Finite Group Databasebeta

← #82607

Subgroups — #82607

Order 384.

1134 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 165 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×#67C211yesmaximal, characteristic, complemented
2C2×C3×#120C211yesmaximal, characteristic, complemented
3C2×#937C211yesmaximal, characteristic, complemented
4C2×#935C211yesmaximal, characteristic, complemented
5#5247C211yesmaximal, characteristic, complemented
6#5223C214yesmaximal, complemented
7#4959C214yesmaximal, complemented
8#4955C211yesmaximal, characteristic, complemented
9#4943C211yesmaximal, characteristic, complemented
10#2613—31nomaximal, Sylow
11C25×C3C2211yescharacteristic
12C24×D3C2211yescharacteristic
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yescomplemented
15C22×#67—24no—
16C22×#67—24no—
17C22×#67—24no—
18C22×#67C2214yes—
19C22×#67C2212yescomplemented
20C23×Dic3C2211yescharacteristic
21C3×#120—24no—
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2212yescomplemented
24C2×#217C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2212yescomplemented
27C2×#217C2214yescomplemented
28#937—24no—
29#935—24no—
30C2×(C2×D3)⋊C4C2212yescomplemented
31C2×(C2×D3)⋊C4C2214yescomplemented
32C2×Dic3⋊C4C2212yescomplemented
33C3⋊#95C2214yescomplemented
34C23×D4—31no—
35C2×#123—31no—
36C2×#120—31no—
37C2×#120D311yescharacteristic, complemented
38#394—34no—
39#392—34no—
40#381—31no—
41#379—31no—
42#379—31no—
43C24×C3—22no—
44C24×C3—22no—
45C24×C3—44no—
46C24×C3D414yescomplemented
47C24×C3C2311yescharacteristic
48C24×C3C2312yes—
49C24×C3C2311yescharacteristic
50C23×D3—22no—
51C23×D3—22no—
52C23×D3—44no—
53C23×D3D414yescomplemented
54C23×D3C2312yes—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4—24no—
59C2×C3×D4—44no—
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C2×#67—232no—
63C2×#67—28no—
64C2×#67—28no—
65C2×#67—24no—
66C2×#67—28no—
67C2×#67—44no—
68C22×Dic3—22no—
69C22×Dic3—22no—
70C22×Dic3D414yescomplemented
71C22×Dic3C2312yes—
72C22×Dic3C2312yes—
73C22×Dic3C2312yes—
74C3×C22⋊C4—22no—
75C3×C22⋊C4—24no—
76#217—22no—
77#217—22no—
78#217—24no—
79#217—44no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—44no—
82Dic3⋊C4—24no—
83C25—31no—
84C25C2×D311yescharacteristic
85C22×D4—34no—
86C22×D4—32no—
87C22×D4—64no—
88C22×D4—64no—
89C22×D4—64no—
90C22×D4C2×D311yescharacteristic
91C22×D4C2×D312yescomplemented
92C23×C4—31no—
93#123—64no—
94#120—24no—
95#120—64no—
96C2×C4⋊C4—32no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—34no—
102C2×C22⋊C4—34no—
103C2×C22⋊C4C2×D311yescharacteristic
104C2×C22⋊C4C2×D312yescomplemented
105#95—34no—
106C23×C3—24no—
107C23×C3—28no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3—24no—
111C23×C3—24no—
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—24no—
116C23×C3—48no—
117C23×C3—48no—
118C23×C3—48no—
119C23×C3C2×D412yes—
120C23×C3C2×D412yes—
121C23×C3C2×D414yes—
122C23×C3C2×D412yes—
123C23×C3C2411yescharacteristic
124C22×D3—28no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—24no—
128C22×D3—24no—
129C22×D3—48no—
130C22×D3—48no—
131C22×D3—48no—
132C22×D3C2×D414yes—
133C3×D4—48no—
134C3×D4—44no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—24no—
140#67—232no—
141#67—48no—
142C2×Dic3—28no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—24no—
148C2×Dic3—24no—
149C2×Dic3C2×D414yes—
150C24—22no—
151C24—22no—
152C24—32no—
153C24—34no—
154C24—44no—
155C24—62no—
156C24—62no—
157C24—124no—
158C24#6714yescomplemented
