Finite Group Databasebeta

← #78639

Subgroups — #78639

Order 384.

806 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#235C211yesmaximal, characteristic, complemented
2C2×#930C211yesmaximal, characteristic, complemented
3C2×#908C212yesmaximal, complemented
4C3×#407C211yesmaximal, characteristic, complemented
5#5133C212yesmaximal, complemented
6#4738C212yesmaximal, complemented
7#4734C214yesmaximal, complemented
8#4732C212yesmaximal, complemented
9#3006—31nomaximal, Sylow
10C23×C3×C4C2211yescharacteristic
11C22×#67—21no—
12C22×#67—21no—
13C2×#235—21no—
14C2×#235—21no—
15C2×#235—28no—
16C2×#235C2212yes—
17C2×#235C2212yescomplemented
18C22×D12C2211yescharacteristic, complemented
19C22×D12C2211yescharacteristic, complemented
20C22×C4×D3—21no—
21C22×Dic6C2211yescharacteristic, complemented
22C2×C3×M16C2212yescomplemented
23#930—21no—
24#930—21no—
25#930C2212yes—
26#930C2212yescomplemented
27C2×(C2×D3)⋊C4—21no—
28C2×C4⋊Dic3C2211yescharacteristic, complemented
29#908—28no—
30C2×D24C2214yescomplemented
31C2×#204C2214yescomplemented
32C3×C22⋊C8C2212yescomplemented
33C3×C22⋊C8C2212yescomplemented
34#821C2214yescomplemented
35#816C2214yescomplemented
36C22×#41—31no—
37C2×AGL(1,ℤ/8)—32no—
38C2×#121—31no—
39#449—34no—
40#448—32no—
41#447—32no—
42#417—32no—
43#407D311yescharacteristic, complemented
44C24×C3D411yescharacteristic
45C23×D3—21no—
46C23×D3—21no—
47C22×C3×C4—24no—
48C22×C3×C4D411yescharacteristic
49C22×C3×C4D411yescharacteristic
50C22×C3×C4D411yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C2×#67—28no—
55C2×#67—22no—
56C2×#67—22no—
57C2×#67—22no—
58C2×#67—22no—
59C2×#67—44no—
60C22×Dic3—21no—
61C22×Dic3—21no—
62#235—28no—
63#235—28no—
64#235—24no—
65#235—24no—
66#235—24no—
67#235D418yescomplemented
68C2×D12—21no—
69C2×D12—24no—
70C2×D12—24no—
71C2×D12—21no—
72C2×D12C2312yes—
73C2×D12C2312yes—
74C2×C4×D3—28no—
75C2×C4×D3—22no—
76C2×C4×D3—22no—
77C2×Dic6—21no—
78C2×Dic6—24no—
79C2×Dic6C2312yes—
80C3×M16—24no—
81C2×C3×C8C2314yes—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—22no—
84C4⋊Dic3—21no—
85C4⋊Dic3C2312yes—
86D24—28no—
87#204—28no—
88C2×#41—32no—
89C2×#41—32no—
90C2×#41—68no—
91C2×#41—61no—
92C2×#41—61no—
93C22×Q8—31no—
94C22×D4—31no—
95C22×D4—31no—
96C22×D4—61no—
97C22×D4—61no—
98C23×C4—61no—
99C23×C4C2×D311yescharacteristic
100AGL(1,ℤ/8)—68no—
101C2×SD16—34no—
102C2×D8—34no—
103C2×M16C2×D312yescomplemented
104#121—32no—
105#121—32no—
106#121—61no—
107#121—61no—
108C2×C4⋊C4—31no—
109C2×C22⋊C4—61no—
110#103—34no—
111#102—34no—
112C22⋊C8C2×D312yescomplemented
113C22⋊C8C2×D312yescomplemented
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—41no—
117C23×C3C2×D411yescharacteristic
118C23×C3C2×D411yescharacteristic
119C23×C3C2×D411yescharacteristic
120C22×D3—24no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—42no—
124C22×D3—42no—
125C22×D3—42no—
126C22×D3—42no—
127C22×D3—42no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2×D414yes—
137C2×C3×C4C2411yescharacteristic
138#67—28no—
139#67—28no—
140#67—44no—
141#67—44no—
142C2×Dic3—24no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—42no—
146D12—24no—
147D12—22no—
148D12—24no—
149D12—42no—
150D12C2×D414yescomplemented
151C4×D3—28no—
152C4×D3—28no—
153Dic6—24no—
154Dic6—22no—
155Dic6C2×D414yescomplemented
156C3×C8—24no—
157C24—61no—
158C24—61no—
159C24D1211yescharacteristic
160#41—38no—
161#41—64no—
162#41—64no—
163#41—68no—
164#41—68no—
165#41—64no—
166C2×Q8—32no—
167C2×Q8—64no—
168C2×Q8—61no—
