Finite Group Databasebeta

← #78670

Subgroups — #78670

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×#1009C211yesmaximal, characteristic, complemented
2#5830C211yesmaximal, characteristic, complemented
3C2×#908C211yesmaximal, characteristic, complemented
4C22×D24C211yesmaximal, characteristic, complemented
5C3×#408C211yesmaximal, characteristic, complemented
6#5131C211yesmaximal, characteristic, complemented
7#5113C211yesmaximal, characteristic, complemented
8#4734C214yesmaximal, complemented
9#4732C214yesmaximal, complemented
10#3008—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12#1009—24no—
13#1009—24no—
14C2×#239—21no—
15C2×D3×D4—21no—
16C2×D3×D4—22no—
17C2×#235C2211yescharacteristic
18C2×#235C2212yescomplemented
19C22×D12C2211yescharacteristic
20C22×D12C2212yescomplemented
21C22×D12C2211yescharacteristic, complemented
22C2×C3×M16C2211yescharacteristic, complemented
23C22×C3×C8C2211yescharacteristic, complemented
24#947—21no—
25#940—21no—
26#937—22no—
27#930C2214yescomplemented
28#926C2211yescharacteristic, complemented
29#908—24no—
30C2×D24—24no—
31C2×D24C2214yescomplemented
32C2×D24C2212yescomplemented
33C2×#204C2212yescomplemented
34C3×C22⋊C8C2214yescomplemented
35#821C2214yescomplemented
36#821C2212yescomplemented
37#816C2212yescomplemented
38C2×H(5,𝔽2)—31no—
39C2×AGL(1,ℤ/8)—31no—
40C22×D8—31no—
41#535—31no—
42#449—34no—
43#447—34no—
44#418—31no—
45#416—31no—
46#408D311yescharacteristic, complemented
47C23×D3—21no—
48C23×D3—22no—
49C23×D3—21no—
50C3×#41—22no—
51C3×#41—22no—
52C2×C3×Q8D411yescharacteristic
53C2×C3×D4D412yes—
54C2×C3×D4D411yescharacteristic
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C2×#67—21no—
58C2×#67—22no—
59C2×#67—24no—
60#239—28no—
61D3×D4—216no—
62D3×D4—28no—
63#235—22no—
64#235—24no—
65#235—24no—
66#235D414yescomplemented
67C2×D12—21no—
68C2×D12—22no—
69C2×D12—22no—
70C2×D12—24no—
71C2×D12—24no—
72C2×D12—22no—
73C2×D12—22no—
74C2×D12—21no—
75C2×D12D414yescomplemented
76C2×D12C2312yes—
77C2×D12C2312yes—
78C2×D12C2311yescharacteristic
79C2×C4×D3—21no—
80C2×C4×D3—22no—
81C2×Dic6C2311yescharacteristic
82C3×M16—22no—
83C2×C3×C8—22no—
84C2×C3×C8C2312yes—
85C2×C3×C8C2312yes—
86#217—21no—
87(C2×D3)⋊C4—24no—
88C4⋊Dic3C2312yes—
89C4×Dic3—21no—
90D24—28no—
91D24—24no—
92#204—24no—
93H(5,𝔽2)—64no—
94H(5,𝔽2)—64no—
95C2×#41—31no—
96C2×#41—32no—
97C2×#41—61no—
98C2×#41C2×D311yescharacteristic
99C22×D4—31no—
100C22×D4—31no—
101C22×D4—32no—
102C22×D4—62no—
103C22×D4—61no—
104AGL(1,ℤ/8)—64no—
105C2×SD16—32no—
106C2×D8—34no—
107C2×D8—32no—
108C2×D8—64no—
109C2×M16C2×D311yescharacteristic, complemented
110C22×C8C2×D311yescharacteristic, complemented
111#127—61no—
112#124—61no—
113#121—34no—
114#120—62no—
115#117—31no—
116#103—32no—
117#102—32no—
118#102—34no—
119C22⋊C8C2×D314yescomplemented
120C23×C3C2×D412yes—
121C23×C3C2×D411yescharacteristic
122C22×D3—21no—
123C22×D3—22no—
124C22×D3—28no—
125C22×D3—24no—
126C22×D3—22no—
127C22×D3—42no—
128C22×D3—82no—
129C22×D3—84no—
130C22×D3—82no—
131C3×Q8—22no—
132C3×D4—24no—
133C3×D4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4C2×D412yes—
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2×D412yes—
139C2×C3×C4C2×D412yes—
140C2×C3×C4C2411yescharacteristic
141#67—28no—
142#67—24no—
143#67—84no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146D12—28no—
147D12—24no—
148D12—21no—
149D12—22no—
150D12—22no—
151D12—22no—
152D12—42no—
153D12C2×D412yes—
154D12C2×D414yes—
155C4×D3—28no—
