Finite Group Databasebeta

← C2×SD16×C3⋊(C3⋊D3) #13179883

Subgroups — C2×SD16×C3⋊(C3⋊D3) #13179883

Order 1728.

4172 conjugacy classes of subgroups; only the 683 normal subgroups are shown.

normal subgroups only

This group has 4172 conjugacy classes of subgroups — too many to draw. Only its 683 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
1C2×Q8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
3C2×C33×SD16C211yesmaximal, characteristic, complemented
4C2×#88862C211yesmaximal, characteristic, complemented
5C2×#88861C211yesmaximal, characteristic, complemented
6SD16×C3⋊(C3⋊D3)C218yesmaximal, direct product factor
7C2×#88851C211yesmaximal, characteristic, complemented
8C2×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
9C2×SD16×C3⋊D3—313nomaximal
10C23×C3⋊(C3⋊D3)—21no—
11C2×C33×Q8C2211yescharacteristic, complemented
12C2×C33×D4C2211yescharacteristic
13C2×#6312—21no—
14Q8×C3⋊(C3⋊D3)—22no—
15Q8×C3⋊(C3⋊D3)C2214yescomplemented
16D4×C3⋊(C3⋊D3)—22no—
17D4×C3⋊(C3⋊D3)C2214yes—
18C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
19C2×C4×C3⋊(C3⋊D3)—21no—
20C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
21C2×#6308—21no—
22C2×#6308C2211yescharacteristic, complemented
23C33×SD16C2214yescomplemented
24C2×C33×C8C2211yescharacteristic, complemented
25#88862C2214yescomplemented
26#88861C2214yescomplemented
27C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
28#88851C2214yescomplemented
29C8×C3⋊(C3⋊D3)C2214yescomplemented
30C2×Q8×C3⋊D3—313no—
31C2×D4×C3⋊D3—313no—
32C2×C32×SD16D3113yescomplemented
33C2×#3755—313no—
34C2×#3754—313no—
35SD16×C3⋊D3—3104no—
36C2×#3744—313no—
37C2×C8×C3⋊D3—313no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C3⋊(C3⋊D3)—24no—
41C22×C3⋊(C3⋊D3)—44no—
42C22×C3⋊(C3⋊D3)D411yescharacteristic
43C33×Q8—21no—
44C33×Q8C2312yescomplemented
45C33×D4—21no—
46C33×D4C2312yes—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49#6312—24no—
50C2×C3⋊(C3⋊Dic3)—21no—
51C2×C3⋊(C3⋊Dic3)D411yescharacteristic
52C4⋊(C3⋊(C3⋊D3))—21no—
53C4⋊(C3⋊(C3⋊D3))C2312yes—
54C4×C3⋊(C3⋊D3)—24no—
55C4×C3⋊(C3⋊D3)D412yes—
56C4×C3⋊(C3⋊D3)C2312yes—
57#6308—21no—
58#6308—24no—
59#6308C2312yescomplemented
60C33×C8C2312yescomplemented
61C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
62C2×D3×SD16—913no—
63C23×C3⋊D3—613no—
64C2×C32×Q8C2×D3113yescomplemented
65C2×C32×D4C2×D3113yes—
66C2×#638—613no—
67Q8×C3⋊D3—352no—
68Q8×C3⋊D3—626no—
69D4×C3⋊D3—352no—
70D4×C3⋊D3—626no—
71C2×C4⋊(C3⋊D3)—313no—
72C2×C4×C3⋊D3—313no—
73C2×C4×C3⋊D3—613no—
74C2×#634—313no—
75C2×#634—613no—
76C32×SD16C2×D3152yescomplemented
77C2×C32×C8C2×D3113yescomplemented
78#3755—352no—
79#3754—352no—
80C2×C3⋊(C3⋊C8)—313no—
81#3744—352no—
82C8×C3⋊D3—352no—
83C22×C33—22no—
84C22×C33—42no—
85C22×C33C2×D411yescharacteristic
86C2×C3⋊(C3⋊D3)—22no—
87C2×C3⋊(C3⋊D3)—42no—
88C2×C3⋊(C3⋊D3)—48no—
89C2×C3⋊(C3⋊D3)SD1614yesdirect product factor
90C2×C3⋊(C3⋊D3)C2×D412yes—
91C33×C4—22no—
92C33×C4C2×D411yescharacteristic
93C33×C4C2411yescharacteristic
