Finite Group Databasebeta

← D3×#63768 #13166239

Subgroups — D3×#63768 #13166239

Order 1728.

456 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 78 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
1D3×C4⋊#157C212yesmaximal
2D3×C8×C3⋊D3C211yesmaximal, characteristic
3#11723093C211yesmaximal, characteristic, complemented
4C3×#63768C211yesmaximal, characteristic, complemented
5#11722714C212yesmaximal, complemented
6C2×#63768—31nomaximal
7C2×D3×#157—22no—
8C4×D3×C3⋊D3C2211yescharacteristic
9C4⋊(C3⋊#157)C2212yes—
10C3×C4⋊#157C2212yes—
11Dic3⋊#157—22no—
12C8×C3⋊(C3⋊D3)C411yescharacteristic, complemented
13C3×C8×C3⋊D3C2211yescharacteristic
14C32×D3×C8C411yescharacteristic, complemented
15#88785C411yescharacteristic, complemented
16C3⋊D3×C3⋊C8C2211yescharacteristic
17D3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
18C2×C4⋊#157—32no—
19C2×C8×C3⋊D3—31no—
20D32×C8—62no—
21#63768—32no—
22#63768—31no—
23#63768D311yescharacteristic, direct product factor
24C2×D3×C3⋊D3D411yescharacteristic
25C2×C3⋊#157—22no—
26C2×C3×#157—22no—
27D3×#157—44no—
28C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
29C3×C4×C3⋊D3C2311yescharacteristic
30C32×C4×D3C2×C411yescharacteristic
31#6289Q811yescharacteristic
32Dic3×C3⋊D3D411yescharacteristic
33D3×C3⋊Dic3Q811yescharacteristic
34C33×C8C2×C411yescharacteristic, complemented
35C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic, complemented
36C3×C3⋊(C3⋊C8)C2×C411yescharacteristic, complemented
37C32×C3⋊C8C2×C411yescharacteristic, complemented
38D3×#106—91nomaximal
39C22×#157—62no—
40C2×C4×C3⋊D3—31no—
41C4×D32—62no—
42C4⋊#157—32no—
43C4⋊#157—62no—
44C4⋊#157C2×D312yes—
45C2×C32×C8—31no—
46C2×C3⋊(C3⋊C8)—31no—
47C8×C3⋊D3—31no—
48C8×C3⋊D3—31no—
49C8×C3⋊D3—31no—
50C8×C3⋊D3—62no—
51C8×C3⋊D3—122no—
52C8×C3⋊D3C2×D311yescharacteristic
53C3×D3×C8—22no—
54C3×D3×C8—62no—
55#3710—62no—
56D3×C3⋊C8—62no—
57D3×C3⋊C8—62no—
58C2×C3⋊(C3⋊D3)C4⋊C411yescharacteristic
59C2×C3×C3⋊D3C2×D411yescharacteristic
60C2×C32×D3C4⋊C411yescharacteristic
61D3×C3⋊D3SD1614yes—
62C3⋊#157—42no—
63C3×#157—42no—
64C33×C4C22×C411yescharacteristic
65C3⋊(C3⋊Dic3)C4⋊C411yescharacteristic
66C3×C3⋊Dic3C2×Q811yescharacteristic
67C32×Dic3C4⋊C411yescharacteristic
68C2×D3×C8—91no—
69C2×D3×C8—182no—
70D3×C4⋊C4—92no—
71C3×#106—91no—
72#817—91no—
73#808—92no—
74C22×C3⋊D3—31no—
75C2×D32—62no—
76C2×#157—22no—
77C2×#157—62no—
78C2×#157—124no—
79C2×C32×C4—31no—
80C2×C3⋊Dic3—31no—
81C4×C3⋊D3—31no—
82C4×C3⋊D3—31no—
83C4×C3⋊D3—31no—
84C4×C3⋊D3—62no—
85C4×C3⋊D3—122no—
86C4×C3⋊D3C22×D311yescharacteristic
87C3×C4×D3—22no—
88C3×C4×D3—62no—
89#624—62no—
90D3×Dic3—62no—
91D3×Dic3—62no—
92C32×C8—22no—
93C32×C8—31no—
94C32×C8—42no—
95C32×C8C4×D311yescharacteristic, complemented
96C3⋊(C3⋊C8)—31no—
97C3⋊(C3⋊C8)—62no—
98C3⋊(C3⋊C8)—122no—
99C3⋊(C3⋊C8)C4×D311yescharacteristic, complemented
100C3×C3⋊C8—22no—
101C3×C3⋊C8—62no—
102C2×#106—271noSylow
103C2×C33C2×C4⋊C411yescharacteristic
104C3⋊(C3⋊D3)#10612yescomplemented
105C3×C3⋊D3C2×SD1612yes—
106C32×D3#10612yescomplemented
107C2×C4×D3—91no—
108C2×C4×D3—182no—
109C2×C4×D3—182no—
110C2×C3×C8—62no—
111C2×C3×C8—91no—
112C3×C4⋊C4—92no—
113C4⋊Dic3—92no—
114Dic3⋊C4—182no—
115C2×C3⋊C8—91no—
116C2×C3⋊C8—182no—
117D3×C8—62no—
118D3×C8—91no—
119D3×C8—91no—
120D3×C8—91no—
121D3×C8—182no—
122D3×C8—182no—
123D3×C8—182no—
124D3×C8—362no—
125D3×C8#15711yescharacteristic, complemented
126C22×C32—31no—
127C2×C3⋊D3—31no—
128C2×C3⋊D3—34no—
129C2×C3⋊D3—62no—
130C2×C3⋊D3—122no—
131C2×C3⋊D3D3×D411yescharacteristic
132C2×C3×D3—22no—
133C2×C3×D3—62no—
134D32—68no—
