Finite Group Databasebeta

← C2×#11720910 #13150044

Subgroups — C2×#11720910 #13150044

Order 1728.

1002 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×Dic18C211yesmaximal, characteristic
2C2×C4×C3⋊Dic9C211yesmaximal, characteristic
3C2×C9×C4⋊Dic3C211yesmaximal, characteristic
4C2×Dic9⋊Dic3C212yesmaximal
5C2×Dic3×Dic9C212yesmaximal
6#11720910C218yesmaximal, direct product factor
7C2×#63843—31nomaximal
8C2×#63449—31nomaximal
9C22×C3×C4×C9C2211yescharacteristic
10C22×C3⋊Dic9C2211yescharacteristic
11C22×C3⋊Dic9C2211yescharacteristic
12C22×C9×Dic3C2212yes—
13C22×C3×Dic9C2212yes—
14C2×C3×Dic18C418yescomplemented
15C2×C3×Dic18C2214yes—
16C4×C3⋊Dic9C2214yes—
17C9×C4⋊Dic3C2214yes—
18Dic9⋊Dic3C2218yes—
19Dic3×Dic9C2218yes—
20C22×C3×Dic6—31no—
21C2×C4×C3⋊Dic3—31no—
22C2×C3×C4⋊Dic3D311yescharacteristic
23C2×Dic3⋊Dic3—32no—
24C2×Dic32—32no—
25#63843—38no—
26C22×Dic18D311yescharacteristic
27C2×C9×C4⋊C4—31no—
28C2×Dic9⋊C4—32no—
29C2×C4×Dic9—31no—
30C2×C4×Dic9—32no—
31C2×C4×Dic9—61no—
32#63449—38no—
33C23×C3×C9C2311yescharacteristic
34C2×C3×C4×C9C2×C414yes—
35C2×C3×C4×C9C2312yes—
36C2×C3⋊Dic9—22no—
37C2×C3⋊Dic9Q814yescomplemented
38C2×C3⋊Dic9C2312yes—
39C2×C3⋊Dic9C2312yes—
40C2×C9×Dic3—24no—
41C2×C9×Dic3C2314yes—
42C2×C3×Dic9C2×C418yes—
43C2×C3×Dic9C2314yes—
44C3×Dic18C2×C4116yescomplemented
45C2×Q8×Dic3—91no—
46C2×#887—91no—
47C22×C32×C4C2×D311yescharacteristic
48C22×C3⋊Dic3—31no—
49C22×C3⋊Dic3—31no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3C2×D312yes—
52C2×C3×Dic6—38no—
53C2×C3×Dic6—34no—
54C4×C3⋊Dic3—34no—
55C3×C4⋊Dic3C2×D314yes—
56Dic3⋊Dic3—38no—
57Dic32—38no—
58C22×C4×C9—21no—
59C22×C4×C9—32no—
60C22×C4×C9C2×D311yescharacteristic
61C22×Dic9—31no—
62C22×Dic9—31no—
63C22×Dic9—61no—
64C22×Dic9—61no—
65C22×Dic9C2×D312yes—
66C2×Dic18Dic318yescomplemented
67C2×Dic18C2×D314yes—
68C9×C4⋊C4—34no—
69Dic9⋊C4—38no—
70C4×Dic9—34no—
71C4×Dic9—38no—
72C4×Dic9—64no—
73C22×C3×C9C22×C412yes—
74C22×C3×C9C2×Q812yes—
75C22×C3×C9#4112yes—
76C22×C3×C9C2411yescharacteristic
77C3×C4×C9C22×C414yes—
78C3⋊Dic9—22no—
79C3⋊Dic9C2×Q814yescomplemented
80C9×Dic3—24no—
81C3×Dic9C22×C418yes—
82C22×C3×Q8—91no—
83C22×Dic6—31no—
84C2×C3×C4⋊C4—31no—
85Q8×Dic3—98no—
86C2×C4⋊Dic3—92no—
87C2×C4⋊Dic3D911yescharacteristic
88C2×Dic3⋊C4—92no—
89C2×C4×Dic3—92no—
90C2×C4×Dic3—91no—
91C2×C4×Dic3—91no—
92C2×C4×Dic3—92no—
93C2×C4×Dic3—181no—
94#887—98no—
95C23×C32C22×D311yescharacteristic
96C2×C32×C4C4×D314yes—
97C2×C32×C4C22×D312yes—
98C2×C3⋊Dic3—34no—
99C2×C3⋊Dic3—32no—
100C2×C3⋊Dic3—32no—
101C2×C3⋊Dic3—62no—
102C2×C3×Dic3—24no—
103C2×C3×Dic3—38no—
104C2×C3×Dic3—34no—
