Finite Group Databasebeta

← #79548

Subgroups — #79548

Order 384.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#908C212yesmaximal, complemented
2#5551C211yesmaximal, characteristic, complemented
3D4×D12C211yesmaximal, characteristic, complemented
4C3×#436C211yesmaximal, characteristic, complemented
5#5134C212yesmaximal, complemented
6#4843C211yesmaximal, characteristic, complemented
7#4841C211yesmaximal, characteristic, complemented
8#4734C212yesmaximal, complemented
9#4731C212yesmaximal, complemented
10#4693C211yesmaximal, characteristic, complemented
11C4×D24C211yesmaximal, characteristic, complemented
12#3272—31nomaximal, Sylow
13C2×D3×D4—21no—
14C2×#235C2212yescomplemented
15C22×D12C2212yes—
16C2×C3×M16C2212yescomplemented
17C3×C4×D4C2211yescharacteristic
18#938—21no—
19#930C2212yes—
20#924C2212yescomplemented
21#908—28no—
22C2×D24C2211yescharacteristic
23C2×D24C2212yescomplemented
24C2×#204C2212yes—
25C2×#204C2212yescomplemented
26C2×#204C2212yescomplemented
27#895—21no—
28D3×C4⋊C4—21no—
29#883—22no—
30#882—22no—
31#874C2211yescharacteristic
32C4×D12C2211yescharacteristic
33C4×D12C2211yescharacteristic, complemented
34#869C2211yescharacteristic, complemented
35C3×C4⋊C8C2211yescharacteristic, complemented
36C3×C22⋊C8C2212yescomplemented
37C3×C4×C8C2211yescharacteristic, complemented
38#821C2211yescharacteristic
39#821C2212yescomplemented
40#821C2211yescharacteristic, complemented
41#818C2211yescharacteristic, complemented
42#816C2212yescomplemented
43C2×AGL(1,ℤ/8)—32no—
44#547—31no—
45D42—31no—
46#492—31no—
47#468—32no—
48#461—31no—
49#459—31no—
50#450—32no—
51#449—32no—
52C4×D8—31no—
53#436D311yescharacteristic, complemented
54C23×D3—22no—
55C2×C3×D4D411yescharacteristic
56C22×C3×C4C2312yes—
57C2×#67—22no—
58C2×#67—22no—
59D3×D4—44no—
60#235—24no—
61#235—42no—
62C2×D12—24no—
63C2×D12—21no—
64C2×D12—22no—
65C2×D12—21no—
66C2×D12—42no—
67C2×D12C2312yes—
68C2×D12C2311yescharacteristic
69C2×D12C2311yescharacteristic
70C2×C4×D3—21no—
71C2×C4×D3—21no—
72C2×C4×D3—22no—
73C2×Dic6C2312yes—
74C3×M16—24no—
75C2×C3×C8C2312yes—
76C2×C3×C8C2311yescharacteristic
77C2×C3×C8C2311yescharacteristic
78C3×C4⋊C4D411yescharacteristic
79C3×C22⋊C4D412yes—
80C3×C42C2311yescharacteristic
81#217—22no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—22no—
85C4⋊Dic3—21no—
86C4⋊Dic3—21no—
87C4⋊Dic3C2311yescharacteristic
88C4⋊Dic3C2311yescharacteristic
89Dic3⋊C4—22no—
90D24—24no—
91D24D414yescomplemented
92#204—24no—
93#204—24no—
94#204—42no—
95C2×#41—32no—
96C22×D4—32no—
97C22×D4—61no—
98AGL(1,ℤ/8)—68no—
99C2×SD16—32no—
100C2×SD16—32no—
101C2×SD16—32no—
102C2×D8—31no—
103C2×D8—32no—
104C2×M16C2×D312yescomplemented
105C4⋊Q8—31no—
106#127—31no—
107#123—62no—
108C22⋊Q8—32no—
109#121—32no—
110#121—61no—
111#121—61no—
112#120—62no—
113C4×D4—31no—
114C4×D4—31no—
115C4×D4C2×D311yescharacteristic
116C2×C4⋊C4—61no—
117#107—31no—
118C4⋊C8C2×D311yescharacteristic, complemented
119#103—32no—
120#102—31no—
121#102—31no—
122#102—32no—
123C22⋊C8C2×D312yescomplemented
124C4×C8C2×D311yescharacteristic, complemented
125C23×C3C2×D412yes—
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—22no—
129C22×D3—44no—
130C22×D3—44no—
131C22×D3—82no—
132C3×D4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4C2×D412yes—
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2411yescharacteristic
139#67—44no—
140#67—44no—
141C2×Dic3—21no—
142C2×Dic3—22no—
143C2×Dic3—21no—
144D12—22no—
145D12—21no—
146D12—21no—
147D12—22no—
148D12—44no—
149D12—42no—
150D12—81no—
151D12C2×D412yes—
152D12C2×D412yescomplemented
153C4×D3—44no—
154C4×D3—42no—
155C4×D3—42no—
156Dic6—22no—
157Dic6—22no—
158C3×C8—22no—
159C3×C8—21no—
160C3×C8C2×D412yescomplemented
