Finite Group Databasebeta

← C32×D3×SD32 #13162349

Subgroups — C32×D3×SD32 #13162349

Order 1728.

644 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 198 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
2C32×D3×D8C211yesmaximal, characteristic
3C33×SD32C211yesmaximal, characteristic, complemented
4C32×#828C211yesmaximal, characteristic, complemented
5C32×#827C211yesmaximal, characteristic, complemented
6C32×#800C211yesmaximal, characteristic, complemented
7C32×D3×C16C211yesmaximal, characteristic, complemented
8C2×C32×SD32—31nomaximal
9C3×D3×SD32C314yesmaximal, direct product factor
10C32×D3×Q8—21no—
11C32×D3×D4—21no—
12C33×Q16C2211yescharacteristic, complemented
13C33×D8C2211yescharacteristic
14C32×#216—21no—
15C32×#213—21no—
16C32×Dic12C2211yescharacteristic, complemented
17C32×D24C2211yescharacteristic
18C32×D3×C8C2211yescharacteristic
19C33×C16C2211yescharacteristic, complemented
20C32×C3⋊C16C2211yescharacteristic, complemented
21C2×C32×Q16—31no—
22C2×C32×D8—31no—
23C3×D3×Q16C2×C314yescomplemented
24C3×D3×D8C2×C314yes—
25C32×SD32—24no—
26C32×SD32—31no—
27C32×SD32—31no—
28C32×SD32—31no—
29C32×SD32D311yescharacteristic, direct product factor
30C32×SD32C2×C314yescomplemented
31C2×C32×C16—31no—
32C3×#828C2×C314yescomplemented
33C3×#827C2×C314yescomplemented
34C3×#800C2×C314yescomplemented
35C3×D3×C16C2×C314yescomplemented
36C22×C32×D3—41no—
37C33×Q8—21no—
38C33×D4—21no—
39C32×#67—41no—
40C32×D12—21no—
41C32×C4×D3—41no—
42C32×C4×D3D411yescharacteristic
43C32×Dic6—21no—
44C32×Dic6—41no—
45C33×C8C2311yescharacteristic
46C32×C3⋊C8D411yescharacteristic
47C2×C3×SD32—34no—
48D3×SD32C3211yescharacteristic, direct product factor
49C2×C32×Q8—61no—
50C2×C32×D4—61no—
51C3×D3×Q8—24no—
52C3×D3×D4—24no—
53C32×Q16—24no—
54C32×Q16—31no—
55C32×Q16—61no—
56C32×Q16C2×D311yescharacteristic, complemented
57C32×Q16C22×C314yescomplemented
58C32×D8—24no—
59C32×D8—31no—
60C32×D8—61no—
61C32×D8C2×D311yescharacteristic
62C32×D8C22×C314yes—
63C2×C32×C8—31no—
64C3×#216—24no—
65C3×#213—24no—
66C3×Dic12C22×C314yescomplemented
67C3×D24C22×C314yes—
68C3×D3×C8C22×C314yes—
69C32×C16—24no—
70C32×C16—31no—
71C32×C16C2×D311yescharacteristic, complemented
72C32×C16C22×C314yescomplemented
73C3×C3⋊C16C22×C314yescomplemented
74C22×C33—41no—
75C2×C32×D3—41no—
76C2×C32×D3—82no—
77C2×C32×D3D811yescharacteristic
78C33×C4—41no—
79C33×C4C2×D411yescharacteristic
80C32×Dic3—41no—
81C32×Dic3D811yescharacteristic
82C2×C3×Q16—34no—
83C2×C3×D8—34no—
84D3×Q16C2×C3211yescharacteristic, complemented
85D3×D8C2×C3211yescharacteristic
86C3×SD32—24no—
87C3×SD32—34no—
88C3×SD32—34no—
89C3×SD32—34no—
90C3×SD32C3×D314yesdirect product factor
91C3×SD32C2×C3211yescharacteristic, complemented
92C2×C3×C16—34no—
93#828C2×C3211yescharacteristic, complemented
94#827C2×C3211yescharacteristic, complemented
95#800C2×C3211yescharacteristic, complemented
96D3×C16C2×C3211yescharacteristic, complemented
97C23×C32—121no—
98C22×C3×D3—44no—
99C32×Q8—21no—
100C32×Q8—24no—
101C32×Q8—44no—
102C32×Q8—61no—
103C32×Q8—121no—
104C32×D4—21no—
105C32×D4—24no—
106C32×D4—44no—
107C32×D4—61no—
108C32×D4—121no—
109C2×C32×C4—31no—
110C2×C32×C4—121no—
111C3×#67—44no—
112C3×D12—24no—
113C3×C4×D3—44no—
114C3×C4×D3C3×D414yes—
115C3×Dic6—24no—
116C3×Dic6—44no—
117C32×C8—24no—
118C32×C8—31no—
119C32×C8C22×D311yescharacteristic
120C32×C8C23×C314yes—
121C3×C3⋊C8C3×D414yes—
122C2×SD32—31noSylow
123C2×C33—81no—
124C2×C33C2×D811yescharacteristic
125C32×D3—81no—
126C32×D3SD3212yesdirect product factor
127C2×C3×Q8—64no—
