Finite Group Databasebeta

← #13158805

Subgroups — #13158805

Order 1728.

820 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 188 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
1C3×C42×C3⋊D3C211yesmaximal, characteristic
2#11722173C211yesmaximal, characteristic, complemented
3C32×#803C211yesmaximal, characteristic, complemented
4#11722051C212yesmaximal, complemented
5#11722023C212yesmaximal
6#10592404—31nomaximal
7#10590262—34nomaximal
8C2×C3×C4×C3⋊D3C411yescharacteristic
9C2×C3×C4×C3⋊D3C411yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2212yes—
13C3×C4×C3⋊Dic3C411yescharacteristic
14C3×C4×C3⋊Dic3C411yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2212yes—
17C42×C3⋊D3D311yescharacteristic
18C3×C42×D3—34no—
19C32×#97—31no—
20#63647—32no—
21#63641—32no—
22#63630—31no—
23#63630—34no—
24#63630—64no—
25C3×#803D314yescomplemented
26Dic3⋊(C3⋊C8)—38no—
27(C2×D3)⋊(C3⋊C8)—38no—
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C33×C4C2×C411yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C3×C4×C3⋊D3—22no—
36C3×C4×C3⋊D3—22no—
37C3×C4×C3⋊D3—22no—
38C3⋊(C3⋊(C3⋊C8))—22no—
39C32×C3⋊C8—22no—
40#4926—91no—
41#4705—94no—
42C2×C4×C3⋊D3Dic311yescharacteristic
43C2×C4×C3⋊D3Dic311yescharacteristic
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×C3×C4×D3—34no—
46C2×C3×C4×D3—34no—
47C2×C3×C4×D3—34no—
48C2×C32×C8—32no—
49C32×C42—24no—
50C32×C42C2×D314yes—
51C32×C42C2×D311yescharacteristic
52C4×C3⋊Dic3Dic311yescharacteristic
53C4×C3⋊Dic3Dic311yescharacteristic
54C4×C3⋊Dic3C2×D311yescharacteristic
55C2×C3⋊(C3⋊C8)—32no—
56C2×C3⋊(C3⋊C8)—38no—
57C2×C3⋊(C3⋊C8)—68no—
58C3×C4×Dic3—34no—
59C3×C4×Dic3—34no—
60C3×C4×Dic3—34no—
61C2×C3×C3⋊C8C2×D318yes—
62C22×C33C22×C411yescharacteristic
63C2×C3×C3⋊D3—21no—
64C2×C3×C3⋊D3—21no—
65C2×C3×C3⋊D3M1612yes—
66C33×C4—21no—
67C33×C4—21no—
68C33×C4#4112yes—
69C3×C3⋊Dic3—21no—
70C3×C3⋊Dic3—21no—
71C3×C3⋊Dic3M1612yes—
72C2×C3×C42—91no—
73C42×D3—34no—
74C3×#97—34no—
75#830—92no—
76(C2×D3)⋊C8—98no—
77Dic3⋊C8—98no—
78C4⋊(C3⋊C8)—92no—
79#803—94no—
80#803—91no—
81#803—184no—
82#803C3⋊D311yescharacteristic, complemented
83C22×C3⋊D3C2×Dic311yescharacteristic
84C22×C3×D3—34no—
85C2×C32×C4—24no—
86C2×C32×C4—24no—
87C2×C32×C4—24no—
88C2×C32×C4C4×D314yes—
89C2×C32×C4C4×D314yes—
90C2×C32×C4C2×Dic311yescharacteristic
91C2×C32×C4C2×Dic311yescharacteristic
92C2×C32×C4C22×D314yes—
93C2×C32×C4C22×D311yescharacteristic
94C2×C3⋊Dic3C2×Dic311yescharacteristic
95C2×C3⋊Dic3C2×Dic311yescharacteristic
96C2×C3⋊Dic3C2×Dic311yescharacteristic
97C4×C3⋊D3—22no—
98C4×C3⋊D3—22no—
99C4×C3⋊D3—22no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—34no—
103C3×C4×D3—68no—
104C3×C4×D3—68no—
105C3×C4×D3—68no—
106C32×C8—62no—
107C3⋊(C3⋊C8)—616no—
108C3⋊(C3⋊C8)—68no—
109C3⋊(C3⋊C8)—62no—
110C3×C3⋊C8—28no—
111#432—271noSylow
112C2×C33#11711yescharacteristic
113C2×C33C2×M1611yescharacteristic
114C2×C33C2×M1611yescharacteristic
115C3×C3⋊D3—22no—
116C22×C3×C4—91no—
117C22×C3×C4—91no—
118C22×C3×C4—91no—
119C2×C4×D3—34no—
120C2×C4×D3—34no—
121C2×C4×D3—34no—
122C2×C3×C8—38no—
123C3×C42—24no—
124C3×C42—91no—
125C3×C42—91no—
126C3×C42—91no—
127C3×C42D3214yes—
128C3×C42C2×C3⋊D311yescharacteristic
129C4×Dic3—34no—
130C4×Dic3—34no—
131C4×Dic3—34no—
132C2×C3⋊C8—98no—
133C2×C3⋊C8—92no—
134C2×C3⋊C8—188no—
135C2×C3⋊C8C2×C3⋊D312yes—
136C22×C32—24no—
137C22×C32C2×C4×D314yes—
138C22×C32C22×Dic311yescharacteristic
139C2×C3⋊D3—21no—
