Finite Group Databasebeta

← #79439

Subgroups — #79439

Order 384.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D12C211yesmaximal, characteristic
2#5543C211yesmaximal, characteristic, complemented
3C3×C8×D4C211yesmaximal, characteristic, complemented
4#5114C212yesmaximal, complemented
5C2×#821C212yesmaximal, complemented
6#4848C211yesmaximal, characteristic, complemented
7#4842C211yesmaximal, characteristic, complemented
8#4735C212yesmaximal, complemented
9#4731C212yesmaximal, complemented
10#4694C211yesmaximal, characteristic, complemented
11C4×#204C211yesmaximal, characteristic, complemented
12#3295—31nomaximal, Sylow
13C2×D3×D4—21no—
14C22×D12C2212yes—
15C22×C3×C8C2212yescomplemented
16C3×C4×D4C2211yescharacteristic
17D4×Dic3—21no—
18#930C2212yes—
19C2×C4⋊Dic3C2212yescomplemented
20#924C2212yescomplemented
21C2×#204C2211yescharacteristic
22C2×#204C2212yescomplemented
23#895—21no—
24#890—21no—
25#882—22no—
26#878—22no—
27#874C2211yescharacteristic
28C4×D12C2211yescharacteristic
29#869C2211yescharacteristic, complemented
30C4×Dic6C2211yescharacteristic, complemented
31C3×C4⋊C8C2211yescharacteristic, complemented
32C3×C22⋊C8C2212yescomplemented
33C3×C4×C8C2211yescharacteristic, complemented
34#821C2211yescharacteristic
35#821C2212yes—
36#821C2212yescomplemented
37#821C2212yescomplemented
38#821C2212yescomplemented
39#817C2211yescharacteristic, complemented
40#817C2212yescomplemented
41#816C2211yescharacteristic, complemented
42#554—31no—
43D42—31no—
44#486—31no—
45#481—32no—
46#476—31no—
47#465—32no—
48#460—31no—
49#450—32no—
50C4×SD16—31no—
51C8×D4D311yescharacteristic, complemented
52C2×#102—32no—
53C23×D3—22no—
54C2×C3×D4D411yescharacteristic
55C22×C3×C4C2312yes—
56C2×#67—22no—
57C22×Dic3—22no—
58D3×D4—44no—
59C2×D12—24no—
60C2×D12—21no—
61C2×D12—22no—
62C2×D12—21no—
63C2×D12—42no—
64C2×D12C2312yes—
65C2×D12C2311yescharacteristic
66C2×C4×D3—21no—
67C2×Dic6—21no—
68C2×Dic6C2311yescharacteristic
69C2×C3×C8—42no—
70C2×C3×C8C2311yescharacteristic
71C2×C3×C8C2312yes—
72C2×C3×C8C2311yescharacteristic
73C3×C4⋊C4D411yescharacteristic
74C3×C22⋊C4D412yes—
75C3×C42C2311yescharacteristic
76#217—22no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—22no—
79C4⋊Dic3—21no—
80C4⋊Dic3—22no—
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2311yescharacteristic
83C4⋊Dic3C2312yes—
84Dic3⋊C4—21no—
85Dic3⋊C4—22no—
86C4×Dic3—21no—
87#204—22no—
88#204—42no—
89C22×D4—32no—
90C22×D4—61no—
91C2×SD16—31no—
92C2×SD16—32no—
93C22×C8C2×D312yescomplemented
94C4⋊Q8—31no—
95#127—31no—
96#125—61no—
97C22⋊Q8—32no—
98C22⋊Q8—62no—
99#121—32no—
100#121—61no—
101#120—62no—
102C4×Q8—31no—
103C4×D4—31no—
104C4×D4—61no—
105C4×D4C2×D311yescharacteristic
106C2×C4⋊C4—32no—
107#106—31no—
108#106—32no—
109C4⋊C8C2×D311yescharacteristic, complemented
110#103—31no—
111#102—31no—
112#102—32no—
113#102—32no—
114#102—32no—
115#102—32no—
116C22⋊C8C2×D312yescomplemented
117C4×C8C2×D311yescharacteristic, complemented
118C23×C3C2×D412yes—
119C22×D3—21no—
120C22×D3—22no—
121C22×D3—44no—
122C22×D3—44no—
123C22×D3—82no—
124C3×D4SD1614yescomplemented
