Finite Group Databasebeta

← C2×#11720906 #13150118

Subgroups — C2×#11720906 #13150118

Order 1728.

1082 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D3×C9C211yesmaximal, characteristic
2C2×C4⋊(C3⋊Dic9)C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
4C2×(C2×D3)⋊Dic9C212yesmaximal
5C2×Dic3×Dic9C212yesmaximal, complemented
6#11720906C218yesmaximal, direct product factor
7C2×#63841—31nomaximal
8C2×#63488—31nomaximal
9C23×D3×C9C2211yescharacteristic
10C22×C3×C4×C9C2211yescharacteristic
11C22×C3⋊Dic9C2212yes—
12C22×C9×Dic3C2211yescharacteristic
13C22×C3×Dic9C2212yes—
14C2×C4×D3×C9C418yescomplemented
15C2×C4×D3×C9C2214yes—
16C4⋊(C3⋊Dic9)C2214yescomplemented
17C3×C4⋊Dic9C2214yescomplemented
18(C2×D3)⋊Dic9C2218yes—
19Dic3×Dic9C2218yescomplemented
20C22×C3×C4×D3D311yescharacteristic
21C2×C4⋊(C3⋊Dic3)—31no—
22C2×C3×C4⋊Dic3—31no—
23C2×(C2×D3)⋊Dic3—32no—
24C2×Dic32—32no—
25#63841—38no—
26C23×C4×C9—31no—
27C2×#3676—32no—
28#63488—38no—
29C2×C4⋊Dic9—31no—
30C2×C4⋊Dic9—61no—
31C2×C4⋊Dic9D311yescharacteristic, complemented
32C2×C4×Dic9—32no—
33C23×C3×C9C2311yescharacteristic
34C22×D3×C9—22no—
35C22×D3×C9—22no—
36C22×D3×C9C2×C414yes—
37C22×D3×C9C2312yes—
38C2×C3×C4×C9C2×C414yes—
39C2×C3×C4×C9C2312yes—
40C2×C3⋊Dic9—24no—
41C2×C3⋊Dic9C2314yes—
42C2×C9×Dic3C2×C414yes—
43C2×C9×Dic3C2312yes—
44C2×C3×Dic9—24no—
45C2×C3×Dic9C2314yes—
46C4×D3×C9C2×C4116yescomplemented
47C2×#926—91no—
48C2×#892—91no—
49C23×C3×D3C2×D311yescharacteristic
50C22×C32×C4C2×D311yescharacteristic
51C22×C3⋊Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3C2×D311yescharacteristic
54C2×C3×C4×D3Dic318yes—
55C2×C3×C4×D3C2×D314yes—
56C24×C9—31no—
57C4⋊(C3⋊Dic3)—34no—
58C3×C4⋊Dic3—34no—
59(C2×D3)⋊Dic3—38no—
60Dic32—38no—
61C22×C4×C9—21no—
62C22×C4×C9—31no—
63C22×C4×C9—38no—
64C22×C4×C9—34no—
65C22×C4×C9C2×D311yescharacteristic
66C22×Dic9—32no—
67C22×Dic9—62no—
68C22×Dic9C2×D312yes—
69#3676—38no—
70C4⋊Dic9—34no—
71C4⋊Dic9—64no—
72C4⋊Dic9C2×D314yescomplemented
73C4×Dic9—38no—
74C22×C3×C9C22×C412yes—
75C22×C3×C9#4112yes—
76C22×C3×C9#4112yes—
77C22×C3×C9C2411yescharacteristic
78C2×D3×C9—28no—
79C2×D3×C9—22no—
80C2×D3×C9—22no—
81C2×D3×C9C22×C414yes—
82C3×C4×C9C22×C414yes—
83C3⋊Dic9—24no—
84C9×Dic3C22×C414yes—
85C3×Dic9—24no—
86C23×C3×C4—31no—
87C22×C4×D3D911yescharacteristic
88C2×C3×C4⋊C4—91no—
89C2×#217—92no—
90C2×(C2×D3)⋊C4—92no—
91#926—98no—
92C2×C4⋊Dic3—31no—
93C2×C4⋊Dic3—91no—
94C2×C4⋊Dic3—91no—
95C2×C4⋊Dic3—181no—
96C2×C4×Dic3—92no—
97C2×C4×Dic3—92no—
98#892—98no—
99C23×C32C22×D311yescharacteristic
100C22×C3×D3—22no—
101C22×C3×D3—22no—
102C22×C3×D3C2×Dic314yes—
103C22×C3×D3C22×D312yes—
104C2×C32×C4C2×Dic314yes—
105C2×C32×C4C22×D312yes—
106C2×C3⋊Dic3—34no—
107C2×C3⋊Dic3—64no—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3C2×Dic314yes—
111C2×C3×Dic3C22×D312yes—
112C3×C4×D3C2×Dic3116yes—
113C23×C9—21no—
