Finite Group Databasebeta

← #13181517

Subgroups — #13181517

Order 1728.

1256 conjugacy classes of subgroups; only the 225 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 225 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
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
2C2×#88994C214yesmaximal, complemented
3D4×C3⋊#157C218yesmaximal, complemented
4#11724514C212yesmaximal, complemented
5#10596642—31nomaximal
6C23×C3×C3⋊D3C2212yes—
7C22×C3⋊#157C2218yescomplemented
8C2×C33×D4C411yescharacteristic, complemented
9C2×C3×#638C412yescomplemented
10C3×D4×C3⋊D3C414yescomplemented
11C3×D4×C3⋊D3C2214yes—
12C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14#88994C2214yes—
15#88994C2218yescomplemented
16C4⋊(C3⋊#157)C2214yescomplemented
17C4×C3⋊#157C2214yescomplemented
18C2×D4×C3⋊D3D311yescharacteristic
19C2×C3×D3×D4—62no—
20C2×C22⋊#157—34no—
21D4×#157—38no—
22#64286—32no—
23C23×C33C2×C412yescomplemented
24C22×C3×C3⋊D3—44no—
25C22×C3×C3⋊D3C2×C412yescomplemented
26C22×C3×C3⋊D3C2×C414yescomplemented
27C22×C3×C3⋊D3C2314yes—
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3⋊#157—48no—
30C2×C3⋊#157C2318yes—
31C33×D4C2×C414yescomplemented
32C2×C33×C4C2×C411yescharacteristic, complemented
33C3×#638C2×C418yescomplemented
34C2×C3×C3⋊Dic3C2×C411yescharacteristic, complemented
35C3×C4⋊(C3⋊D3)C2×C414yescomplemented
36C3×C4×C3⋊D3C2×C412yescomplemented
37C3×C4×C3⋊D3C2312yes—
38#5798—91nomaximal
39C23×C3⋊D3C2×D312yes—
40C23×C3×D3—64no—
41C22×#157—38no—
42C2×C32×D4—22no—
43C2×C32×D4—42no—
44C2×C32×D4Dic311yescharacteristic, complemented
45C2×#638Dic312yescomplemented
46D4×C3⋊D3Dic314yescomplemented
47D4×C3⋊D3C2×D314yes—
48C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C2×C3×#67—64no—
51C3×D3×D4—68no—
52C3×D3×D4—68no—
53C2×C3×D12—62no—
54C2×C3×C4×D3—62no—
55C22⋊#157—34no—
56C22⋊#157—38no—
57C4⋊#157—34no—
58C4×#157—34no—
59C22×C33—42no—
60C22×C33C22×C411yescharacteristic
61C22×C33C22×C414yes—
62C2×C3×C3⋊D3—22no—
63C2×C3×C3⋊D3—28no—
64C2×C3×C3⋊D3—44no—
65C2×C3×C3⋊D3—42no—
66C2×C3×C3⋊D3C22×C411yescharacteristic
67C2×C3×C3⋊D3C22×C414yes—
68C2×C3×C3⋊D3C2411yescharacteristic
69C3⋊#157—28no—
70C33×C4C22×C412yes—
71C3×C3⋊Dic3C22×C412yes—
72C22×C3×D4—91no—
73C2×D3×D4—62no—
74C2×#217—94no—
75D4×Dic3—98no—
76#926—92no—
77C23×C32—24no—
78C23×C32—44no—
79C23×C32C2×Dic312yescomplemented
80C22×C3⋊D3—44no—
81C22×C3⋊D3C2×Dic312yescomplemented
82C22×C3⋊D3C2×Dic314yescomplemented
83C22×C3⋊D3C22×D314yes—
84C22×C3⋊D3C22×D311yescharacteristic
85C22×C3×D3—62no—
86C22×C3×D3—64no—
87C22×C3×D3—68no—
88C22×C3×D3—68no—
89C22×C3×D3—1216no—
90C2×#157—38no—
91C2×#157—128no—
92C32×D4—28no—
93C32×D4—48no—
94C32×D4C2×Dic314yescomplemented
95C2×C32×C4—22no—
96C2×C32×C4—42no—
97C2×C32×C4C2×Dic311yescharacteristic, complemented
98#638C2×Dic318yescomplemented
