Finite Group Databasebeta

← C22×C3×#3722 #13175078

Subgroups — C22×C3×#3722 #13175078

Order 1728.

4050 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C3×C3⋊D3C211yesmaximal, characteristic
2C23×C32×Dic3C212yesmaximal, complemented
3C2×C3×#3722C2112yesmaximal, direct product factor
4C22×C3×(C2×D3)⋊C4—32nomaximal
5C22×#3722C311yesmaximal, characteristic, direct product factor
6C24×C33C2211yescharacteristic
7C23×C3×C3⋊D3—28no—
8C23×C3×C3⋊D3C418yescomplemented
9C23×C3×C3⋊D3C2216yes—
10C22×C32×Dic3—28no—
11C22×C32×Dic3C22112yescomplemented
12C3×#3722C22116yesdirect product factor
13C24×C3⋊D3C2×C311yescharacteristic
14C24×C3×D3—32no—
15C24×C3×D3—61no—
16C23×C32×C4—32no—
17C23×C3×Dic3—22no—
18C23×C3×Dic3D312yescomplemented
19C23×C3×Dic3C2×C312yescomplemented
20C2×C3×(C2×D3)⋊C4—324no—
21C2×#3722C2×C3112yesdirect product factor
22C23×C33C2×C414yes—
23C23×C33D418yes—
24C23×C33C2313yes—
25C22×C3×C3⋊D3—232no—
26C22×C3×C3⋊D3—224no—
27C22×C3×C3⋊D3C2×C4124yescomplemented
28C22×C3×C3⋊D3C2314yes—
29C2×C32×Dic3—224no—
30C2×C32×Dic3C2318yescomplemented
31C22×C3×C22⋊C4—91no—
32C22×(C2×D3)⋊C4—32no—
33C24×C32—21no—
34C24×C32—22no—
35C24×C32—41no—
36C24×C32C2×D312yes—
37C24×C32C22×C311yescharacteristic
38C23×C3⋊D3—28no—
39C23×C3⋊D3C3×C418yescomplemented
40C23×C3⋊D3C22×C316yes—
41C23×C3×D3—316no—
42C23×C3×D3—312no—
43C23×C3×D3—68no—
44C23×C3×D3—66no—
45C23×C3×D3—616no—
46C23×C3×D3—68no—
47C23×C3×D3—68no—
48C22×C32×C4—312no—
49C22×C32×C4—68no—
50C22×C3×Dic3—216no—
51C22×C3×Dic3—212no—
52C22×C3×Dic3—28no—
53C22×C3×Dic3—28no—
54C22×C3×Dic3C2×D3112yescomplemented
55C22×C3×Dic3C22×C3112yescomplemented
56C3×(C2×D3)⋊C4—332no—
57#3722C22×C3116yesdirect product factor
58C22×C33C22⋊C4116yescomplemented
59C22×C33C22×C416yes—
60C22×C33C2×D4112yes—
61C22×C33C2411yescharacteristic
62C2×C3×C3⋊D3—248no—
63C2×C3×C3⋊D3—28no—
64C2×C3×C3⋊D3C22×C418yescomplemented
65C32×Dic3—28no—
66C25×C3—91no—
67C24×D3—32no—
68C24×D3—61no—
69C23×C3×C4—32no—
70C23×C3×C4—32no—
71C23×C3×C4—62no—
72C23×Dic3C3×D312yescomplemented
73C2×C3×C22⋊C4—912no—
74C2×(C2×D3)⋊C4—324no—
75C23×C32—24no—
76C23×C32—28no—
77C23×C32—28no—
78C23×C32—28no—
79C23×C32—28no—
80C23×C32—23no—
81C23×C32—26no—
82C23×C32—44no—
83C23×C32—48no—
84C23×C32—43no—
85C23×C32C4×D318yes—
86C23×C32D1218yes—
87C23×C32#6718yes—
88C23×C32C2×C3×C414yes—
89C23×C32C3×D418yes—
90C23×C32C22×D316yes—
91C23×C32C23×C313yes—
92C22×C3⋊D3—232no—
93C22×C3⋊D3—224no—
94C22×C3⋊D3C2×C3×C4124yescomplemented
95C22×C3⋊D3C23×C314yes—
96C22×C3×D3—348no—
97C22×C3×D3—38no—
98C22×C3×D3—624no—
99C22×C3×D3—64no—
100C22×C3×D3—664no—
101C22×C3×D3—648no—
102C22×C3×D3—664no—
103C22×C3×D3—624no—
104C22×C3×D3—624no—
105C2×C32×C4—38no—
106C2×C32×C4—624no—
107C2×C3×Dic3—248no—
108C2×C3×Dic3—224no—
109C2×C3×Dic3—224no—
110C2×C3×Dic3—28no—
111C2×C3×Dic3C22×D318yescomplemented
112C2×C3×Dic3C23×C318yescomplemented
113C22×C22⋊C4—91noSylow
114C2×C33C2×C22⋊C4112yescomplemented
115C2×C33C23×C411yescharacteristic
116C2×C33C22×D412yes—
117C3×C3⋊D3—28no—
118C24×C3—22no—
119C24×C3—21no—
