Finite Group Databasebeta

← C22×Dic3×C22⋊C9 #13164641

Subgroups — C22×Dic3×C22⋊C9 #13164641

Order 1728.

1338 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 1338 conjugacy classes of subgroups — too many to draw. Only its 215 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C22⋊C9C211yesmaximal, characteristic
2C2×Dic3×C22⋊C9C216yesmaximal, direct product factor
3C24×C3×Dic3C311yesmaximal, characteristic
4C22×C4×C22⋊C9—31nomaximal
5C22×C3×C22⋊C9C414yescomplemented
6C22×C3×C22⋊C9C2213yes—
7C22×C9×Dic3—41nomaximal
8Dic3×C22⋊C9C2214yesdirect product factor
9C25×C32C2×C311yescharacteristic
10C23×C3×Dic3—31no—
11C23×C3×Dic3—31no—
12C23×C3×Dic3—36no—
13C23×C3×Dic3C2×C316yes—
14C23×C22⋊C9—21no—
15C23×C22⋊C9D311yescharacteristic
16C2×C4×C22⋊C9—36no—
17C23×C3×C9—41no—
18C2×C3×C22⋊C9C2×C416yescomplemented
19C2×C3×C22⋊C9C2311yescharacteristic
20C2×C9×Dic3—46no—
21C24×C3×C4—31no—
22C24×Dic3C911yescharacteristic, complemented
23C24×C32—31no—
24C24×C32—33no—
25C24×C32—34no—
26C24×C32C3×C414yes—
27C24×C32C22×C313yes—
28C22×C3×Dic3—36no—
29C22×C3×Dic3—38no—
30C22×C3×Dic3—31no—
31C22×C3×Dic3—36no—
32C22×C3×Dic3—36no—
33C22×C3×Dic3—36no—
34C22×C3×Dic3—312no—
35C22×C3×Dic3A411yescharacteristic
36C22×C3×Dic3C22×C314yes—
37C22×C22⋊C9—24no—
38C22×C22⋊C9—23no—
39C22×C22⋊C9Dic314yesdirect product factor
40C22×C22⋊C9C2×D313yes—
41C22×C4×C9—121no—
42C4×C22⋊C9—34no—
43C22×C3×C9—44no—
44C22×C3×C9—43no—
45C3×C22⋊C9C22×C411yescharacteristic, complemented
46C9×Dic3—44no—
47C25×C3—21no—
48C25×C3C2×C911yescharacteristic
49C25×C3C3×D311yescharacteristic
50C23×C3×C4—36no—
51C23×C3×C4—91no—
52C23×C3×C4—96no—
53C23×C3×C4—91no—
54C23×Dic3—31no—
55C23×Dic3—31no—
56C23×Dic3—36no—
57C23×Dic3C2×C916yescomplemented
58C23×C32—312no—
59C23×C32—36no—
60C23×C32—33no—
61C23×C32—312no—
62C23×C32—32no—
63C23×C32—34no—
64C23×C32—34no—
65C23×C32—33no—
66C23×C32—33no—
67C23×C32C2×C3×C416yes—
68C23×C32C2×A411yescharacteristic
69C23×C32C23×C311yescharacteristic
70C2×C3×Dic3—312no—
71C2×C3×Dic3—36no—
72C2×C3×Dic3—312no—
73C2×C3×Dic3—34no—
74C2×C3×Dic3—34no—
75C2×C3×Dic3C2×A416yes—
76C23×C9—41no—
77C23×C9—81no—
78C2×C22⋊C9—26no—
79C2×C22⋊C9—21no—
80C2×C22⋊C9C2×Dic316yesdirect product factor
81C2×C22⋊C9C22×D311yescharacteristic
82C2×C4×C9—126no—
83C24×C4—31noSylow
84C2×C3×C9—46no—
85C2×C3×C9—41no—
86C24×C3—24no—
87C24×C3—23no—
88C24×C3—34no—
89C24×C3—33no—
90C24×C3—34no—
91C24×C3—33no—
92C24×C3—31no—
93C24×C3—31no—
94C24×C3—61no—
95C24×C3—63no—
96C24×C3—64no—
97C24×C3C4×C914yescomplemented
98C24×C3C22×C913yes—
99C24×C3C3×Dic314yes—
100C24×C3C2×C3×D313yes—
101C22×C3×C4—31no—
102C22×C3×C4—34no—
103C22×C3×C4—96no—
104C22×C3×C4—96no—
