Finite Group Databasebeta

← #74016

Subgroups — #74016

Order 384.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 189 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×D12C211yesmaximal, characteristic, complemented
2C2×C4×#67C211yesmaximal, characteristic, complemented
3C2×#895C212yesmaximal, complemented
4C2×#882C212yesmaximal, complemented
5C3×#380C211yesmaximal, characteristic, complemented
6#4988C212yesmaximal, complemented
7#4955C212yesmaximal, complemented
8#4672C212yesmaximal, complemented
9#4671C212yesmaximal, complemented
10#2408—31nomaximal, Sylow
11C24×D3C2212yes—
12C23×C3×C4C2211yescharacteristic
13C22×#67C2211yescharacteristic
14C22×D12—24no—
15C22×D12—28no—
16C22×D12C2212yes—
17C22×D12C2212yes—
18C22×D12C2214yescomplemented
19C22×C4×D3C2211yescharacteristic, complemented
20C2×C3×C4⋊C4C2212yescomplemented
21C2×C3×C22⋊C4C2212yes—
22C2×#217C2211yescharacteristic
23C4×#67—24no—
24C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
25C2×(C2×D3)⋊C4C2214yescomplemented
26C2×(C2×D3)⋊C4C2214yescomplemented
27C2×(C2×D3)⋊C4C2214yescomplemented
28C2×Dic3⋊C4C2211yescharacteristic, complemented
29C2×C4×Dic3C2211yescharacteristic, complemented
30#895—28no—
31#882—28no—
32C3×#95C2212yescomplemented
33C23×D4—31no—
34C2×#121—32no—
35C2×#120—32no—
36C2×C4×D4—31no—
37#394—32no—
38#392—32no—
39#390—32no—
40#380D311yescharacteristic, complemented
41#379—32no—
42C24×C3C2311yescharacteristic
43C23×D3—24no—
44C23×D3—28no—
45C23×D3—24no—
46C23×D3—24no—
47C23×D3—44no—
48C23×D3C2314yes—
49C23×D3C2311yescharacteristic
50C22×C3×C4—24no—
51C22×C3×C4D414yescomplemented
52C22×C3×C4C2314yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C2×#67—22no—
56C2×#67—22no—
57C2×#67D414yescomplemented
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C2×D12—232no—
62C2×D12—28no—
63C2×D12—24no—
64C2×D12—24no—
65C2×D12—24no—
66C2×D12—28no—
67C2×D12—28no—
68C2×D12—44no—
69C2×D12D418yescomplemented
70C2×C4×D3—24no—
71C2×C4×D3—22no—
72C3×C4⋊C4—24no—
73C3×C22⋊C4—24no—
74#217—22no—
75(C2×D3)⋊C4—28no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—28no—
78(C2×D3)⋊C4—44no—
79Dic3⋊C4—22no—
80C4×Dic3—22no—
81C25—32no—
82C22×D4—31no—
83C22×D4—32no—
84C22×D4—32no—
85C22×D4—34no—
86C22×D4—68no—
87C22×D4—64no—
88C23×C4—31no—
89C23×C4C2×D311yescharacteristic
90#121—68no—
91#120—68no—
92C4×D4—64no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4C2×D312yescomplemented
95C2×C22⋊C4—31no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—34no—
98C2×C22⋊C4—34no—
99C2×C22⋊C4—34no—
100C2×C22⋊C4C2×D312yes—
101C2×C42—31no—
102#95C2×D312yescomplemented
103C23×C3—22no—
104C23×C3—22no—
105C23×C3C2×D412yes—
106C23×C3C2×D412yes—
107C23×C3#4112yes—
108C23×C3C2411yescharacteristic
109C22×D3—216no—
110C22×D3—21no—
111C22×D3—24no—
112C22×D3—22no—
113C22×D3—24no—
114C22×D3—24no—
115C22×D3—21no—
116C22×D3—24no—
117C22×D3—24no—
118C22×D3—22no—
119C22×D3—24no—
120C22×D3—416no—
121C22×D3—416no—
122C22×D3—48no—
123C22×D3—48no—
124C22×D3C2×D418yes—
125C22×D3C2×D412yescomplemented
126C2×C3×C4—22no—
127C2×C3×C4—28no—
128C2×C3×C4—24no—
129C2×C3×C4—24no—
130C2×C3×C4—24no—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D414yes—
134#67—28no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3C2×D412yescomplemented
142D12—216no—
143D12—216no—
144D12—416no—
145C4×D3—44no—
146C24—34no—
147C24—31no—
148C24—68no—
149C24—64no—
150C24—64no—
151C24—64no—
152C24—124no—
153C24C22×D311yescharacteristic
154C2×D4—34no—
155C2×D4—38no—
156C2×D4—68no—
157C2×D4—632no—
158C2×D4—68no—
159C2×D4—68no—
160C2×D4—62no—
161C2×D4—62no—
162C2×D4—64no—
