Finite Group Databasebeta

← C3⋊D3×#828 #13160225

Subgroups — C3⋊D3×#828 #13160225

Order 1728.

644 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 104 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×Q16×C3⋊D3C211yesmaximal, characteristic, complemented
2C3⋊D3×D24C211yesmaximal, characteristic
3#11722198C211yesmaximal, characteristic, complemented
4C32×#828C211yesmaximal, characteristic, complemented
5#11722012C211yesmaximal, characteristic, complemented
6#11722007C211yesmaximal, characteristic, complemented
7C3⋊D3×C3⋊C16C211yesmaximal, characteristic, complemented
8C3⋊D3×SD32—31nomaximal
9D3×#828—34nomaximal
10C3×Q8×C3⋊D3—21no—
11C3⋊D3×D12—21no—
12C33×Q16C2211yescharacteristic, complemented
13#88852C2211yescharacteristic
14C3×#3756—21no—
15C3×#3746C2211yescharacteristic, complemented
16C3×C8×C3⋊D3C2211yescharacteristic
17C32×D24C2211yescharacteristic
18#88790—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic, complemented
20C32×C3⋊C16C2211yescharacteristic, complemented
21Q16×C3⋊D3D311yescharacteristic, complemented
22D8×C3⋊D3—31no—
23C3×D3×Q16—34no—
24D3×D24—34no—
25C32×SD32—31no—
26#63655—31no—
27#63655—34no—
28#63655—64no—
29#63654—31no—
30#63627—31no—
31C16×C3⋊D3—31no—
32C3×#828D314yescomplemented
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—41no—
40C3×C4×C3⋊D3D411yescharacteristic
41C3×#634—21no—
42C3×#634—41no—
43C32×D12—21no—
44#6291—41no—
45C33×C8C2311yescharacteristic
46C3×C3⋊(C3⋊C8)D411yescharacteristic
47C2×#828—91no—
48D3×SD32—94no—
49Q8×C3⋊D3—21no—
50D4×C3⋊D3—61no—
51C3×D3×Q8—64no—
52D3×D12—64no—
53C32×Q16—24no—
54C32×Q16C2×D314yescomplemented
55C32×Q16C2×D311yescharacteristic, complemented
56C32×D8—31no—
57#3756—21no—
58#3753—61no—
59#3746C2×D311yescharacteristic, complemented
60#3745—31no—
61#3745—34no—
62#3745—64no—
63C8×C3⋊D3C2×D311yescharacteristic
64C3×#216—64no—
65C3×Dic12—34no—
66C3×D24C2×D314yes—
67C3×D3×C8—34no—
68#3714—64no—
69C32×C16—31no—
70C3⋊(C3⋊C16)—31no—
71C3⋊(C3⋊C16)—34no—
72C3⋊(C3⋊C16)—64no—
73C3×C3⋊C16C2×D314yescomplemented
74C2×C3⋊(C3⋊D3)—41no—
75C2×C3×C3⋊D3D811yescharacteristic
76C2×C32×D3—41no—
77D3×C3⋊D3—82no—
78C33×C4—41no—
79C33×C4C2×D411yescharacteristic
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3D811yescharacteristic
82C2×C3×Q16—91no—
83D3×Q16—34no—
84D3×D8—94no—
85C2×D24—91no—
86C3×SD32—34no—
87#828—94no—
88#828—91no—
89#828—91no—
90#828—91no—
91#828—184no—
92#828C3⋊D311yescharacteristic, direct product factor
93#827—94no—
94C2×C3⋊C16—91no—
95#800—94no—
96D3×C16—94no—
97C22×C3⋊D3—121no—
98C2×D32—124no—
99C32×Q8—21no—
100C32×Q8—24no—
101C32×Q8—44no—
102C32×D4—61no—
103#638—121no—
104C4⋊(C3⋊D3)—61no—
105C4⋊(C3⋊D3)—64no—
106C4⋊(C3⋊D3)—124no—
107C4×C3⋊D3—41no—
108C4×C3⋊D3#6711yescharacteristic
109#634—21no—
110#634—41no—
111C3×D12—24no—
112C3×C4×D3—34no—
113C3×C4×D3—124no—
114C3×Dic6—64no—
115C3×Dic6—124no—
116#626—124no—
117C32×C8—24no—
118C32×C8C22×D314yes—
119C32×C8C22×D311yescharacteristic
120C3⋊(C3⋊C8)#6711yescharacteristic
121C3×C3⋊C8—34no—
122C2×SD32—271noSylow
123C2×C33C2×D811yescharacteristic
124C3⋊(C3⋊D3)—81no—
125C3×C3⋊D3SD3212yescomplemented
126C32×D3—81no—
127C2×C3×Q8—181no—
128D3×Q8—64no—
129D3×D4—184no—
130C2×D12—181no—
131C3×Q16—24no—
132C3×Q16—91no—
133C3×Q16—181no—
134C3×Q16D3214yescomplemented
135C3×Q16C2×C3⋊D311yescharacteristic, complemented
136C3×D8—34no—
137C2×C3×C8—91no—
138#216—64no—
139#213—184no—
140Dic12—34no—
141D24—94no—
142D24—91no—
143D24—184no—
144D24—181no—
145D24C2×C3⋊D311yescharacteristic
146D3×C8—34no—