156C4×D3—24no—
157Dic6—21no—
158Dic6C2×D412yes—
159C3×C8—22no—
160C3×C8—22no—
161C24—62no—
162C24—61no—
163C24—61no—
164#41—22no—
165#41—22no—
166#41—34no—
167#41—64no—
168#41—64no—
169#41—68no—
170#41—62no—
171C2×Q8—31no—
172C2×Q8D1211yescharacteristic
173C2×D4—34no—
174C2×D4—32no—
175C2×D4—31no—
176C2×D4—32no—
177C2×D4—64no—
178C2×D4—64no—
179C2×D4—616no—
180C2×D4—68no—
181C2×D4—62no—
182C2×D4—61no—
183C2×D4—64no—
184C2×D4—62no—
185C2×D4—61no—
186C2×D4—61no—
187C2×D4—62no—
188C2×D4—62no—
189C2×D4—62no—
190C2×D4D1212yes—
191C2×D4D1211yescharacteristic
192C22×C4—62no—
193C22×C4—61no—
194C22×C4C22×D312yes—
195C22×C4C22×D311yescharacteristic
196SD16—64no—
197D8—68no—
198D8—64no—
199M16—22no—
200C2×C8—22no—
201C2×C8C22×D312yes—
202C2×C8C22×D312yes—
203C4⋊C4—32no—
204C22⋊C4—61no—
205C22⋊C4—64no—
206C42—61no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—24no—
210C22×C3#12012yes—
211C22×C3C22×D411yescharacteristic
212C2×D3—24no—
213C2×D3—22no—
214C2×D3—44no—
215C2×D3—42no—
216C2×D3—42no—
217C2×D3—44no—
218C2×D3—84no—
219C2×D3—88no—
220C2×D3—84no—
221C3×C4—22no—
222C3×C4#12012yes—
223C3×C4C22×D411yescharacteristic
224C3×C4C22×D411yescharacteristic
225Dic3—22no—
226Dic3—42no—
227C23—68no—
228C23—62no—
229C23—61no—
230C23—64no—
231C23—62no—
232C23—122no—
233C23—242no—
234C23—242no—
235C23—244no—
236C23C2×D1212yes—
237C23C2×D1211yescharacteristic
238Q8—22no—
239Q8—32no—
240Q8—61no—
241D4—24no—
242D4—22no—
243D4—34no—
244D4—32no—
245D4—68no—
246D4—68no—
247D4—64no—
248D4—64no—
249D4—61no—
250D4—62no—
251D4—62no—
252D4—62no—
253D4—122no—
254D4—244no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—68no—
258C2×C4—61no—
259C2×C4—64no—
260C2×C4—62no—
261C2×C4C2×D1212yes—
262C2×C4C2×D1211yescharacteristic
263C2×C4D3×D412yes—
264C2×C4D3×D412yes—
265C2×C4C23×D311yescharacteristic
266C8—22no—
267C8—22no—
268C2×C3—22no—
269C2×C3—42no—
270C2×C3C2×#12011yescharacteristic
271C2×C3#57612yes—
272D3—44no—
273D3—42no—
274D3—82no—
275C22—22no—
276C22—22no—
277C22—24no—
278C22—64no—
279C22—62no—
280C22—124no—
281C22—122no—
282C22—122no—
283C22—124no—
284C22—244no—
285C22—244no—
286C22—248no—
287C22#88212yes—
288C22C22×D1211yescharacteristic
289C4—22no—
290C4—62no—
291C4—122no—
292C4#88212yes—
293C4C2×D3×D411yescharacteristic
294C4C2×D3×D411yescharacteristic
295C3#300811yescharacteristic, complemented, Sylow
296C2—22no—
297C2—42no—
298C2—124no—
299C2—122no—
300C2—242no—
301C2C2×#88211yescharacteristic
302C2#575212yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#408C2
socleC22×C3C22×D4
hypercenter#408D3
hypocenterC3#3008
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#1009C2semidirect
2. #5830C2semidirect
3. C2×#908C2semidirect
4. C22×D24C2semidirect
5. C3×#408C2semidirect
6. #5131C2semidirect
7. #5113C2semidirect
8. #4734C2semidirect
9. #4732C2semidirect
18. C2×#235C22semidirect
20. C22×D12C22semidirect
21. C22×D12C22semidirect
22. C2×C3×M16C22semidirect
23. C22×C3×C8C22semidirect
27. #930C22semidirect
28. #926C22semidirect
31. C2×D24C22semidirect
32. C2×D24C22semidirect
33. C2×#204C22semidirect
34. C3×C22⋊C8C22semidirect
35. #821C22semidirect
36. #821C22semidirect
37. #816C22semidirect
46. #408D3semidirect
66. #235D4semidirect
75. C2×D12D4semidirect
109. C2×M16C2×D3semidirect
110. C22×C8C2×D3semidirect
119. C22⋊C8C2×D3semidirect
295. C3#3008semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
#58302
C2×#9082
C22×D242
C3×#4082
#51312
#51132
#47342
#47322
#30083