Finite Group Databasebeta

← C2×#11720937 #13150071

Subgroups — C2×#11720937 #13150071

Order 1728.

786 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C3⋊Dic9C211yesmaximal, characteristic
2C2×C9×C4⋊Dic3C211yesmaximal, characteristic
3C2×C3×C4⋊Dic9C211yesmaximal, characteristic
4C2×Dic3⋊Dic9C212yesmaximal
5C2×Dic9⋊Dic3C212yesmaximal
6#11720937C218yesmaximal, direct product factor
7C2×#63872—31nomaximal
8C2×#63452—31nomaximal
9C22×C3×C4×C9C2211yescharacteristic
10C22×C3⋊Dic9C2211yescharacteristic
11C22×C3⋊Dic9C2211yescharacteristic
12C22×C9×Dic3C2212yes—
13C22×C3×Dic9C2212yes—
14C4×C3⋊Dic9C2214yes—
15C9×C4⋊Dic3C2214yes—
16C3×C4⋊Dic9C2214yes—
17Dic3⋊Dic9C2218yes—
18Dic9⋊Dic3C2218yes—
19C2×C4×C3⋊Dic3—31no—
20C2×C3×C4⋊Dic3—31no—
21C2×C3×C4⋊Dic3D311yescharacteristic
22C2×Dic3⋊Dic3—32no—
23C2×Dic3⋊Dic3—32no—
24#63872—38no—
25C2×C9×C4⋊C4—31no—
26C2×C4⋊Dic9—32no—
27C2×C4⋊Dic9D311yescharacteristic
28C2×Dic9⋊C4—32no—
29C2×C4×Dic9—31no—
30C2×C4×Dic9—61no—
31#63452—38no—
32C23×C3×C9C2311yescharacteristic
33C2×C3×C4×C9Q814yes—
34C2×C3×C4×C9C2312yes—
35C2×C3⋊Dic9—22no—
36C2×C3⋊Dic9—22no—
37C2×C3⋊Dic9C2312yes—
38C2×C3⋊Dic9C2312yes—
39C2×C9×Dic3—24no—
40C2×C9×Dic3C2314yes—
41C2×C3×Dic9—24no—
42C2×C3×Dic9C2314yes—
43C2×#890—91no—
44C2×#890—91no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C3⋊Dic3—31no—
47C22×C3⋊Dic3—31no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3C2×D312yes—
50C4×C3⋊Dic3—34no—
51C3×C4⋊Dic3—34no—
52C3×C4⋊Dic3C2×D314yes—
53Dic3⋊Dic3—38no—
54Dic3⋊Dic3—38no—
55C22×C4×C9—21no—
56C22×C4×C9—32no—
57C22×C4×C9C2×D311yescharacteristic
58C22×Dic9—31no—
59C22×Dic9—31no—
60C22×Dic9—61no—
61C22×Dic9—61no—
62C22×Dic9C2×D312yes—
63C9×C4⋊C4—34no—
64C4⋊Dic9—38no—
65C4⋊Dic9C2×D314yes—
66Dic9⋊C4—38no—
67C4×Dic9—34no—
68C4×Dic9—64no—
69C22×C3×C9C2×Q812yes—
70C22×C3×C9#4112yes—
71C22×C3×C9#4112yes—
72C22×C3×C9C2411yescharacteristic
73C3×C4×C9C2×Q814yes—
74C3⋊Dic9—22no—
75C3⋊Dic9—22no—
76C9×Dic3—24no—
77C3×Dic9—24no—
78C2×C3×C4⋊C4—31no—
79C2×C3×C4⋊C4—91no—
80C2×C4⋊Dic3—31no—
81C2×C4⋊Dic3—92no—
82C2×C4⋊Dic3—92no—
83C2×C4⋊Dic3D911yescharacteristic
84C2×Dic3⋊C4—92no—
85C2×Dic3⋊C4—92no—
86C2×C4×Dic3—91no—
87C2×C4×Dic3—91no—
88C2×C4×Dic3—181no—
89#890—98no—
90#890—98no—
91C23×C32C22×D311yescharacteristic
92C2×C32×C4Dic614yes—
93C2×C32×C4C22×D312yes—
94C2×C3⋊Dic3—32no—
95C2×C3⋊Dic3—32no—
96C2×C3⋊Dic3—62no—
97C2×C3⋊Dic3—62no—
98C2×C3×Dic3—24no—
99C2×C3×Dic3—34no—
