Finite Group Databasebeta

← C2×#11720931 #13150074

Subgroups — C2×#11720931 #13150074

Order 1728.

1226 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic
2C2×C4×C9×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
4C2×#88445C212yesmaximal
5C2×#88442C212yesmaximal, complemented
6#11720931C218yesmaximal, direct product factor
7C2×#63874—31nomaximal
8C2×#63434—31nomaximal
9C23×C3⋊D9C2211yescharacteristic
10C22×C3×C4×C9C2211yescharacteristic
11C22×C3⋊Dic9C2211yescharacteristic
12C2×C4×C3⋊D9C418yescomplemented
13C2×C4×C3⋊D9C2214yes—
14C22×C9×Dic3C2212yes—
15C22×C3×Dic9C2212yes—
16C4×C9×Dic3C2214yescomplemented
17C3×C4×Dic9C2214yescomplemented
18#88445C2218yes—
19#88442C2218yescomplemented
20C22×C4×C3⋊D3—31no—
21C2×C3×C4×Dic3—31no—
22C2×C3×C4×Dic3D311yescharacteristic
23C2×#3724—32no—
24C2×#3722—32no—
25#63874—38no—
26C22×C4×D9—31no—
27C22×C4×D9—61no—
28C2×C42×C9—31no—
29C2×(C2×D9)⋊C4—32no—
30C2×Dic9⋊C4—32no—
31C2×C4×Dic9D311yescharacteristic, complemented
32#63434—38no—
33C23×C3×C9C2311yescharacteristic
34C22×C3⋊D9—24no—
35C22×C3⋊D9C2×C414yes—
36C22×C3⋊D9C2312yes—
37C2×C3×C4×C9C2×C414yes—
38C2×C3×C4×C9C2312yes—
39C2×C3⋊Dic9C2×C414yes—
40C2×C3⋊Dic9C2312yes—
41C4×C3⋊D9C2×C4116yescomplemented
42C2×C9×Dic3—24no—
43C2×C9×Dic3C2314yes—
44C2×C3×Dic9—24no—
45C2×C3×Dic9C2314yes—
46C2×#872—91no—
47C2×#872—91no—
48C23×C3⋊D3—31no—
49C22×C32×C4C2×D311yescharacteristic
50C22×C3⋊Dic3—31no—
51C2×C4×C3⋊D3—38no—
52C2×C4×C3⋊D3—34no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3C2×D312yes—
55C23×D9—31no—
56C23×D9—61no—
57C3×C4×Dic3—34no—
58C3×C4×Dic3C2×D314yes—
59#3724—38no—
60#3722—38no—
61C22×C4×C9—21no—
62C22×C4×C9—32no—
63C22×C4×C9C2×D311yescharacteristic
64C22×Dic9—31no—
65C22×Dic9—61no—
66C22×Dic9C2×D312yes—
67C2×C4×D9—38no—
68C2×C4×D9—34no—
69C2×C4×D9—68no—
70C2×C4×D9—64no—
71C42×C9—34no—
72(C2×D9)⋊C4—38no—
73Dic9⋊C4—38no—
74C4×Dic9C2×D314yescomplemented
75C22×C3×C9C22×C412yes—
76C22×C3×C9#4114yes—
77C22×C3×C9C2411yescharacteristic
78C2×C3⋊D9—28no—
79C2×C3⋊D9—24no—
80C2×C3⋊D9C22×C414yes—
81C3×C4×C9C22×C414yes—
82C3⋊Dic9C22×C414yes—
83C9×Dic3—24no—
84C3×Dic9—24no—
85C22×C4×D3—91no—
86C22×C4×D3—91no—
87C22×C4×D3—181no—
88C2×C3×C42—31no—
89C2×C3×C42—91no—
90C2×(C2×D3)⋊C4—92no—
91C2×(C2×D3)⋊C4—92no—
92C2×Dic3⋊C4—92no—
93C2×Dic3⋊C4—92no—
94C2×C4×Dic3—31no—
95C2×C4×Dic3D911yescharacteristic, complemented
96#872—98no—
97#872—98no—
98C23×C32C22×D311yescharacteristic
99C22×C3⋊D3—34no—
100C22×C3⋊D3—32no—
101C22×C3⋊D3—64no—
102C2×C32×C4C4×D314yes—
103C2×C32×C4C22×D312yes—
104C2×C3⋊Dic3—34no—
105C2×C3⋊Dic3—32no—
106C4×C3⋊D3—316no—
107C2×C3×Dic3—24no—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3C22×D314yes—
