Finite Group Databasebeta

← C4×#88807 #13160581

Subgroups — C4×#88807 #13160581

Order 1728.

1596 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3C2×C32×C4×Dic3C211yesmaximal, characteristic
4C2×#88807C212yesmaximal
5#11722061C212yesmaximal
6C4×#3750—31nomaximal
7C4×Dic3⋊Dic3—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C4×C3⋊(C3⋊Dic3)—22no—
14C3×C4×C3⋊Dic3C414yescomplemented
15C32×C4×Dic3—22no—
16#88807C418yesdirect product factor
17C2×C32×C42—31no—
18C2×#3750—32no—
19C2×C4×C3⋊Dic3—31no—
20C2×C4×C3⋊Dic3—34no—
21C2×C4×C3⋊Dic3—64no—
22C2×C4×C3⋊Dic3D311yescharacteristic
23C2×C3×C4×Dic3—34no—
24C2×C3×C4×Dic3D314yes—
25C2×Dic3⋊Dic3—38no—
26#63658—32no—
27#63579—38no—
28C23×C33C2311yescharacteristic
29C2×C33×C4C2×C412yes—
30C2×C33×C4D412yes—
31C2×C33×C4Q812yes—
32C2×C3⋊(C3⋊Dic3)—22no—
33C2×C3⋊(C3⋊Dic3)—22no—
34C2×C3⋊(C3⋊Dic3)C2×C414yes—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3C2×C412yes—
37C2×C3×C3⋊Dic3C2×C414yes—
38C2×C3×C3⋊Dic3C2×C412yes—
39C2×C32×Dic3—22no—
40C2×C32×Dic3—22no—
41C2×C32×Dic3C2×C414yes—
42C4×C4⋊Dic3—91no—
43C4×Dic3⋊C4—94no—
44C22×C32×C4—24no—
45C22×C32×C4—32no—
46C22×C32×C4C2×D314yes—
47C22×C32×C4C2×D311yescharacteristic
48C22×C3⋊Dic3—32no—
49C22×C3⋊Dic3—38no—
50C22×C3⋊Dic3—68no—
51C22×C3⋊Dic3C2×D311yescharacteristic
52C22×C3⋊Dic3C2×D311yescharacteristic
53C22×C3×Dic3—34no—
54C22×C3×Dic3—34no—
55C22×C3×Dic3C2×D318yes—
56C32×C42—62no—
57#3750—38no—
58C4×C3⋊Dic3—616no—
59C4×C3⋊Dic3—62no—
60C4×C3⋊Dic3—68no—
61C4×C3⋊Dic3Dic314yescomplemented
62C3×C4×Dic3—28no—
63C3×C4×Dic3—316no—
64Dic3⋊Dic3—332no—
65C22×C33C22×C411yescharacteristic
66C22×C33C22×C412yes—
67C22×C33C2×D411yescharacteristic
68C22×C33C2×Q811yescharacteristic
69C22×C33#4112yes—
70C33×C4C4⋊C414yescomplemented
71C3⋊(C3⋊Dic3)—24no—
72C3×C3⋊Dic3—22no—
73C3×C3⋊Dic3C4214yescomplemented
74C32×Dic3—24no—
75C2×C3×C42—34no—
76C2×C3×C42—91no—
77C2×C4⋊Dic3—92no—
78C2×Dic3⋊C4—98no—
79C2×C4×Dic3—34no—
80C2×C4×Dic3—94no—
81C2×C4×Dic3—91no—
82C2×C4×Dic3—184no—
83C2×C4×Dic3C3⋊D311yescharacteristic
84C3⋊#95—98no—
85C3⋊#95—92no—
86C23×C32—24no—
87C23×C32C22×D314yes—
88C23×C32C22×D311yescharacteristic
89C2×C32×C4—28no—
90C2×C32×C4—28no—
91C2×C32×C4—28no—
92C2×C32×C4—34no—
93C2×C32×C4—62no—
94C2×C32×C4—62no—
95C2×C32×C4Dic618yes—
96C2×C32×C4Dic612yes—
97C2×C32×C4C4×D318yes—
98C2×C32×C4D1212yes—
99C2×C32×C4C2×Dic312yes—
100C2×C32×C4#6718yes—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—34no—
103C2×C3⋊Dic3—316no—
104C2×C3⋊Dic3—616no—
105C2×C3⋊Dic3—616no—
106C2×C3⋊Dic3—616no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—68no—
109C2×C3⋊Dic3—62no—
110C2×C3⋊Dic3—68no—
111C2×C3⋊Dic3C4×D314yes—
112C2×C3⋊Dic3C2×Dic312yes—
