Finite Group Databasebeta

← C2×#11723437 #13176811

Subgroups — C2×#11723437 #13176811

Order 1728.

2412 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×C32×C4×Dic3C211yesmaximal, characteristic, complemented
4C2×#88807C212yesmaximal, complemented
5C2×#88803C212yesmaximal
6#11723437C218yesmaximal, direct product factor
7C2×#64081—31nomaximal
8C2×#63855—34nomaximal
9C23×C3×C3⋊D3C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)C2212yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C2×C3×C4×C3⋊D3C418yescomplemented
14C2×C3×C4×C3⋊D3C2214yes—
15C22×C32×Dic3C2212yes—
16C4×C3⋊(C3⋊Dic3)C2214yescomplemented
17C32×C4×Dic3C2214yescomplemented
18#88807C2218yescomplemented
19#88803C2218yes—
20C22×C4×C3⋊D3D311yescharacteristic
21C22×C3×C4×D3—34no—
22C2×C32×C42—31no—
23C2×#3752—32no—
24C2×#3750—32no—
25C2×C4×C3⋊Dic3—31no—
26C2×C4×C3⋊Dic3—34no—
27C2×C4×C3⋊Dic3—64no—
28#64081—38no—
29C2×C3×C4×Dic3D314yescomplemented
30C2×Dic3⋊Dic3—38no—
31C2×(C2×D3)⋊Dic3—38no—
32#63855—332no—
33C23×C33C2311yescharacteristic
34C22×C3×C3⋊D3—24no—
35C22×C3×C3⋊D3C2×C414yes—
36C22×C3×C3⋊D3C2312yes—
37C2×C33×C4C2×C414yes—
38C2×C33×C4C2312yes—
39C2×C3⋊(C3⋊Dic3)—24no—
40C2×C3⋊(C3⋊Dic3)C2314yes—
41C2×C3×C3⋊Dic3C2×C414yes—
42C2×C3×C3⋊Dic3C2312yes—
43C3×C4×C3⋊D3C2×C4116yescomplemented
44C2×C32×Dic3—24no—
45C2×C32×Dic3C2314yes—
46C2×#926—91no—
47C2×#872—94no—
48C23×C3⋊D3C2×D311yescharacteristic
49C23×C3×D3—34no—
50C22×C32×C4—24no—
51C22×C32×C4—32no—
52C22×C32×C4C2×D314yes—
53C22×C32×C4C2×D311yescharacteristic
54C22×C3⋊Dic3—32no—
55C22×C3⋊Dic3—38no—
56C22×C3⋊Dic3—68no—
57C22×C3⋊Dic3C2×D311yescharacteristic
58C2×C4×C3⋊D3Dic318yescomplemented
59C2×C4×C3⋊D3C2×D314yes—
60C22×C3×Dic3—34no—
61C22×C3×Dic3C2×D318yes—
62C2×C3×C4×D3—332no—
63C2×C3×C4×D3—316no—
64C32×C42—34no—
65#3752—38no—
66#3750—38no—
67C4×C3⋊Dic3—34no—
68C4×C3⋊Dic3—316no—
69C4×C3⋊Dic3—616no—
70C3×C4×Dic3C2×D3116yescomplemented
71Dic3⋊Dic3—332no—
72(C2×D3)⋊Dic3—332no—
73C22×C33C22×C412yes—
74C22×C33#4114yes—
75C22×C33C2411yescharacteristic
76C2×C3×C3⋊D3—28no—
77C2×C3×C3⋊D3—24no—
78C2×C3×C3⋊D3C22×C414yes—
79C33×C4C22×C414yes—
80C3⋊(C3⋊Dic3)—24no—
81C3×C3⋊Dic3C22×C414yes—
82C32×Dic3—24no—
83C23×C3×C4—91no—
84C22×C4×D3—34no—
85C2×C3×C42—34no—
86C2×#217—92no—
87C2×(C2×D3)⋊C4—98no—
88#926—98no—
89C2×C4⋊Dic3—92no—
90C2×Dic3⋊C4—98no—
91C2×C4×Dic3—94no—
92C2×C4×Dic3—91no—
93C2×C4×Dic3—184no—
94C2×C4×Dic3C3⋊D311yescharacteristic, complemented
95#872—932no—
96C23×C32—24no—
97C23×C32C22×D314yes—
98C23×C32C22×D311yescharacteristic
99C22×C3⋊D3—24no—
100C22×C3⋊D3C2×Dic314yes—
101C22×C3⋊D3C22×D312yes—
102C22×C3×D3—316no—
103C22×C3×D3—38no—
104C22×C3×D3—616no—
105C2×C32×C4—216no—
106C2×C32×C4—28no—
