Finite Group Databasebeta

← Dic3×#3792 #13166575

Subgroups — Dic3×#3792 #13166575

Order 1728.

608 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 112 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×#88993C211yesmaximal, characteristic
2C2×C3×#3792C211yesmaximal, characteristic
3C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
4#11722724C212yesmaximal, complemented
5Dic3×#622C212yesmaximal, complemented
6C4×#3792—31nomaximal
7C22×C3⋊(C3⋊Dic3)C411yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C32×Dic3C411yescharacteristic
10#88993C414yescomplemented
11C2×C3⋊#622C2212yes—
12C3×#3792C414yescomplemented
13C2×C3×#622C2212yes—
14Dic3×C3⋊Dic3C411yescharacteristic
15Dic3×C3⋊Dic3C411yescharacteristic
16Dic3×C3⋊Dic3C2211yescharacteristic
17Dic3×C3⋊Dic3C2211yescharacteristic
18C2×#3792—31no—
19C2×#3792D311yescharacteristic
20C2×C4×C3⋊Dic3—31no—
21C2×Dic32—62no—
22#63777—32no—
23C4×#622—32no—
24C23×C33C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)—22no—
26C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
27C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2×C411yescharacteristic
29C2×C3×C3⋊Dic3C2×C412yes—
30C2×C3×C3⋊Dic3C2×C412yes—
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C2×C32×Dic3—22no—
33C2×C32×Dic3C2×C411yescharacteristic
34C2×C32×Dic3C2×C411yescharacteristic
35C3⋊#622C2×C414yescomplemented
36C3×#622C2×C414yescomplemented
37Dic3×M16—91nomaximal
38C22×C32×C4—31no—
39C22×C3⋊Dic3—31no—
40C22×C3⋊Dic3—62no—
41C22×C3⋊Dic3—122no—
42C22×C3⋊Dic3C2×D311yescharacteristic
43C22×C3×Dic3—22no—
44C22×C3×Dic3—62no—
45#3792—34no—
46#3792Dic314yesdirect product factor
47C2×#622—32no—
48C2×#622C2×D312yes—
49C4×C3⋊Dic3—31no—
50C4×C3⋊Dic3—31no—
51C4×C3⋊Dic3—31no—
52C4×C3⋊Dic3—31no—
53Dic32—62no—
54Dic32—62no—
55Dic32—62no—
56Dic32—62no—
57C22×C33—21no—
58C22×C33—21no—
59C22×C33C4212yes—
60C22×C33C22×C411yescharacteristic
61C3⋊(C3⋊Dic3)—21no—
62C3⋊(C3⋊Dic3)M1612yescomplemented
63C3×C3⋊Dic3C4212yes—
64C3×C3⋊Dic3C22×C411yescharacteristic
65C3×C3⋊Dic3C22×C411yescharacteristic
66C32×Dic3—21no—
67C32×Dic3M1612yescomplemented
68C2×C3×M16—91no—
69C2×C4×Dic3—91no—
70C2×C4×Dic3—182no—
71C2×#208—91no—
72#815—92no—
73C8×Dic3—92no—
74C23×C32—22no—
75C23×C32—42no—
76C23×C32C4×D311yescharacteristic
77C2×C32×C4—31no—
78C2×C32×C4—31no—
79C2×C32×C4—62no—
80C2×C3⋊Dic3—31no—
81C2×C3⋊Dic3—31no—
82C2×C3⋊Dic3—62no—
83C2×C3⋊Dic3—62no—
84C2×C3⋊Dic3—62no—
85C2×C3⋊Dic3—68no—
86C2×C3⋊Dic3—128no—
87C2×C3⋊Dic3—122no—
88C2×C3⋊Dic3—122no—
89C2×C3⋊Dic3C4×D311yescharacteristic
90C2×C3⋊Dic3C4×D312yes—
91C2×C3⋊Dic3C2×Dic312yes—
92C2×C3⋊Dic3C22×D311yescharacteristic
93C2×C3×Dic3—28no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—64no—
99C2×C3×Dic3—64no—
100#622—34no—
101#622C2×Dic314yescomplemented
102C4×M16—271noSylow
103C2×C33—22no—
104C2×C33C2×C4211yescharacteristic
105C2×C33C2×M1611yescharacteristic
106C2×C33C2×M1611yescharacteristic
107C22×C3×C4—62no—
108C22×C3×C4—91no—
109C22×Dic3—62no—
110C22×Dic3—91no—
111C22×Dic3—182no—
112C22×Dic3—362no—
113C22×Dic3#15711yescharacteristic
114C3×M16—94no—
115C2×C3×C8—92no—
116C4×Dic3—91no—
117C4×Dic3—91no—
118C4×Dic3—91no—
119C4×Dic3—91no—
120C4×Dic3—182no—
121C4×Dic3—182no—
122C4×Dic3—182no—
123C4×Dic3—182no—
124#208—94no—
125C2×C3⋊C8—92no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—24no—
129C22×C32—24no—
130C22×C32—22no—
131C22×C32—24no—
132C22×C32—44no—
133C22×C32—44no—
134C22×C32—44no—
135C22×C32—42no—
136C22×C32C4×Dic312yes—
137C22×C32C2×C4×D311yescharacteristic
138C32×C4—32no—
139C32×C4—61no—
140C3⋊Dic3—32no—
