Finite Group Databasebeta

← #82401

Subgroups — #82401

Order 384.

696 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×C4×#67C212yesmaximal, complemented
3C2×#872C211yesmaximal, characteristic, complemented
4#5223C211yesmaximal, characteristic
5#5222C211yesmaximal, characteristic, complemented
6C4×#217C212yesmaximal, complemented
7#4992C211yesmaximal, characteristic, complemented
8#4983C211yesmaximal, characteristic, complemented
9#4961C212yesmaximal, complemented
10#4948C212yesmaximal, complemented
11C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
12#2392—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2212yes—
15C22×#67C2212yes—
16C22×C4×D3C2211yescharacteristic
17C3×C4×D4—24no—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C42C2211yescharacteristic
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24C4×#67—28no—
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2212yes—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2211yescharacteristic
32C2×C4×Dic3C2212yes—
33C2×C4×Dic3C2211yescharacteristic
34#872—24no—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×C4×D4—32no—
40C2×C4×D4D311yescharacteristic, complemented
41C2×#117—31no—
42#398—31no—
43#397—31no—
44#396—32no—
45#394—31no—
46#394—32no—
47#392—31no—
48C4×C4⋊C4—31no—
49C4×C22⋊C4—32no—
50C24×C3C2312yes—
51C23×D3C2311yescharacteristic
52C2×C3×D4—24no—
53C2×C3×D4—22no—
54C22×C3×C4—24no—
55C22×C3×C4—24no—
56C22×C3×C4—24no—
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C2×#67—24no—
62C2×#67—24no—
63C2×#67—24no—
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69C2×C4×D3—22no—
70C2×C4×D3—22no—
71C2×C4×D3D414yescomplemented
72C3×C4⋊C4—22no—
73C3×C22⋊C4—24no—
74C3×C42D414yescomplemented
75#217—24no—
76#217—24no—
77#217—24no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81C4⋊Dic3—24no—
82Dic3⋊C4—22no—
83Dic3⋊C4—22no—
84Dic3⋊C4—24no—
85C4×Dic3—22no—
86C4×Dic3—24no—
87C22×D4—32no—
88C22×D4C2×D311yescharacteristic
89C23×C4—31no—
90C23×C4C2×D312yes—
91C4×D4—24no—
92C4×D4—68no—
93#117—64no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4—32no—
98C2×C4⋊C4C2×D311yescharacteristic
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—31no—
103C2×C22⋊C4—31no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4C2×D312yes—
106C2×C42—31no—
107C2×C42—32no—
108C2×C42C2×D311yescharacteristic
109#95—32no—
110#95—32no—
111#95—31no—
112#95—31no—
113C23×C3—22no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3#4114yes—
120C23×C3C2411yescharacteristic
121C22×D3—21no—
122C22×D3—21no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3C2×D412yes—
126C3×D4—28no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—48no—
133C2×C3×C4C2×D412yes—
134C2×C3×C4C2×D412yes—
135C2×C3×C4C2×D412yes—
136C2×C3×C4C2×D412yes—
137C2×C3×C4#4112yes—
138#67—216no—
139C2×Dic3—22no—
140C2×Dic3—21no—
141C2×Dic3—21no—
142C2×Dic3—22no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3C2×D412yes—
151C2×Dic3#4114yes—
152C4×D3—28no—
153C24—31no—
154C24C22×D312yes—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—64no—
158C2×D4—64no—
159C2×D4—64no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—32no—
164C22×C4—31no—
165C22×C4—34no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—31no—
169C22×C4—62no—
170C22×C4—62no—
171C22×C4C22×D312yes—
172C22×C4C22×D311yescharacteristic
173C22×C4C22×D311yescharacteristic
174C22×C4C22×D311yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—64no—
177C4⋊C4—64no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C22⋊C4—24no—
181C22⋊C4—64no—
182C22⋊C4—64no—
183C22⋊C4—64no—
184C22⋊C4—64no—
185C22⋊C4—62no—
186C22⋊C4—62no—
187C42—64no—
188C42—62no—
189C42#6714yescomplemented
190C22×C3—24no—
191C22×C3—24no—
192C22×C3—24no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3C22×D411yescharacteristic
198C22×C3C22×D411yescharacteristic
199C22×C3C2×#4112yes—
200C22×C3C2×#4111yescharacteristic
201C22×C3C2×#4112yes—
202C2×D3—22no—
203C2×D3—24no—
204C2×D3—41no—
205C2×D3—42no—
206C2×D3—41no—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—42no—
210C3×C4#12114yes—
211Dic3—22no—
212Dic3—24no—
213Dic3—41no—
214Dic3—42no—
215Dic3—41no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—32no—
223C23—62no—
224C23—61no—
225C23—62no—
226C23—61no—
227C23#23514yes—
228C23C23×D311yescharacteristic
229D4—28no—
230D4—616no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—32no—
237C2×C4—34no—
238C2×C4—48no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—68no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4D3×D412yes—
252C2×C4#23712yes—
253C2×C4C2×#6712yes—
254C2×C4C2×#6712yes—
255C2×C4C2×#6712yes—
256C2×C3—44no—
257C2×C3C2×#12111yescharacteristic
258C2×C3#52511yescharacteristic
259C2×C3#52512yes—
260C2×C3#52912yes—
261C2×C3#53211yescharacteristic
262D3—42no—
263C22—24no—
264C22—24no—
265C22—24no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—62no—
271C22—64no—
272C22—121no—
273C22—122no—
274C22—121no—
275C22C2×#23512yes—
276C22C2×D3×D411yescharacteristic
277C22C2×#23711yescharacteristic
278C22D3×#4112yes—
279C22C22×#6711yescharacteristic
280C4—22no—
281C4—22no—
282C4—42no—
283C4—62no—
284C4—64no—
285C4—121no—
286C4—122no—
287C4—121no—
288C4#93814yes—
289C3#239211yescharacteristic, complemented, Sylow
290C2—44no—
291C2—122no—
292C2#553812yes—
293C2#554712yes—
294C2#579111yescharacteristic
295C2C2×#93811yescharacteristic
296C2#583211yescharacteristic
subgroupG/H
centerC22×C4C22×D3
commutator, socle, cosocleC23×C3C24
FrattiniC23C23×D3
FittingC2×C3×C4×D4C2
prime radical#5223C2
hypercenterC2×C4×D4D3
hypocenterC3#2392
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
2. C2×C4×#67C2semidirect
3. C2×#872C2semidirect
5. #5222C2semidirect
6. C4×#217C2semidirect
7. #4992C2semidirect
8. #4983C2semidirect
9. #4961C2semidirect
10. #4948C2semidirect
11. C4×C4⋊Dic3C2semidirect
40. C2×C4×D4D3semidirect
71. C2×C4×D3D4semidirect
74. C3×C42D4semidirect
189. C42#67semidirect
289. C3#2392semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×C4×#672
C2×#8722
#52232
#52222
C4×#2172
#49922
#49832
#49612
#49482
C4×C4⋊Dic32
#23923