Finite Group Databasebeta

← D3×#63655 #13160227

Subgroups — D3×#63655 #13160227

Order 1728.

644 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 128 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
1C32×D3×Q16C211yesmaximal, characteristic, complemented
2D3×#3745C211yesmaximal, characteristic
3#11722198C211yesmaximal, characteristic, complemented
4C3×#63655C211yesmaximal, characteristic, complemented
5#11722010C211yesmaximal, characteristic, complemented
6#11722008C211yesmaximal, characteristic, complemented
7D3×C3⋊(C3⋊C16)C211yesmaximal, characteristic, complemented
8C2×#63655—31nomaximal
9D3×#828—34nomaximal
10C32×D3×Q8—21no—
11D3×C4⋊(C3⋊D3)—21no—
12C33×Q16C2211yescharacteristic, complemented
13#88852C2211yescharacteristic
14C3×#3745C2211yescharacteristic
15C32×#216—21no—
16C32×Dic12C2211yescharacteristic, complemented
17C32×D3×C8C2211yescharacteristic
18#88791—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic, complemented
20C3×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
21C2×C32×Q16—31no—
22C2×#3745—31no—
23C3×D3×Q16D314yescomplemented
24D3×D24—34no—
25#63655—31no—
26#63655—31no—
27#63655—31no—
28#63655—34no—
29#63655—64no—
30#63655D311yescharacteristic, direct product factor
31C2×C3⋊(C3⋊C16)—31no—
32C3×#828—34no—
33#63549—34no—
34#63547—34no—
35D3×C3⋊C16—34no—
36C2×D3×C3⋊D3—41no—
37C33×Q8—21no—
38C4⋊(C3⋊(C3⋊D3))—21no—
39C3×C4⋊(C3⋊D3)—21no—
40C32×C4×D3—41no—
41C32×C4×D3D411yescharacteristic
42C32×Dic6—21no—
43C32×Dic6—41no—
44#6292—41no—
45C33×C8C2311yescharacteristic
46C32×C3⋊C8D411yescharacteristic
47C2×#828—94no—
48D3×SD32—91no—
49C2×C32×Q8—61no—
50C2×C4⋊(C3⋊D3)—61no—
51C3×D3×Q8—24no—
52D3×D12—64no—
53C32×Q16—24no—
54C32×Q16—31no—
55C32×Q16—61no—
56C32×Q16C2×D314yescomplemented
57C32×Q16C2×D311yescharacteristic, complemented
58C2×C32×C8—31no—
59#3745—31no—
60#3745—34no—
61#3745—64no—
62#3745—61no—
63#3745C2×D311yescharacteristic
64C3×#216—24no—
65C3×Dic12C2×D314yescomplemented
66C3×D24—34no—
67C3×D3×C8C2×D314yes—
68#3714—64no—
69C3⋊(C3⋊C16)—31no—
70C3⋊(C3⋊C16)—34no—
71C3⋊(C3⋊C16)—64no—
72C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
73C3×C3⋊C16—34no—
74C2×C3⋊(C3⋊D3)—41no—
75C2×C3×C3⋊D3—41no—
76C2×C32×D3D811yescharacteristic
77D3×C3⋊D3—82no—
78C33×C4—41no—
79C33×C4C2×D411yescharacteristic
80C32×Dic3—41no—
81C32×Dic3D811yescharacteristic
82C2×C3×Q16—34no—
83D3×Q16C3⋊D311yescharacteristic, complemented
84D3×D8—91no—
85C2×D24—94no—
86C3×SD32—91no—
87#828—34no—
88#828—94no—
89#828—94no—
90#828—94no—
91#828—91no—
92#828—184no—
93#827—91no—
94C2×C3⋊C16—94no—
95#800—91no—
96D3×C16—91no—
97C22×C3⋊D3—121no—
98C2×D32—124no—
99C32×Q8—21no—
100C32×Q8—24no—
101C32×Q8—44no—
102C32×Q8—61no—
103C32×Q8—121no—
104C2×C32×C4—31no—
105C2×C32×C4—121no—
106C4⋊(C3⋊D3)—21no—
107C4⋊(C3⋊D3)—61no—
108C4⋊(C3⋊D3)—64no—
109C4⋊(C3⋊D3)—124no—
110C4⋊(C3⋊D3)—121no—
111C3×D12—64no—
112C3×C4×D3—44no—
113C3×C4×D3#6714yes—
114C3×Dic6—24no—
115C3×Dic6—44no—
116#626—124no—
117C32×C8—24no—
118C32×C8—31no—
119C32×C8C22×D314yes—
120C32×C8C22×D311yescharacteristic
121C3×C3⋊C8#6714yes—
122C2×SD32—271noSylow
123C2×C33C2×D811yescharacteristic
124C3⋊(C3⋊D3)—81no—
125C3×C3⋊D3—81no—
126C32×D3SD3212yescomplemented
127C2×C3×Q8—64no—
128D3×Q8—21no—
129D3×D4—181no—
130C2×D12—184no—
131C3×Q16—24no—
132C3×Q16—34no—
133C3×Q16—64no—
134C3×Q16D3214yescomplemented
135C3×Q16C2×C3⋊D311yescharacteristic, complemented
136C3×D8—91no—
137C2×C3×C8—34no—
138#216—21no—
139#213—181no—
140Dic12C2×C3⋊D311yescharacteristic, complemented
141D24—34no—
142D24—94no—
143D24—91no—
144D24—184no—
145D24—184no—
146D3×C8C2×C3⋊D311yescharacteristic
147C3×C16—91no—
148C3⋊C16—34no—
149C3⋊C16—94no—
150C3⋊C16—91no—