161C24—62no—
162#41—64no—
163#41—122no—
164C2×Q8—32no—
165C2×D4—31no—
166C2×D4—31no—
167C2×D4—32no—
168C2×D4—64no—
169C2×D4—62no—
170C2×D4—62no—
171C2×D4—62no—
172C2×D4—61no—
173C2×D4—61no—
174C2×D4—122no—
175C2×D4—124no—
176C2×D4D1211yescharacteristic
177C22×C4—61no—
178C22×C4—62no—
179C22×C4—61no—
180C22×C4C22×D312yes—
181SD16—64no—
182SD16—64no—
183SD16—122no—
184D8—34no—
185D8—64no—
186M16—24no—
187C2×C8C22×D311yescharacteristic
188C2×C8C22×D312yes—
189C2×C8C22×D311yescharacteristic
190C4⋊C4—31no—
191C4⋊C4—31no—
192C4⋊C4—61no—
193C4⋊C4—62no—
194C4⋊C4—61no—
195C4⋊C4D1211yescharacteristic
196C22⋊C4—62no—
197C22⋊C4—61no—
198C22⋊C4—61no—
199C22⋊C4—62no—
200C22⋊C4D1212yes—
201C42C22×D311yescharacteristic
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3C22×D411yescharacteristic
206C2×D3—22no—
207C2×D3—22no—
208C2×D3—41no—
209C2×D3—41no—
210C2×D3—41no—
211C2×D3—41no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—42no—
215C2×D3—88no—
216C3×C4—22no—
217C3×C4—21no—
218C3×C4AGL(1,ℤ/8)12yescomplemented
219C3×C4C22×D411yescharacteristic
220C3×C4H(5,𝔽2)11yescharacteristic
221Dic3—42no—
222Dic3—41no—
223Dic3—41no—
224C23—61no—
225C23—61no—
226C23—62no—
227C23—124no—
228C23—124no—
229C23—242no—
230C23C2×D1212yes—
231Q8—62no—
232Q8—62no—
233D4—22no—
234D4—32no—
235D4—32no—
236D4—62no—
237D4—61no—
238D4—61no—
239D4—62no—
240D4—124no—
241D4—124no—
242D4—124no—
243D4—122no—
244D4—241no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—62no—
250C2×C4—124no—
251C2×C4—122no—
252C2×C4—122no—
253C2×C4C2×D1211yescharacteristic
254C2×C4C2×D1212yes—
255C2×C4C2×D1211yescharacteristic
256C2×C4C23×D311yescharacteristic
257C8—22no—
258C8—21no—
259C8D3×D412yescomplemented
260C2×C3—42no—
261C2×C3D4211yescharacteristic
262C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
263C2×C3#57711yescharacteristic
264D3—42no—
265D3—42no—
266D3—82no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—62no—
271C22—62no—
272C22—121no—
273C22—121no—
274C22—121no—
275C22—121no—
276C22—122no—
277C22—122no—
278C22—122no—
279C22—248no—
280C22C22×D1211yescharacteristic
281C4—21no—
282C4—22no—
283C4—121no—
284C4—121no—
285C4—122no—
286C4#90812yescomplemented
287C4C2×D3×D411yescharacteristic
288C4#100911yescharacteristic
289C3#327211yescharacteristic, complemented, Sylow
290C2—42no—
291C2—122no—
292C2—122no—
293C2—242no—
294C2D4×D1211yescharacteristic
295C2C2×#90811yescharacteristic
296C2#575111yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#436C2
socleC22×C3C22×D4
hypercenter#436D3
hypocenterC3#3272
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#908C2semidirect
2. #5551C2semidirect
3. D4×D12C2semidirect
4. C3×#436C2semidirect
5. #5134C2semidirect
6. #4843C2semidirect
7. #4841C2semidirect
8. #4734C2semidirect
9. #4731C2semidirect
10. #4693C2semidirect
11. C4×D24C2semidirect
14. C2×#235C22semidirect
16. C2×C3×M16C22semidirect
20. #924C22semidirect
23. C2×D24C22semidirect
25. C2×#204C22semidirect
26. C2×#204C22semidirect
33. C4×D12C22semidirect
34. #869C22semidirect
35. C3×C4⋊C8C22semidirect
36. C3×C22⋊C8C22semidirect
37. C3×C4×C8C22semidirect
39. #821C22semidirect
40. #821C22semidirect
41. #818C22semidirect
42. #816C22semidirect
53. #436D3semidirect
91. D24D4semidirect
104. C2×M16C2×D3semidirect
118. C4⋊C8C2×D3semidirect
123. C22⋊C8C2×D3semidirect
124. C4×C8C2×D3semidirect
152. D12C2×D4semidirect
160. C3×C8C2×D4semidirect
218. C3×C4AGL(1,ℤ/8)semidirect
259. C8D3×D4semidirect
286. C4#908semidirect
289. C3#3272semidirect

Maximal subgroups

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

subgroupindex
C2×#9082
#55512
D4×D122
C3×#4362
#51342
#48432
#48412
#47342
#47312
#46932
C4×D242
#32723