120C24×C3—41no—
121C24×C3—98no—
122C24×C3—96no—
123C24×C3—188no—
124C24×C3D3211yescharacteristic
125C24×C3C2×C3×D312yes—
126C23×D3—316no—
127C23×D3—312no—
128C23×D3—68no—
129C23×D3—66no—
130C23×D3—616no—
131C23×D3—68no—
132C23×D3—68no—
133C22×C3×C4—312no—
134C22×C3×C4—312no—
135C22×C3×C4—612no—
136C22×C3×C4—616no—
137C22×C3×C4—68no—
138C22×C3×C4—68no—
139C22×Dic3—28no—
140C22×Dic3C2×C3×D3112yescomplemented
141C3×C22⋊C4—916no—
142(C2×D3)⋊C4—332no—
143C22×C32—21no—
144C22×C32—22no—
145C22×C32—216no—
146C22×C32—232no—
147C22×C32—26no—
148C22×C32—212no—
149C22×C32—212no—
150C22×C32—212no—
151C22×C32—212no—
152C22×C32—416no—
153C22×C32—46no—
154C22×C32—412no—
155C22×C32—41no—
156C22×C32(C2×D3)⋊C4132yescomplemented
157C22×C32C3×C22⋊C4116yescomplemented
158C22×C32C2×C4×D3112yes—
159C22×C32C2×D12112yes—
160C22×C32C2×#67112yes—
161C22×C32C22×C3×C416yes—
162C22×C32C2×C3×D4112yes—
163C22×C32C23×D312yes—
164C22×C32C24×C311yescharacteristic
165C2×C3⋊D3—248no—
166C2×C3⋊D3—28no—
167C2×C3⋊D3C22×C3×C418yescomplemented
168C2×C3×D3—316no—
169C2×C3×D3—68no—
170C2×C3×D3—616no—
171C2×C3×D3—696no—
172C2×C3×D3—696no—
173C2×C3×D3—68no—
174C2×C3×D3—68no—
175C32×C4—68no—
176C3×Dic3—216no—
177C3×Dic3—28no—
178C3×Dic3—28no—
179C25—91no—
180C23×C4—32no—
181C2×C22⋊C4—912no—
182C33C22×C22⋊C411yescharacteristic, complemented, Sylow
183C23×C3—28no—
184C23×C3—24no—
185C23×C3—28no—
186C23×C3—28no—
187C23×C3—28no—
188C23×C3—26no—
189C23×C3—23no—
190C23×C3—44no—
191C23×C3—48no—
192C23×C3—43no—
193C23×C3—94no—
194C23×C3—924no—
195C23×C3—1832no—
196C23×C3—1824no—
197C23×C3#62414yes—
198C23×C3#62618yes—
199C23×C3C3×C4×D318yes—
200C23×C3C3×D1218yes—
201C23×C3C3×#6718yes—
202C23×C3C2×D3213yes—
203C23×C3C22×C3×D316yes—
204C22×D3—38no—
205C22×D3—348no—
206C22×D3—624no—
207C22×D3—64no—
208C22×D3—648no—
209C22×D3—664no—
210C22×D3—664no—
211C22×D3—624no—
212C22×D3—624no—
213C2×C3×C4—38no—
214C2×C3×C4—38no—
215C2×C3×C4—68no—
216C2×C3×C4—648no—
217C2×C3×C4—624no—
218C2×C3×C4—624no—
219C2×Dic3—224no—
220C2×Dic3C22×C3×D318yescomplemented
221C2×C32—224no—
222C2×C32—21no—
223C2×C32—22no—
224C2×C32—22no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—212no—
228C2×C32—412no—
229C2×C32—41no—
230C2×C32—42no—
231C2×C32C2×(C2×D3)⋊C4124yescomplemented
232C2×C32C2×C3×C22⋊C4112yescomplemented
233C2×C32C22×C4×D312yes—
234C2×C32C22×D1212yes—
235C2×C32C22×#6712yes—
236C2×C32C23×C3×C411yescharacteristic
237C2×C32C22×C3×D412yes—
238C3⋊D3—28no—
239C3×D3—616no—
240C3×D3—616no—
241C24—98no—
242C24—96no—
243C24—188no—
244C24C3×D3211yescharacteristic
245C22×C4—312no—
246C22×C4—68no—
247C22⋊C4—916no—
248C22×C3—22no—
249C22×C3—21no—
250C22×C3—26no—
251C22×C3—212no—
252C22×C3—212no—
253C22×C3—212no—
254C22×C3—212no—
255C22×C3—216no—
256C22×C3—232no—
257C22×C3—41no—
258C22×C3—46no—
259C22×C3—412no—
260C22×C3—416no—
261C22×C3—98no—
262C22×C3—188no—
263C22×C3—1848no—
264C22×C3#3722116yesdirect product factor
265C22×C3C3×(C2×D3)⋊C4132yescomplemented
266C22×C3C2×#62416yes—
267C22×C3C2×#626112yes—
268C22×C3C2×C3×C4×D3112yes—
