C4(C2×D3)⋊C412yes—
133C32×C4#21711yescharacteristic
134C3⋊Dic3—21no—
135C3×Dic3—62no—
136C3×Dic3—62no—
137C22×C8—91no—
138C2×C16—92no—
139C33C22⋊C1611yescharacteristic, complemented, Sylow
140C23×C3—91no—
141C22×D3—32no—
142C22×D3—61no—
143C2×C3×C4—22no—
144C2×C3×C4—21no—
145C2×C3×C4—41no—
146C2×C3×C4—91no—
147C2×C3×C4—181no—
148C2×C3×C4—181no—
149C2×C3×C4D3×Dic312yes—
150C2×C3×C4#62411yescharacteristic
151C2×Dic3—32no—
152C2×Dic3—61no—
153C4×D3—62no—
154C4×D3—62no—
155C4×D3—62no—
156C4×D3—62no—
157C3×C8—22no—
158C3×C8—22no—
159C3×C8—22no—
160C3×C8—42no—
161C3×C8—181no—
162C3×C8#62512yes—
163C3×C8#62612yes—
164C3×C8#62612yes—
165C3⋊C8—62no—
166C3⋊C8—62no—
167C2×C32—22no—
168C2×C32—21no—
169C2×C32—22no—
170C2×C32—21no—
171C2×C32—22no—
172C2×C32—21no—
173C2×C32—41no—
174C2×C32—41no—
175C2×C32—41no—
176C2×C32D3×C1612yes—
177C2×C32#79812yes—
178C2×C32C2×C3⋊C1611yescharacteristic
179C2×C32#81211yescharacteristic
180C2×C32(C2×D3)⋊C812yes—
181C2×C32#83011yescharacteristic
182C3⋊D3—22no—
183C3×D3—64no—
184C3×D3—64no—
185C22×C4—91no—
186C2×C8—91no—
187C2×C8—182no—
188C2×C8#111311yescharacteristic
189C16—182no—
190C22×C3—21no—
191C22×C3—22no—
192C22×C3—41no—
193C22×C3—92no—
194C22×C3—181no—
195C22×C3—181no—
196C22×C3D3×C3⋊C812yes—
197C22×C3#371011yescharacteristic
198C2×D3—34no—
199C2×D3—62no—
200C2×D3—62no—
201C2×D3—62no—
202C2×D3—62no—
203C2×D3—62no—
204C3×C4—21no—
205C3×C4—22no—
206C3×C4—21no—
207C3×C4—22no—
208C3×C4—41no—
209C3×C4—41no—
210C3×C4—181no—
211C3×C4#371112yes—
212C3×C4#371211yescharacteristic
213C3×C4(C2×D3)⋊Dic312yes—
214C3×C4#372211yescharacteristic
215Dic3—62no—
216Dic3—62no—
217C32—22no—
218C32—21no—
219C32—41no—
220C32(C2×D3)⋊C1612yescomplemented
221C32#454911yescharacteristic, complemented
222C23—91no—
223C2×C4—91no—
224C2×C4—181no—
225C2×C4—181no—
226C2×C4#629411yescharacteristic
227C8—181no—
228C8#629512yes—
229C2×C3—21no—
230C2×C3—22no—
231C2×C3—21no—
232C2×C3—22no—
233C2×C3—21no—
234C2×C3—22no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—182no—
239C2×C3D3×C3⋊C1612yes—
240C2×C3#6354211yescharacteristic
241C2×C3#6354312yes—
242C2×C3#6354411yescharacteristic
243C2×C3(C2×D3)⋊(C3⋊C8)12yes—
244C2×C3#6355711yescharacteristic
245D3—64no—
246D3—64no—
247C22—92no—
248C22—181no—
249C22—181no—
250C22#8880811yescharacteristic
251C4—181no—
252C4#8880911yescharacteristic
253C4#8881411yescharacteristic
254C3—21no—
255C3—22no—
256C3—41no—
257C3(C2×D3)⋊(C3⋊C16)12yescomplemented
258C3#1058910811yescharacteristic, complemented
259C2—182no—
260C2#1172206511yescharacteristic
261C2#1172206611yescharacteristic
262C2#1172207211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C8#1113
commutatorC2×C33C2×C16
Fitting, cosocleC2×C33×C8C22
socleC22×C33C2×C8
prime radicalC22×C3×C3⋊D3C8
hypocenterC33C22⋊C16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊(C3⋊C16)C2semidirect
13. C2×C3⋊(C3⋊C16)D3semidirect
48. C2×C3×C3⋊D3C16semidirect
117. C2×C3⋊D3C3⋊C16semidirect
139. C33C22⋊C16semidirect
220. C32(C2×D3)⋊C16semidirect
221. C32#4549semidirect
257. C3(C2×D3)⋊(C3⋊C16)semidirect
258. C3#10589108semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×C3⋊D32
C2×C3×C3⋊(C3⋊C16)2
#105891083
(C2×D3)⋊(C3⋊C16)3