105C22×C3×C4—98no—
106C22×C3×C4—96no—
107C22×C3×C4—91no—
108C22×C3×C4—912no—
109C22×C3×C4—96no—
110C22×Dic3—36no—
111C22×Dic3—38no—
112C22×Dic3—31no—
113C22×Dic3—36no—
114C22×Dic3—36no—
115C22×Dic3—36no—
116C22×Dic3—312no—
117C22×Dic3C22⋊C911yescharacteristic, direct product factor
118C22×Dic3C22×C914yescomplemented
119C22×C32—36no—
120C22×C32—31no—
121C22×C32—38no—
122C22×C32—36no—
123C22×C32—312no—
124C22×C32—36no—
125C22×C32—36no—
126C22×C32—33no—
127C22×C32—31no—
128C22×C32C4×A414yes—
129C22×C32C22×C3×C411yescharacteristic
130C22×C32C22×A413yes—
131C3×Dic3—34no—
132C3×Dic3C22×A414yes—
133C22×C9—43no—
134C22×C9—44no—
135C22×C9—83no—
136C22×C9—84no—
137C22⋊C9—21no—
138C22⋊C9C22×Dic311yescharacteristic, direct product factor
139C4×C9—124no—
140C25D3×C911yescharacteristic
141C23×C4—36no—
142C23×C4—91no—
143C23×C4—96no—
144C23×C4—91no—
145C3×C9—41noSylow
146C23×C3—26no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—33no—
150C23×C3—33no—
151C23×C3—312no—
152C23×C3—36no—
153C23×C3—312no—
154C23×C3—36no—
155C23×C3—32no—
156C23×C3—32no—
157C23×C3—312no—
158C23×C3—312no—
159C23×C3—33no—
160C23×C3—33no—
161C23×C3—34no—
162C23×C3—34no—
163C23×C3—33no—
164C23×C3—33no—
165C23×C3—34no—
166C23×C3—34no—
167C23×C3—64no—
168C23×C3—63no—
169C23×C3—64no—
170C23×C3—63no—
171C23×C3—63no—
172C23×C3—612no—
173C23×C3—66no—
174C23×C3—62no—
175C23×C3—612no—
176C23×C3C2×C4×C916yescomplemented
177C23×C3C2×C22⋊C911yescharacteristic
178C23×C3C23×C911yescharacteristic
179C23×C3C2×C3×Dic316yes—
180C23×C3D3×A411yescharacteristic
181C23×C3C22×C3×D311yescharacteristic
182C2×C3×C4—36no—
183C2×C3×C4—94no—
184C2×C3×C4—912no—
185C2×C3×C4—912no—
186C2×C3×C4—96no—
187C2×C3×C4—94no—
188C2×Dic3—312no—
189C2×Dic3—36no—
190C2×Dic3—312no—
191C2×Dic3—34no—
192C2×Dic3—34no—
193C2×Dic3C2×C22⋊C916yesdirect product factor
194C2×C32—36no—
195C2×C32—31no—
196C2×C32—31no—
197C2×C32C2×C4×A416yes—
198C2×C32C23×A411yescharacteristic
199C2×C9—46no—
200C2×C9—41no—
201C2×C9—86no—
202C2×C9—81no—
203C24—33no—
204C24—34no—
205C24—31no—
206C24C9×Dic314yescomplemented
207C24C2×D3×C913yes—
208C22×C4—31no—
209C22×C4—34no—
210C22×C4—96no—
211C22×C4—96no—
212C22×C4—98no—
213C22×C4—96no—
214C22×C4—91no—
215C22×C4—912no—
216C22×C4—96no—
217C22×C3—24no—
218C22×C3—23no—
219C22×C3—21no—
220C22×C3—36no—
221C22×C3—31no—
222C22×C3—36no—
223C22×C3—31no—
224C22×C3—36no—
225C22×C3—38no—
226C22×C3—36no—
227C22×C3—38no—
228C22×C3—36no—
229C22×C3—312no—
230C22×C3—36no—
231C22×C3—31no—
232C22×C3—33no—
233C22×C3—31no—
234C22×C3—33no—
235C22×C3—36no—
236C22×C3—312no—
237C22×C3—36no—
238C22×C3—61no—
239C22×C3—66no—
240C22×C3—63no—
241C22×C3—612no—
242C22×C3—66no—
