Finite Group Databasebeta

← #78928

Subgroups — #78928

Order 384.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#1004C211yesmaximal, characteristic, complemented
2C22×#213C211yesmaximal, characteristic, complemented
3C2×#908C211yesmaximal, characteristic, complemented
4#5525C211yesmaximal, characteristic, complemented
5C3×#418C211yesmaximal, characteristic, complemented
6#5129C211yesmaximal, characteristic, complemented
7#5002C211yesmaximal, characteristic, complemented
8#4781C214yesmaximal, complemented
9#4773C214yesmaximal, complemented
10#3008—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×#67—22no—
13#1004—24no—
14#1004—24no—
15C2×#237—21no—
16C2×#237C2212yescomplemented
17C2×D3×D4—21no—
18C2×D3×D4C2212yescomplemented
19C2×#235C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C22×D12C2211yescharacteristic, complemented
22C2×C3×M16C2211yescharacteristic, complemented
23C3×#117C2211yescharacteristic, complemented
24C2×#213—24no—
25C2×#213C2214yescomplemented
26C22×C3⋊C8C2211yescharacteristic, complemented
27#908—24no—
28C2×D24C2212yescomplemented
29C2×#204C2212yescomplemented
30#895C2214yescomplemented
31#882—22no—
32#875—21no—
33#874—21no—
34C3×#102C2214yescomplemented
35(C2×D3)⋊C8C2214yescomplemented
36#810C2212yescomplemented
37#809C2212yescomplemented
38C2×H(5,𝔽2)—31no—
39C2×AGL(1,ℤ/8)—31no—
40C22×D8—31no—
41#535—31no—
42#449—34no—
43#447—34no—
44#418D311yescharacteristic, complemented
45#416—31no—
46#408—31no—
47C24×C3—21no—
48C23×D3—22no—
49C23×D3—21no—
50C2×C3×D4—21no—
51C2×C3×D4—22no—
52C2×C3×D4D414yescomplemented
53C2×C3×D4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C2×#67—216no—
56C2×#67—21no—
57C2×#67—22no—
58C2×#67—22no—
59C2×#67D412yes—
60C22×Dic3—22no—
61#237—28no—
62#237—24no—
63#237—24no—
64D3×D4—28no—
65D3×D4—24no—
66D3×D4—24no—
67#235—22no—
68#235—22no—
69#235—22no—
70#235D414yescomplemented
71C2×D12—22no—
72C2×D12—22no—
73C2×D12—21no—
74C2×D12—24no—
75C2×D12D411yescharacteristic
76C2×D12C2312yes—
77C2×D12C2311yescharacteristic
78C2×C4×D3—21no—
79C2×C4×D3C2312yes—
80C2×Dic6D411yescharacteristic
81C2×Dic6C2311yescharacteristic
82C3×M16—22no—
83C2×C3×C8C2312yes—
84C3×C4⋊C4C2312yes—
85C3×C22⋊C4—21no—
86C3×C42—21no—
87#213—28no—
88(C2×D3)⋊C4—24no—
89C2×C3⋊C8—22no—
90C2×C3⋊C8C2312yes—
91D24—24no—
92#204—24no—
93H(5,𝔽2)—64no—
94H(5,𝔽2)—64no—
95C2×#41—31no—
96C2×#41—32no—
97C2×#41—31no—
98C2×#41—61no—
99C22×D4—32no—
100C22×D4—31no—
101C22×D4—61no—
102C22×D4—62no—
103C22×D4C2×D311yescharacteristic
104AGL(1,ℤ/8)—64no—
105C2×SD16—32no—
106C2×D8—34no—
107C2×D8—32no—
108C2×D8—64no—
109C2×M16C2×D311yescharacteristic, complemented
110C22×C8—31no—
111#127—61no—
112#124—61no—
113#121—34no—
114#120—62no—
115#117C2×D311yescharacteristic, complemented
116#103—32no—
117#102—32no—
118#102C2×D314yescomplemented
119C22⋊C8—34no—
120C23×C3—22no—
121C23×C3—24no—
122C23×C3—82no—
123C23×C3C2×D411yescharacteristic
124C22×D3—21no—
125C22×D3—28no—
126C22×D3—22no—
127C22×D3—42no—
128C22×D3—84no—
129C22×D3—82no—
130C22×D3C2×D412yes—
131C3×D4—24no—
132C3×D4—22no—
133C3×D4C2×D414yes—
134C2×C3×C4—22no—
135C2×C3×C4C2×D412yes—
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2×D412yes—
138C2×C3×C4C2411yescharacteristic
139#67—28no—
140#67—28no—
141#67—24no—
142#67—24no—
143C2×Dic3—28no—
144C2×Dic3—21no—
145C2×Dic3C2×D412yes—
146D12—22no—
147D12—21no—
148D12—22no—
149D12—22no—
150D12—42no—
151D12—84no—
152D12C2×D412yes—
153C4×D3—24no—
154C4×D3—22no—
155C4×D3—22no—
156Dic6—22no—