107C2×C32×C4—34no—
108C2×C32×C4—64no—
109C2×C32×C4C4×D3116yes—
110C2×C32×C4C2×Dic314yes—
111C2×C32×C4C22×D318yes—
112C2×C32×C4C22×D312yes—
113C2×C3⋊Dic3—34no—
114C2×C3⋊Dic3—316no—
115C2×C3⋊Dic3—616no—
116C2×C3⋊Dic3—632no—
117C2×C3⋊Dic3—616no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3C2×Dic314yes—
120C2×C3⋊Dic3C22×D312yes—
121C4×C3⋊D3C2×Dic3116yescomplemented
122C2×C3×Dic3—216no—
123C2×C3×Dic3—316no—
124C2×C3×Dic3—38no—
125C2×C3×Dic3C22×D3116yes—
126C3×C4×D3—364no—
127C2×#117—271noSylow
128C2×C33#11714yescomplemented
129C2×C33C23×C411yescharacteristic
130C2×C33C2×#4112yes—
131C3×C3⋊D3—24no—
132C24×C3—91no—
133C23×D3—34no—
134C22×C3×C4—24no—
135C22×C3×C4—38no—
136C22×C3×C4—91no—
137C22×C3×C4—98no—
138C22×C3×C4—94no—
139C22×C3×C4D3214yes—
140C22×C3×C4C2×C3⋊D311yescharacteristic
141C22×Dic3—34no—
142C22×Dic3—98no—
143C22×Dic3—92no—
144C22×Dic3—188no—
145C22×Dic3C2×C3⋊D312yes—
146C2×C4×D3—332no—
147C2×C4×D3—316no—
148C3×C42—316no—
149#217—98no—
150(C2×D3)⋊C4—932no—
151C4⋊Dic3—98no—
152Dic3⋊C4—932no—
153C4×Dic3—916no—
154C4×Dic3—94no—
155C4×Dic3—1816no—
156C4×Dic3C2×C3⋊D314yescomplemented
157C22×C32—24no—
158C22×C32—216no—
159C22×C32—28no—
160C22×C32C2×C4×D318yes—
161C22×C32#235116yes—
162C22×C32#23514yes—
163C22×C32C22×Dic312yes—
164C22×C32C23×D314yes—
165C22×C32C23×D311yescharacteristic
166C2×C3⋊D3—28no—
167C2×C3⋊D3—24no—
168C2×C3⋊D3C22×Dic314yes—
169C2×C3×D3—316no—
170C2×C3×D3—632no—
171C2×C3×D3—616no—
172C32×C4—216no—
173C32×C4—64no—
174C32×C4C2×C4×D3116yes—
175C32×C4C22×Dic314yes—
176C3⋊Dic3—632no—
177C3⋊Dic3—616no—
178C3⋊Dic3—64no—
179C3⋊Dic3C22×Dic314yes—
180C3×Dic3—216no—
181C3×Dic3—316no—
182C23×C4—91no—
183#117—278no—
184C2×C4⋊C4—272no—
185C2×C22⋊C4—272no—
186C2×C42—31no—
187C2×C42—271no—
188C33C2×#11711yescharacteristic, complemented, Sylow
189C23×C3—24no—
190C23×C3—92no—
191C23×C3—94no—
192C23×C3—184no—
193C23×C3C2×D3214yes—
194C23×C3C22×C3⋊D311yescharacteristic
195C22×D3—38no—
196C22×D3—316no—
197C22×D3—616no—
198C2×C3×C4—216no—
199C2×C3×C4—28no—
200C2×C3×C4—316no—
201C2×C3×C4—616no—
202C2×C3×C4—92no—
203C2×C3×C4—94no—
204C2×C3×C4—916no—
205C2×C3×C4D3×Dic3116yes—
206C2×C3×C4C4×C3⋊D314yes—
207C2×C3×C4C2×D3218yes—
208C2×C3×C4C22×C3⋊D312yes—
209C2×Dic3—24no—
210C2×Dic3—38no—
211C2×Dic3—316no—
212C2×Dic3—916no—
213C2×Dic3—94no—
214C2×Dic3—1816no—
215C2×Dic3—1832no—
216C2×Dic3—1816no—
217C2×Dic3—184no—
218C2×Dic3C22×C3⋊D314yes—
219C4×D3—364no—
220C2×C32—28no—
221C2×C32—24no—
222C2×C32—216no—
223C2×C32#872116yescomplemented
224C2×C32#92614yescomplemented
225C2×C32C22×C4×D314yes—
226C2×C32C2×#23518yes—
227C2×C32C2×#23512yes—
228C2×C32C23×Dic311yescharacteristic
229C3⋊D3—24no—
230C3×D3—616no—