113C2×C3⋊Dic3C2×Dic312yes—
114C2×C3×Dic3—28no—
115C2×C3×Dic3—28no—
116C2×C3×Dic3—316no—
117C2×C3×Dic3—38no—
118C2×C3×Dic3—38no—
119C2×C3×Dic3—68no—
120C2×C3×Dic3C4×D3116yes—
121C4×C4⋊C4—271noSylow
122C2×C33C2×C4211yescharacteristic
123C2×C33C2×C4⋊C411yescharacteristic
124C2×C33#11711yescharacteristic
125C2×C33C4×D412yes—
126C2×C33C4×Q812yes—
127C22×C3×C4—24no—
128C22×C3×C4—38no—
129C22×C3×C4—91no—
130C22×C3×C4—91no—
131C22×C3×C4D3214yes—
132C22×C3×C4C2×C3⋊D311yescharacteristic
133C22×Dic3—34no—
134C22×Dic3—34no—
135C22×Dic3—98no—
136C22×Dic3—92no—
137C22×Dic3—188no—
138C22×Dic3C2×C3⋊D312yes—
139C3×C42—68no—
140C3×C42—94no—
141C4⋊Dic3—98no—
142Dic3⋊C4—932no—
143C4×Dic3—22no—
144C4×Dic3—316no—
145C4×Dic3—1816no—
146C4×Dic3—188no—
147C4×Dic3—182no—
148C22×C32—28no—
149C22×C32—28no—
150C22×C32—24no—
151C22×C32—24no—
152C22×C32—24no—
153C22×C32C2×Dic614yes—
154C22×C32C2×Dic611yescharacteristic
155C22×C32C2×C4×D318yes—
156C22×C32C2×C4×D314yes—
157C22×C32C2×C4×D312yes—
158C22×C32C2×D1211yescharacteristic
159C22×C32#23518yes—
160C22×C32#23512yes—
161C22×C32C22×Dic311yescharacteristic
162C22×C32C2×#6714yes—
163C32×C4—216no—
164C32×C4—64no—
165C32×C4Dic3⋊C4116yescomplemented
166C32×C4C4⋊Dic314yescomplemented
167C3⋊Dic3—22no—
168C3⋊Dic3—632no—
169C3⋊Dic3—64no—
170C3⋊Dic3—616no—
171C3⋊Dic3C4×Dic314yescomplemented
172C3×Dic3—216no—
173C3×Dic3—316no—
174C3×Dic3—68no—
175C2×C4⋊C4—272no—
176C2×C42—31no—
177C2×C42—91no—
178C2×C42—271no—
179#95—272no—
180C33C4×C4⋊C411yescharacteristic, complemented, Sylow
181C23×C3—24no—
182C23×C3C2×D3214yes—
183C23×C3C22×C3⋊D311yescharacteristic
184C2×C3×C4—28no—
185C2×C3×C4—28no—
186C2×C3×C4—28no—
187C2×C3×C4—316no—
188C2×C3×C4—68no—
189C2×C3×C4—68no—
190C2×C3×C4—94no—
191C2×C3×C4—92no—
192C2×C3×C4—92no—
193C2×C3×C4—182no—
194C2×C3×C4D3×Dic318yes—
195C2×C3×C4#62618yes—
196C2×C3×C4#62718yes—
197C2×C3×C4#63412yes—
198C2×C3×C4C4×C3⋊D312yes—
199C2×C3×C4#63812yes—
200C2×Dic3—22no—
201C2×Dic3—22no—
202C2×Dic3—316no—
203C2×Dic3—38no—
204C2×Dic3—38no—
205C2×Dic3—68no—
206C2×Dic3—916no—
207C2×Dic3—94no—
208C2×Dic3—1816no—
209C2×Dic3—1816no—
210C2×Dic3—1816no—
211C2×Dic3—188no—
212C2×Dic3—188no—
213C2×Dic3—182no—
214C2×Dic3—182no—
215C2×Dic3C4×C3⋊D314yes—
216C2×C32—24no—
217C2×C32—24no—
218C2×C32—28no—
219C2×C32—28no—
220C2×C32—24no—
221C2×C32C4×Dic618yes—
222C2×C32C4×Dic612yes—
223C2×C32C42×D314yes—
224C2×C32#87214yes—
225C2×C32C4×D1212yes—
226C2×C32C2×C4×Dic311yescharacteristic
227C2×C32C2×Dic3⋊C414yes—
228C2×C32C2×C4⋊Dic311yescharacteristic
229C2×C32#92611yescharacteristic
230C2×C32C4×#6718yes—
231C22×C4—32no—
232C22×C4—91no—
233C22×C4—91no—
234C22×C4—272no—
235C22×C4D3×C3⋊D311yescharacteristic
