Finite Group Databasebeta

← #13173372

Subgroups — #13173372

Order 1728.

412 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 89 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#11724514C211yesmaximal, characteristic
2C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic
3C3×M16×C3⋊D3C211yesmaximal, characteristic
4#11723094C212yesmaximal, complemented
5#11723093C212yesmaximal, complemented
6#10593774—31nomaximal
7C22×C3⋊#157—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9#88994—21no—
10C4⋊(C3⋊#157)—21no—
11C4⋊(C3⋊#157)C2211yescharacteristic
12C4⋊(C3⋊#157)C2212yes—
13C4⋊(C3⋊#157)C2211yescharacteristic
14C4×C3⋊#157—21no—
15C33×M16C411yescharacteristic, complemented
16C3×#3748C411yescharacteristic, complemented
17C3×#3743C412yescomplemented
18C3×C8×C3⋊D3C2212yes—
19#64286—31no—
20C2×C4⋊#157—31no—
21M16×C3⋊D3D311yescharacteristic
22C3×D3×M16—62no—
23#63769—32no—
24#63768—32no—
25C22×C3×C3⋊D3D411yescharacteristic
26C2×C3⋊#157—22no—
27C2×C3⋊#157—21no—
28C2×C3⋊#157—21no—
29C2×C3⋊#157—42no—
30C2×C33×C4C2×C411yescharacteristic
31C2×C3×C3⋊Dic3Q811yescharacteristic
32C3×C4×C3⋊D3C2×C411yescharacteristic
33C3×C4×C3⋊D3D411yescharacteristic
34C3×C4×C3⋊D3Q811yescharacteristic
35C3×C4×C3⋊D3C2311yescharacteristic
36C33×C8C2×C412yescomplemented
37C3×C3⋊(C3⋊C8)C2×C412yescomplemented
38#5121—91nomaximal
39C22×#157—61no—
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×C4×D3—62no—
42C22⋊#157—61no—
43C4⋊#157—31no—
44C4⋊#157—32no—
45C4⋊#157—31no—
46C4⋊#157—61no—
47C4×#157—61no—
48C32×M16—22no—
49C32×M16—42no—
50C32×M16Dic311yescharacteristic, complemented
51#3748Dic311yescharacteristic, complemented
52#3743Dic312yescomplemented
53C8×C3⋊D3C2×D312yes—
54C3×#208—62no—
55C3×ΓL(1,𝔽25)—64no—
56C3×D3×C8—64no—
57C22×C33C4⋊C411yescharacteristic
58C2×C3×C3⋊D3—21no—
59C2×C3×C3⋊D3—21no—
60C2×C3×C3⋊D3C4⋊C411yescharacteristic
61C2×C3×C3⋊D3C2×D411yescharacteristic
62C3⋊#157—41no—
63C3⋊#157—42no—
64C3⋊#157—41no—
65C33×C4C4⋊C411yescharacteristic
66C33×C4C22×C411yescharacteristic
67C3×C3⋊Dic3C4⋊C411yescharacteristic
68C3×C3⋊Dic3C2×Q811yescharacteristic
69C2×C3×M16—91no—
70#926—91no—
71C2×C4⋊Dic3—91no—
72D3×M16—62no—
73#818—92no—
74#817—92no—
75C22×C3⋊D3D1211yescharacteristic
76C22×C3×D3—62no—
77C2×#157—62no—
78C2×#157—61no—
79C2×#157—61no—
80C2×#157—122no—
81C2×C32×C4—22no—
82C2×C32×C4—42no—
83C2×C32×C4C2×Dic311yescharacteristic
84C2×C3⋊Dic3Dic611yescharacteristic
85C4×C3⋊D3Dic611yescharacteristic
86C4×C3⋊D3D1211yescharacteristic
87C4×C3⋊D3C2×Dic311yescharacteristic
88C4×C3⋊D3C22×D311yescharacteristic
89C2×C3×Dic3—62no—
90C3×C4×D3—62no—
91C3×C4×D3—62no—
92C3×C4×D3—62no—
93C3×C4×D3—62no—
94C32×C8—24no—
95C32×C8—44no—
96C32×C8C2×Dic312yescomplemented
97C3⋊(C3⋊C8)C2×Dic312yescomplemented
98C3×C3⋊C8—64no—
99#428—271noSylow
100C2×C33—21no—
101C2×C33C2×C4⋊C411yescharacteristic
102C3×C3⋊D3—21no—
103C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
104C3×C3⋊D3#13711yescharacteristic
105C22×C3×C4—91no—
106C22×Dic3—181no—
107C2×C4×D3—62no—
108C3×M16—22no—
109C3×M16—42no—
110C3×M16—91no—
111C3×M16—92no—
112C3×M16#15711yescharacteristic, complemented
113C2×C3×C8—92no—
114#217—181no—
115C4⋊Dic3—91no—
116C4⋊Dic3—92no—
117C4⋊Dic3—91no—
118C4⋊Dic3—181no—
119C4×Dic3—181no—
120#208—62no—
121ΓL(1,𝔽25)—64no—
122D3×C8—64no—
123C22×C32—22no—
124C22×C32—42no—
125C22×C32C4⋊Dic311yescharacteristic
126C2×C3⋊D3—21no—
127C2×C3⋊D3—21no—
128C2×C3⋊D3C4⋊Dic311yescharacteristic
129C2×C3⋊D3C2×D1211yescharacteristic
130C2×C3×D3—62no—
131C2×C3×D3—62no—
132C2×C3×D3—122no—
133C2×C3×D3—122no—
134#157—121no—
135#157—122no—