231C24—91no—
232C22×C4—32no—
233C22×C4—91no—
234C22×C4—98no—
235C22×C4—94no—
236C22×C4—272no—
237C22×C4D3×C3⋊D311yescharacteristic
238C4⋊C4—278no—
239C22⋊C4—278no—
240C42—34no—
241C42—274no—
242C22×C3—24no—
243C22×C3—216no—
244C22×C3—28no—
245C22×C3—94no—
246C22×C3—188no—
247C22×C3—184no—
248C22×C3#3798116yes—
249C22×C3C2×D3×Dic318yes—
250C22×C3C2×C4×C3⋊D312yes—
251C22×C3#382814yes—
252C22×C3C22×D3214yes—
253C22×C3C23×C3⋊D311yescharacteristic
254C2×D3—316no—
255C2×D3—616no—
256C2×D3—632no—
257C3×C4—216no—
258C3×C4—616no—
259C3×C4—94no—
260C3×C4C2×D3×Dic3116yes—
261C3×C4C2×C4×C3⋊D314yes—
262Dic3—24no—
263Dic3—316no—
264Dic3—1832no—
265Dic3—1816no—
266Dic3—184no—
267C32—24no—
268C32C2×#87214yescomplemented
269C32C2×#92611yescharacteristic, complemented
270C23—92no—
271C23—94no—
272C23—184no—
273C23C2×D3×C3⋊D311yescharacteristic
274C2×C4—34no—
275C2×C4—64no—
276C2×C4—92no—
277C2×C4—94no—
278C2×C4—916no—
279C2×C4—274no—
280C2×C4—544no—
281C2×C4Dic3×C3⋊D314yes—
282C2×C4C2×D3×C3⋊D312yes—
283C2×C3—28no—
284C2×C3—24no—
285C2×C3—216no—
286C2×C3—184no—
287C2×C3#63855116yescomplemented
288C2×C3#6408114yescomplemented
289C2×C3C2×#379818yes—
290C2×C3C22×D3×Dic314yes—
291C2×C3C22×C4×C3⋊D311yescharacteristic
292C2×C3C2×#382812yes—
293D3—616no—
294C22—94no—
295C22—184no—
296C22—188no—
297C22#8902014yes—
298C22C2×Dic3×C3⋊D312yes—
299C22C22×D3×C3⋊D311yescharacteristic
300C4—64no—
301C4—94no—
302C4—544no—
303C4C2×Dic3×C3⋊D314yes—
304C3—24no—
305C3C2×#6385514yescomplemented
306C3C2×#6408111yescharacteristic, complemented
307C2—184no—
308C2#1172343714yesdirect product factor
309C2C2×#8902012yes—
310C2C22×Dic3×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC22×C4D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C33C24
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊(C3⋊Dic3)C2semidirect
3. C2×C32×C4×Dic3C2semidirect
4. C2×#88807C2semidirect
6. #11723437C2direct (N × H)
13. C2×C3×C4×C3⋊D3C4semidirect
16. C4×C3⋊(C3⋊Dic3)C22semidirect
17. C32×C4×Dic3C22semidirect
18. #88807C22semidirect
29. C2×C3×C4×Dic3D3semidirect
43. C3×C4×C3⋊D3C2×C4semidirect
58. C2×C4×C3⋊D3Dic3semidirect
70. C3×C4×Dic3C2×D3semidirect
94. C2×C4×Dic3C3⋊D3semidirect
121. C4×C3⋊D3C2×Dic3semidirect
128. C2×C33#117semidirect
156. C4×Dic3C2×C3⋊D3semidirect
188. C33C2×#117semidirect
223. C2×C32#872semidirect
224. C2×C32#926semidirect
268. C32C2×#872semidirect
269. C32C2×#926semidirect
287. C2×C3#63855semidirect
288. C2×C3#64081semidirect
305. C3C2×#63855semidirect
306. C3C2×#64081semidirect
308. C2#11723437direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C4×C3⋊(C3⋊Dic3)2
C2×C32×C4×Dic32
C2×#888072
C2×#888032
#117234372
C2×#640813
C2×#638553