99C2×C3⋊Dic3C2×Dic311yescharacteristic, complemented
100C4⋊(C3⋊D3)C2×Dic314yescomplemented
101C4×C3⋊D3C2×Dic312yescomplemented
102C4×C3⋊D3C22×D312yes—
103C3×#67—616no—
104C2×C3×Dic3—62no—
105C3×D12—68no—
106C3×C4×D3—64no—
107C3×C4×D3—64no—
108#518—271noSylow
109C2×C33—21no—
110C2×C33—24no—
111C2×C33C23×C411yescharacteristic
112C3×C3⋊D3—21no—
113C3×C3⋊D3—24no—
114C3×C3⋊D3H(5,𝔽2)12yes—
115C24×C3—92no—
116C23×D3—64no—
117C2×C3×D4—22no—
118C2×C3×D4—42no—
119C2×C3×D4—92no—
120C2×C3×D4—94no—
121C2×C3×D4—94no—
122C2×C3×D4—91no—
123C2×C3×D4#15711yescharacteristic, complemented
124C22×C3×C4—91no—
125C2×#67—64no—
126C22×Dic3—98no—
127D3×D4—68no—
128D3×D4—68no—
129C2×D12—62no—
130C2×C4×D3—62no—
131#217—94no—
132#217—98no—
133C4⋊Dic3—94no—
134C4×Dic3—94no—
135C22×C32—22no—
136C22×C32—28no—
137C22×C32—42no—
138C22×C32—48no—
139C22×C32—42no—
140C22×C32—48no—
141C22×C32—88no—
142C22×C32C22×Dic311yescharacteristic
143C22×C32C22×Dic314yes—
144C2×C3⋊D3—22no—
145C2×C3⋊D3—28no—
146C2×C3⋊D3—44no—
147C2×C3⋊D3—42no—
148C2×C3⋊D3C22×Dic311yescharacteristic
149C2×C3⋊D3C22×Dic314yes—
150C2×C3⋊D3C23×D311yescharacteristic
151C2×C3×D3—62no—
152C2×C3×D3—62no—
153C2×C3×D3—68no—
154C2×C3×D3—68no—
155C2×C3×D3—1216no—
156C2×C3×D3—1216no—
157C2×C3×D3—128no—
158#157—68no—
159C32×C4—24no—
160C32×C4—44no—
161C32×C4C22×Dic312yes—
162C3⋊Dic3C22×Dic312yes—
163C3×Dic3—64no—
164C22×D4—91no—
165C4×D4—278no—
166#117—272no—
167C2×C22⋊C4—274no—
168C33#51811yescharacteristic, complemented, Sylow
169C23×C3—24no—
170C23×C3—44no—
171C23×C3—91no—
172C23×C3—94no—
173C23×C3—94no—
174C23×C3—92no—
175C23×C3—364no—
176C23×C3C2×#15712yescomplemented
177C22×D3—62no—
178C22×D3—64no—
179C22×D3—68no—
180C22×D3—68no—
181C22×D3—1216no—
182C3×D4—28no—
183C3×D4—48no—
184C3×D4—98no—
185C3×D4—94no—
186C3×D4C2×#15714yescomplemented
187C2×C3×C4—22no—
188C2×C3×C4—42no—
189C2×C3×C4—91no—
190C2×C3×C4—92no—
191C2×C3×C4—92no—
192C2×C3×C4C2×#15711yescharacteristic, complemented
193#67—616no—
194C2×Dic3—62no—
195C2×Dic3—98no—
196C2×Dic3—368no—
197D12—68no—
198C4×D3—64no—
199C4×D3—64no—
200C2×C32—21no—
201C2×C32—24no—
202C2×C32—22no—
203C2×C32—24no—
204C2×C32—44no—
205C2×C32—42no—
206C2×C32—48no—
207C2×C32—88no—
208C2×C32C23×Dic311yescharacteristic
209C3⋊D3—21no—
210C3⋊D3—24no—
211C3⋊D3#100412yes—
212C3×D3—64no—
213C3×D3—64no—
214C3×D3—128no—
215C24—92no—
216C2×D4—92no—
217C2×D4—94no—
218C2×D4—94no—
219C2×D4—91no—
220C2×D4C3⋊#15711yescharacteristic, complemented
221C22×C4—91no—
222C22×C4—278no—
223C4⋊C4—274no—
224C22⋊C4—274no—
225C22⋊C4—278no—
226C42—274no—
227C22×C3—22no—
228C22×C3—28no—
229C22×C3—42no—
230C22×C3—48no—
231C22×C3—42no—
232C22×C3—48no—
233C22×C3—88no—
234C22×C3—91no—
235C22×C3—91no—
236C22×C3—94no—
237C22×C3—188no—
238C22×C3—182no—
239C22×C3—362no—
240C22×C3—364no—
241C22×C3C22×#15711yescharacteristic
