Finite Group Databasebeta

← D3×#63550 #13155211

Subgroups — D3×#63550 #13155211

Order 1728.

414 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 86 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×D3×Dic12C212yesmaximal
2#11722069C211yesmaximal, characteristic, complemented
3#11722013C212yesmaximal, complemented
4D3×C3⋊(C3⋊C16)C211yesmaximal, characteristic
5C3×#63550C211yesmaximal, characteristic, complemented
6D3×#829—32nomaximal
7C2×#63550—31nomaximal
8C3×D3×Dic6—22no—
9C3×#3746C2212yes—
10C32×Dic12C2212yes—
11C32×D3×C8C2211yescharacteristic
12C3×#3719—22no—
13C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic
14C3×C3⋊(C3⋊C16)C2211yescharacteristic
15C3×D3×Q16—32no—
16C2×C3×Dic12—32no—
17D3×Dic12D312yes—
18#63656—32no—
19C2×C3⋊(C3⋊C16)—31no—
20C3×#829—32no—
21#63551—32no—
22#63550—32no—
23#63550—32no—
24#63550—31no—
25#63550—62no—
26#63550D311yescharacteristic, direct product factor
27D3×C3⋊C16—32no—
28D3×C3⋊C16—61no—
29C3×#634—22no—
30C32×C4×D3D411yescharacteristic
31C32×Dic6—22no—
32C3×#627—42no—
33C3×D3×Dic3—42no—
34C33×C8C2311yescharacteristic
35C32×C3⋊C8D411yescharacteristic
36C2×#829—92no—
37D3×Q32—91no—
38C3×D3×Q8—62no—
39C2×C3×Dic6—62no—
40D3×Dic6—22no—
41C32×Q16—32no—
42C2×C32×C8—31no—
43#3746C2×D312yes—
44C3×#216—62no—
45C3×Dic12—22no—
46C3×Dic12—32no—
47C3×Dic12—32no—
48C3×Dic12—62no—
49C3×Dic12—62no—
50C3×Dic12C2×D312yes—
51C3×Dic12C2×D312yes—
52C3×D3×C8—21no—
53C3×D3×C8C2×D312yes—
54#3719—22no—
55C3⋊(C3⋊C16)—32no—
56C3⋊(C3⋊C16)—31no—
57C3⋊(C3⋊C16)—61no—
58C3⋊(C3⋊C16)—62no—
59C3⋊(C3⋊C16)—121no—
60C3⋊(C3⋊C16)C2×D311yescharacteristic
61C3×C3⋊C16—32no—
62C3×C3⋊C16—61no—
63C2×C32×D3D811yescharacteristic
64C33×C4C2×D411yescharacteristic
65C3×C3⋊Dic3—42no—
66C32×Dic3—42no—
67C32×Dic3D811yescharacteristic
68C2×C3×Q16—92no—
69D3×Q16—32no—
70C2×Dic12—32no—
71C3×Q32—91no—
72#829—32no—
73#829—92no—
74#829—92no—
75#829—92no—
76#829—92no—
77#829—182no—
78C2×C3⋊C16—92no—
79C2×C3⋊C16—181no—
80Dic24—91no—
81D3×C16—91no—
82C32×Q8—62no—
83C2×C32×C4—31no—
84#634—22no—
85C2×C3×Dic3—122no—
86C3×C4×D3—21no—
87C3×C4×D3—122no—
88C3×C4×D3#6712yes—
89C3×Dic6—22no—
90C3×Dic6—22no—
91C3×Dic6—42no—
92C3×Dic6—62no—
93C3×Dic6—62no—
94C3×Dic6—122no—
95C3×Dic6—122no—
96C3×Dic6—122no—
97#627—42no—
98D3×Dic3—42no—
99C32×C8—21no—
100C32×C8—22no—
101C32×C8—31no—
102C32×C8—41no—
103C32×C8C22×D311yescharacteristic
104C32×C8C22×D312yes—
105C3×C3⋊C8—21no—
106C3×C3⋊C8#6712yes—
107C2×Q32—271noSylow
108C2×C33C2×D811yescharacteristic
109C32×D3Q3212yescomplemented
110C2×C3×Q8—182no—
111D3×Q8—62no—
112C2×Dic6—62no—
113C3×Q16—32no—
114C3×Q16—32no—
115C3×Q16—62no—
116C3×Q16—92no—
117C3×Q16—182no—
118C2×C3×C8—32no—
119C2×C3×C8—61no—
120#216—62no—
121Dic12—32no—
122Dic12—32no—
123Dic12—62no—
124Dic12—62no—
125Dic12D3212yes—
126D3×C8D3211yescharacteristic
127C3×C16—91no—
128C3⋊C16—32no—
129C3⋊C16—61no—
130C3⋊C16—91no—
131C3⋊C16—92no—
132C3⋊C16—181no—