169C2×D4—32no—
170C2×D4—32no—
171C2×D4—64no—
172C2×D4—68no—
173C2×D4—62no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4—62no—
177C2×D4—61no—
178C2×D4—61no—
179C2×D4—64no—
180C2×D4—124no—
181C22×C4—24no—
182C22×C4—68no—
183C22×C4—61no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4—61no—
187C22×C4D1211yescharacteristic
188C22×C4D1211yescharacteristic
189C22×C4D1211yescharacteristic
190C22×C4C22×D311yescharacteristic
191C22×C4C22×D311yescharacteristic
192C22×C4C22×D311yescharacteristic
193SD16—68no—
194D8—68no—
195M16—24no—
196C2×C8C22×D314yes—
197C4⋊C4—32no—
198C4⋊C4—61no—
199C22⋊C4—62no—
200C22⋊C4—62no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—44no—
209C22×C3#12012yes—
210C22×C3AGL(1,ℤ/8)12yescomplemented
211C22×C3C22×D411yescharacteristic
212C2×D3—24no—
213C2×D3—48no—
214C2×D3—42no—
215C2×D3—42no—
216C2×D3—42no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—84no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4#12012yes—
223C3×C4C22×D412yes—
224Dic3—24no—
225Dic3—42no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23—64no—
230C23—62no—
231C23—62no—
232C23—122no—
233C23—122no—
234C23—122no—
235C23—122no—
236C23—122no—
237C23C2×D1211yescharacteristic
238C23C2×D1211yescharacteristic
239C23C2×D1211yescharacteristic
240Q8—34no—
241Q8—64no—
242Q8—62no—
243D4—34no—
244D4—64no—
245D4—68no—
246D4—68no—
247D4—62no—
248D4—64no—
249D4—124no—
250D4—124no—
251D4—122no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—68no—
258C2×C4—64no—
259C2×C4—62no—
260C2×C4—68no—
261C2×C4—62no—
262C2×C4—122no—
263C2×C4C2×D1211yescharacteristic
264C2×C4C2×D1211yescharacteristic
265C2×C4C2×D1211yescharacteristic
266C2×C4D3×D414yes—
267C2×C4C23×D311yescharacteristic
268C8—24no—
269C2×C3—22no—
270C2×C3—22no—
271C2×C3—41no—
272C2×C3C2×#12011yescharacteristic
273C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
274C2×C3#57511yescharacteristic
275D3—44no—
276D3—82no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—44no—
285C22—64no—
286C22—128no—
287C22—122no—
288C22—122no—
289C22—122no—
290C22—122no—
291C22—122no—
292C22—244no—
293C22#88212yes—
294C22#90812yescomplemented
295C22C22×D1211yescharacteristic
296C4—21no—
297C4—21no—
298C4—64no—
299C4—122no—
300C4#88212yes—
301C4C2×D3×D412yes—
302C3#300611yescharacteristic, complemented, Sylow
303C2—22no—
304C2—22no—
305C2—41no—
306C2—124no—
307C2—242no—
308C2C2×#88211yescharacteristic
309C2C2×#90811yescharacteristic
310C2#574811yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#407C2
socleC22×C3C22×D4
hypercenter#407D3
hypocenterC3#3006
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#235C2semidirect
2. C2×#930C2semidirect
3. C2×#908C2semidirect
4. C3×#407C2semidirect
5. #5133C2semidirect
6. #4738C2semidirect
7. #4734C2semidirect
8. #4732C2semidirect
17. C2×#235C22semidirect
18. C22×D12C22semidirect
19. C22×D12C22semidirect
21. C22×Dic6C22semidirect
22. C2×C3×M16C22semidirect
26. #930C22semidirect
28. C2×C4⋊Dic3C22semidirect
30. C2×D24C22semidirect
31. C2×#204C22semidirect
32. C3×C22⋊C8C22semidirect
33. C3×C22⋊C8C22semidirect
34. #821C22semidirect
35. #816C22semidirect
43. #407D3semidirect
67. #235D4semidirect
103. C2×M16C2×D3semidirect
112. C22⋊C8C2×D3semidirect
113. C22⋊C8C2×D3semidirect
150. D12C2×D4semidirect
155. Dic6C2×D4semidirect
210. C22×C3AGL(1,ℤ/8)semidirect
294. C22#908semidirect
302. C3#3006semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C2×#9302
C2×#9082
C3×#4072
#51332
#47382
#47342
#47322
#30063