242C22×C3C22×#15714yes—
243C2×D3—62no—
244C2×D3—62no—
245C2×D3—68no—
246C2×D3—68no—
247C2×D3—1216no—
248C2×D3—1216no—
249C2×D3—128no—
250C3×C4—24no—
251C3×C4—44no—
252C3×C4—92no—
253C3×C4C22×#15712yes—
254Dic3—64no—
255Dic3—188no—
256C32—22no—
257C32—42no—
258C32#579811yescharacteristic, complemented
259C23—91no—
260C23—94no—
261C23—94no—
262C23—92no—
263C23—364no—
264C23C2×C3⋊#15712yescomplemented
265D4—98no—
266D4—94no—
267D4C2×C3⋊#15714yescomplemented
268C2×C4—91no—
269C2×C4—92no—
270C2×C4—92no—
271C2×C4—278no—
272C2×C4—1088no—
273C2×C4C2×C3⋊#15711yescharacteristic, complemented
274C2×C3—24no—
275C2×C3—21no—
276C2×C3—24no—
277C2×C3—22no—
278C2×C3—48no—
279C2×C3—44no—
280C2×C3—42no—
281C2×C3—88no—
282C2×C3—92no—
283C2×C3—181no—
284C2×C3—184no—
285C2×C3C23×#15711yescharacteristic
286D3—64no—
287D3—64no—
288D3—128no—
289C22—42no—
290C22—91no—
291C22—91no—
292C22—94no—
293C22—188no—
294C22—182no—
295C22—362no—
296C22—364no—
297C22C22×C3⋊#15711yescharacteristic
298C22C22×C3⋊#15714yes—
299C4—92no—
300C4—548no—
301C4C22×C3⋊#15712yes—
302C3—22no—
303C3—42no—
304C3#1059664211yescharacteristic, complemented
305C2—21no—
306C2—24no—
307C2—92no—
308C2—181no—
309C2—184no—
310C2C23×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC2C23×C3⋊#157
commutator, socleC2×C33C23×C4
FittingC2×C33×D4C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC2×C3×C3⋊D3C24
hypercenterC2×D4C3⋊#157
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88994C2semidirect
3. D4×C3⋊#157C2semidirect
4. #11724514C2semidirect
7. C22×C3⋊#157C22semidirect
8. C2×C33×D4C4semidirect
9. C2×C3×#638C4semidirect
10. C3×D4×C3⋊D3C4semidirect
12. C2×C3×C4⋊(C3⋊D3)C4semidirect
15. #88994C22semidirect
16. C4⋊(C3⋊#157)C22semidirect
17. C4×C3⋊#157C22semidirect
23. C23×C33C2×C4semidirect
25. C22×C3×C3⋊D3C2×C4semidirect
26. C22×C3×C3⋊D3C2×C4semidirect
31. C33×D4C2×C4semidirect
32. C2×C33×C4C2×C4semidirect
33. C3×#638C2×C4semidirect
34. C2×C3×C3⋊Dic3C2×C4semidirect
35. C3×C4⋊(C3⋊D3)C2×C4semidirect
36. C3×C4×C3⋊D3C2×C4semidirect
44. C2×C32×D4Dic3semidirect
45. C2×#638Dic3semidirect
46. D4×C3⋊D3Dic3semidirect
48. C2×C4⋊(C3⋊D3)Dic3semidirect
79. C23×C32C2×Dic3semidirect
81. C22×C3⋊D3C2×Dic3semidirect
82. C22×C3⋊D3C2×Dic3semidirect
94. C32×D4C2×Dic3semidirect
97. C2×C32×C4C2×Dic3semidirect
98. #638C2×Dic3semidirect
99. C2×C3⋊Dic3C2×Dic3semidirect
100. C4⋊(C3⋊D3)C2×Dic3semidirect
101. C4×C3⋊D3C2×Dic3semidirect
123. C2×C3×D4#157semidirect
168. C33#518semidirect
176. C23×C3C2×#157semidirect
186. C3×D4C2×#157semidirect
192. C2×C3×C4C2×#157semidirect
220. C2×D4C3⋊#157semidirect
258. C32#5798semidirect
264. C23C2×C3⋊#157semidirect
267. D4C2×C3⋊#157semidirect
273. C2×C4C2×C3⋊#157semidirect
304. C3#10596642semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×#889942
D4×C3⋊#1572
#117245142
#105966423
#57989