Finite Group Databasebeta

← C22×#88994 #13181528

Subgroups — C22×#88994 #13181528

Order 1728.

3112 conjugacy classes of subgroups; only the 341 normal subgroups are shown.

normal subgroups only

This group has 3112 conjugacy classes of subgroups — too many to draw. Only its 341 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
1C24×C3×C3⋊D3C211yesmaximal, characteristic
2C23×C3⋊#157C212yesmaximal, complemented
3C2×#88994C2112yesmaximal, direct product factor
4C22×C22⋊#157—31nomaximal
5C24×C33C411yescharacteristic, complemented
6C23×C3×C3⋊D3—28no—
7C23×C3×C3⋊D3C416yescomplemented
8C23×C3×C3⋊D3C411yescharacteristic, complemented
9C23×C3×C3⋊D3C2216yes—
10C23×C3×C3⋊D3C2211yescharacteristic
11C22×C3⋊#157—28no—
12C22×C3⋊#157C22112yescomplemented
13#88994C22116yesdirect product factor
14C24×C3⋊D3D311yescharacteristic
15C24×C3×D3—62no—
16C23×#157—32no—
17C2×C22⋊#157—312no—
18C23×C33—24no—
19C23×C33C2×C411yescharacteristic
20C23×C33C2×C416yescomplemented
21C22×C3×C3⋊D3—24no—
22C22×C3×C3⋊D3—224no—
23C22×C3×C3⋊D3—224no—
24C22×C3×C3⋊D3C2×C413yes—
25C22×C3×C3⋊D3C2×C4112yescomplemented
26C22×C3×C3⋊D3C2×C416yescomplemented
27C22×C3×C3⋊D3D418yes—
28C22×C3×C3⋊D3C2314yes—
29C22×C3×C3⋊D3C2313yes—
30C2×C3⋊#157—224no—
31C2×C3⋊#157C2318yescomplemented
32C22×#217—91nomaximal
33C24×C32—22no—
34C24×C32—42no—
35C24×C32Dic311yescharacteristic, complemented
36C23×C3⋊D3—28no—
37C23×C3⋊D3Dic316yescomplemented
38C23×C3⋊D3Dic311yescharacteristic, complemented
39C23×C3⋊D3C2×D316yes—
40C23×C3⋊D3C2×D311yescharacteristic
41C23×C3×D3—612no—
42C23×C3×D3—62no—
43C23×C3×D3—62no—
44C23×C3×D3—612no—
45C23×C3×D3—632no—
46C22×#157—312no—
47C22×#157—68no—
48C22⋊#157—316no—
49C22×C33—212no—
50C22×C33C22⋊C414yescomplemented
51C22×C33C22×C413yes—
52C22×C33C22×C414yescomplemented
53C2×C3×C3⋊D3—212no—
54C2×C3×C3⋊D3—224no—
55C2×C3×C3⋊D3—28no—
56C2×C3×C3⋊D3C22⋊C4112yescomplemented
57C2×C3×C3⋊D3C22×C413yes—
58C2×C3×C3⋊D3C22×C414yescomplemented
59C2×C3×C3⋊D3C2×D4112yes—
60C2×C3×C3⋊D3C2411yescharacteristic
61C3⋊#157—28no—
62C25×C3—91no—
63C24×D3—62no—
64C23×Dic3—92no—
65C2×#217—912no—
66C23×C32—24no—
67C23×C32—216no—
68C23×C32—22no—
69C23×C32—212no—
70C23×C32—416no—
71C23×C32—42no—
72C23×C32—412no—
73C23×C32C2×Dic311yescharacteristic
74C23×C32C2×Dic316yescomplemented
75C22×C3⋊D3—24no—
76C22×C3⋊D3—224no—
77C22×C3⋊D3—224no—
78C22×C3⋊D3C2×Dic313yes—
79C22×C3⋊D3C2×Dic3112yescomplemented
80C22×C3⋊D3C2×Dic316yescomplemented
81C22×C3⋊D3#6718yes—
82C22×C3⋊D3C22×D314yes—
83C22×C3⋊D3C22×D313yes—
84C22×C3×D3—66no—
85C22×C3×D3—612no—
86C22×C3×D3—624no—
87C22×C3×D3—616no—
88C22×C3×D3—66no—
89C22×C3×D3—68no—
90C22×C3×D3—696no—
91C22×C3×D3—696no—
92C22×C3×D3—616no—
93C2×#157—38no—