133C3⋊C16—182no—
134C3⋊C16—361no—
135C22×C32—31no—
136C2×C3×D3—21no—
137C2×C3×D3#21312yes—
138C32×C4—21no—
139C32×C4—22no—
140C32×C4—31no—
141C32×C4—41no—
142C32×C4—122no—
143C32×C4D3×D411yescharacteristic
144C32×C4C2×#6712yes—
145C3⋊Dic3—42no—
146C3×Dic3—21no—
147C3×Dic3—42no—
148C3×Dic3—42no—
149C3×Dic3—82no—
150C3×Dic3—122no—
151C3×Dic3—122no—
152C3×Dic3—242no—
153C3×Dic3#21312yes—
154C2×Q16—92no—
155Q32—91no—
156Q32—271no—
157Q32—272no—
158C2×C16—271no—
159C33C2×Q3211yescharacteristic, complemented, Sylow
160C3×Q8—62no—
161C3×Q8—62no—
162C3×Q8—122no—
163C3×Q8—182no—
164C3×Q8—362no—
165C2×C3×C4—32no—
166C2×C3×C4—61no—
167C2×C3×C4—362no—
168C2×Dic3—122no—
169C4×D3—122no—
170C4×D3#62511yescharacteristic
171Dic6—22no—
172Dic6—62no—
173Dic6—62no—
174Dic6—122no—
175Dic6—122no—
176Dic6—122no—
177C3×C8—21no—
178C3×C8—22no—
179C3×C8—32no—
180C3×C8—41no—
181C3×C8—61no—
182C3×C8C2×D3211yescharacteristic
183C3×C8C2×D3212yes—
184C3⋊C8#62511yescharacteristic
185C2×C32—21no—
186C2×C32—22no—
187C2×C32—32no—
188C2×C32—41no—
189C2×C32D3×D811yescharacteristic
190C2×C32C2×#21312yes—
191C3×D3—22no—
192C3×D3#82914yescomplemented
193C2×Q8—182no—
194Q16—32no—
195Q16—92no—
196Q16—182no—
197C2×C8—31no—
198C16—91no—
199C16—271no—
200C22×C3—32no—
201C22×C3—61no—
202C2×D3#371311yescharacteristic
203C3×C4—21no—
204C3×C4—22no—
205C3×C4—32no—
206C3×C4—41no—
207C3×C4—61no—
208C3×C4—122no—
209C3×C4—122no—
210C3×C4—242no—
211C3×C4—362no—
212C3×C4C2×#62511yescharacteristic
213C3×C4D3×#6712yes—
214Dic3—42no—
215Dic3—122no—
216Dic3—122no—
217Dic3—242no—
218Dic3#371311yescharacteristic
219C32—22no—
220C32—21no—
221C32—41no—
222C32D3×Q3211yescharacteristic, complemented
223C32C2×#82912yescomplemented
224Q8—62no—
225Q8—182no—
226Q8—362no—
227C2×C4—31no—
228C2×C4—362no—
229C8—31no—
230C8D3311yescharacteristic
231C2×C3—22no—
232C2×C3—21no—
233C2×C3—34no—
234C2×C3—41no—
235C2×C3—62no—
236C2×C3C2×#371311yescharacteristic
237C2×C3D3×#21312yes—
238D3#6355012yesdirect product factor
239C22—31no—
240C4—31no—
241C4—122no—
242C4—362no—
243C4D3×#62511yescharacteristic
244C3—21no—
245C3—22no—
246C3—41no—
247C3C2×#6355011yescharacteristic, complemented
248C3D3×#82912yescomplemented
249C2—32no—
250C2D3×#371311yescharacteristic
subgroupG/H
centerC2D3×#3713
commutator, Fitting, cosocleC33×C8C23
Frattini, hypercenterC8D33
socleC2×C33C2×D8
prime radicalC2×C32×D3D8
hypocenterC33C2×Q32
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11722069C2semidirect
3. #11722013C2semidirect
5. C3×#63550C2semidirect
26. #63550D3direct (N × H)
109. C32×D3Q32semidirect
159. C33C2×Q32semidirect
192. C3×D3#829semidirect
222. C32D3×Q32semidirect
223. C32C2×#829semidirect
238. D3#63550direct (N × H)
247. C3C2×#63550semidirect
248. C3D3×#829semidirect

Maximal subgroups

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

subgroupindex
C3×D3×Dic122
#117220692
#117220132
D3×C3⋊(C3⋊C16)2
C3×#635502
D3×#8293
C2×#635503