141C3⋊Dic3—64no—
142C3⋊Dic3—61no—
143C3⋊Dic3—64no—
144C3⋊Dic3—124no—
145C3⋊Dic3—124no—
146C3⋊Dic3C4×Dic312yes—
147C3⋊Dic3C2×C4×D311yescharacteristic
148C3⋊Dic3C22×Dic311yescharacteristic
149C3×Dic3—24no—
150C3×Dic3—24no—
151C3×Dic3—64no—
152C3×Dic3—62no—
153C3×Dic3—62no—
154C2×M16—91no—
155C2×M16—271no—
156C2×C42—271no—
157#97—272no—
158C4×C8—272no—
159C33C4×M1611yescharacteristic, complemented, Sylow
160C23×C3—22no—
161C23×C3—42no—
162C23×C3C2×#15711yescharacteristic
163C2×C3×C4—62no—
164C2×C3×C4—62no—
165C2×C3×C4—68no—
166C2×C3×C4—91no—
167C2×C3×C4—91no—
168C2×C3×C4—92no—
169C2×C3×C4—92no—
170C2×Dic3—22no—
171C2×Dic3—62no—
172C2×Dic3—64no—
173C2×Dic3—64no—
174C2×Dic3—62no—
175C2×Dic3—91no—
176C2×Dic3—91no—
177C2×Dic3—182no—
178C2×Dic3—182no—
179C2×Dic3—182no—
180C2×Dic3—188no—
181C2×Dic3—362no—
182C2×Dic3—362no—
183C2×Dic3—368no—
184C2×Dic3C2×#15711yescharacteristic
185C2×Dic3C2×#15711yescharacteristic
186C3×C8—94no—
187C3⋊C8—94no—
188C2×C32—22no—
189C2×C32—28no—
190C2×C32—22no—
191C2×C32—22no—
192C2×C32—22no—
193C2×C32—48no—
194C2×C32—42no—
195C2×C32—42no—
196C2×C32—42no—
197C2×C32D3×M1611yescharacteristic
198C2×C32D3×M1611yescharacteristic
199C2×C32C2×C4×Dic311yescharacteristic
200C22×C4—31no—
201C22×C4—91no—
202C22×C4—271no—
203M16—94no—
204M16—274no—
205C2×C8—92no—
206C2×C8—272no—
207C42—271no—
208C42—271no—
209C42—271no—
210C42—271no—
211C22×C3—24no—
212C22×C3—24no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—22no—
216C22×C3—24no—
217C22×C3—44no—
218C22×C3—44no—
219C22×C3—42no—
220C22×C3—44no—
221C22×C3C4×#15712yes—
222C22×C3C22×#15711yescharacteristic
223C3×C4—64no—
224C3×C4—64no—
225C3×C4—91no—
226C3×C4—91no—
227C3×C4—92no—
228Dic3—21no—
229Dic3—64no—
230Dic3—62no—
231Dic3—62no—
232Dic3—92no—
233Dic3—184no—
234Dic3—184no—
235Dic3—181no—
236Dic3—364no—
237Dic3—364no—
238Dic3#379212yesdirect product factor
239C32—22no—
240C32—42no—
241C32Dic3×M1611yescharacteristic, complemented
242C23D3×#15711yescharacteristic
243C2×C4—31no—
244C2×C4—31no—
245C2×C4—62no—
246C2×C4—91no—
247C2×C4—91no—
248C2×C4—92no—
249C2×C4—92no—
250C2×C4—271no—
251C2×C4—271no—
252C2×C4—542no—
253C8—94no—
254C8—274no—
255C2×C3—28no—
256C2×C3—22no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—22no—
260C2×C3—48no—
261C2×C3—42no—
262C2×C3—42no—
263C2×C3—42no—
264C2×C3#6428311yescharacteristic
265C2×C3C2×C4×#15711yescharacteristic
266C2×C3C2×#379211yescharacteristic
267C22—21no—
268C22—21no—
269C22Dic3×#15712yes—
270C22C2×D3×#15711yescharacteristic
271C4—32no—
272C4—61no—
273C4—91no—
274C4—91no—
275C4—92no—
276C4—272no—
277C4—541no—
278C3—22no—
279C3—42no—
280C3C4×#379211yescharacteristic, complemented
281C2—22no—
282C2C2×Dic3×#15711yescharacteristic
283C2D3×#379211yescharacteristic
284C2#1172426911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutatorC2×C33C2×C42
Fitting, prime radicalC23×C33C2×C4
socleC22×C33C22×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D3×#157
hypocenterC33C4×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722724C2semidirect
5. Dic3×#622C2semidirect
10. #88993C4semidirect
12. C3×#3792C4semidirect
35. C3⋊#622C2×C4semidirect
36. C3×#622C2×C4semidirect
46. #3792Dic3direct (N × H)
62. C3⋊(C3⋊Dic3)M16semidirect
67. C32×Dic3M16semidirect
101. #622C2×Dic3semidirect
159. C33C4×M16semidirect
238. Dic3#3792direct (N × H)
241. C32Dic3×M16semidirect
280. C3C4×#3792semidirect

Maximal subgroups

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

subgroupindex
C2×#889932
C2×C3×#37922
C2×Dic3×C3⋊Dic32
#117227242
Dic3×#6222
C4×#37923
Dic3×M169