Finite Group Databasebeta

← #74384

Subgroups — #74384

Order 384.

632 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Dic3⋊C4C211yesmaximal, characteristic, complemented
2C2×#886C211yesmaximal, characteristic
3C2×#67⋊C4C211yesmaximal, characteristic
4C3×#385C211yesmaximal, characteristic, complemented
5#5213C211yesmaximal, characteristic
6#4989C212yesmaximal, complemented
7#4972C212yesmaximal, complemented
8#4943C211yesmaximal, characteristic, complemented
9Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
10#4669C212yesmaximal, complemented
11#4654C212yesmaximal, complemented
12#2322—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C23×Dic3C2211yescharacteristic
17C22×C4×D3C2211yescharacteristic
18C2×C3×C4⋊C4C2212yes—
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×#217C2211yescharacteristic
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×C4⋊Dic3C2212yes—
26C2×Dic3⋊C4—24no—
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×C4×Dic3C2211yescharacteristic
31C2×C4×Dic3C2212yes—
32#886C418yescomplemented
33#67⋊C4—24no—
34C3×#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×#123—31no—
40C2×C4×D4—31no—
41C22×C4⋊C4—31no—
42#389—32no—
43#388—32no—
44#387—32no—
45#386—31no—
46#385—32no—
47#385D311yescharacteristic, complemented
48#381—31no—
49C4×C22⋊C4—31no—
50C24×C3C2311yescharacteristic
51C23×D3C2311yescharacteristic
52C22×C3×C4—22no—
53C22×C3×C4—42no—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×C3×C4C2312yes—
58C2×#67—22no—
59C2×#67—22no—
60C2×#67C2×C414yes—
61C22×Dic3—22no—
62C22×Dic3—22no—
63C22×Dic3—42no—
64C22×Dic3C2×C414yescomplemented
65C22×Dic3D414yes—
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2311yescharacteristic
71C2×C4×D3—22no—
72C2×C4×D3—42no—
73C3×C4⋊C4—42no—
74C3×C22⋊C4—22no—
75C3×C22⋊C4C2×C414yescomplemented
76#217—22no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—42no—
79(C2×D3)⋊C4C2×C418yescomplemented
80C4⋊Dic3C2×C418yescomplemented
81Dic3⋊C4—22no—
82Dic3⋊C4—24no—
83Dic3⋊C4—24no—
84C4×Dic3—22no—
85C4×Dic3—42no—
86C22×D4—31no—
87C23×C4—31no—
88C23×C4—31no—
89C23×C4—31no—
90C23×C4C2×D311yescharacteristic
91#123—38no—
92C4×D4—64no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4—64no—
98C2×C4⋊C4C2×D312yes—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—31no—
101C2×C22⋊C4—31no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4C2×D311yescharacteristic
104C2×C22⋊C4C2×D311yescharacteristic
105C2×C42—31no—
106C2×C42—32no—
107#95—32no—
108#95—32no—
109#95—32no—
110#95—32no—
111#95C2×D312yes—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3C22×C412yes—
115C23×C3C2×D412yes—
116C23×C3#4112yes—
117C23×C3C2411yescharacteristic
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3C22×C412yes—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—44no—
133C2×C3×C4C22×C414yes—
134#67—28no—
135C2×Dic3—28no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—44no—
146C2×Dic3C22×C412yes—
147C2×Dic3C22×C414yes—
148C2×Dic3C2×D412yes—
149C2×Dic3C2×D412yes—
150C4×D3—44no—
151C24—31no—