105C2×C3×Dic3C22×D314yes—
106C3×Dic6—316no—
107C23×C9—21no—
108C23×C9C22×D311yescharacteristic
109C2×C4×C9—24no—
110C2×C4×C9—22no—
111C2×C4×C9—34no—
112C2×C4×C9—64no—
113C2×C4×C9C2×Dic314yes—
114C2×C4×C9C22×D312yes—
115C2×Dic9—34no—
116C2×Dic9—32no—
117C2×Dic9—32no—
118C2×Dic9—62no—
119C2×Dic9—62no—
120C2×Dic9—64no—
121C2×Dic9—64no—
122C2×Dic9—62no—
123C2×Dic9C2×Dic318yes—
124C2×Dic9C22×D314yes—
125Dic18C2×Dic3116yescomplemented
126C2×C4×Q8—271noSylow
127C2×C3×C9C4×Q814yescomplemented
128C2×C3×C9C23×C411yescharacteristic
129C2×C3×C9C22×Q811yescharacteristic
130C2×C3×C9C2×#4111yescharacteristic
131C2×C3×Q8—98no—
132C2×C3×Q8—94no—
133C22×C3×C4—21no—
134C22×C3×C4—32no—
135C22×C3×C4—92no—
136C22×C3×C4C2×D911yescharacteristic
137C22×C3×C4D3211yescharacteristic
138C22×Dic3—32no—
139C22×Dic3—91no—
140C22×Dic3—91no—
141C22×Dic3—91no—
142C22×Dic3—91no—
143C22×Dic3—181no—
144C22×Dic3—181no—
145C22×Dic3C2×D912yes—
146C2×Dic6—38no—
147C2×Dic6—34no—
148C3×C4⋊C4—34no—
149C4⋊Dic3—98no—
150C4⋊Dic3C2×D914yes—
151Dic3⋊C4—98no—
152C4×Dic3—98no—
153C4×Dic3—94no—
154C4×Dic3—94no—
155C4×Dic3—98no—
156C4×Dic3—184no—
157C22×C32C2×C4×D312yes—
158C22×C32#23712yes—
159C22×C32D3×Q812yes—
160C22×C32C23×D311yescharacteristic
161C32×C4C2×C4×D314yes—
162C3⋊Dic3—34no—
163C3⋊Dic3—62no—
164C3×Dic3—24no—
165C3×Dic3—38no—
166C22×C9—21no—
167C22×C9—22no—
168C22×C9—22no—
169C22×C9—22no—
170C22×C9D3×Q812yes—
171C22×C9#23912yes—
172C22×C9C22×Dic312yes—
173C22×C9C23×D311yescharacteristic
174C4×C9—24no—
175C4×C9—64no—
176C4×C9C22×Dic314yes—
177Dic9—34no—
178Dic9—64no—
179Dic9—64no—
180Dic9—62no—
181Dic9C22×Dic318yes—
182C22×Q8—91no—
183C4×Q8—278no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—272no—
186C2×C42—272no—
187C2×C42—271no—
188C3×C9C2×C4×Q811yescharacteristic, complemented, Sylow
189C23×C3—21no—
190C23×C3C22×D911yescharacteristic
191C23×C3C2×D3211yescharacteristic
192C3×Q8—916no—
193C2×C3×C4—24no—
194C2×C3×C4—22no—
195C2×C3×C4—34no—
196C2×C3×C4—64no—
197C2×C3×C4—98no—
198C2×C3×C4—94no—
199C2×C3×C4C4×D914yes—
200C2×C3×C4C22×D912yes—
201C2×C3×C4D3×Dic314yes—
202C2×C3×C4C2×D3212yes—
203C2×Dic3—24no—
204C2×Dic3—34no—
205C2×Dic3—38no—
206C2×Dic3—92no—
207C2×Dic3—94no—
208C2×Dic3—92no—
209C2×Dic3—94no—
210C2×Dic3—92no—
211C2×Dic3—92no—
212C2×Dic3—182no—
213C2×Dic3—182no—
214C2×Dic3—184no—
215C2×Dic3—184no—
216C2×Dic3—182no—
217C2×Dic3—182no—
218C2×Dic3C22×D914yes—
219Dic6—316no—
220C2×C32#88714yes—
221C2×C32C22×C4×D311yescharacteristic
222C2×C32C2×#23711yescharacteristic
223C2×C32C2×D3×Q811yescharacteristic
224C2×C9—21no—
225C2×C9—21no—
226C2×C9—21no—
227C2×C9—24no—
