Finite Group Databasebeta

← C22×C3×#3724 #13175081

Subgroups — C22×C3×#3724 #13175081

Order 1728.

2514 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C23×C32×Dic3C212yesmaximal
3C2×C3×#3724C2112yesmaximal, direct product factor
4C22×C3×Dic3⋊C4—32nomaximal
5C22×#3724C311yesmaximal, characteristic, direct product factor
6C24×C33C2211yescharacteristic
7C22×C3×C3⋊Dic3C418yescomplemented
8C22×C3×C3⋊Dic3C2216yes—
9C22×C32×Dic3—28no—
10C22×C32×Dic3C22112yes—
11C3×#3724C22116yesdirect product factor
12C23×C32×C4—32no—
13C23×C3⋊Dic3C2×C311yescharacteristic
14C23×C3×Dic3—22no—
15C23×C3×Dic3—32no—
16C23×C3×Dic3—61no—
17C23×C3×Dic3D312yes—
18C23×C3×Dic3C2×C312yes—
19C2×C3×Dic3⋊C4—324no—
20C2×#3724C2×C3112yesdirect product factor
21C23×C33C2×C414yes—
22C23×C33D414yes—
23C23×C33Q814yes—
24C23×C33C2313yes—
25C2×C3×C3⋊Dic3C2×C4124yescomplemented
26C2×C3×C3⋊Dic3C2314yes—
27C2×C32×Dic3—224no—
28C2×C32×Dic3C2318yes—
29C22×C3×C4⋊C4—91no—
30C22×Dic3⋊C4—32no—
31C24×C32—21no—
32C24×C32—22no—
33C24×C32—41no—
34C24×C32C2×D312yes—
35C24×C32C22×C311yescharacteristic
36C22×C32×C4—312no—
37C22×C32×C4—68no—
38C22×C3⋊Dic3C3×C418yescomplemented
39C22×C3⋊Dic3C22×C316yes—
40C22×C3×Dic3—216no—
41C22×C3×Dic3—212no—
42C22×C3×Dic3—28no—
43C22×C3×Dic3—28no—
44C22×C3×Dic3—316no—
45C22×C3×Dic3—312no—
46C22×C3×Dic3—68no—
47C22×C3×Dic3—66no—
48C22×C3×Dic3C2×D3112yes—
49C22×C3×Dic3C22×C3112yes—
50C3×Dic3⋊C4—332no—
51#3724C22×C3116yesdirect product factor
52C22×C33C4⋊C4116yescomplemented
53C22×C33C22×C416yes—
54C22×C33C2×D416yes—
55C22×C33C2×Q816yes—
56C22×C33C2411yescharacteristic
57C3×C3⋊Dic3C22×C418yescomplemented
58C32×Dic3—28no—
59C23×C3×C4—32no—
60C23×C3×C4—32no—
61C23×C3×C4—62no—
62C23×C3×C4—91no—
63C23×Dic3—32no—
64C23×Dic3—61no—
65C23×Dic3C3×D312yes—
66C2×C3×C4⋊C4—912no—
67C2×Dic3⋊C4—324no—
68C23×C32—24no—
69C23×C32—24no—
70C23×C32—24no—
71C23×C32—28no—
72C23×C32—28no—
73C23×C32—28no—
74C23×C32—23no—
75C23×C32—26no—
76C23×C32—44no—
77C23×C32—44no—
78C23×C32—44no—
79C23×C32—43no—
80C23×C32Dic618yes—
81C23×C32C4×D318yes—
82C23×C32#6718yes—
83C23×C32C2×C3×C414yes—
84C23×C32C3×D414yes—
85C23×C32C3×Q814yes—
86C23×C32C22×D316yes—
87C23×C32C23×C313yes—
88C2×C32×C4—38no—
89C2×C32×C4—624no—
90C2×C3⋊Dic3C2×C3×C4124yescomplemented
91C2×C3⋊Dic3C23×C314yes—
92C2×C3×Dic3—248no—
93C2×C3×Dic3—224no—
94C2×C3×Dic3—224no—
95C2×C3×Dic3—28no—
96C2×C3×Dic3—348no—
97C2×C3×Dic3—38no—
98C2×C3×Dic3—624no—
99C2×C3×Dic3—64no—
100C2×C3×Dic3C22×D318yes—
101C2×C3×Dic3C23×C318yes—
102C22×C4⋊C4—91noSylow
103C2×C33C2×C4⋊C4112yescomplemented
104C2×C33C23×C411yescharacteristic
105C2×C33C22×D411yescharacteristic
106C2×C33C22×Q811yescharacteristic
107C24×C3—22no—
108C24×C3—21no—
109C24×C3—41no—
