Finite Group Databasebeta

← #80863

Subgroups — #80863

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×#912C211yesmaximal, characteristic, complemented
2C2×D3×M16C211yesmaximal, characteristic, complemented
3#5606C212yesmaximal, complemented
4C3×#469C211yesmaximal, characteristic, complemented
5#5159C211yesmaximal, characteristic, complemented
6#5137C211yesmaximal, characteristic, complemented
7#5042C212yesmaximal, complemented
8#4867C211yesmaximal, characteristic, complemented
9#4860C211yesmaximal, characteristic, complemented
10#4780C212yesmaximal, complemented
11#4772C212yesmaximal, complemented
12#3054—31nomaximal, Sylow
13C2×#237C2212yes—
14C22×C4×D3C2211yescharacteristic
15C2×C3×D8C2211yescharacteristic, complemented
16C2×C3×M16C2211yescharacteristic
17C3×#121C2212yes—
18#938C2212yes—
19#937—22no—
20D4×Dic3—22no—
21C2×#214C2212yescomplemented
22#924C2212yes—
23C2×#208C2211yescharacteristic
24#912—24no—
25D3×M16—24no—
26C2×Dic12C2211yescharacteristic, complemented
27C2×ΓL(1,𝔽25)C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2211yescharacteristic
29C2×D3×C8C2211yescharacteristic
30C2×D3×C8C2211yescharacteristic
31#897C2212yes—
32#892C2212yes—
33#883—22no—
34#67⋊C4—22no—
35C3×#106C2211yescharacteristic, complemented
36C3×#102C2212yescomplemented
37#832C2212yescomplemented
38#817C2211yescharacteristic, complemented
39#816C2212yescomplemented
40#810C2212yescomplemented
41#807C2212yescomplemented
42C2×#135—31no—
43C22×M16—31no—
44#525—32no—
45#470—31no—
46#470—31no—
47#469—31no—
48#469D311yescharacteristic, complemented
49#468—32no—
50#468—32no—
51#427—31no—
52#416—32no—
53C23×D3D411yescharacteristic
54C2×C3×D4—22no—
55C2×C3×D4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C2×#67—22no—
58C2×#67—22no—
59C22×Dic3—22no—
60C22×Dic3D411yescharacteristic
61#237—24no—
62#237—24no—
63C2×C4×D3—22no—
64C2×C4×D3—22no—
65C2×C4×D3D411yescharacteristic
66C2×C4×D3D411yescharacteristic
67C2×C4×D3C2311yescharacteristic
68C2×C4×D3C2311yescharacteristic
69C2×Dic6C2312yes—
70C3×D8—22no—
71C3×M16—22no—
72C2×C3×C8C2311yescharacteristic
73C2×C3×C8C2311yescharacteristic
74C3×C4⋊C4C2312yes—
75C3×C22⋊C4—22no—
76#217—22no—
77#217—22no—
78#214—24no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81C4⋊Dic3—22no—
82C4⋊Dic3C2312yes—
83Dic3⋊C4—22no—
84C4×Dic3—22no—
85#208—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88Dic12—22no—
89ΓL(1,𝔽25)—22no—
90ΓL(1,𝔽25)—22no—
91D3×C8—22no—
92D3×C8D414yescomplemented
93C2×#41—32no—
94C23×C4—31no—
95#135—64no—
96C2×Q16—31no—
97C2×SD16—32no—
98C2×D8C2×D311yescharacteristic, complemented
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⋊Q8—32no—
109#121—32no—
110#121C2×D312yes—
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#103—32no—
120#102—32no—
121#102C2×D312yescomplemented
122C23×C3—22no—
123C23×C3C2×D411yescharacteristic
124C22×D3—22no—
125C22×D3—42no—
126C22×D3C2×D411yescharacteristic
127C22×D3C2×D411yescharacteristic
128C3×D4—22no—
129C3×D4—22no—
130C3×D4—44no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2411yescharacteristic
136#67—44no—
137#67—44no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—44no—
142C2×Dic3C2×D411yescharacteristic
143C2×Dic3C2×D411yescharacteristic
144C4×D3—22no—
145C4×D3—21no—
146C4×D3—21no—
147C4×D3—22no—
148C4×D3C2×D412yes—
149C4×D3#4112yes—
150Dic6—22no—
151Dic6—22no—
152C3×C8—21no—
153C3×C8C2×D412yes—
154C3⋊C8—21no—
155C3⋊C8C2×D412yes—
156C24—31no—
157#41—64no—
158#41—64no—
159C2×Q8—32no—
160C2×D4—22no—
161C2×D4—62no—
162C2×D4—62no—
163C2×D4C22×D312yes—
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—62no—
174SD16—64no—
175D8—22no—
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—32no—
187C4⋊C4—62no—
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×C3—42no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3C22×D411yescharacteristic
202C2×D3—22no—
203C2×D3—21no—
204C2×D3—22no—
205C2×D3—44no—
206C2×D3#12112yes—
207C3×C4—21no—
208C3×C4—42no—
209C3×C4C22×D411yescharacteristic
210C3×C4C2×#4111yescharacteristic
211Dic3—21no—
212Dic3—42no—
213Dic3—42no—
214Dic3#12112yes—
215C23—22no—
216C23—31no—
217C23—31no—
218C23—62no—
219C23—122no—
220C23D3×D411yescharacteristic
221Q8—62no—
222Q8—62no—
223D4—22no—
224D4—22no—
225D4—44no—
226D4—124no—
227D4—124no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—31no—
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—31no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—124no—
243C2×C4D3×D411yescharacteristic
244C2×C4C23×D311yescharacteristic
245C8—21no—
246C8—32no—
247C8—61no—
248C8D3×D412yes—
249C2×C3—41no—
250C2×C3—82no—
251C2×C3C2×#12111yescharacteristic
252C2×C3#57512yes—
253D3—22no—
254D3—41no—
255C22—22no—
256C22—21no—
257C22—32no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—62no—
262C22—62no—
263C22—61no—
264C22—124no—
265C22C2×D3×D411yescharacteristic
266C4—21no—
267C4—32no—
268C4—42no—
269C4—61no—
270C4—122no—
271C4—122no—
272C4C2×D3×D411yescharacteristic
273C4D3×#4111yescharacteristic
274C3#305411yescharacteristic, complemented, Sylow
275C2—41no—
276C2—62no—
277C2—82no—
278C2—121no—
279C2D3×#12111yescharacteristic
280C2#577312yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3054
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#912C2semidirect
2. C2×D3×M16C2semidirect
3. #5606C2semidirect
4. C3×#469C2semidirect
5. #5159C2semidirect
6. #5137C2semidirect
7. #5042C2semidirect
8. #4867C2semidirect
9. #4860C2semidirect
10. #4780C2semidirect
11. #4772C2semidirect
15. C2×C3×D8C22semidirect
21. C2×#214C22semidirect
26. C2×Dic12C22semidirect
35. C3×#106C22semidirect
36. C3×#102C22semidirect
37. #832C22semidirect
38. #817C22semidirect
39. #816C22semidirect
40. #810C22semidirect
41. #807C22semidirect
48. #469D3semidirect
92. D3×C8D4semidirect
98. C2×D8C2×D3semidirect
117. #106C2×D3semidirect
121. #102C2×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×#9122
C2×D3×M162
#56062
C3×#4692
#51592
#51372
#50422
#48672
#48602
#47802
#47722
#30543