228C2×C9Q8×Dic314yescomplemented
229C2×C9C2×D3×Q811yescharacteristic
230C2×C9C2×#23911yescharacteristic
231C2×C9C23×Dic311yescharacteristic
232C2×Q8—98no—
233C2×Q8—94no—
234C22×C4—32no—
235C22×C4—92no—
236C22×C4—271no—
237C22×C4—271no—
238C22×C4D3×D911yescharacteristic
239C4⋊C4—34no—
240C4⋊C4—278no—
241C42—278no—
242C42—274no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3C2×C4×D912yes—
248C22×C3#369912yes—
249C22×C3Q8×D912yes—
250C22×C3C23×D911yescharacteristic
251C22×C3#379612yes—
252C22×C3#379712yes—
253C22×C3C2×D3×Dic312yes—
254C22×C3C22×D3211yescharacteristic
255C3×C4—24no—
256C3×C4—64no—
257C3×C4—98no—
258C3×C4C2×C4×D914yes—
259C3×C4C2×D3×Dic314yes—
260Dic3—24no—
261Dic3—38no—
262Dic3—94no—
263Dic3—94no—
264Dic3—184no—
265Dic3—184no—
266Dic3—182no—
267Dic3—182no—
268C32C2×#88711yescharacteristic
269C9—21no—
270C9C2×Q8×Dic311yescharacteristic, complemented
271C23C2×D3×D911yescharacteristic
272Q8—916no—
273C2×C4—34no—
274C2×C4—64no—
275C2×C4—94no—
276C2×C4—98no—
277C2×C4—274no—
278C2×C4—272no—
279C2×C4—272no—
280C2×C4—542no—
281C2×C4Dic3×D914yes—
282C2×C4C2×D3×D912yes—
283C2×C3—21no—
284C2×C3—24no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3#6344914yescomplemented
288C2×C3C22×C4×D911yescharacteristic
289C2×C3C2×#369911yescharacteristic
290C2×C3C2×Q8×D911yescharacteristic
291C2×C3#6384314yes—
292C2×C3C2×#379611yescharacteristic
293C2×C3C2×#379711yescharacteristic
294C2×C3C22×D3×Dic311yescharacteristic
295C22(C3⋊D9)⋊Q812yes—
296C22#8863912yes—
297C22C2×Dic3×D912yes—
298C22C22×D3×D911yescharacteristic
299C4—64no—
300C4—98no—
301C4—274no—
302C4—542no—
303C4C2×Dic3×D914yes—
304C3—21no—
305C3C2×#6344911yescharacteristic, complemented
306C3C2×#6384311yescharacteristic
307C2#1172091014yesdirect product factor
308C2C2×(C3⋊D9)⋊Q811yescharacteristic
309C2C2×#8863911yescharacteristic
310C2C22×Dic3×D911yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socle, prime radicalC23×C32C22×D3
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×C4×Q8
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11720910C2direct (N × H)
14. C2×C3×Dic18C4semidirect
37. C2×C3⋊Dic9Q8semidirect
44. C3×Dic18C2×C4semidirect
66. C2×Dic18Dic3semidirect
79. C3⋊Dic9C2×Q8semidirect
125. Dic18C2×Dic3semidirect
127. C2×C3×C9C4×Q8semidirect
188. C3×C9C2×C4×Q8semidirect
228. C2×C9Q8×Dic3semidirect
270. C9C2×Q8×Dic3semidirect
287. C2×C3#63449semidirect
305. C3C2×#63449semidirect
307. C2#11720910direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Dic182
C2×C4×C3⋊Dic92
C2×C9×C4⋊Dic32
C2×Dic9⋊Dic32
C2×Dic3×Dic92
#117209102
C2×#638433
C2×#634493