110C24×C3D3211yescharacteristic
111C24×C3C2×C3×D312yes—
112C22×C3×C4—312no—
113C22×C3×C4—312no—
114C22×C3×C4—612no—
115C22×C3×C4—616no—
116C22×C3×C4—68no—
117C22×C3×C4—68no—
118C22×C3×C4—98no—
119C22×C3×C4—96no—
120C22×Dic3—28no—
121C22×Dic3—316no—
122C22×Dic3—312no—
123C22×Dic3—68no—
124C22×Dic3—66no—
125C22×Dic3C2×C3×D3112yes—
126C3×C4⋊C4—916no—
127Dic3⋊C4—332no—
128C22×C32—21no—
129C22×C32—22no—
130C22×C32—216no—
131C22×C32—232no—
132C22×C32—26no—
133C22×C32—26no—
134C22×C32—26no—
135C22×C32—212no—
136C22×C32—212no—
137C22×C32—212no—
138C22×C32—416no—
139C22×C32—46no—
140C22×C32—46no—
141C22×C32—46no—
142C22×C32—41no—
143C22×C32Dic3⋊C4132yescomplemented
144C22×C32C3×C4⋊C4116yescomplemented
145C22×C32C2×Dic6112yes—
146C22×C32C2×C4×D3112yes—
147C22×C32C2×#67112yes—
148C22×C32C22×C3×C416yes—
149C22×C32C2×C3×D416yes—
150C22×C32C2×C3×Q816yes—
151C22×C32C23×D312yes—
152C22×C32C24×C311yescharacteristic
153C32×C4—68no—
154C3⋊Dic3C22×C3×C418yescomplemented
155C3×Dic3—216no—
156C3×Dic3—28no—
157C3×Dic3—28no—
158C3×Dic3—316no—
159C3×Dic3—68no—
160C23×C4—32no—
161C23×C4—91no—
162C2×C4⋊C4—912no—
163C33C22×C4⋊C411yescharacteristic, complemented, Sylow
164C23×C3—28no—
165C23×C3—24no—
166C23×C3—28no—
167C23×C3—24no—
168C23×C3—28no—
169C23×C3—24no—
170C23×C3—26no—
171C23×C3—23no—
172C23×C3—44no—
173C23×C3—44no—
174C23×C3—44no—
175C23×C3—43no—
176C23×C3#62414yes—
177C23×C3#62514yes—
178C23×C3#62714yes—
179C23×C3C3×Dic618yes—
180C23×C3C3×C4×D318yes—
181C23×C3C3×#6718yes—
182C23×C3C2×D3213yes—
183C23×C3C22×C3×D316yes—
184C2×C3×C4—38no—
185C2×C3×C4—38no—
186C2×C3×C4—68no—
187C2×C3×C4—648no—
188C2×C3×C4—624no—
189C2×C3×C4—624no—
190C2×C3×C4—94no—
191C2×C3×C4—924no—
192C2×Dic3—224no—
193C2×Dic3—38no—
194C2×Dic3—348no—
195C2×Dic3—624no—
196C2×Dic3—64no—
197C2×Dic3C22×C3×D318yes—
198C2×C32—224no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—22no—
203C2×C32—22no—
204C2×C32—22no—
205C2×C32—212no—
206C2×C32—412no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32C2×Dic3⋊C4124yescomplemented
211C2×C32C2×C3×C4⋊C4112yescomplemented
212C2×C32C22×Dic612yes—
213C2×C32C22×C4×D312yes—
214C2×C32C22×#6712yes—
215C2×C32C23×C3×C411yescharacteristic
216C2×C32C22×C3×D411yescharacteristic
217C2×C32C22×C3×Q811yescharacteristic
218C24C3×D3211yescharacteristic
219C22×C4—312no—
220C22×C4—68no—
221C22×C4—98no—
222C22×C4—96no—
223C4⋊C4—916no—
224C22×C3—21no—
225C22×C3—22no—
226C22×C3—26no—
227C22×C3—212no—
228C22×C3—212no—
229C22×C3—26no—
230C22×C3—212no—
231C22×C3—26no—
232C22×C3—232no—
233C22×C3—216no—
234C22×C3—41no—
235C22×C3—46no—
236C22×C3—46no—
237C22×C3—46no—
238C22×C3—416no—
239C22×C3#3724116yesdirect product factor
240C22×C3C3×Dic3⋊C4132yescomplemented