157Dic6—21no—
158Dic6C2×D412yes—
159C3×C8—22no—
160C3⋊C8—22no—
161C24—21no—
162C24—62no—
163C24—61no—
164#41—34no—
165#41—64no—
166#41—68no—
167#41—64no—
168#41—62no—
169#41—62no—
170#41—62no—
171C2×Q8—31no—
172C2×Q8—31no—
173C2×D4—21no—
174C2×D4—22no—
175C2×D4—32no—
176C2×D4—31no—
177C2×D4—31no—
178C2×D4—32no—
179C2×D4—64no—
180C2×D4—62no—
181C2×D4—64no—
182C2×D4—68no—
183C2×D4—62no—
184C2×D4—616no—
185C2×D4—61no—
186C2×D4—64no—
187C2×D4—62no—
188C2×D4—62no—
189C2×D4—61no—
190C2×D4D1214yescomplemented
191C2×D4C22×D312yes—
192C22×C4—32no—
193C22×C4—62no—
194C22×C4—61no—
195C22×C4C22×D311yescharacteristic
196SD16—64no—
197D8—64no—
198D8—68no—
199M16—22no—
200C2×C8—32no—
201C2×C8—62no—
202C2×C8C22×D312yes—
203C4⋊C4C22×D312yes—
204C22⋊C4—21no—
205C22⋊C4—64no—
206C42—21no—
207C22×C3—22no—
208C22×C3—24no—
209C22×C3—44no—
210C22×C3—84no—
211C22×C3#12012yes—
212C22×C3C22×D411yescharacteristic
213C2×D3—22no—
214C2×D3—22no—
215C2×D3—24no—
216C2×D3—42no—
217C2×D3—42no—
218C2×D3—44no—
219C2×D3—88no—
220C2×D3—84no—
221C3×C4—42no—
222C3×C4#12012yes—
223C3×C4C22×D411yescharacteristic
224C3×C4C22×D411yescharacteristic
225Dic3—22no—
226Dic3—22no—
227C23—22no—
228C23—24no—
229C23—32no—
230C23—68no—
231C23—61no—
232C23—62no—
233C23—82no—
234C23—122no—
235C23—242no—
236C23—244no—
237C23D3×D411yescharacteristic
238Q8—32no—
239Q8—62no—
240Q8—61no—
241D4—24no—
242D4—22no—
243D4—32no—
244D4—68no—
245D4—68no—
246D4—64no—
247D4—64no—
248D4—62no—
249D4—61no—
250D4—62no—
251D4—62no—
252D4—122no—
253D4—244no—
254D4C2×D1214yes—
255C2×C4—22no—
256C2×C4—32no—
257C2×C4—68no—
258C2×C4—61no—
259C2×C4—64no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4C2×D1212yes—
263C2×C4D3×D411yescharacteristic
264C2×C4D3×D412yes—
265C2×C4C23×D311yescharacteristic
266C8—22no—
267C8—62no—
268C2×C3—22no—
269C2×C3—44no—
270C2×C3C2×#12011yescharacteristic
271C2×C3#57612yes—
272D3—42no—
273D3—42no—
274D3—82no—
275C22—22no—
276C22—24no—
277C22—44no—
278C22—62no—
279C22—62no—
280C22—64no—
281C22—84no—
282C22—122no—
283C22—122no—
284C22—124no—
285C22—244no—
286C22—248no—
287C22#88212yes—
288C22C2×D3×D411yescharacteristic
289C4—42no—
290C4—62no—
291C4—62no—
292C4#88212yes—
293C4C22×D1211yescharacteristic
294C4C2×D3×D411yescharacteristic
295C3#300811yescharacteristic, complemented, Sylow
296C2—22no—
297C2—44no—
298C2—122no—
299C2—122no—
300C2—242no—
301C2C2×#88211yescharacteristic
302C2#577412yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#418C2
socleC22×C3C22×D4
hypercenter#418D3
hypocenterC3#3008
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#1004C2semidirect
2. C22×#213C2semidirect
3. C2×#908C2semidirect
4. #5525C2semidirect
5. C3×#418C2semidirect
6. #5129C2semidirect
7. #5002C2semidirect
8. #4781C2semidirect
9. #4773C2semidirect
16. C2×#237C22semidirect
18. C2×D3×D4C22semidirect
21. C22×D12C22semidirect
22. C2×C3×M16C22semidirect
23. C3×#117C22semidirect
25. C2×#213C22semidirect
26. C22×C3⋊C8C22semidirect
28. C2×D24C22semidirect
29. C2×#204C22semidirect
30. #895C22semidirect
34. C3×#102C22semidirect
35. (C2×D3)⋊C8C22semidirect
36. #810C22semidirect
37. #809C22semidirect
44. #418D3semidirect
52. C2×C3×D4D4semidirect
70. #235D4semidirect
109. C2×M16C2×D3semidirect
115. #117C2×D3semidirect
118. #102C2×D3semidirect
190. C2×D4D12semidirect
295. C3#3008semidirect

Maximal subgroups

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

subgroupindex
C2×#10042
C22×#2132
C2×#9082
#55252
C3×#4182
#51292
#50022
#47812
#47732
#30083