100C2×C3×Dic3—64no—
101C2×C3×Dic3C22×D314yes—
102C23×C9—21no—
103C23×C9C22×D311yescharacteristic
104C2×C4×C9—24no—
105C2×C4×C9—22no—
106C2×C4×C9—34no—
107C2×C4×C9—64no—
108C2×C4×C9Dic614yes—
109C2×C4×C9C22×D312yes—
110C2×Dic9—24no—
111C2×Dic9—32no—
112C2×Dic9—32no—
113C2×Dic9—62no—
114C2×Dic9—62no—
115C2×Dic9—64no—
116C2×Dic9—64no—
117C2×Dic9—62no—
118C2×Dic9—62no—
119C2×Dic9C22×D314yes—
120C2×#125—271noSylow
121C2×C3×C9#12514yescomplemented
122C2×C3×C9C22×Q811yescharacteristic
123C2×C3×C9C2×#4111yescharacteristic
124C2×C3×C9C2×#4111yescharacteristic
125C22×C3×C4—21no—
126C22×C3×C4—32no—
127C22×C3×C4—92no—
128C22×C3×C4C2×D911yescharacteristic
129C22×C3×C4D3211yescharacteristic
130C22×Dic3—32no—
131C22×Dic3—91no—
132C22×Dic3—91no—
133C22×Dic3—91no—
134C22×Dic3—91no—
135C22×Dic3—181no—
136C22×Dic3—181no—
137C22×Dic3C2×D912yes—
138C3×C4⋊C4—34no—
139C3×C4⋊C4—94no—
140C4⋊Dic3—34no—
141C4⋊Dic3—98no—
142C4⋊Dic3—98no—
143C4⋊Dic3C2×D914yes—
144Dic3⋊C4—98no—
145Dic3⋊C4—98no—
146C4×Dic3—94no—
147C4×Dic3—94no—
148C4×Dic3—184no—
149C22×C32C2×Dic612yes—
150C22×C32#23712yes—
151C22×C32#23912yes—
152C22×C32C23×D311yescharacteristic
153C32×C4C2×Dic614yes—
154C3⋊Dic3—62no—
155C3⋊Dic3—62no—
156C3×Dic3—24no—
157C3×Dic3—64no—
158C22×C9—21no—
159C22×C9—22no—
160C22×C9—22no—
161C22×C9—22no—
162C22×C9C2×Dic612yes—
163C22×C9#23712yes—
164C22×C9#23912yes—
165C22×C9C23×D311yescharacteristic
166C4×C9—24no—
167C4×C9—64no—
168C4×C9C2×Dic614yes—
169Dic9—24no—
170Dic9—64no—
171Dic9—64no—
172Dic9—62no—
173Dic9—62no—
174#125—278no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—91no—
177C2×C4⋊C4—272no—
178C2×C4⋊C4—272no—
179C2×C42—271no—
180C3×C9C2×#12511yescharacteristic, complemented, Sylow
181C23×C3—21no—
182C23×C3C22×D911yescharacteristic
183C23×C3C2×D3211yescharacteristic
184C2×C3×C4—24no—
185C2×C3×C4—22no—
186C2×C3×C4—34no—
187C2×C3×C4—64no—
188C2×C3×C4—94no—
189C2×C3×C4—184no—
190C2×C3×C4Dic1814yes—
191C2×C3×C4C22×D912yes—
192C2×C3×C4#62714yes—
193C2×C3×C4C2×D3212yes—
194C2×Dic3—24no—
195C2×Dic3—34no—
196C2×Dic3—64no—
197C2×Dic3—92no—
198C2×Dic3—92no—
199C2×Dic3—92no—
200C2×Dic3—92no—
201C2×Dic3—182no—
202C2×Dic3—182no—
203C2×Dic3—184no—
204C2×Dic3—184no—
205C2×Dic3—182no—
206C2×Dic3—182no—
207C2×Dic3—182no—
208C2×Dic3—182no—
209C2×Dic3C22×D914yes—
210C2×C32#89014yes—
211C2×C32C22×Dic611yescharacteristic
212C2×C32C2×#23711yescharacteristic
213C2×C32C2×#23911yescharacteristic
214C2×C9—21no—