241C22×C3C2×#62416yes—
242C22×C3C2×#62516yes—
243C22×C3C2×#62716yes—
244C22×C3C2×C3×Dic6112yes—
245C22×C3C2×C3×C4×D3112yes—
246C22×C3C2×C3×#67112yes—
247C22×C3C22×D3211yescharacteristic
248C22×C3C23×C3×D312yes—
249C3×C4—616no—
250C3×C4—68no—
251C3×C4—68no—
252C3×C4—98no—
253Dic3—28no—
254Dic3—316no—
255Dic3—68no—
256C32—22no—
257C32—21no—
258C32—41no—
259C32C22×Dic3⋊C412yescomplemented
260C32C22×C3×C4⋊C411yescharacteristic, complemented
261C23C3×#62414yes—
262C23C3×#62514yes—
263C23C3×#62714yes—
264C23C2×C3×D3213yes—
265C2×C4—38no—
266C2×C4—624no—
267C2×C4—94no—
268C2×C4—924no—
269C2×C3—21no—
270C2×C3—22no—
271C2×C3—22no—
272C2×C3—21no—
273C2×C3—22no—
274C2×C3—21no—
275C2×C3—224no—
276C2×C3—212no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—412no—
281C2×C3C2×#3724112yesdirect product factor
282C2×C3C2×C3×Dic3⋊C4124yescomplemented
283C2×C3C22×#62411yescharacteristic
284C2×C3C22×#62511yescharacteristic
285C2×C3C22×#62711yescharacteristic
286C2×C3C22×C3×Dic612yes—
287C2×C3C22×C3×C4×D312yes—
288C2×C3C22×C3×#6712yes—
289C22C3×#3724116yesdirect product factor
290C22C2×C3×#62416yes—
291C22C2×C3×#62516yes—
292C22C2×C3×#62716yes—
293C22C22×C3×D3211yescharacteristic
294C4—68no—
295C4—98no—
296C3—22no—
297C3—21no—
298C3—41no—
299C3C22×#372411yescharacteristic, direct product factor
300C3C22×C3×Dic3⋊C412yescomplemented
301C2C2×C3×#3724112yesdirect product factor
302C2C22×C3×#62411yescharacteristic
303C2C22×C3×#62511yescharacteristic
304C2C22×C3×#62711yescharacteristic
subgroupG/H
center, hypercenterC24×C3D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
Fitting, socle, prime radicalC24×C33C22
cosocleC22×C32C24×C3
hypocenterC32C22×C3×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3724C2direct (N × H)
5. C22×#3724C3direct (N × H)
7. C22×C3×C3⋊Dic3C4semidirect
11. C3×#3724C22direct (N × H)
20. C2×#3724C2×C3direct (N × H)
25. C2×C3×C3⋊Dic3C2×C4semidirect
38. C22×C3⋊Dic3C3×C4semidirect
51. #3724C22×C3direct (N × H)
52. C22×C33C4⋊C4semidirect
57. C3×C3⋊Dic3C22×C4semidirect
90. C2×C3⋊Dic3C2×C3×C4semidirect
103. C2×C33C2×C4⋊C4semidirect
143. C22×C32Dic3⋊C4semidirect
144. C22×C32C3×C4⋊C4semidirect
154. C3⋊Dic3C22×C3×C4semidirect
163. C33C22×C4⋊C4semidirect
210. C2×C32C2×Dic3⋊C4semidirect
211. C2×C32C2×C3×C4⋊C4semidirect
239. C22×C3#3724direct (N × H)
240. C22×C3C3×Dic3⋊C4semidirect
259. C32C22×Dic3⋊C4semidirect
260. C32C22×C3×C4⋊C4semidirect
281. C2×C3C2×#3724direct (N × H)
282. C2×C3C2×C3×Dic3⋊C4semidirect
289. C22C3×#3724direct (N × H)
299. C3C22×#3724direct (N × H)
300. C3C22×C3×Dic3⋊C4semidirect
301. C2C2×C3×#3724direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C23×C32×Dic32
C2×C3×#37242
C22×C3×Dic3⋊C43
C22×#37243