128C2×C3×D4—64no—
129D3×Q8—21no—
130D3×D4—21no—
131C3×Q16—24no—
132C3×Q16—34no—
133C3×Q16—64no—
134C3×Q16C2×C3×D314yescomplemented
135C3×Q16C22×C3211yescharacteristic, complemented
136C3×D8—24no—
137C3×D8—34no—
138C3×D8—64no—
139C3×D8C2×C3×D314yes—
140C3×D8C22×C3211yescharacteristic
141C2×C3×C8—34no—
142#216—21no—
143#213—21no—
144Dic12C22×C3211yescharacteristic, complemented
145D24C22×C3211yescharacteristic
146D3×C8C22×C3211yescharacteristic
147C3×C16—24no—
148C3×C16—34no—
149C3×C16C2×C3×D314yescomplemented
150C3×C16C22×C3211yescharacteristic, complemented
151C3⋊C16C22×C3211yescharacteristic, complemented
152C22×C32—31no—
153C22×C32—41no—
154C22×C32—44no—
155C22×C32—84no—
156C22×C32—121no—
157C22×C32—242no—
158C2×C3×D3—44no—
159C2×C3×D3—88no—
160C2×C3×D3C3×D814yes—
161C32×C4—24no—
162C32×C4—31no—
163C32×C4—41no—
164C32×C4—44no—
165C32×C4—84no—
166C32×C4—121no—
167C32×C4D3×D411yescharacteristic
168C32×C4C2×C3×D414yes—
169C3×Dic3—44no—
170C3×Dic3C3×D814yes—
171C2×Q16—31no—
172C2×D8—31no—
173SD32—31no—
174SD32—31no—
175SD32—31no—
176SD32C32×D311yescharacteristic, direct product factor
177C2×C16—31no—
178C33C2×SD3211yescharacteristic, complemented, Sylow
179C23×C3—124no—
180C22×D3—41no—
181C3×Q8—24no—
182C3×Q8—21no—
183C3×Q8—44no—
184C3×Q8—64no—
185C3×Q8—124no—
186C3×D4—24no—
187C3×D4—21no—
188C3×D4—44no—
189C3×D4—64no—
190C3×D4—124no—
191C2×C3×C4—34no—
192C2×C3×C4—124no—
193#67—41no—
194D12—21no—
195C4×D3—41no—
196C4×D3C32×D411yescharacteristic
197Dic6—21no—
198Dic6—41no—
199C3×C8—24no—
200C3×C8—34no—
201C3×C8C22×C3×D314yes—
202C3×C8C23×C3211yescharacteristic
203C3⋊C8C32×D411yescharacteristic
204C2×C32—24no—
205C2×C32—32no—
206C2×C32—81no—
207C2×C32—84no—
208C2×C32—164no—
209C2×C32—241no—
210C2×C32D3×D811yescharacteristic
211C2×C32C2×C3×D814yes—
212C3×D3—84no—
213C3×D3C3×SD3218yesdirect product factor
214C2×Q8—61no—
215C2×D4—61no—
216Q16—31no—
217Q16—61no—
218Q16C2×C32×D311yescharacteristic, complemented
219D8—31no—
220D8—61no—
221D8C2×C32×D311yescharacteristic
222C2×C8—31no—
223C16—31no—
224C16C2×C32×D311yescharacteristic, complemented
225C22×C3—34no—
226C22×C3—44no—
227C22×C3—41no—
228C22×C3—84no—
229C22×C3—124no—
230C22×C3—248no—
231C2×D3—41no—
232C2×D3—82no—
233C2×D3C32×D811yescharacteristic
234C3×C4—24no—
235C3×C4—34no—
236C3×C4—44no—
237C3×C4—41no—
238C3×C4—84no—
239C3×C4—124no—
240C3×C4C3×D3×D414yes—
241C3×C4C2×C32×D411yescharacteristic
242Dic3—41no—
243Dic3C32×D811yescharacteristic
244C32—24no—
245C32D3×SD3211yescharacteristic, direct product factor
246C32C2×C3×SD3214yescomplemented
247C23—121no—
248Q8—21no—
249Q8—61no—
250Q8—121no—
251D4—21no—
252D4—61no—
253D4—121no—
254C2×C4—31no—
255C2×C4—121no—
256C8—31no—
257C8C22×C32×D311yescharacteristic
258C2×C3—24no—
259C2×C3—38no—
260C2×C3—84no—
261C2×C3—81no—
262C2×C3—164no—
263C2×C3—244no—
264C2×C3C3×D3×D814yes—
265C2×C3C2×C32×D811yescharacteristic
266D3—81no—
267D3C32×SD3212yesdirect product factor
268C22—31no—
269C22—41no—
270C22—121no—
271C22—242no—
272C4—31no—
273C4—41no—
274C4—121no—
275C4C32×D3×D411yescharacteristic
276C3—24no—
277C3C3×D3×SD3214yesdirect product factor
278C3C2×C32×SD3211yescharacteristic, complemented
279C2—32no—
280C2—81no—
281C2—241no—
282C2C32×D3×D811yescharacteristic
subgroupG/H
centerC2×C32D3×D8
commutator, cosocleC3×C8C23×C32
FrattiniC8C22×C32×D3
FittingC33×SD32C2
socleC2×C33C2×D8
prime radicalC32×D3×D8C2
hypercenterC32×SD32D3
hypocenterC3C2×C32×SD32
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C32×C4 ◁ C32×C8 ◁ C32×SD32