94C3⋊(C3⋊Dic3)—22no—
95C3⋊(C3⋊Dic3)C2×D412yes—
96C2×D3×Q8—913no—
97C2×D3×D4—913no—
98C2×C3×SD16C3⋊D3113yescomplemented
99C2×#215—913no—
100C2×#214—913no—
101D3×SD16—9104no—
102C2×#204—913no—
103C2×D3×C8—913no—
104C23×C32—213no—
105C22×C3⋊D3—313no—
106C22×C3⋊D3—652no—
107C22×C3⋊D3—613no—
108C22×C3⋊D3—1252no—
109C32×Q8—213no—
110C32×Q8C22×D3126yescomplemented
111C32×D4—213no—
112C32×D4C22×D3126yes—
113C2×C32×C4—213no—
114C2×C32×C4C22×D3113yes—
115#638—652no—
116C2×C3⋊Dic3—313no—
117C2×C3⋊Dic3—613no—
118C4⋊(C3⋊D3)—326no—
119C4⋊(C3⋊D3)—613no—
120C4×C3⋊D3—326no—
121C4×C3⋊D3—326no—
122C4×C3⋊D3—652no—
123#634—326no—
124#634—652no—
125#634—613no—
126C32×C8C22×D3126yescomplemented
127C3⋊(C3⋊C8)—326no—
128C22×SD16—271noSylow
129C2×C33—42no—
130C2×C33C2×SD1612yescomplemented
131C2×C33C22×D411yescharacteristic
132C3⋊(C3⋊D3)—42no—
133C3⋊(C3⋊D3)C2×SD1614yesdirect product factor
134C23×D3—1813no—
135C2×C3×Q8C2×C3⋊D3113yescomplemented
136C2×C3×D4C2×C3⋊D3113yes—
137C2×#67—1813no—
138D3×Q8—952no—
139D3×Q8—1826no—
140D3×D4—952no—
141D3×D4—1826no—
142C2×D12—913no—
143C2×C4×D3—913no—
144C2×C4×D3—1813no—
145C2×Dic6—913no—
146C2×Dic6—1813no—
147C3×SD16C2×C3⋊D3152yescomplemented
148C2×C3×C8C2×C3⋊D3113yescomplemented
149#215—952no—
150#214—952no—
151C2×C3⋊C8—913no—
152#204—952no—
153D3×C8—952no—
154C22×C32—226no—
155C22×C32—426no—
156C22×C32D3×D4113yes—
157C2×C3⋊D3—352no—
158C2×C3⋊D3—326no—
159C2×C3⋊D3—626no—
160C2×C3⋊D3—12104no—
161C2×C3⋊D3—1226no—
162C32×C4—226no—
163C32×C4D3×D4113yes—
164C32×C4C23×D3113yes—
165C3⋊Dic3—326no—
166C3⋊Dic3—626no—
167C22×Q8—271no—
168C22×D4—271no—
169C2×SD16—271no—
170C2×SD16—271no—
171C2×SD16—278no—
172C2×SD16—271no—
173C2×SD16C3⋊(C3⋊D3)11yescharacteristic, direct product factor
174C22×C8—271no—
175C33C22×SD1611yescharacteristic, complemented, Sylow
176C23×C3—213no—
177C22×D3—913no—
178C22×D3—1852no—
179C22×D3—1813no—
180C22×D3—3652no—
181C3×Q8—213no—
182C3×Q8C22×C3⋊D3126yescomplemented
183C3×D4—213no—
184C3×D4C22×C3⋊D3126yes—
185C2×C3×C4—213no—
186C2×C3×C4C22×C3⋊D3113yes—
187#67—1852no—
188C2×Dic3—913no—
189C2×Dic3—1813no—
190D12—926no—
191D12—1813no—
192C4×D3—926no—
193C4×D3—926no—
194C4×D3—1852no—
195Dic6—926no—
196Dic6—1852no—
197Dic6—1813no—
198C3×C8C22×C3⋊D3126yescomplemented
199C3⋊C8—926no—
200C2×C32—426no—
201C2×C32D3×SD16126yescomplemented
202C2×C32C2×D3×D4113yes—
203C3⋊D3—352no—
204C3⋊D3—1226no—
205C24—541no—
206C2×Q8—274no—
207C2×Q8—271no—
208C2×Q8—541no—
209C2×Q8—542no—
210C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
211C2×D4—274no—
212C2×D4—271no—
213C2×D4—542no—
214C2×D4—541no—
215C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
216C22×C4—271no—
217C22×C4—541no—
218SD16—274no—
219SD16—274no—
220SD16—274no—
221SD16C2×C3⋊(C3⋊D3)14yesdirect product factor
222C2×C8—271no—
223C2×C8—274no—
224C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
225C22×C3—226no—
226C22×C3—426no—
227C22×C3D4×C3⋊D3113yes—
228C2×D3—926no—
229C2×D3—952no—
230C2×D3—1826no—