152C24C22×D311yescharacteristic
153C2×D4—34no—
154C2×D4—62no—
155C2×D4—62no—
156C22×C4—22no—
157C22×C4—34no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—32no—
161C22×C4—31no—
162C22×C4—34no—
163C22×C4—32no—
164C22×C4—42no—
165C22×C4—62no—
166C22×C4—62no—
167C22×C4—62no—
168C22×C4—122no—
169C22×C4—122no—
170C22×C4C22×D311yescharacteristic
171C22×C4C22×D311yescharacteristic
172C22×C4C22×D312yes—
173C22×C4C22×D312yes—
174C4⋊C4—38no—
175C4⋊C4—42no—
176C4⋊C4—64no—
177C4⋊C4—64no—
178C4⋊C4—62no—
179C22⋊C4—22no—
180C22⋊C4—38no—
181C22⋊C4—62no—
182C22⋊C4—62no—
183C22⋊C4—122no—
184C22⋊C4C4×D314yescomplemented
185C42—62no—
186C42—122no—
187C22×C3—24no—
188C22×C3—22no—
189C22×C3—22no—
190C22×C3—24no—
191C22×C3C4×D414yes—
192C22×C3C23×C411yescharacteristic
193C22×C3C22×D411yescharacteristic
194C22×C3C2×#4111yescharacteristic
195C22×C3H(5,𝔽2)12yes—
196C22×C32−1+412yes—
197C2×D3—22no—
198C2×D3—24no—
199C2×D3—41no—
200C2×D3—41no—
201C2×D3—42no—
202C3×C4—24no—
203C3×C4—41no—
204C3×C4—42no—
205C3×C4—41no—
206Dic3—24no—
207Dic3—22no—
208Dic3—22no—
209Dic3—22no—
210Dic3—42no—
211C23—22no—
212C23—22no—
213C23—32no—
214C23—61no—
215C23—62no—
216C23—61no—
217C23—62no—
218C23C2×C4×D312yes—
219C23D3×D412yes—
220C23#23912yes—
221C23C23×D311yescharacteristic
222D4—68no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—34no—
233C2×C4—32no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—44no—
237C2×C4—68no—
238C2×C4—61no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—124no—
248C2×C4—124no—
249C2×C4C2×C4×D314yes—
250C2×C3—24no—
251C2×C3C2×C4×D411yescharacteristic
252C2×C3#52012yes—
253C2×C3#53411yescharacteristic
254C2×C3#53611yescharacteristic
255C2×C3#53912yes—
256D3—42no—
257C22—24no—
258C22—22no—
259C22—24no—
260C22—22no—
261C22—62no—
262C22—64no—
263C22—122no—
264C22—121no—
265C22—121no—
266C22#89314yes—
267C22C22×C4×D311yescharacteristic
268C22C2×D3×D411yescharacteristic
269C22#100412yes—
270C22C2×#23911yescharacteristic
271C22#101012yes—
272C4—24no—
273C4—41no—
274C4—42no—
275C4—41no—
276C4—64no—
277C4—62no—
278C4—62no—
279C4—62no—
280C4—122no—
281C3#232211yescharacteristic, complemented, Sylow
282C2—24no—
283C2—122no—
284C2C2×#89311yescharacteristic
285C2#554612yes—
286C2#558911yescharacteristic
287C2#564512yes—
288C2#564811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#385C2
socle, cosocleC23×C3C24
prime radicalC22×#67C22
hypercenter#385D3
hypocenterC3#2322
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×Dic3⋊C4C2semidirect
4. C3×#385C2semidirect
6. #4989C2semidirect
7. #4972C2semidirect
8. #4943C2semidirect
9. Dic3×C22⋊C4C2semidirect
10. #4669C2semidirect
11. #4654C2semidirect
32. #886C4semidirect
47. #385D3semidirect
64. C22×Dic3C2×C4semidirect
75. C3×C22⋊C4C2×C4semidirect
79. (C2×D3)⋊C4C2×C4semidirect
80. C4⋊Dic3C2×C4semidirect
184. C22⋊C4C4×D3semidirect
281. C3#2322semidirect

Maximal subgroups

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

subgroupindex
C22×Dic3⋊C42
C2×#8862
C2×#67⋊C42
C3×#3852
#52132
#49892
#49722
#49432
Dic3×C22⋊C42
#46692
#46542
#23223