135#157—42no—
136#157—122no—
137C32×C4—22no—
138C32×C4—31no—
139C32×C4—42no—
140C32×C4C2×C4×D311yescharacteristic
141C3⋊Dic3—31no—
142C3⋊Dic3—62no—
143C3⋊Dic3—122no—
144C3⋊Dic3D3×Q811yescharacteristic
145C3×Dic3—22no—
146C3×Dic3—62no—
147C22×C8—271no—
148C2×C4⋊C4—272no—
149#106—91no—
150#106—272no—
151#106—271no—
152C33C2×#10611yescharacteristic, complemented, Sylow
153C22×D3—91no—
154C22×D3—182no—
155C2×C3×C4—62no—
156C2×C3×C4—91no—
157C2×C3×C4—182no—
158C2×Dic3—91no—
159C2×Dic3—182no—
160C2×Dic3—182no—
161C4×D3—62no—
162C4×D3—91no—
163C4×D3—91no—
164C4×D3—91no—
165C4×D3—182no—
166C4×D3—182no—
167C4×D3—182no—
168C4×D3—362no—
169C4×D3—364no—
170C4×D3C2×#15711yescharacteristic
171C3×C8—22no—
172C3×C8—42no—
173C3×C8—62no—
174C3×C8—91no—
175C3×C8C2×#15711yescharacteristic, complemented
176C3⋊C8—62no—
177C3⋊C8—91no—
178C3⋊C8—182no—
179C3⋊C8—362no—
180C3⋊C8C2×#15711yescharacteristic, complemented
181C2×C32—22no—
182C2×C32—32no—
183C2×C32—42no—
184C2×C32D3×C4⋊C411yescharacteristic
185C3⋊D3—32no—
186C3⋊D3—64no—
187C3⋊D3—124no—
188C3⋊D3D3×SD1612yes—
189C3×D3—24no—
190C3×D3—64no—
191C22×C4—271no—
192C22×C4—542no—
193C2×C8—31no—
194C2×C8—91no—
195C2×C8—271no—
196C2×C8—271no—
197C2×C8—271no—
198C2×C8—271no—
199C4⋊C4—92no—
200C4⋊C4—272no—
201C4⋊C4—542no—
202C22×C3—62no—
203C22×C3—91no—
204C2×D3—62no—
205C2×D3—91no—
206C2×D3—94no—
207C2×D3—188no—
208C2×D3—182no—
209C2×D3—362no—
210C2×D3C4⋊#15711yescharacteristic
211C3×C4—22no—
212C3×C4—42no—
213C3×C4—62no—
214C3×C4—91no—
215C3×C4—362no—
216C3×C4C22×#15711yescharacteristic
217Dic3—62no—
218Dic3—91no—
219Dic3—182no—
220Dic3—362no—
221Dic3—362no—
222Dic3C4⋊#15711yescharacteristic
223C32—22no—
224C32—42no—
225C32D3×#10611yescharacteristic, complemented
226C23—271no—
227C2×C4—31no—
228C2×C4—91no—
229C2×C4—182no—
230C2×C4—271no—
231C2×C4—271no—
232C2×C4—271no—
233C2×C4—271no—
234C2×C4—542no—
235C2×C4—1084no—
236C8—31no—
237C8—91no—
238C8—271no—
239C8D3×#15711yescharacteristic, complemented
240C2×C3—22no—
241C2×C3—42no—
242C2×C3—64no—
243C2×C3—92no—
244C2×C3C2×C4⋊#15711yescharacteristic
245D3—64no—
246D3—92no—
247D3—184no—
248D3—364no—
249D3#6376812yesdirect product factor
250C22—31no—
251C22—91no—
252C22—274no—
253C22—271no—
254C4—31no—
255C4—91no—
256C4—271no—
257C4—362no—
258C4—1082no—
259C4C2×D3×#15711yescharacteristic
260C3—22no—
261C3—42no—
262C3C2×#6376811yescharacteristic, complemented
263C2—32no—
264C2—92no—
265C2—272no—
266C2D3×C4⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C4⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C2×C4⋊C4
prime radicalC2×D3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterC8D3×#157
hypocenterC33C2×#106
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
3. #11723093C2semidirect
4. C3×#63768C2semidirect
5. #11722714C2semidirect
12. C8×C3⋊(C3⋊D3)C4semidirect
14. C32×D3×C8C4semidirect
15. #88785C4semidirect
17. D3×C3⋊(C3⋊C8)C4semidirect
23. #63768D3direct (N × H)
34. C33×C8C2×C4semidirect
35. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
36. C3×C3⋊(C3⋊C8)C2×C4semidirect
37. C32×C3⋊C8C2×C4semidirect
95. C32×C8C4×D3semidirect
99. C3⋊(C3⋊C8)C4×D3semidirect
104. C3⋊(C3⋊D3)#106semidirect
106. C32×D3#106semidirect
125. D3×C8#157semidirect
152. C33C2×#106semidirect
175. C3×C8C2×#157semidirect
180. C3⋊C8C2×#157semidirect
225. C32D3×#106semidirect
239. C8D3×#157semidirect
249. D3#63768direct (N × H)
262. C3C2×#63768semidirect

Maximal subgroups

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

subgroupindex
D3×C4⋊#1572
D3×C8×C3⋊D32
#117230932
C3×#637682
#117227142
C2×#637683
D3×#1069