269C22×C3C2×C3×D12112yes—
270C22×C3C2×C3×#67112yes—
271C22×C3C22×D3211yescharacteristic
272C22×C3C23×C3×D312yes—
273C2×D3—316no—
274C2×D3—68no—
275C2×D3—616no—
276C2×D3—696no—
277C2×D3—696no—
278C2×D3—68no—
279C2×D3—68no—
280C3×C4—616no—
281C3×C4—68no—
282C3×C4—68no—
283Dic3—28no—
284C32—22no—
285C32—21no—
286C32—41no—
287C32C22×(C2×D3)⋊C412yescomplemented
288C32C22×C3×C22⋊C411yescharacteristic, complemented
289C23—94no—
290C23—924no—
291C23—1824no—
292C23—1832no—
293C23C3×#62414yes—
294C23C3×#62618yes—
295C23C2×C3×D3213yes—
296C2×C4—38no—
297C2×C4—624no—
298C2×C3—21no—
299C2×C3—22no—
300C2×C3—22no—
301C2×C3—22no—
302C2×C3—22no—
303C2×C3—212no—
304C2×C3—224no—
305C2×C3—41no—
306C2×C3—42no—
307C2×C3—412no—
308C2×C3—188no—
309C2×C3C2×#3722112yesdirect product factor
310C2×C3C2×C3×(C2×D3)⋊C4124yescomplemented
311C2×C3C22×#62411yescharacteristic
312C2×C3C22×#62612yes—
313C2×C3C22×C3×C4×D312yes—
314C2×C3C22×C3×D1212yes—
315C2×C3C22×C3×#6712yes—
316D3—616no—
317D3—616no—
318C22—98no—
319C22—188no—
320C22—1848no—
321C22C3×#3722116yesdirect product factor
322C22C2×C3×#62416yes—
323C22C2×C3×#626112yes—
324C22C22×C3×D3211yescharacteristic
325C4—68no—
326C3—21no—
327C3—22no—
328C3—41no—
329C3C22×#372211yescharacteristic, direct product factor
330C3C22×C3×(C2×D3)⋊C412yescomplemented
331C2—188no—
332C2C2×C3×#3722112yesdirect product factor
333C2C22×C3×#62411yescharacteristic
334C2C22×C3×#62612yes—
subgroupG/H
center, hypercenterC24×C3D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
Fitting, socleC24×C33C22
prime radicalC24×C3×C3⋊D3C2
cosocleC22×C32C24×C3
hypocenterC32C22×C3×C22⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C23×C32×Dic3C2semidirect
3. C2×C3×#3722C2direct (N × H)
5. C22×#3722C3direct (N × H)
8. C23×C3×C3⋊D3C4semidirect
11. C22×C32×Dic3C22semidirect
12. C3×#3722C22direct (N × H)
18. C23×C3×Dic3D3semidirect
19. C23×C3×Dic3C2×C3semidirect
21. C2×#3722C2×C3direct (N × H)
27. C22×C3×C3⋊D3C2×C4semidirect
30. C2×C32×Dic3C23semidirect
39. C23×C3⋊D3C3×C4semidirect
54. C22×C3×Dic3C2×D3semidirect
55. C22×C3×Dic3C22×C3semidirect
57. #3722C22×C3direct (N × H)
58. C22×C33C22⋊C4semidirect
64. C2×C3×C3⋊D3C22×C4semidirect
72. C23×Dic3C3×D3semidirect
94. C22×C3⋊D3C2×C3×C4semidirect
111. C2×C3×Dic3C22×D3semidirect
112. C2×C3×Dic3C23×C3semidirect
114. C2×C33C2×C22⋊C4semidirect
140. C22×Dic3C2×C3×D3semidirect
156. C22×C32(C2×D3)⋊C4semidirect
157. C22×C32C3×C22⋊C4semidirect
167. C2×C3⋊D3C22×C3×C4semidirect
182. C33C22×C22⋊C4semidirect
220. C2×Dic3C22×C3×D3semidirect
231. C2×C32C2×(C2×D3)⋊C4semidirect
232. C2×C32C2×C3×C22⋊C4semidirect
264. C22×C3#3722direct (N × H)
265. C22×C3C3×(C2×D3)⋊C4semidirect
287. C32C22×(C2×D3)⋊C4semidirect
288. C32C22×C3×C22⋊C4semidirect
309. C2×C3C2×#3722direct (N × H)
310. C2×C3C2×C3×(C2×D3)⋊C4semidirect
321. C22C3×#3722direct (N × H)
329. C3C22×#3722direct (N × H)
330. C3C22×C3×(C2×D3)⋊C4semidirect
332. C2C2×C3×#3722direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C3⋊D32
C23×C32×Dic32
C2×C3×#37222
C22×C3×(C2×D3)⋊C43
C22×#37223