114C23×C9—34no—
115C23×C9—32no—
116C23×C9—62no—
117C23×C9—62no—
118C23×C9C22×D311yescharacteristic
119C2×C4×C9—24no—
120C2×C4×C9—22no—
121C2×C4×C9—34no—
122C2×C4×C9—32no—
123C2×C4×C9—316no—
124C2×C4×C9C4×D314yes—
125C2×C4×C9C22×D312yes—
126C2×Dic9—24no—
127C2×Dic9—34no—
128C2×Dic9—64no—
129C2×Dic9—64no—
130C2×Dic9—124no—
131C2×Dic9C22×D314yes—
132C2×#117—271noSylow
133C2×C3×C9#11714yescomplemented
134C2×C3×C9C23×C411yescharacteristic
135C2×C3×C9C2×#4111yescharacteristic
136C2×C3×C9C2×#4111yescharacteristic
137D3×C9—24no—
138C24×C3—31no—
139C23×D3C2×D911yescharacteristic
140C22×C3×C4—21no—
141C22×C3×C4—31no—
142C22×C3×C4—38no—
143C22×C3×C4—34no—
144C22×C3×C4—92no—
145C22×C3×C4C2×D911yescharacteristic
146C22×C3×C4D3211yescharacteristic
147C22×Dic3—32no—
148C22×Dic3—92no—
149C22×Dic3—92no—
150C22×Dic3—182no—
151C22×Dic3C2×D911yescharacteristic
152C2×C4×D3Dic918yescomplemented
153C2×C4×D3C2×D914yes—
154C3×C4⋊C4—94no—
155#217—98no—
156(C2×D3)⋊C4—98no—
157C4⋊Dic3—34no—
158C4⋊Dic3—94no—
159C4⋊Dic3—94no—
160C4⋊Dic3—184no—
161C4×Dic3—98no—
162C4×Dic3—98no—
163C22×C32#23512yes—
164C22×C32#23512yes—
165C22×C32C22×Dic312yes—
166C22×C32C23×D311yescharacteristic
167C2×C3×D3—28no—
168C2×C3×D3—22no—
169C2×C3×D3—22no—
170C2×C3×D3C22×Dic314yes—
171C32×C4C22×Dic314yes—
172C3⋊Dic3—64no—
173C3×Dic3—64no—
174C3×Dic3C22×Dic314yes—
175C22×C9—21no—
176C22×C9—22no—
177C22×C9—22no—
178C22×C9—22no—
179C22×C9—34no—
180C22×C9—68no—
181C22×C9—62no—
182C22×C9—62no—
183C22×C9C2×C4×D312yes—
184C22×C9#23712yes—
185C22×C9#23912yes—
186C22×C9C23×D311yescharacteristic
187C4×C9—24no—
188C4×C9—34no—
189C4×C9C2×C4×D314yes—
190Dic9—24no—
191Dic9—64no—
192Dic9—124no—
193C23×C4—31no—
194#117—278no—
195C2×C4⋊C4—91no—
196C2×C4⋊C4—271no—
197C2×C22⋊C4—272no—
198C2×C42—272no—
199C3×C9C2×#11711yescharacteristic, complemented, Sylow
200C23×C3—21no—
201C23×C3—32no—
202C23×C3—34no—
203C23×C3—62no—
204C23×C3—62no—
205C23×C3C22×D911yescharacteristic
206C23×C3C2×D3211yescharacteristic
207C22×D3—22no—
208C22×D3—22no—
209C22×D3C2×Dic914yes—
210C22×D3C22×D912yes—
211C2×C3×C4—24no—
212C2×C3×C4—22no—
213C2×C3×C4—32no—
214C2×C3×C4—34no—
215C2×C3×C4—316no—
216C2×C3×C4—94no—
217C2×C3×C4—184no—
218C2×C3×C4C2×Dic914yes—
219C2×C3×C4C22×D912yes—
220C2×C3×C4D3×Dic314yes—
221C2×C3×C4C2×D3212yes—
222C2×Dic3—34no—
223C2×Dic3—64no—
224C2×Dic3—94no—
225C2×Dic3—94no—
226C2×Dic3—184no—
227C2×Dic3—184no—
228C2×Dic3—184no—
229C2×Dic3—364no—
230C2×Dic3C2×Dic914yes—
231C2×Dic3C22×D912yes—
232C4×D3C2×Dic9116yescomplemented
233C2×C32#92614yes—
234C2×C32C2×#23511yescharacteristic
235C2×C32C2×#23511yescharacteristic
236C2×C32C23×Dic311yescharacteristic
237C3×D3—24no—
238C2×C9—21no—
239C2×C9—21no—
240C2×C9—21no—
241C2×C9—24no—
242C2×C9—64no—
243C2×C9#89214yescomplemented
244C2×C9C22×C4×D311yescharacteristic
245C2×C9C2×#23711yescharacteristic