136#157—121no—
137C32×C4—22no—
138C32×C4—22no—
139C32×C4—42no—
140C32×C4—42no—
141C32×C4C4⋊Dic311yescharacteristic
142C32×C4C22×Dic311yescharacteristic
143C3⋊Dic3C4⋊Dic311yescharacteristic
144C3⋊Dic3C2×Dic611yescharacteristic
145C3×Dic3—62no—
146C3×Dic3—62no—
147C2×M16—91no—
148#117—271no—
149C2×C4⋊C4—271no—
150#107—272no—
151#106—272no—
152C33#42811yescharacteristic, complemented, Sylow
153C23×C3—91no—
154C22×D3—62no—
155C2×C3×C4—22no—
156C2×C3×C4—42no—
157C2×C3×C4—91no—
158C2×C3×C4—91no—
159C2×C3×C4—91no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4C2×#15711yescharacteristic
163C2×Dic3—62no—
164C2×Dic3—181no—
165C2×Dic3—181no—
166C2×Dic3—182no—
167C2×Dic3—362no—
168C4×D3—62no—
169C4×D3—62no—
170C4×D3—62no—
171C4×D3—62no—
172C3×C8—24no—
173C3×C8—44no—
174C3×C8—92no—
175C3×C8C2×#15712yescomplemented
176C3⋊C8—64no—
177C2×C32—22no—
178C2×C32—21no—
179C2×C32—42no—
180C2×C32—42no—
181C2×C32—82no—
182C2×C32C2×C4⋊Dic311yescharacteristic
183C3⋊D3—21no—
184C3⋊D3#90811yescharacteristic
185C3⋊D3#90911yescharacteristic
186C3×D3—62no—
187C3×D3—62no—
188C3×D3—122no—
189C22×C4—91no—
190C22×C4—541no—
191M16—91no—
192M16—92no—
193M16C3⋊#15711yescharacteristic, complemented
194C2×C8—92no—
195C4⋊C4—271no—
196C4⋊C4—272no—
197C4⋊C4—271no—
198C4⋊C4—541no—
199C22⋊C4—541no—
200C42—541no—
201C22×C3—22no—
202C22×C3—42no—
203C22×C3—91no—
204C22×C3—91no—
205C22×C3—181no—
206C22×C3—181no—
207C22×C3C4⋊#15711yescharacteristic
208C2×D3—62no—
209C2×D3—62no—
210C2×D3—122no—
211C2×D3—122no—
212C3×C4—22no—
213C3×C4—22no—
214C3×C4—42no—
215C3×C4—42no—
216C3×C4—91no—
217C3×C4—91no—
218C3×C4C4⋊#15711yescharacteristic
219C3×C4C22×#15711yescharacteristic
220Dic3—62no—
221Dic3—62no—
222Dic3—362no—
223Dic3—361no—
224Dic3—361no—
225C32—22no—
226C32—42no—
227C32#512111yescharacteristic, complemented
228C23—91no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4—91no—
232C2×C4—91no—
233C2×C4—91no—
234C2×C4—541no—
235C2×C4—541no—
236C2×C4—542no—
237C2×C4—1082no—
238C2×C4C2×C3⋊#15711yescharacteristic
239C8—92no—
240C8C2×C3⋊#15712yescomplemented
241C2×C3—21no—
242C2×C3—22no—
243C2×C3—42no—
244C2×C3—42no—
245C2×C3—82no—
246C2×C3—91no—
247C2×C3—91no—
248C2×C3—181no—
249C2×C3C2×C4⋊#15711yescharacteristic
250D3—62no—
251D3—62no—
252D3—122no—
253C22—91no—
254C22—91no—
255C22—181no—
256C22—181no—
257C22C4⋊(C3⋊#157)11yescharacteristic
258C4—91no—
259C4—91no—
260C4—1082no—
261C4—1081no—
262C4—1081no—
263C4C4⋊(C3⋊#157)11yescharacteristic
264C4C22×C3⋊#15711yescharacteristic
265C3—22no—
266C3—42no—
267C3#1059377411yescharacteristic, complemented
268C2—21no—
269C2—91no—
270C2—91no—
271C2—181no—
272C2C2×C4⋊(C3⋊#157)11yescharacteristic
subgroupG/H
centerC2C2×C4⋊(C3⋊#157)
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC33×M16C4
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterM16C3⋊#157
hypocenterC33#428
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
4. #11723094C2semidirect
5. #11723093C2semidirect
15. C33×M16C4semidirect
16. C3×#3748C4semidirect
17. C3×#3743C4semidirect
36. C33×C8C2×C4semidirect
37. C3×C3⋊(C3⋊C8)C2×C4semidirect
50. C32×M16Dic3semidirect
51. #3748Dic3semidirect
52. #3743Dic3semidirect
96. C32×C8C2×Dic3semidirect
97. C3⋊(C3⋊C8)C2×Dic3semidirect
112. C3×M16#157semidirect
152. C33#428semidirect
175. C3×C8C2×#157semidirect
193. M16C3⋊#157semidirect
227. C32#5121semidirect
240. C8C2×C3⋊#157semidirect
267. C3#10593774semidirect

Maximal subgroups

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

subgroupindex
#117245142
C2×C4⋊(C3⋊#157)2
C3×M16×C3⋊D32
#117230942
#117230932
#105937743
#51219