140C2×C3⋊D3—21no—
141C2×C3⋊D3#20812yes—
142C2×C3×D3—38no—
143C2×C3×D3—64no—
144C2×C3×D3—64no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—28no—
148C32×C4—24no—
149C32×C4—24no—
150C32×C4—44no—
151C32×C4—44no—
152C32×C4#23518yes—
153C32×C4#23512yes—
154C3⋊Dic3—21no—
155C3⋊Dic3—21no—
156C3⋊Dic3#20812yes—
157C3×Dic3—38no—
158C3×Dic3—64no—
159C3×Dic3—64no—
160C2×C42—91no—
161C4⋊C8—272no—
162C22⋊C8—272no—
163#97—31no—
164#97—271no—
165C33#43211yescharacteristic, complemented, Sylow
166C23×C3—91no—
167C22×D3—34no—
168C2×C3×C4—24no—
169C2×C3×C4—24no—
170C2×C3×C4—24no—
171C2×C3×C4—91no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—182no—
175C2×C3×C4—182no—
176C2×C3×C4—182no—
177C2×C3×C4D3×Dic314yes—
178C2×C3×C4D3×Dic314yes—
179C2×C3×C4C4×C3⋊D311yescharacteristic
180C2×C3×C4C4×C3⋊D311yescharacteristic
181C2×C3×C4C2×D3214yes—
182C2×C3×C4C22×C3⋊D311yescharacteristic
183C2×Dic3—34no—
184C2×Dic3—34no—
185C2×Dic3—34no—
186C4×D3—68no—
187C4×D3—68no—
188C4×D3—68no—
189C3×C8—68no—
190C3⋊C8—22no—
191C3⋊C8—1816no—
192C3⋊C8—188no—
193C3⋊C8—182no—
194C2×C32—24no—
195C2×C32—24no—
196C2×C32—24no—
197C2×C32#87214yes—
198C2×C32D3×M1614yes—
199C2×C32D3×M1614yes—
200C2×C32C2×#20811yescharacteristic
201C2×C32C2×#20811yescharacteristic
202C2×C32#92611yescharacteristic
203C3⋊D3—22no—
204C3×D3—68no—
205C22×C4—91no—
206C22×C4—91no—
207C22×C4—91no—
208C2×C8—32no—
209C2×C8—272no—
210C42—91no—
211C42—91no—
212C42—91no—
213C42D3×C3⋊D311yescharacteristic
214C22×C3—24no—
215C22×C3—92no—
216C22×C3—181no—
217C22×C3—181no—
218C22×C3C2×D3×Dic314yes—
219C22×C3C2×C4×C3⋊D311yescharacteristic
220C2×D3—38no—
221C2×D3—64no—
222C2×D3—64no—
223C3×C4—24no—
224C3×C4—24no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—28no—
228C3×C4—44no—
229C3×C4—44no—
230C3×C4—92no—
231C3×C4—181no—
232C3×C4—181no—
233C3×C4#379818yes—
234C3×C4#382812yes—
235Dic3—38no—
236Dic3—64no—
237Dic3—64no—
238C32—24no—
239C32#470514yescomplemented
240C32#492611yescharacteristic, complemented
241C23—91no—
242C2×C4—91no—
243C2×C4—91no—
244C2×C4—91no—
245C2×C4—182no—
246C2×C4—182no—
247C2×C4—182no—
248C2×C4Dic3×C3⋊D311yescharacteristic
249C2×C4Dic3×C3⋊D311yescharacteristic
250C2×C4C2×D3×C3⋊D311yescharacteristic
251C8—62no—
252C8—542no—
253C2×C3—24no—
254C2×C3—24no—
255C2×C3—24no—
256C2×C3—182no—
257C2×C3D3×#20814yes—
258C2×C3D3×#20814yes—
259C2×C3#6385514yes—
260C2×C3#6408111yescharacteristic
261C2×C3M16×C3⋊D311yescharacteristic
262C2×C3M16×C3⋊D311yescharacteristic
263D3—68no—
264C22—92no—
265C22—181no—
266C22—181no—
267C22C2×Dic3×C3⋊D311yescharacteristic
268C4—21no—
269C4—21no—
270C4—92no—
271C4—181no—
272C4—181no—
273C4#8902012yes—
274C3—24no—
275C3#1059026214yescomplemented
276C3#1059240411yescharacteristic, complemented
277C2—182no—
278C2C3⋊D3×#20811yescharacteristic
279C2C3⋊D3×#20811yescharacteristic
280C2#1172343711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC33×C42C22
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#432
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11722173C2semidirect
3. C32×#803C2semidirect
4. #11722051C2semidirect
25. C3×#803D3semidirect
82. #803C3⋊D3semidirect
165. C33#432semidirect
239. C32#4705semidirect
240. C32#4926semidirect
275. C3#10590262semidirect
276. C3#10592404semidirect

Maximal subgroups

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

subgroupindex
C3×C42×C3⋊D32
#117221732
C32×#8032
#117220512
#117220232
#105924043
#105902623