243C22×C3—66no—
244C22×C3—61no—
245C22×C3—66no—
246C22×C3—68no—
247C22×C3C4×C22⋊C914yescomplemented
248C22×C3C22×C4×C911yescharacteristic, complemented
249C22×C3C22×C22⋊C913yes—
250C22×C3Dic3×A414yes—
251C22×C3C22×C3×Dic311yescharacteristic
252C22×C3C2×D3×A413yes—
253C3×C4—34no—
254C3×C4—94no—
255Dic3—34no—
256Dic3C22×C22⋊C914yesdirect product factor
257C32C22×C4×A411yescharacteristic
258C9—41no—
259C9—81no—
260C23—33no—
261C23—312no—
262C23—36no—
263C23—32no—
264C23—312no—
265C23—34no—
266C23—34no—
267C23—33no—
268C23—33no—
269C23C2×C9×Dic316yescomplemented
270C23D3×C22⋊C911yescharacteristic
271C23C22×D3×C911yescharacteristic
272C2×C4—36no—
273C2×C4—94no—
274C2×C4—912no—
275C2×C4—912no—
276C2×C4—96no—
277C2×C4—94no—
278C2×C3—21no—
279C2×C3—26no—
280C2×C3—31no—
281C2×C3—36no—
282C2×C3—31no—
283C2×C3—31no—
284C2×C3—36no—
285C2×C3—31no—
286C2×C3—61no—
287C2×C3—66no—
288C2×C3—61no—
289C2×C3C2×C4×C22⋊C916yescomplemented
290C2×C3C23×C22⋊C911yescharacteristic
291C2×C3C2×Dic3×A416yes—
292C2×C3C22×D3×A411yescharacteristic
293C22—36no—
294C22—31no—
295C22—36no—
296C22—38no—
297C22—31no—
298C22—33no—
299C22—36no—
300C22—312no—
301C22—36no—
302C22Dic3×C22⋊C914yesdirect product factor
303C22C22×C9×Dic311yescharacteristic, complemented
304C22C2×D3×C22⋊C913yes—
305C4—34no—
306C4—94no—
307C3—21no—
308C3C22×C4×C22⋊C911yescharacteristic, complemented
309C3C22×Dic3×A411yescharacteristic
310C2—31no—
311C2—36no—
312C2—31no—
313C2C2×Dic3×C22⋊C916yesdirect product factor
314C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
center, hypercenterC23×C3D3×A4
commutator, hypocenterC22×C3C22×C4×C9
FrattiniC2×C3C22×D3×A4
Fitting, socle, prime radicalC25×C32C2×C3
cosocleC23×C32C23×C3
LCSG ▷ C22×C3
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×C22⋊C9C2direct (N × H)
5. C22×C3×C22⋊C9C4semidirect
8. Dic3×C22⋊C9C22direct (N × H)
18. C2×C3×C22⋊C9C2×C4semidirect
22. C24×Dic3C9semidirect
39. C22×C22⋊C9Dic3direct (N × H)
45. C3×C22⋊C9C22×C4semidirect
57. C23×Dic3C2×C9semidirect
80. C2×C22⋊C9C2×Dic3direct (N × H)
97. C24×C3C4×C9semidirect
117. C22×Dic3C22⋊C9direct (N × H)
118. C22×Dic3C22×C9semidirect
138. C22⋊C9C22×Dic3direct (N × H)
176. C23×C3C2×C4×C9semidirect
193. C2×Dic3C2×C22⋊C9direct (N × H)
206. C24C9×Dic3semidirect
247. C22×C3C4×C22⋊C9semidirect
248. C22×C3C22×C4×C9semidirect
256. Dic3C22×C22⋊C9direct (N × H)
269. C23C2×C9×Dic3semidirect
289. C2×C3C2×C4×C22⋊C9semidirect
302. C22Dic3×C22⋊C9direct (N × H)
303. C22C22×C9×Dic3semidirect
308. C3C22×C4×C22⋊C9semidirect
313. C2C2×Dic3×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C22⋊C92
C2×Dic3×C22⋊C92
C24×C3×Dic33
C22×C4×C22⋊C93
C22×C9×Dic34