159C24C22×D311yescharacteristic
160C24C22×D312yes—
161C24C22×D311yescharacteristic
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—24no—
166C2×D4—44no—
167C2×D4—68no—
168C2×D4—68no—
169C2×D4—632no—
170C2×D4—68no—
171C2×D4—64no—
172C2×D4—124no—
173C22×C4—34no—
174C22×C4—32no—
175C22×C4—32no—
176C22×C4—32no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4C22×D312yes—
180C22×C4C22×D311yescharacteristic
181C4⋊C4—64no—
182C22⋊C4—22no—
183C22⋊C4—24no—
184C22⋊C4—64no—
185C22⋊C4—64no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C22⋊C4—124no—
189C22⋊C4—124no—
190C22×C3—24no—
191C22×C3—24no—
192C22×C3—24no—
193C22×C3—24no—
194C22×C3—24no—
195C22×C3—24no—
196C22×C3—416no—
197C22×C3—44no—
198C22×C3—44no—
199C22×C3—41no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3#12014yes—
204C22×C3C22×D411yescharacteristic
205C22×C3C22×D411yescharacteristic
206C22×C3C22×D411yescharacteristic
207C22×C3H(5,𝔽2)14yes—
208C2×D3—24no—
209C2×D3—24no—
210C2×D3—24no—
211C2×D3—416no—
212C2×D3—44no—
213C2×D3—44no—
214C3×C4—41no—
215C3×C4—42no—
216Dic3—24no—
217Dic3—42no—
218Dic3—42no—
219C23—24no—
220C23—28no—
221C23—22no—
222C23—22no—
223C23—24no—
224C23—24no—
225C23—21no—
226C23—21no—
227C23—24no—
228C23—21no—
229C23—34no—
230C23—48no—
231C23—48no—
232C23—48no—
233C23—68no—
234C23—62no—
235C23—64no—
236C23—64no—
237C23—62no—
238C23—128no—
239C23—128no—
240C23—128no—
241C23D3×D412yes—
242C23D3×D412yes—
243C23C2×#6714yes—
244C23C2×#6712yes—
245C23C23×D311yescharacteristic
246D4—48no—
247D4—44no—
248D4—632no—
249D4—128no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—24no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—34no—
256C2×C4—68no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—64no—
262C2×C4—64no—
263C2×C3—24no—
264C2×C3—44no—
265C2×C3—81no—
266C2×C3C2×#12011yescharacteristic
267C2×C3#53412yes—
268C2×C3#53412yes—
269C2×C3#53512yes—
270D3—44no—
271C22—24no—
272C22—24no—
273C22—24no—
274C22—24no—
275C22—24no—
276C22—24no—
277C22—416no—
278C22—44no—
279C22—44no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—64no—
285C22—64no—
286C22—64no—
287C22—1216no—
288C22—124no—
289C22—124no—
290C22#93714yes—
291C22C2×D3×D411yescharacteristic
292C22C2×D3×D411yescharacteristic
293C22#100414yes—
294C22C22×#6711yescharacteristic
295C4—41no—
296C4—42no—
297C4—64no—
298C4—122no—
299C4—122no—
300C3#261311yescharacteristic, complemented, Sylow
301C2—24no—
302C2—44no—
303C2—81no—
304C2—124no—
305C2#558912yes—
306C2#559212yes—
307C2C2×#93711yescharacteristic
308C2#580412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#120C2
hypercenterC2×#120D3
hypocenterC3#2613
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C23×#67C2semidirect
2. C2×C3×#120C2semidirect
3. C2×#937C2semidirect
4. C2×#935C2semidirect
5. #5247C2semidirect
6. #5223C2semidirect
7. #4959C2semidirect
8. #4955C2semidirect
9. #4943C2semidirect
14. C22×C3×D4C22semidirect
19. C22×#67C22semidirect
23. C2×C3×C22⋊C4C22semidirect
26. C2×#217C22semidirect
27. C2×#217C22semidirect
30. C2×(C2×D3)⋊C4C22semidirect
31. C2×(C2×D3)⋊C4C22semidirect
32. C2×Dic3⋊C4C22semidirect
33. C3⋊#95C22semidirect
37. C2×#120D3semidirect
46. C24×C3D4semidirect
53. C23×D3D4semidirect
70. C22×Dic3D4semidirect
91. C22×D4C2×D3semidirect
104. C2×C22⋊C4C2×D3semidirect
158. C24#67semidirect
300. C3#2613semidirect

Maximal subgroups

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

subgroupindex
C23×#672
C2×C3×#1202
C2×#9372
C2×#9352
#52472
#52232
#49592
#49552
#49432
#26133