147C3×C16—34no—
148C3⋊C16—94no—
149C3⋊C16—91no—
150C3⋊C16—184no—
151C3⋊C16C2×C3⋊D311yescharacteristic, complemented
152C22×C32—121no—
153C2×C3⋊D3—121no—
154C2×C3⋊D3—124no—
155C2×C3⋊D3—244no—
156C2×C3⋊D3—242no—
157C2×C3⋊D3#21311yescharacteristic
158C2×C3×D3—34no—
159C2×C3×D3—44no—
160D32—248no—
161C32×C4—24no—
162C32×C4—41no—
163C32×C4—44no—
164C32×C4—84no—
165C32×C4D3×D414yes—
166C32×C4C2×#6711yescharacteristic
167C3⋊Dic3—41no—
168C3⋊Dic3#21311yescharacteristic
169C3×Dic3—34no—
170C3×Dic3—124no—
171C2×Q16—91no—
172C2×D8—271no—
173SD32—31no—
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—181no—
185C3×Q8—361no—
186C3×D4—64no—
187C2×C3×C4—91no—
188C2×C3×C4—361no—
189#67—364no—
190D12—21no—
191D12—184no—
192D12—181no—
193D12—364no—
194D12—361no—
195C4×D3—34no—
196C4×D3—124no—
197Dic6—64no—
198Dic6—124no—
199C3×C8—24no—
200C3×C8—91no—
201C3×C8C2×D3214yes—
202C3×C8C22×C3⋊D311yescharacteristic
203C3⋊C8—34no—
204C2×C32—24no—
205C2×C32—241no—
206C2×C32D3×D814yes—
207C2×C32C2×#21311yescharacteristic
208C3⋊D3—241no—
209C3⋊D3—244no—
210C3⋊D3—484no—
211C3⋊D3#82812yesdirect product factor
212C3×D3—38no—
213C3×D3—84no—
214C2×Q8—181no—
215C2×D4—541no—
216Q16—91no—
217Q16—181no—
218Q16D3×C3⋊D311yescharacteristic, complemented
219D8—31no—
220D8—271no—
221D8—541no—
222C2×C8—91no—
223C16—31no—
224C16—271no—
225C22×C3—91no—
226C22×C3—124no—
227C2×D3—34no—
228C2×D3—41no—
229C2×D3—361no—
230C2×D3—364no—
231C2×D3—724no—
232C2×D3—728no—
233C2×D3—722no—
234C3×C4—24no—
235C3×C4—41no—
236C3×C4—44no—
237C3×C4—84no—
238C3×C4—91no—
239C3×C4—361no—
240C3×C4D3×#6714yes—
241C3×C4D4×C3⋊D311yescharacteristic
242Dic3—34no—
243Dic3—124no—
244C32—24no—
245C32D3×SD3214yescomplemented
246C32C2×#82811yescharacteristic, complemented
247C23—1081no—
248Q8—21no—
249Q8—181no—
250Q8—361no—
251D4—61no—
252D4—541no—
253D4—1081no—
254C2×C4—91no—
255C2×C4—361no—
256C8—91no—
257C8C2×D3×C3⋊D311yescharacteristic
258C2×C3—24no—
259C2×C3—92no—
260C2×C3—244no—
261C2×C3D3×#21314yes—
262C2×C3D8×C3⋊D311yescharacteristic
263D3—38no—
264D3—81no—
265D3—724no—
266D3—721no—
267D3—1444no—
268C22—91no—
269C22—121no—
270C22—1081no—
271C22—2162no—
272C4—41no—
273C4—91no—
274C4—361no—
275C4C3⋊D3×#6711yescharacteristic
276C3—24no—
277C3D3×#82814yescomplemented
278C3C3⋊D3×SD3211yescharacteristic, complemented
279C2—92no—
280C2—241no—
281C2—2161no—
282C2C3⋊D3×#21311yescharacteristic
subgroupG/H
centerC2C3⋊D3×#213
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C2×D8
prime radicalC3⋊D3×D24C2
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. C3×Q16×C3⋊D3C2semidirect
3. #11722198C2semidirect
4. C32×#828C2semidirect
5. #11722012C2semidirect
6. #11722007C2semidirect
7. C3⋊D3×C3⋊C16C2semidirect
12. C33×Q16C22semidirect
15. C3×#3746C22semidirect
19. C3⋊(C3⋊(C3⋊C16))C22semidirect
20. C32×C3⋊C16C22semidirect
21. Q16×C3⋊D3D3semidirect
32. C3×#828D3semidirect
54. C32×Q16C2×D3semidirect
55. C32×Q16C2×D3semidirect
59. #3746C2×D3semidirect
73. C3×C3⋊C16C2×D3semidirect
92. #828C3⋊D3direct (N × H)
125. C3×C3⋊D3SD32semidirect
134. C3×Q16D32semidirect
135. C3×Q16C2×C3⋊D3semidirect
151. C3⋊C16C2×C3⋊D3semidirect
178. C33C2×SD32semidirect
211. C3⋊D3#828direct (N × H)
218. Q16D3×C3⋊D3semidirect
245. C32D3×SD32semidirect
246. C32C2×#828semidirect
277. C3D3×#828semidirect
278. C3C3⋊D3×SD32semidirect

Maximal subgroups

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

subgroupindex
C3×Q16×C3⋊D32
C3⋊D3×D242
#117221982
C32×#8282
#117220122
#117220072
C3⋊D3×C3⋊C162
C3⋊D3×SD323
D3×#8283