Finite Group Databasebeta

← C3×C3⋊D3×C4⋊C8 #13162873

Subgroups — C3×C3⋊D3×C4⋊C8 #13162873

Order 1728.

980 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C8×C3⋊D3C212yesmaximal
2C3×C42×C3⋊D3C211yesmaximal, characteristic
3C33×C4⋊C8C211yesmaximal, characteristic, complemented
4C3×#63641C212yesmaximal, complemented
5C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6C3⋊D3×C4⋊C8C311yesmaximal, characteristic, direct product factor
7C3×D3×C4⋊C8—34nomaximal
8C2×C3×C4×C3⋊D3C411yescharacteristic
9C2×C3×C4×C3⋊D3C411yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C33×C8C2212yes—
12C33×C42C2211yescharacteristic
13C3×C4×C3⋊Dic3C411yescharacteristic
14C3×C4×C3⋊Dic3C411yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C3×C8×C3⋊D3—24no—
18C2×C8×C3⋊D3C2×C312yes—
19C42×C3⋊D3C2×C311yescharacteristic
20C2×C3×D3×C8—38no—
21C3×C42×D3—34no—
22C32×C4⋊C8—24no—
23C32×C4⋊C8D314yescomplemented
24C32×C4⋊C8C2×C311yescharacteristic, complemented
25#63641C2×C312yescomplemented
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
27C3×Dic3⋊C8—38no—
28C3×C4⋊(C3⋊C8)—34no—
29C22×C3×C3⋊D3C2×C411yescharacteristic
30C2×C33×C4C2×C411yescharacteristic
31C2×C33×C4C2×C411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C3×C4×C3⋊D3—22no—
37C3×C4×C3⋊D3C814yescomplemented
38C3×C4×C3⋊D3D412yes—
39C3×C4×C3⋊D3Q812yes—
40C33×C8—22no—
41C3×C3⋊(C3⋊C8)—22no—
42C2×C3×C4⋊C8—91no—
43D3×C4⋊C8—34no—
44C2×C4×C3⋊D3C3×C411yescharacteristic
45C2×C4×C3⋊D3C3×C411yescharacteristic
46C2×C4×C3⋊D3C22×C311yescharacteristic
47C2×C3×C4×D3—34no—
48C2×C3×C4×D3—34no—
49C2×C3×C4×D3—34no—
50C2×C32×C8—28no—
51C2×C32×C8C2×D318yes—
52C2×C32×C8C22×C312yes—
53C32×C42—24no—
54C32×C42C2×D314yes—
55C32×C42C22×C311yescharacteristic
56C4×C3⋊Dic3C3×C411yescharacteristic
57C4×C3⋊Dic3C3×C411yescharacteristic
58C4×C3⋊Dic3C22×C311yescharacteristic
59C2×C3⋊(C3⋊C8)C22×C312yes—
60C8×C3⋊D3—24no—
61C3×C4×Dic3—34no—
62C3×C4×Dic3—34no—
63C3×C4×Dic3—34no—
64C2×C3×C3⋊C8—38no—
65C3×D3×C8—616no—
66C22×C33C22×C411yescharacteristic
67C2×C3×C3⋊D3C4⋊C412yes—
68C2×C3×C3⋊D3C2×C812yes—
69C2×C3×C3⋊D3M1612yes—
70C33×C4—21no—
71C33×C4C2×C812yescomplemented
72C33×C4C2×D411yescharacteristic
73C33×C4C2×Q811yescharacteristic
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3C4⋊C412yes—
76C3×C3⋊Dic3C2×C812yescomplemented
77C22×C3×C8—92no—
78C2×C3×C42—91no—
79C2×D3×C8—38no—
80C42×D3—34no—
81C3×C4⋊C8—24no—
82C3×C4⋊C8—92no—
83C3×C4⋊C8—91no—
84C3×C4⋊C8C3×D314yescomplemented
85C3×C4⋊C8C3⋊D311yescharacteristic, direct product factor
86Dic3⋊C8—38no—
87C4⋊(C3⋊C8)—34no—
88C22×C3⋊D3C2×C3×C411yescharacteristic
89C22×C3×D3—34no—
90C2×C32×C4—24no—
91C2×C32×C4—24no—
92C2×C32×C4—24no—
93C2×C32×C4C4×D314yes—
94C2×C32×C4C4×D314yes—
95C2×C32×C4C2×C3×C411yescharacteristic
96C2×C32×C4C2×C3×C411yescharacteristic
97C2×C32×C4C22×D314yes—
98C2×C32×C4C23×C311yescharacteristic
99C2×C3⋊Dic3C2×C3×C411yescharacteristic
100C2×C3⋊Dic3C2×C3×C411yescharacteristic
101C2×C3⋊Dic3C2×C3×C411yescharacteristic
102C4×C3⋊D3—22no—
103C4×C3⋊D3C3×C814yescomplemented
104C4×C3⋊D3C3×D412yes—
105C4×C3⋊D3C3×Q812yes—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—34no—
109C3×C4×D3—316no—
110C3×C4×D3—38no—
111C3×C4×D3—38no—
112C3×C4×D3—68no—
113C32×C8—22no—
114C32×C8—28no—
115C32×C8—48no—
116C3⋊(C3⋊C8)—22no—
117C3×C3⋊C8—68no—
118C2×C4⋊C8—91noSylow
119C2×C33C2×C4⋊C411yescharacteristic
120C2×C33C22×C811yescharacteristic
121C2×C33C2×M1611yescharacteristic
122C3×C3⋊D3C4⋊C814yesdirect product factor
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C22×C3×C4—91no—
126C2×C4×D3—34no—
127C2×C4×D3—34no—
128C2×C4×D3—34no—
129C2×C3×C8—28no—
130C2×C3×C8—92no—
131C2×C3×C8—184no—
132C2×C3×C8C2×C3×D318yes—
133C2×C3×C8C2×C3⋊D312yes—
134C3×C42—24no—
135C3×C42—91no—
136C3×C42—91no—
137C3×C42—91no—
138C3×C42C2×C3×D314yes—
139C3×C42C2×C3⋊D311yescharacteristic
140C4×Dic3—34no—
141C4×Dic3—34no—