111C23×C9—21no—
112C23×C9C22×D311yescharacteristic
113C22×D9—34no—
114C22×D9—32no—
115C22×D9—64no—
116C22×D9—62no—
117C22×D9—68no—
118C22×D9—64no—
119C2×C4×C9—24no—
120C2×C4×C9—22no—
121C2×C4×C9—34no—
122C2×C4×C9—64no—
123C2×C4×C9C4×D314yes—
124C2×C4×C9C22×D312yes—
125C2×Dic9—24no—
126C2×Dic9—34no—
127C2×Dic9—32no—
128C2×Dic9—64no—
129C2×Dic9—62no—
130C2×Dic9C22×D314yes—
131C4×D9—316no—
132C4×D9—616no—
133C2×#117—271noSylow
134C2×C3×C9#11714yescomplemented
135C2×C3×C9C23×C411yescharacteristic
136C2×C3×C9C2×#4112yes—
137C3⋊D9—24no—
138C23×D3—91no—
139C23×D3—91no—
140C23×D3—181no—
141C22×C3×C4—21no—
142C22×C3×C4—32no—
143C22×C3×C4—92no—
144C22×C3×C4C2×D911yescharacteristic
145C22×C3×C4D3211yescharacteristic
146C22×Dic3—32no—
147C22×Dic3—91no—
148C22×Dic3—91no—
149C22×Dic3—181no—
150C22×Dic3C2×D912yes—
151C2×C4×D3—98no—
152C2×C4×D3—94no—
153C2×C4×D3—98no—
154C2×C4×D3—94no—
155C2×C4×D3—188no—
156C2×C4×D3—184no—
157C3×C42—34no—
158C3×C42—94no—
159(C2×D3)⋊C4—98no—
160(C2×D3)⋊C4—98no—
161Dic3⋊C4—98no—
162Dic3⋊C4—98no—
163C4×Dic3—34no—
164C4×Dic3C2×D914yescomplemented
165C22×C32C2×C4×D312yes—
166C22×C32#23514yes—
167C22×C32C23×D311yescharacteristic
168C2×C3⋊D3—34no—
169C2×C3⋊D3—68no—
170C2×C3⋊D3—64no—
171C32×C4C2×C4×D314yes—
172C3⋊Dic3—34no—
173C3×Dic3—24no—
174C3×Dic3—64no—
175C22×C9—21no—
176C22×C9—24no—
177C22×C9—22no—
178C22×C9C2×C4×D312yes—
179C22×C9#23514yes—
180C22×C9C23×D311yescharacteristic
181C2×D9—34no—
182C2×D9—64no—
183C2×D9—68no—
184C2×D9—616no—
185C2×D9—68no—
186C2×D9—64no—
187C4×C9—24no—
188C4×C9—64no—
189C4×C9C2×C4×D314yes—
190Dic9—24no—
191Dic9—34no—
192Dic9—64no—
193C23×C4—271no—
194#117—278no—
195C2×C4⋊C4—272no—
196C2×C22⋊C4—272no—
197C2×C42—31no—
198C2×C42—91no—
199C3×C9C2×#11711yescharacteristic, complemented, Sylow
200C23×C3—21no—
201C23×C3C22×D911yescharacteristic
202C23×C3C2×D3211yescharacteristic
203C22×D3—92no—
204C22×D3—94no—
205C22×D3—92no—
206C22×D3—94no—
207C22×D3—182no—
208C22×D3—184no—
209C22×D3—188no—
210C22×D3—184no—
211C22×D3—184no—
212C2×C3×C4—24no—
213C2×C3×C4—22no—
214C2×C3×C4—34no—
215C2×C3×C4—64no—
216C2×C3×C4—94no—
217C2×C3×C4—184no—
218C2×C3×C4C4×D914yes—
219C2×C3×C4C22×D912yes—
220C2×C3×C4#62414yes—
221C2×C3×C4C2×D3212yes—
222C2×Dic3—24no—
223C2×Dic3—34no—
224C2×Dic3—64no—
225C2×Dic3—92no—
226C2×Dic3—94no—
227C2×Dic3—92no—
228C2×Dic3—94no—
229C2×Dic3—182no—
230C2×Dic3—184no—
231C2×Dic3C22×D914yes—
232C4×D3—916no—
233C4×D3—916no—
234C4×D3—1816no—
235C2×C32#87214yes—
236C2×C32C22×C4×D311yescharacteristic
237C2×C32C2×#23512yes—
238C3⋊D3—64no—
239C2×C9—22no—
240C2×C9—21no—
241C2×C9—24no—
242C2×C9#87214yescomplemented