215C2×C9—21no—
216C2×C9—21no—
217C2×C9—24no—
218C2×C9#89014yescomplemented
219C2×C9C22×Dic611yescharacteristic
220C2×C9C2×#23711yescharacteristic
221C2×C9C2×#23911yescharacteristic
222C22×C4—32no—
223C22×C4—92no—
224C22×C4—271no—
225C22×C4—271no—
226C22×C4D3×D911yescharacteristic
227C4⋊C4—34no—
228C4⋊C4—94no—
229C4⋊C4—278no—
230C4⋊C4—278no—
231C42—274no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—21no—
235C22×C3—22no—
236C22×C3C2×Dic1812yes—
237C22×C3#369912yes—
238C22×C3#370112yes—
239C22×C3C23×D911yescharacteristic
240C22×C3#379612yes—
241C22×C3#379612yes—
242C22×C3C2×#62712yes—
243C22×C3C22×D3211yescharacteristic
244C3×C4—24no—
245C3×C4—64no—
246C3×C4—184no—
247C3×C4C2×Dic1814yes—
248C3×C4C2×#62714yes—
249Dic3—24no—
250Dic3—64no—
251Dic3—184no—
252Dic3—184no—
253Dic3—182no—
254Dic3—182no—
255Dic3—182no—
256Dic3—182no—
257C32C2×#89011yescharacteristic
258C9—21no—
259C9C2×#89011yescharacteristic, complemented
260C23C2×D3×D911yescharacteristic
261C2×C4—34no—
262C2×C4—64no—
263C2×C4—94no—
264C2×C4—184no—
265C2×C4—272no—
266C2×C4—272no—
267C2×C4—542no—
268C2×C4—542no—
269C2×C4#618914yes—
270C2×C4C2×D3×D912yes—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—24no—
275C2×C3#6345214yescomplemented
276C2×C3C22×Dic1811yescharacteristic
277C2×C3C2×#369911yescharacteristic
278C2×C3C2×#370111yescharacteristic
279C2×C3#6387214yes—
280C2×C3C2×#379611yescharacteristic
281C2×C3C2×#379611yescharacteristic
282C2×C3C22×#62711yescharacteristic
283C22#8863412yes—
284C22#8863912yes—
285C22C2×#618912yes—
286C22C22×D3×D911yescharacteristic
287C4—64no—
288C4—184no—
289C4—542no—
290C4—542no—
291C4C2×#618914yes—
292C3—21no—
293C3C2×#6345211yescharacteristic, complemented
294C3C2×#6387211yescharacteristic
295C2#1172093714yesdirect product factor
296C2C2×#8863411yescharacteristic
297C2C2×#8863911yescharacteristic
298C2C22×#618911yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socle, prime radicalC23×C32C22×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#125
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11720937C2direct (N × H)
121. C2×C3×C9#125semidirect
180. C3×C9C2×#125semidirect
218. C2×C9#890semidirect
259. C9C2×#890semidirect
275. C2×C3#63452semidirect
293. C3C2×#63452semidirect
295. C2#11720937direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊Dic92
C2×C9×C4⋊Dic32
C2×C3×C4⋊Dic92
C2×Dic3⋊Dic92
C2×Dic9⋊Dic32
#117209372
C2×#638723
C2×#634523