Finite Group Databasebeta

← #80903

Subgroups — #80903

Order 384.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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×#919C211yesmaximal, characteristic, complemented
2C2×D3×M16C211yesmaximal, characteristic, complemented
3#5627C212yesmaximal, complemented
4C3×#470C211yesmaximal, characteristic, complemented
5#5189C211yesmaximal, characteristic, complemented
6#5135C211yesmaximal, characteristic, complemented
7#5049C212yesmaximal, complemented
8#4865C211yesmaximal, characteristic, complemented
9#4860C211yesmaximal, characteristic, complemented
10#4812C212yesmaximal, complemented
11#4804C212yesmaximal, complemented
12#3054—31nomaximal, Sylow
13C2×#239C2212yes—
14C22×C4×D3C2211yescharacteristic
15C2×C3×Q16C2211yescharacteristic, complemented
16C2×C3×M16C2211yescharacteristic
17C3×C22⋊Q8C2212yes—
18#946C2212yes—
19C2×#215C2212yescomplemented
20#930C2212yes—
21C2×#208C2211yescharacteristic
22#919—24no—
23D3×M16—24no—
24C2×D24C2211yescharacteristic, complemented
25C2×ΓL(1,𝔽25)C2211yescharacteristic
26C2×ΓL(1,𝔽25)C2211yescharacteristic
27C2×D3×C8C2211yescharacteristic
28C2×D3×C8C2211yescharacteristic
29#895C2212yes—
30#894—22no—
31#893—22no—
32#892C2212yes—
33#882—22no—
34#67⋊C4—22no—
35C3×#106C2211yescharacteristic, complemented
36C3×#103C2212yescomplemented
37#835C2212yescomplemented
38#821C2212yescomplemented
39#817C2211yescharacteristic, complemented
40#809C2212yescomplemented
41#807C2212yescomplemented
42C2×#135—31no—
43C22×M16—31no—
44#525—32no—
45#470—31no—
46#470D311yescharacteristic, complemented
47#469—31no—
48#469—31no—
49#468—32no—
50#468—32no—
51#427—31no—
52#416—32no—
53C23×D3D411yescharacteristic
54C2×C3×Q8C2312yes—
55C22×C3×C4C2311yescharacteristic
56C2×#67—22no—
57C22×Dic3D411yescharacteristic
58#239—24no—
59#239—24no—
60C2×D12—22no—
61C2×D12—22no—
62C2×D12C2312yes—
63C2×C4×D3—22no—
64C2×C4×D3—22no—
65C2×C4×D3—22no—
66C2×C4×D3D411yescharacteristic
67C2×C4×D3D411yescharacteristic
68C2×C4×D3C2311yescharacteristic
69C2×C4×D3C2311yescharacteristic
70C3×Q16—22no—
71C3×M16—22no—
72C2×C3×C8C2311yescharacteristic
73C2×C3×C8C2311yescharacteristic
74C3×C4⋊C4—22no—
75C3×C4⋊C4C2312yes—
76C3×C22⋊C4—22no—
77#215—24no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82C4⋊Dic3C2312yes—
83Dic3⋊C4—22no—
84C4×Dic3—22no—
85#208—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88D24—22no—
89ΓL(1,𝔽25)—22no—
90ΓL(1,𝔽25)—22no—
91D3×C8—22no—
92D3×C8D414yescomplemented
93C2×#41—32no—
94C23×C4—31no—
95#135—64no—
96C2×Q16C2×D311yescharacteristic, complemented
97C2×SD16—32no—
98C2×D8—31no—
99C2×M16—31no—
100C2×M16—31no—
101C2×M16—31no—
102C2×M16—64no—
103C2×M16C2×D311yescharacteristic
104C22×C8—31no—
105C22×C8—31no—
106#123—62no—
107C22⋊Q8—32no—
108C22⋊Q8C2×D312yes—
109#121—32no—
110#121—32no—
111#120—62no—
112C4×D4—62no—
113C4×D4—62no—
114#117—32no—
115#107—32no—
116#106—31no—
117#106C2×D311yescharacteristic, complemented
118#103—32no—
119#103C2×D312yescomplemented
120#102—32no—
121#102—32no—
122C23×C3C2×D411yescharacteristic
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—42no—
126C22×D3C2×D411yescharacteristic
127C22×D3C2×D411yescharacteristic
128C3×Q8—22no—
129C3×Q8—22no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2411yescharacteristic