236C4⋊C4—278no—
237C42—62no—
238C42—94no—
239C42—542no—
240C22×C3—28no—
241C22×C3—28no—
242C22×C3—24no—
243C22×C3—24no—
244C22×C3—24no—
245C22×C3#379818yes—
246C22×C3C4×D3218yes—
247C22×C3C2×D3×Dic314yes—
248C22×C3C2×#62614yes—
249C22×C3C2×#62714yes—
250C22×C3C2×#63411yescharacteristic
251C22×C3C2×C4×C3⋊D312yes—
252C22×C3C2×C4×C3⋊D311yescharacteristic
253C22×C3#382812yes—
254C22×C3C2×#63811yescharacteristic
255C3×C4—216no—
256C3×C4—616no—
257C3×C4—94no—
258C3×C4—182no—
259C3×C4Dic3⋊Dic3116yescomplemented
260C3×C4#375014yescomplemented
261Dic3—24no—
262Dic3—316no—
263Dic3—68no—
264Dic3—1832no—
265Dic3—184no—
266Dic3—1816no—
267C32—24no—
268C32C4×Dic3⋊C414yescomplemented
269C32C4×C4⋊Dic311yescharacteristic, complemented
270C23C2×D3×C3⋊D311yescharacteristic
271C2×C4—34no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—94no—
275C2×C4—92no—
276C2×C4—92no—
277C2×C4—182no—
278C2×C4—274no—
279C2×C4—542no—
280C2×C4—542no—
281C2×C4Dic3×C3⋊D312yes—
282C2×C4#629112yes—
283C2×C4#629312yes—
284C2×C3—28no—
285C2×C3—24no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3—28no—
289C2×C3#6385514yes—
290C2×C3C4×D3×Dic314yes—
291C2×C3C4×#62618yes—
292C2×C3C4×#62718yes—
293C2×C3C2×Dic3⋊Dic314yes—
294C2×C3C4×#63412yes—
295C2×C3C42×C3⋊D311yescharacteristic
296C2×C3#6408111yescharacteristic
297C2×C3C2×#375011yescharacteristic
298C2×C3C4×#63812yes—
299C22#8902012yes—
300C22C4×D3×C3⋊D312yes—
301C22C2×Dic3×C3⋊D311yescharacteristic
302C22C2×#629111yescharacteristic
303C22C2×#629311yescharacteristic
304C4—64no—
305C4—94no—
306C4—182no—
307C4—544no—
308C4#8880714yesdirect product factor
309C3—24no—
310C3C4×Dic3⋊Dic314yescomplemented
311C3C4×#375011yescharacteristic, complemented
312C2#1172343711yescharacteristic
313C2C4×Dic3×C3⋊D311yescharacteristic
314C2C4×#629112yes—
315C2C4×#629312yes—
316C2C2×#8880711yescharacteristic
subgroupG/H
center, hypercenterC22×C4D3×C3⋊D3
commutatorC2×C33C2×C42
FrattiniC23C2×D3×C3⋊D3
FittingC22×C33×C4C22
socle, prime radical, cosocleC23×C33C23
hypocenterC33C4×C4⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
14. C3×C4×C3⋊Dic3C4semidirect
16. #88807C4direct (N × H)
61. C4×C3⋊Dic3Dic3semidirect
70. C33×C4C4⋊C4semidirect
73. C3×C3⋊Dic3C42semidirect
165. C32×C4Dic3⋊C4semidirect
166. C32×C4C4⋊Dic3semidirect
171. C3⋊Dic3C4×Dic3semidirect
180. C33C4×C4⋊C4semidirect
259. C3×C4Dic3⋊Dic3semidirect
260. C3×C4#3750semidirect
268. C32C4×Dic3⋊C4semidirect
269. C32C4×C4⋊Dic3semidirect
308. C4#88807direct (N × H)
310. C3C4×Dic3⋊Dic3semidirect
311. C3C4×#3750semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊(C3⋊Dic3)2
C2×C3×C4×C3⋊Dic32
C2×C32×C4×Dic32
C2×#888072
#117220612
C4×#37503
C4×Dic3⋊Dic33