94C2×#157—624no—
95C22×C22⋊C4—271noSylow
96C2×C33—24no—
97C2×C33C2×C22⋊C416yescomplemented
98C2×C33C23×C411yescharacteristic
99C3×C3⋊D3—24no—
100C3×C3⋊D3C2×C22⋊C416yescomplemented
101C3×C3⋊D3C22×D412yes—
102C24×C3—22no—
103C24×C3—42no—
104C24×C3—91no—
105C24×C3—91no—
106C24×C3—96no—
107C24×C3—96no—
108C24×C3—188no—
109C24×C3#15711yescharacteristic, complemented
110C23×D3—62no—
111C23×D3—612no—
112C23×D3—62no—
113C23×D3—612no—
114C23×D3—632no—
115C22×Dic3—912no—
116C22×Dic3—188no—
117#217—916no—
118C22×C32—212no—
119C22×C32—248no—
120C22×C32—26no—
121C22×C32—28no—
122C22×C32—28no—
123C22×C32—448no—
124C22×C32—48no—
125C22×C32—46no—
126C22×C32—48no—
127C22×C32#21714yescomplemented
128C22×C32C22×Dic313yes—
129C22×C32C22×Dic314yescomplemented
130C2×C3⋊D3—212no—
131C2×C3⋊D3—224no—
132C2×C3⋊D3—28no—
133C2×C3⋊D3#217112yescomplemented
134C2×C3⋊D3C22×Dic313yes—
135C2×C3⋊D3C22×Dic314yescomplemented
136C2×C3⋊D3C2×#67112yes—
137C2×C3⋊D3C23×D311yescharacteristic
138C2×C3×D3—624no—
139C2×C3×D3—66no—
140C2×C3×D3—68no—
141C2×C3×D3—624no—
142C2×C3×D3—62no—
143C2×C3×D3—696no—
144C2×C3×D3—632no—
145C2×C3×D3—648no—
146#157—68no—
147C25—91no—
148C23×C4—272no—
149C2×C22⋊C4—2712no—
150C33C22×C22⋊C411yescharacteristic, complemented, Sylow
151C23×C3—216no—
152C23×C3—24no—
153C23×C3—22no—
154C23×C3—212no—
155C23×C3—416no—
156C23×C3—42no—
157C23×C3—412no—
158C23×C3—93no—
159C23×C3—93no—
160C23×C3—98no—
161C23×C3—96no—
162C23×C3—94no—
163C23×C3—912no—
164C23×C3—1824no—
165C23×C3—1824no—
166C23×C3—184no—
167C23×C3C2×#15711yescharacteristic
168C23×C3C2×#15716yescomplemented
169C22×D3—66no—
170C22×D3—612no—
171C22×D3—624no—
172C22×D3—616no—
173C22×D3—66no—
174C22×D3—68no—
175C22×D3—696no—
176C22×D3—696no—
177C22×D3—616no—
178C2×Dic3—98no—
179C2×Dic3—1824no—
180C2×C32—24no—
181C2×C32—216no—
182C2×C32—212no—
183C2×C32—22no—
184C2×C32—416no—
185C2×C32—412no—
186C2×C32—42no—
187C2×C32C2×#21716yescomplemented
188C2×C32C23×Dic311yescharacteristic
189C3⋊D3—24no—
190C3⋊D3C2×#21716yescomplemented
191C3⋊D3C22×#6712yes—
192C3×D3—612no—
193C3×D3—64no—
194C3×D3—616no—
195C24—91no—
196C24—91no—
197C24—96no—
198C24—96no—
199C24—188no—
200C24C3⋊#15711yescharacteristic, complemented
201C22×C4—2712no—
202C22×C4—548no—
203C22⋊C4—2716no—
204C22×C3—248no—
205C22×C3—212no—
206C22×C3—26no—
207C22×C3—28no—
208C22×C3—28no—
209C22×C3—448no—
210C22×C3—46no—
211C22×C3—48no—
212C22×C3—48no—
213C22×C3—912no—
214C22×C3—912no—
215C22×C3—91no—
216C22×C3—93no—
217C22×C3—94no—
218C22×C3—188no—
219C22×C3—1824no—
220C22×C3—1812no—
221C22×C3C22⋊#15714yescomplemented
222C22×C3C22×#15713yes—
223C22×C3C22×#15714yescomplemented
224C2×D3—624no—
225C2×D3—66no—
226C2×D3—68no—
227C2×D3—624no—