246C2×C9C2×#23911yescharacteristic
247C24—31no—
248C22×C4—31no—
249C22×C4—38no—
250C22×C4—34no—
251C22×C4—92no—
252C22×C4—272no—
253C22×C4D3×D911yescharacteristic
254C4⋊C4—94no—
255C4⋊C4—274no—
256C22⋊C4—278no—
257C42—278no—
258C22×C3—22no—
259C22×C3—21no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—34no—
263C22×C3—62no—
264C22×C3—62no—
265C22×C3—68no—
266C22×C3#369712yes—
267C22×C3#369712yes—
268C22×C3C22×Dic912yes—
269C22×C3C23×D911yescharacteristic
270C22×C3#379512yes—
271C22×C3#379912yes—
272C22×C3C2×D3×Dic312yes—
273C22×C3C22×D3211yescharacteristic
274C2×D3—28no—
275C2×D3—22no—
276C2×D3—22no—
277C2×D3C22×Dic914yes—
278C3×C4—24no—
279C3×C4—34no—
280C3×C4—184no—
281C3×C4C22×Dic914yes—
282C3×C4C2×D3×Dic314yes—
283Dic3—64no—
284Dic3—184no—
285Dic3—184no—
286Dic3—364no—
287Dic3C22×Dic914yes—
288C32C2×#92611yescharacteristic
289C9—21no—
290C9C2×#89211yescharacteristic, complemented
291C23—32no—
292C23—34no—
293C23—62no—
294C23—62no—
295C23C2×D3×D911yescharacteristic
296C2×C4—32no—
297C2×C4—34no—
298C2×C4—316no—
299C2×C4—94no—
300C2×C4—184no—
301C2×C4—274no—
302C2×C4—544no—
303C2×C4D3×Dic914yes—
304C2×C4C2×D3×D912yes—
305C2×C3—21no—
306C2×C3—24no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—64no—
310C2×C3#6348814yescomplemented
311C2×C3C2×#369711yescharacteristic
312C2×C3C2×#369711yescharacteristic
313C2×C3C23×Dic911yescharacteristic
314C2×C3#6384114yes—
315C2×C3C2×#379511yescharacteristic
316C2×C3C2×#379911yescharacteristic
317C2×C3C22×D3×Dic311yescharacteristic
318D3—24no—
319C22—34no—
320C22—68no—
321C22—62no—
322C22—62no—
323C22#8864112yes—
324C22#8864212yes—
325C22C2×D3×Dic912yes—
326C22C22×D3×D911yescharacteristic
327C4—34no—
328C4—184no—
329C4—544no—
330C4C2×D3×Dic914yes—
331C3—21no—
332C3C2×#6348811yescharacteristic, complemented
333C3C2×#6384111yescharacteristic
334C2—64no—
335C2#1172090614yesdirect product factor
336C2C2×#8864111yescharacteristic
337C2C2×#8864211yescharacteristic
338C2C22×D3×Dic911yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3×D3C2×D3
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#117
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C3⋊Dic9)C2semidirect
3. C2×C3×C4⋊Dic9C2semidirect
5. C2×Dic3×Dic9C2semidirect
6. #11720906C2direct (N × H)
14. C2×C4×D3×C9C4semidirect
16. C4⋊(C3⋊Dic9)C22semidirect
17. C3×C4⋊Dic9C22semidirect
19. Dic3×Dic9C22semidirect
31. C2×C4⋊Dic9D3semidirect
46. C4×D3×C9C2×C4semidirect
72. C4⋊Dic9C2×D3semidirect
133. C2×C3×C9#117semidirect
152. C2×C4×D3Dic9semidirect
199. C3×C9C2×#117semidirect
232. C4×D3C2×Dic9semidirect
243. C2×C9#892semidirect
290. C9C2×#892semidirect
310. C2×C3#63488semidirect
332. C3C2×#63488semidirect
335. C2#11720906direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D3×C92
C2×C4⋊(C3⋊Dic9)2
C2×C3×C4⋊Dic92
C2×(C2×D3)⋊Dic92
C2×Dic3×Dic92
#117209062
C2×#638413
C2×#634883