142C4×Dic3—34no—
143C2×C3⋊C8—38no—
144D3×C8—616no—
145C22×C32—24no—
146C22×C32C2×C4×D314yes—
147C22×C32C22×C3×C411yescharacteristic
148C2×C3⋊D3C3×C4⋊C412yes—
149C2×C3⋊D3C2×C3×C812yes—
150C2×C3⋊D3C3×M1612yes—
151C2×C3×D3—38no—
152C2×C3×D3—38no—
153C2×C3×D3—38no—
154C32×C4—21no—
155C32×C4—28no—
156C32×C4—24no—
157C32×C4—24no—
158C32×C4—24no—
159C32×C4—44no—
160C32×C4D3×C818yescomplemented
161C32×C4C2×C3×C812yescomplemented
162C32×C4D3×D414yes—
163C32×C4D3×Q814yes—
164C32×C4C2×C3×D411yescharacteristic
165C32×C4C2×C3×Q811yescharacteristic
166C3⋊Dic3—21no—
167C3⋊Dic3C3×C4⋊C412yes—
168C3⋊Dic3C2×C3×C812yescomplemented
169C3×Dic3—38no—
170C3×Dic3—38no—
171C3×Dic3—64no—
172C22×C8—92no—
173C2×C42—91no—
174C4⋊C8—92no—
175C4⋊C8—91no—
176C4⋊C8C3×C3⋊D311yescharacteristic, direct product factor
177C33C2×C4⋊C811yescharacteristic, complemented, Sylow
178C23×C3—91no—
179C22×D3—34no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—24no—
183C2×C3×C4—91no—
184C2×C3×C4—91no—
185C2×C3×C4—94no—
186C2×C3×C4—92no—
187C2×C3×C4—92no—
188C2×C3×C4—91no—
189C2×C3×C4—182no—
190C2×C3×C4C3×C4×D314yes—
191C2×C3×C4C3×C4×D314yes—
192C2×C3×C4C4×C3⋊D311yescharacteristic
193C2×C3×C4C4×C3⋊D311yescharacteristic
194C2×C3×C4C22×C3×D314yes—
195C2×C3×C4C22×C3⋊D311yescharacteristic
196C2×Dic3—34no—
197C2×Dic3—34no—
198C2×Dic3—34no—
199C4×D3—316no—
200C4×D3—38no—
201C4×D3—38no—
202C4×D3—68no—
203C3×C8—28no—
204C3×C8—22no—
205C3×C8—48no—
206C3×C8—182no—
207C3⋊C8—68no—
208C2×C32—24no—
209C2×C32—24no—
210C2×C32—24no—
211C2×C32D3×C4⋊C414yes—
212C2×C32C2×D3×C814yes—
213C2×C32D3×M1614yes—
214C2×C32C2×C3×C4⋊C411yescharacteristic
215C2×C32C22×C3×C811yescharacteristic
216C2×C32C2×C3×M1611yescharacteristic
217C3⋊D3C3×C4⋊C814yesdirect product factor
218C3×D3—316no—
219C22×C4—91no—
220C22×C4—91no—
221C22×C4—91no—
222C2×C8—92no—
223C2×C8—184no—
224C2×C8C2×C3×C3⋊D312yes—
225C42—91no—
226C42—91no—
227C42—91no—
228C42C2×C3×C3⋊D311yescharacteristic
229C22×C3—24no—
230C22×C3—92no—
231C22×C3—92no—
232C22×C3—92no—
233C22×C3C2×C3×C4×D314yes—
234C22×C3C2×C4×C3⋊D311yescharacteristic
235C2×D3—38no—
236C2×D3—38no—
237C2×D3—38no—
238C3×C4—24no—
239C3×C4—21no—
240C3×C4—28no—
241C3×C4—24no—
242C3×C4—24no—
243C3×C4—44no—
244C3×C4—92no—
245C3×C4—92no—
246C3×C4—181no—
247C3×C4C3×D3×C818yescomplemented
248C3×C4C8×C3⋊D312yescomplemented
249C3×C4C3×D3×D414yes—
250C3×C4C3×D3×Q814yes—
251C3×C4D4×C3⋊D311yescharacteristic
252C3×C4Q8×C3⋊D311yescharacteristic
253Dic3—38no—
254Dic3—38no—
255Dic3—64no—
256C32—24no—
257C32D3×C4⋊C814yescomplemented
258C32C2×C3×C4⋊C811yescharacteristic, complemented
259C23—91no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—94no—
263C2×C4—92no—
264C2×C4—92no—
265C2×C4—91no—
266C2×C4—182no—
267C2×C4C3×C4×C3⋊D311yescharacteristic
268C2×C4C3×C4×C3⋊D311yescharacteristic
269C2×C4C22×C3×C3⋊D311yescharacteristic
270C8—22no—
271C8—182no—
272C2×C3—24no—
273C2×C3—24no—
274C2×C3—24no—
275C2×C3—94no—
276C2×C3C3×D3×C4⋊C414yes—
277C2×C3C2×C3×D3×C814yes—
278C2×C3C3×D3×M1614yes—
279C2×C3C4⋊C4×C3⋊D311yescharacteristic
280C2×C3C2×C8×C3⋊D311yescharacteristic
281C2×C3M16×C3⋊D311yescharacteristic
282D3—316no—
283C22—92no—
284C22—92no—
285C22—92no—
286C22C2×C3×C4×C3⋊D311yescharacteristic
287C4—21no—
288C4—92no—
289C4—92no—
290C4—181no—
291C4C3×C8×C3⋊D312yescomplemented
292C4C3×D4×C3⋊D311yescharacteristic
293C4C3×Q8×C3⋊D311yescharacteristic
294C3—24no—
295C3C3×D3×C4⋊C814yescomplemented
296C3C3⋊D3×C4⋊C811yescharacteristic, direct product factor
297C2—94no—
298C2C3×C4⋊C4×C3⋊D311yescharacteristic
299C2C2×C3×C8×C3⋊D311yescharacteristic
300C2C3×M16×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3×C4C22×C3⋊D3
commutatorC2×C32C22×C3×C8
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4⋊C8C2
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C8C3⋊D3
hypocenterC32C2×C3×C4⋊C8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C4⋊C8