Complements

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

NH ≅ G/Ntype
1. C32×D3×Q16C2semidirect
3. C33×SD32C2semidirect
4. C32×#828C2semidirect
5. C32×#827C2semidirect
6. C32×#800C2semidirect
7. C32×D3×C16C2semidirect
9. C3×D3×SD32C3direct (N × H)
12. C33×Q16C22semidirect
16. C32×Dic12C22semidirect
19. C33×C16C22semidirect
20. C32×C3⋊C16C22semidirect
23. C3×D3×Q16C2×C3semidirect
29. C32×SD32D3direct (N × H)
30. C32×SD32C2×C3semidirect
32. C3×#828C2×C3semidirect
33. C3×#827C2×C3semidirect
34. C3×#800C2×C3semidirect
35. C3×D3×C16C2×C3semidirect
48. D3×SD32C32direct (N × H)
56. C32×Q16C2×D3semidirect
57. C32×Q16C22×C3semidirect
66. C3×Dic12C22×C3semidirect
71. C32×C16C2×D3semidirect
72. C32×C16C22×C3semidirect
73. C3×C3⋊C16C22×C3semidirect
84. D3×Q16C2×C32semidirect
90. C3×SD32C3×D3direct (N × H)
91. C3×SD32C2×C32semidirect
93. #828C2×C32semidirect
94. #827C2×C32semidirect
95. #800C2×C32semidirect
96. D3×C16C2×C32semidirect
126. C32×D3SD32direct (N × H)
134. C3×Q16C2×C3×D3semidirect
135. C3×Q16C22×C32semidirect
144. Dic12C22×C32semidirect
149. C3×C16C2×C3×D3semidirect
150. C3×C16C22×C32semidirect
151. C3⋊C16C22×C32semidirect
176. SD32C32×D3direct (N × H)
178. C33C2×SD32semidirect
213. C3×D3C3×SD32direct (N × H)
218. Q16C2×C32×D3semidirect
224. C16C2×C32×D3semidirect
245. C32D3×SD32direct (N × H)
246. C32C2×C3×SD32semidirect
267. D3C32×SD32direct (N × H)
277. C3C3×D3×SD32direct (N × H)
278. C3C2×C32×SD32semidirect

Maximal subgroups

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

subgroupindex
C32×D3×Q162
C32×D3×D82
C33×SD322
C32×#8282
C32×#8272
C32×#8002
C32×D3×C162
C2×C32×SD323
C3×D3×SD323