136#67—44no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3C2×D411yescharacteristic
140C2×Dic3C2×D411yescharacteristic
141D12—22no—
142D12—22no—
143D12—44no—
144D12—44no—
145C4×D3—22no—
146C4×D3—21no—
147C4×D3—21no—
148C4×D3—22no—
149C4×D3—44no—
150C4×D3C2×D412yes—
151C4×D3#4112yes—
152C3×C8—21no—
153C3×C8C2×D412yes—
154C3⋊C8—21no—
155C3⋊C8C2×D412yes—
156C24—31no—
157#41—64no—
158#41—64no—
159C2×Q8C22×D312yes—
160C2×D4—32no—
161C2×D4—62no—
162C2×D4—62no—
163C2×D4—62no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—62no—
170C22×C4—62no—
171C22×C4—62no—
172C22×C4C22×D311yescharacteristic
173Q16—22no—
174SD16—64no—
175D8—62no—
176M16—22no—
177M16—62no—
178M16—62no—
179M16—62no—
180C2×C8—31no—
181C2×C8—34no—
182C2×C8—31no—
183C2×C8—62no—
184C2×C8C22×D311yescharacteristic
185C2×C8C22×D311yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—32no—
188C4⋊C4—62no—
189C4⋊C4C22×D312yes—
190C22⋊C4—22no—
191C22⋊C4—62no—
192C22⋊C4—62no—
193C22⋊C4—62no—
194C22⋊C4—62no—
195C42—62no—
196C22×C3—21no—
197C22×C3—22no—
198C22×C3C22×D411yescharacteristic
199C2×D3—22no—
200C2×D3—21no—
201C2×D3—22no—
202C2×D3—44no—
203C2×D3—42no—
204C2×D3—42no—
205C2×D3—42no—
206C2×D3#12112yes—
207C3×C4—21no—
208C3×C4—42no—
209C3×C4—42no—
210C3×C4C22×D411yescharacteristic
211C3×C4C2×#4111yescharacteristic
212Dic3—21no—
213Dic3—42no—
214Dic3#12112yes—
215C23—31no—
216C23—31no—
217C23—62no—
218C23—62no—
219C23—122no—
220C23D3×D411yescharacteristic
221Q8—22no—
222Q8—22no—
223D4—62no—
224D4—62no—
225D4—124no—
226D4—124no—
227D4—124no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—31no—
233C2×C4—32no—
234C2×C4—32no—
235C2×C4—31no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—62no—
242C2×C4—124no—
243C2×C4D3×D411yescharacteristic
244C2×C4C23×D311yescharacteristic
245C8—21no—
246C8—32no—
247C8—61no—
248C8D3×D412yes—
249C2×C3—41no—
250C2×C3C2×#12111yescharacteristic
251C2×C3#57512yes—
252D3—22no—
253D3—41no—
254D3—82no—
255C22—22no—
256C22—21no—
257C22—32no—
258C22—62no—
259C22—62no—
260C22—61no—
261C22—124no—
262C22—122no—
263C22—122no—
264C22—122no—
265C22C2×D3×D411yescharacteristic
266C4—21no—
267C4—32no—
268C4—42no—
269C4—42no—
270C4—61no—
271C4—122no—
272C4C2×D3×D411yescharacteristic
273C4D3×#4111yescharacteristic
274C3#305411yescharacteristic, complemented, Sylow
275C2—41no—
276C2—62no—
277C2—121no—
278C2—242no—
279C2D3×#12111yescharacteristic
280C2#577712yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#470C2
socleC22×C3C22×D4
hypercenter#470D3
hypocenterC3#3054
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#919C2semidirect
2. C2×D3×M16C2semidirect
3. #5627C2semidirect
4. C3×#470C2semidirect
5. #5189C2semidirect
6. #5135C2semidirect
7. #5049C2semidirect
8. #4865C2semidirect
9. #4860C2semidirect
10. #4812C2semidirect
11. #4804C2semidirect
15. C2×C3×Q16C22semidirect
19. C2×#215C22semidirect
24. C2×D24C22semidirect
35. C3×#106C22semidirect
36. C3×#103C22semidirect
37. #835C22semidirect
38. #821C22semidirect
39. #817C22semidirect
40. #809C22semidirect
41. #807C22semidirect
46. #470D3semidirect
92. D3×C8D4semidirect
96. C2×Q16C2×D3semidirect
117. #106C2×D3semidirect
119. #103C2×D3semidirect
274. C3#3054semidirect

Maximal subgroups

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

subgroupindex
C2×#9192
C2×D3×M162
#56272
C3×#4702
#51892
#51352
#50492
#48652
#48602
#48122
#48042
#30543