Complements

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

NH ≅ G/Ntype
3. C33×C4⋊C8C2semidirect
4. C3×#63641C2semidirect
5. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
6. C3⋊D3×C4⋊C8C3direct (N × H)
23. C32×C4⋊C8D3semidirect
24. C32×C4⋊C8C2×C3semidirect
25. #63641C2×C3semidirect
26. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
37. C3×C4×C3⋊D3C8semidirect
71. C33×C4C2×C8semidirect
76. C3×C3⋊Dic3C2×C8semidirect
84. C3×C4⋊C8C3×D3semidirect
85. C3×C4⋊C8C3⋊D3direct (N × H)
103. C4×C3⋊D3C3×C8semidirect
122. C3×C3⋊D3C4⋊C8direct (N × H)
160. C32×C4D3×C8semidirect
161. C32×C4C2×C3×C8semidirect
168. C3⋊Dic3C2×C3×C8semidirect
176. C4⋊C8C3×C3⋊D3direct (N × H)
177. C33C2×C4⋊C8semidirect
217. C3⋊D3C3×C4⋊C8direct (N × H)
247. C3×C4C3×D3×C8semidirect
248. C3×C4C8×C3⋊D3semidirect
257. C32D3×C4⋊C8semidirect
258. C32C2×C3×C4⋊C8semidirect
291. C4C3×C8×C3⋊D3semidirect
295. C3C3×D3×C4⋊C8semidirect
296. C3C3⋊D3×C4⋊C8direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C8×C3⋊D32
C3×C42×C3⋊D32
C33×C4⋊C82
C3×#636412
C3×C4⋊(C3⋊(C3⋊C8))2
C3⋊D3×C4⋊C83
C3×D3×C4⋊C83