243C2×C9C22×C4×D311yescharacteristic
244C2×C9C2×#23512yes—
245D9—68no—
246D9—64no—
247C24—271no—
248C22×C4—32no—
249C22×C4—92no—
250C22×C4—271no—
251C22×C4—278no—
252C22×C4—274no—
253C22×C4D3×D911yescharacteristic
254C4⋊C4—278no—
255C22⋊C4—278no—
256C42—34no—
257C42—94no—
258C22×C3—24no—
259C22×C3—21no—
260C22×C3—22no—
261C22×C3C2×C4×D912yes—
262C22×C3#369714yes—
263C22×C3C23×D911yescharacteristic
264C22×C3#379814yes—
265C22×C3C2×#62412yes—
266C22×C3C22×D3211yescharacteristic
267C2×D3—94no—
268C2×D3—94no—
269C2×D3—184no—
270C2×D3—188no—
271C2×D3—1816no—
272C2×D3—184no—
273C2×D3—188no—
274C2×D3—188no—
275C2×D3—184no—
276C3×C4—24no—
277C3×C4—64no—
278C3×C4—184no—
279C3×C4C2×C4×D914yes—
280C3×C4C2×#62414yes—
281Dic3—24no—
282Dic3—64no—
283Dic3—94no—
284Dic3—94no—
285Dic3—184no—
286C32C2×#87211yescharacteristic
287C9—21no—
288C9C2×#87211yescharacteristic, complemented
289C23—272no—
290C23—274no—
291C23—544no—
292C23C2×D3×D911yescharacteristic
293C2×C4—34no—
294C2×C4—64no—
295C2×C4—94no—
296C2×C4—184no—
297C2×C4—272no—
298C2×C4—274no—
299C2×C4—2716no—
300C2×C4(C3⋊D9)⋊C414yes—
301C2×C4C2×D3×D912yes—
302C2×C3—21no—
303C2×C3—22no—
304C2×C3—24no—
305C2×C3#6343414yescomplemented
306C2×C3C22×C4×D911yescharacteristic
307C2×C3C2×#369712yes—
308C2×C3#6387414yes—
309C2×C3C2×#379812yes—
310C2×C3C22×#62411yescharacteristic
311D3—188no—
312D3—184no—
313D3—184no—
314C22—274no—
315C22—544no—
316C22—548no—
317C22#8864014yes—
318C22C2×(C3⋊D9)⋊C412yes—
319C22C22×D3×D911yescharacteristic
320C4—64no—
321C4—184no—
322C4—274no—
323C4C2×(C3⋊D9)⋊C414yes—
324C3—21no—
325C3C2×#6343411yescharacteristic, complemented
326C3C2×#6387411yescharacteristic
327C2—544no—
328C2#1172093114yesdirect product factor
329C2C2×#8864012yes—
330C2C22×(C3⋊D9)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC22×C4D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
cosocleC22×C3×C9C24
hypocenterC3×C9C2×#117
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C9×Dic3C2semidirect
3. C2×C3×C4×Dic9C2semidirect
5. C2×#88442C2semidirect
6. #11720931C2direct (N × H)
12. C2×C4×C3⋊D9C4semidirect
16. C4×C9×Dic3C22semidirect
17. C3×C4×Dic9C22semidirect
19. #88442C22semidirect
31. C2×C4×Dic9D3semidirect
41. C4×C3⋊D9C2×C4semidirect
74. C4×Dic9C2×D3semidirect
95. C2×C4×Dic3D9semidirect
134. C2×C3×C9#117semidirect
164. C4×Dic3C2×D9semidirect
199. C3×C9C2×#117semidirect
242. C2×C9#872semidirect
288. C9C2×#872semidirect
305. C2×C3#63434semidirect
325. C3C2×#63434semidirect
328. C2#11720931direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C2×C4×C9×Dic32
C2×C3×C4×Dic92
C2×#884452
C2×#884422
#117209312
C2×#638743
C2×#634343