231C2×D3—36104no—
232C2×D3—3626no—
233C3×C4—226no—
234C3×C4D4×C3⋊D3113yes—
235C3×C4C23×C3⋊D3113yes—
236Dic3—926no—
237Dic3—1826no—
238C32C2×D3×SD16113yescomplemented
239C23—21no—
240C23—271no—
241C23—544no—
242C23—541no—
243C23—1084no—
244Q8—21no—
245Q8—272no—
246Q8—544no—
247Q8—541no—
248Q8C22×C3⋊(C3⋊D3)12yescomplemented
249D4—21no—
250D4—272no—
251D4—544no—
252D4—541no—
253D4C22×C3⋊(C3⋊D3)12yes—
254C2×C4—21no—
255C2×C4—271no—
256C2×C4—272no—
257C2×C4—272no—
258C2×C4—544no—
259C2×C4—541no—
260C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
261C8—272no—
262C8C22×C3⋊(C3⋊D3)12yescomplemented
263C2×C3—426no—
264C2×C3SD16×C3⋊D3126yescomplemented
265C2×C3C2×D4×C3⋊D3113yes—
266D3—952no—
267D3—3626no—
268C22—22no—
269C22—42no—
270C22—272no—
271C22—274no—
272C22—542no—
273C22—1088no—
274C22—1082no—
275C22D4×C3⋊(C3⋊D3)11yescharacteristic
276C4—22no—
277C4—272no—
278C4—542no—
279C4D4×C3⋊(C3⋊D3)11yescharacteristic
280C4C23×C3⋊(C3⋊D3)11yescharacteristic
281C3C2×SD16×C3⋊D3113yescomplemented
282C2—42no—
283C2—274no—
284C2—1082no—
285C2SD16×C3⋊(C3⋊D3)12yesdirect product factor
286C2C2×D4×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC33×C4C24
FrattiniC4C23×C3⋊(C3⋊D3)
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×D4×C3⋊(C3⋊D3)C2
hypercenterC2×SD16C3⋊(C3⋊D3)
hypocenterC33C22×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C3⋊(C3⋊D3)C2semidirect
3. C2×C33×SD16C2semidirect
4. C2×#88862C2semidirect
5. C2×#88861C2semidirect
6. SD16×C3⋊(C3⋊D3)C2direct (N × H)
7. C2×#88851C2semidirect
8. C2×C8×C3⋊(C3⋊D3)C2semidirect
11. C2×C33×Q8C22semidirect
15. Q8×C3⋊(C3⋊D3)C22semidirect
22. C2×#6308C22semidirect
23. C33×SD16C22semidirect
24. C2×C33×C8C22semidirect
25. #88862C22semidirect
26. #88861C22semidirect
27. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
28. #88851C22semidirect
29. C8×C3⋊(C3⋊D3)C22semidirect
32. C2×C32×SD16D3semidirect
44. C33×Q8C23semidirect
59. #6308C23semidirect
60. C33×C8C23semidirect
61. C3⋊(C3⋊(C3⋊C8))C23semidirect
64. C2×C32×Q8C2×D3semidirect
76. C32×SD16C2×D3semidirect
77. C2×C32×C8C2×D3semidirect
89. C2×C3⋊(C3⋊D3)SD16direct (N × H)
98. C2×C3×SD16C3⋊D3semidirect
110. C32×Q8C22×D3semidirect
126. C32×C8C22×D3semidirect
130. C2×C33C2×SD16semidirect
133. C3⋊(C3⋊D3)C2×SD16direct (N × H)
135. C2×C3×Q8C2×C3⋊D3semidirect
147. C3×SD16C2×C3⋊D3semidirect
148. C2×C3×C8C2×C3⋊D3semidirect
173. C2×SD16C3⋊(C3⋊D3)direct (N × H)
175. C33C22×SD16semidirect
182. C3×Q8C22×C3⋊D3semidirect
198. C3×C8C22×C3⋊D3semidirect
201. C2×C32D3×SD16semidirect
210. C2×Q8C2×C3⋊(C3⋊D3)semidirect
221. SD16C2×C3⋊(C3⋊D3)direct (N × H)
224. C2×C8C2×C3⋊(C3⋊D3)semidirect
238. C32C2×D3×SD16semidirect
248. Q8C22×C3⋊(C3⋊D3)semidirect
262. C8C22×C3⋊(C3⋊D3)semidirect
264. C2×C3SD16×C3⋊D3semidirect
281. C3C2×SD16×C3⋊D3semidirect
285. C2SD16×C3⋊(C3⋊D3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊(C3⋊D3)2
C2×D4×C3⋊(C3⋊D3)2
C2×C33×SD162
C2×#888622
C2×#888612
SD16×C3⋊(C3⋊D3)2
C2×#888512
C2×C8×C3⋊(C3⋊D3)2
C2×SD16×C3⋊D33