228C2×D3—62no—
229C2×D3—696no—
230C2×D3—632no—
231C2×D3—648no—
232Dic3—188no—
233C32—22no—
234C32—42no—
235C32C22×#21711yescharacteristic, complemented
236C23—24no—
237C23—93no—
238C23—93no—
239C23—98no—
240C23—96no—
241C23—94no—
242C23—912no—
243C23—1824no—
244C23—1824no—
245C23—184no—
246C23C2×C3⋊#15711yescharacteristic
247C23C2×C3⋊#15716yescomplemented
248C2×C4—278no—
249C2×C4—5424no—
250C2×C3—216no—
251C2×C3—24no—
252C2×C3—22no—
253C2×C3—212no—
254C2×C3—416no—
255C2×C3—42no—
256C2×C3—412no—
257C2×C3—92no—
258C2×C3—96no—
259C2×C3—184no—
260C2×C3C2×C22⋊#15716yescomplemented
261C2×C3C23×#15711yescharacteristic
262D3—612no—
263D3—64no—
264D3—616no—
265C22—212no—
266C22—91no—
267C22—93no—
268C22—912no—
269C22—912no—
270C22—94no—
271C22—188no—
272C22—1824no—
273C22—1812no—
274C22#8899414yesdirect product factor
275C22C22×C3⋊#15713yes—
276C22C22×C3⋊#15714yescomplemented
277C4—548no—
278C3—22no—
279C3—42no—
280C3C22×C22⋊#15711yescharacteristic, complemented
281C2—24no—
282C2—92no—
283C2—96no—
284C2—184no—
285C2C2×#8899416yesdirect product factor
286C2C23×C3⋊#15711yescharacteristic
subgroupG/H
centerC23C2×C3⋊#157
commutatorC2×C33C23×C4
FrattiniC2C23×C3⋊#157
FittingC24×C33C4
socleC23×C33C2×C4
prime radicalC24×C3×C3⋊D3C2
cosocleC2×C3×C3⋊D3C24
hypercenterC24C3⋊#157
hypocenterC33C22×C22⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×C3⋊#157C2semidirect
3. C2×#88994C2direct (N × H)
5. C24×C33C4semidirect
7. C23×C3×C3⋊D3C4semidirect
8. C23×C3×C3⋊D3C4semidirect
12. C22×C3⋊#157C22semidirect
13. #88994C22direct (N × H)
20. C23×C33C2×C4semidirect
25. C22×C3×C3⋊D3C2×C4semidirect
26. C22×C3×C3⋊D3C2×C4semidirect
31. C2×C3⋊#157C23semidirect
35. C24×C32Dic3semidirect
37. C23×C3⋊D3Dic3semidirect
38. C23×C3⋊D3Dic3semidirect
50. C22×C33C22⋊C4semidirect
52. C22×C33C22×C4semidirect
56. C2×C3×C3⋊D3C22⋊C4semidirect
58. C2×C3×C3⋊D3C22×C4semidirect
74. C23×C32C2×Dic3semidirect
79. C22×C3⋊D3C2×Dic3semidirect
80. C22×C3⋊D3C2×Dic3semidirect
97. C2×C33C2×C22⋊C4semidirect
100. C3×C3⋊D3C2×C22⋊C4semidirect
109. C24×C3#157semidirect
127. C22×C32#217semidirect
129. C22×C32C22×Dic3semidirect
133. C2×C3⋊D3#217semidirect
135. C2×C3⋊D3C22×Dic3semidirect
150. C33C22×C22⋊C4semidirect
168. C23×C3C2×#157semidirect
187. C2×C32C2×#217semidirect
190. C3⋊D3C2×#217semidirect
200. C24C3⋊#157semidirect
221. C22×C3C22⋊#157semidirect
223. C22×C3C22×#157semidirect
235. C32C22×#217semidirect
247. C23C2×C3⋊#157semidirect
260. C2×C3C2×C22⋊#157semidirect
274. C22#88994direct (N × H)
276. C22C22×C3⋊#157semidirect
280. C3C22×C22⋊#157semidirect
285. C2C2×#88994direct (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×C3⋊#1572
C2×#889942
C22×C22⋊#1573
C22×#2179