151C3⋊C16—184no—
152C22×C32—31no—
153C2×C3⋊D3—41no—
154C2×C3⋊D3—121no—
155C2×C3⋊D3—124no—
156C2×C3⋊D3—244no—
157C2×C3⋊D3—242no—
158C2×C3×D3—124no—
159C2×C3×D3#21314yes—
160D32—248no—
161C32×C4—24no—
162C32×C4—31no—
163C32×C4—41no—
164C32×C4—44no—
165C32×C4—84no—
166C32×C4—121no—
167C32×C4D3×D411yescharacteristic
168C32×C4C2×#6714yes—
169C3×Dic3—44no—
170C3×Dic3#21314yes—
171C2×Q16—31no—
172C2×D8—271no—
173SD32—91no—
174SD32—271no—
175SD32—271no—
176SD32—271no—
177C2×C16—271no—
178C33C2×SD3211yescharacteristic, complemented, Sylow
179C22×D3—364no—
180C22×D3—361no—
181C3×Q8—24no—
182C3×Q8—21no—
183C3×Q8—44no—
184C3×Q8—64no—
185C3×Q8—124no—
186C3×D4—181no—
187C2×C3×C4—34no—
188C2×C3×C4—124no—
189#67—361no—
190D12—64no—
191D12—184no—
192D12—181no—
193D12—364no—
194D12—364no—
195C4×D3—41no—
196C4×D3#63811yescharacteristic
197Dic6—21no—
198Dic6—41no—
199C3×C8—24no—
200C3×C8—34no—
201C3×C8C2×D3214yes—
202C3×C8C22×C3⋊D311yescharacteristic
203C3⋊C8#63811yescharacteristic
204C2×C32—24no—
205C2×C32—32no—
206C2×C32D3×D811yescharacteristic
207C2×C32C2×#21314yes—
208C3⋊D3—81no—
209C3⋊D3—241no—
210C3⋊D3—244no—
211C3⋊D3—484no—
212C3×D3—244no—
213C3×D3#82818yescomplemented
214C2×Q8—61no—
215C2×D4—541no—
216Q16—31no—
217Q16—61no—
218Q16D3×C3⋊D311yescharacteristic, complemented
219D8—91no—
220D8—271no—
221D8—541no—
222C2×C8—31no—
223C16—91no—
224C16—271no—
225C22×C3—34no—
226C22×C3—361no—
227C2×D3—124no—
228C2×D3—361no—
229C2×D3—364no—
230C2×D3—724no—
231C2×D3—728no—
232C2×D3—722no—
233C2×D3#375311yescharacteristic
234C3×C4—24no—
235C3×C4—34no—
236C3×C4—41no—
237C3×C4—44no—
238C3×C4—84no—
239C3×C4—124no—
240C3×C4D3×#6714yes—
241C3×C4C2×#63811yescharacteristic
242Dic3—41no—
243Dic3#375311yescharacteristic
244C32—24no—
245C32D3×SD3211yescharacteristic, complemented
246C32C2×#82814yescomplemented
247C23—1081no—
248Q8—21no—
249Q8—61no—
250Q8—121no—
251D4—181no—
252D4—541no—
253D4—1081no—
254C2×C4—31no—
255C2×C4—121no—
256C8—31no—
257C8C2×D3×C3⋊D311yescharacteristic
258C2×C3—24no—
259C2×C3—38no—
260C2×C3—721no—
261C2×C3D3×#21314yes—
262C2×C3C2×#375311yescharacteristic
263D3—244no—
264D3—724no—
265D3—721no—
266D3—1444no—
267D3#6365512yesdirect product factor
268C22—31no—
269C22—361no—
270C22—1081no—
271C22—2162no—
272C4—31no—
273C4—41no—
274C4—121no—
275C4D3×#63811yescharacteristic
276C3—24no—
277C3D3×#82814yescomplemented
278C3C2×#6365511yescharacteristic, complemented
279C2—32no—
280C2—721no—
281C2—2161no—
282C2D3×#375311yescharacteristic
subgroupG/H
centerC2D3×#3753
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C2×D8
prime radicalD3×#3745C2
hypercenterQ16D3×C3⋊D3
hypocenterC33C2×SD32
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. C32×D3×Q16C2semidirect
3. #11722198C2semidirect
4. C3×#63655C2semidirect
5. #11722010C2semidirect
6. #11722008C2semidirect
7. D3×C3⋊(C3⋊C16)C2semidirect
12. C33×Q16C22semidirect
16. C32×Dic12C22semidirect
19. C3⋊(C3⋊(C3⋊C16))C22semidirect
20. C3×C3⋊(C3⋊C16)C22semidirect
23. C3×D3×Q16D3semidirect
30. #63655D3direct (N × H)
56. C32×Q16C2×D3semidirect
57. C32×Q16C2×D3semidirect
65. C3×Dic12C2×D3semidirect
72. C3⋊(C3⋊C16)C2×D3semidirect
83. D3×Q16C3⋊D3semidirect
126. C32×D3SD32semidirect
134. C3×Q16D32semidirect
135. C3×Q16C2×C3⋊D3semidirect
140. Dic12C2×C3⋊D3semidirect
178. C33C2×SD32semidirect
213. C3×D3#828semidirect
218. Q16D3×C3⋊D3semidirect
245. C32D3×SD32semidirect
246. C32C2×#828semidirect
267. D3#63655direct (N × H)
277. C3D3×#828semidirect
278. C3C2×#63655semidirect

Maximal subgroups

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

subgroupindex
C32×D3×Q162
D3×#37452
#117221982
C3×#636552
#117220102
#117220082
D3×C3⋊(C3⋊C16)2
C2×#636553
D3×#8283