163C2×D4—64no—
164C2×D4—64no—
165C2×D4—124no—
166C22×C4—24no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—62no—
171C22×C4—64no—
172C22×C4D1214yescomplemented
173C22×C4C22×D314yes—
174C22×C4C22×D311yescharacteristic
175C22×C4C22×D311yescharacteristic
176C4⋊C4—24no—
177C4⋊C4—62no—
178C22⋊C4—24no—
179C22⋊C4—68no—
180C22⋊C4—68no—
181C22⋊C4—62no—
182C22⋊C4—62no—
183C22⋊C4—124no—
184C42—62no—
185C22×C3—22no—
186C22×C3—22no—
187C22×C3—24no—
188C22×C3—24no—
189C22×C3#12114yescomplemented
190C22×C3C22×D411yescharacteristic
191C22×C3C22×D412yes—
192C22×C3C22×D411yescharacteristic
193C22×C3C2×#4111yescharacteristic
194C22×C3H(5,𝔽2)12yes—
195C2×D3—28no—
196C2×D3—22no—
197C2×D3—24no—
198C2×D3—28no—
199C2×D3—28no—
200C2×D3—432no—
201C2×D3—41no—
202C2×D3—44no—
203C2×D3—41no—
204C2×D3—44no—
205C2×D3—44no—
206C2×D3—42no—
207C2×D3—44no—
208C3×C4—22no—
209C3×C4—44no—
210C3×C4#12014yescomplemented
211Dic3—22no—
212Dic3—41no—
213Dic3—41no—
214C23—22no—
215C23—22no—
216C23—38no—
217C23—32no—
218C23—616no—
219C23—61no—
220C23—64no—
221C23—64no—
222C23—62no—
223C23—64no—
224C23—61no—
225C23—64no—
226C23—64no—
227C23—62no—
228C23—64no—
229C23—1216no—
230C23—1216no—
231C23—128no—
232C23—128no—
233C23C2×D1212yes—
234C23D3×D412yes—
235C23#23912yes—
236C23C23×D311yescharacteristic
237D4—616no—
238D4—68no—
239D4—616no—
240D4—1216no—
241C2×C4—22no—
242C2×C4—28no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—32no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—124no—
254C2×C4C2×D1212yes—
255C2×C4C2×D1212yes—
256C2×C4D3×D414yes—
257C2×C3—24no—
258C2×C3C2×#12011yescharacteristic
259C2×C3C2×#12111yescharacteristic
260C2×C3#53511yescharacteristic
261C2×C3D4212yes—
262C2×C3#54612yes—
263D3—42no—
264D3—48no—
265C22—22no—
266C22—22no—
267C22—24no—
268C22—24no—
269C22—68no—
270C22—62no—
271C22—64no—
272C22—68no—
273C22—68no—
274C22—1232no—
275C22—121no—
276C22—124no—
277C22—124no—
278C22—121no—
279C22—124no—
280C22—124no—
281C22—122no—
282C22#89514yescomplemented
283C22C22×D1211yescharacteristic
284C22C2×D3×D412yes—
285C22C2×D3×D411yescharacteristic
286C22C2×#23911yescharacteristic
287C22#100912yes—
288C4—22no—
289C4—44no—
290C4—62no—
291C4—121no—
292C4—121no—
293C4#88214yescomplemented
294C3#240811yescharacteristic, complemented, Sylow
295C2—24no—
296C2—128no—
297C2—122no—
298C2C2×#88211yescharacteristic
299C2C2×#89511yescharacteristic
300C2#552511yescharacteristic
301C2#560912yes—
302C2#563012yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#380C2
hypercenter#380D3
hypocenterC3#2408
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×D12C2semidirect
2. C2×C4×#67C2semidirect
3. C2×#895C2semidirect
4. C2×#882C2semidirect
5. C3×#380C2semidirect
6. #4988C2semidirect
7. #4955C2semidirect
8. #4672C2semidirect
9. #4671C2semidirect
18. C22×D12C22semidirect
19. C22×C4×D3C22semidirect
20. C2×C3×C4⋊C4C22semidirect
24. C2×(C2×D3)⋊C4C22semidirect
25. C2×(C2×D3)⋊C4C22semidirect
26. C2×(C2×D3)⋊C4C22semidirect
27. C2×(C2×D3)⋊C4C22semidirect
28. C2×Dic3⋊C4C22semidirect
29. C2×C4×Dic3C22semidirect
32. C3×#95C22semidirect
40. #380D3semidirect
51. C22×C3×C4D4semidirect
57. C2×#67D4semidirect
69. C2×D12D4semidirect
94. C2×C4⋊C4C2×D3semidirect
102. #95C2×D3semidirect
125. C22×D3C2×D4semidirect
141. C2×Dic3C2×D4semidirect
172. C22×C4D12semidirect
189. C22×C3#121semidirect
210. C3×C4#120semidirect
282. C22#895semidirect
293. C4#882semidirect
294. C3#2408semidirect

Maximal subgroups

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

subgroupindex
C23×D122
C2×C4×#672
C2×#8952
C2×#8822
C3×#3802
#49882
#49552
#46722
#46712
#24083