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4C2×D412yes—
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2411yescharacteristic
131#67—44no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3—82no—
137D12—22no—
138D12—21no—
139D12—22no—
140D12—44no—
141D12—42no—
142D12—81no—
143D12#4112yes—
144C4×D3—42no—
145Dic6—21no—
146Dic6—21no—
147Dic6—81no—
148C3×C8—21no—
149C3×C8—22no—
150C3×C8—21no—
151C24—62no—
152C2×Q8—31no—
153C2×Q8—61no—
154C2×D4—31no—
155C2×D4—32no—
156C2×D4—64no—
157C2×D4—62no—
158C2×D4—62no—
159C2×D4—61no—
160C2×D4—61no—
161C2×D4—122no—
162C2×D4—124no—
163C2×D4D1211yescharacteristic
164C22×C4—62no—
165C22×C4—61no—
166C22×C4C22×D312yes—
167SD16—62no—
168SD16—122no—
169C2×C8—42no—
170C2×C8C22×D311yescharacteristic
171C2×C8C22×D311yescharacteristic
172C2×C8C22×D312yes—
173C4⋊C4—31no—
174C4⋊C4—32no—
175C4⋊C4—31no—
176C4⋊C4—62no—
177C4⋊C4—61no—
178C4⋊C4—61no—
179C4⋊C4—62no—
180C4⋊C4D1211yescharacteristic
181C22⋊C4—62no—
182C22⋊C4—61no—
183C22⋊C4—62no—
184C22⋊C4D1212yes—
185C42—61no—
186C42C22×D311yescharacteristic
187C22×C3—22no—
188C22×C3—22no—
189C22×C3C2×SD1614yes—
190C22×C3C22×D411yescharacteristic
191C2×D3—22no—
192C2×D3—41no—
193C2×D3—41no—
194C2×D3—42no—
195C2×D3—42no—
196C2×D3—42no—
197C2×D3—88no—
198C3×C4—22no—
199C3×C4—21no—
200C3×C4C2×SD1612yes—
201C3×C4C2×#4111yescharacteristic
202C3×C4H(5,𝔽2)11yescharacteristic
203Dic3—41no—
204Dic3—41no—
205Dic3—42no—
206Dic3—41no—
207C23—61no—
208C23—62no—
209C23—124no—
210C23—124no—
211C23—242no—
212C23C2×D1212yes—
213Q8—61no—
214Q8—61no—
215Q8—241no—
216D4—32no—
217D4—61no—
218D4—62no—
219D4—62no—
220D4—124no—
221D4—124no—
222D4—122no—
223D4—241no—
224D4#20414yescomplemented
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—61no—
230C2×C4—62no—
231C2×C4—122no—
232C2×C4—242no—
233C2×C4C2×D1211yescharacteristic
234C2×C4C2×D1212yes—
235C2×C4C2×D1211yescharacteristic
236C2×C4C23×D311yescharacteristic
237C8—21no—
238C8—22no—
239C8—21no—
240C2×C3—24no—
241C2×C3#54611yescharacteristic
242C2×C3C22×SD1611yescharacteristic
243C2×C3#57611yescharacteristic
244D3—42no—
245D3—82no—
246C22—22no—
247C22—22no—
248C22—62no—
249C22—121no—
250C22—121no—
251C22—122no—
252C22—122no—
253C22—122no—
254C22—248no—
255C22C2×#20414yes—
256C22C22×D1211yescharacteristic
257C4—21no—
258C4—22no—
259C4—121no—
260C4—121no—
261C4—122no—
262C4—121no—
263C4C2×#20412yes—
264C4#100411yescharacteristic
265C4D3×#4111yescharacteristic
266C3#329511yescharacteristic, complemented, Sylow
267C2—24no—
268C2—122no—
269C2—242no—
270C2#555411yescharacteristic
271C2C22×#20411yescharacteristic
272C2#575211yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C8×D4C2
socleC22×C3C22×D4
prime radicalD4×D12C2
hypercenterC8×D4D3
hypocenterC3#3295
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
2. #5543C2semidirect
3. C3×C8×D4C2semidirect
4. #5114C2semidirect
5. C2×#821C2semidirect
6. #4848C2semidirect
7. #4842C2semidirect
8. #4735C2semidirect
9. #4731C2semidirect
10. #4694C2semidirect
11. C4×#204C2semidirect
15. C22×C3×C8C22semidirect
19. C2×C4⋊Dic3C22semidirect
20. #924C22semidirect
22. C2×#204C22semidirect
29. #869C22semidirect
30. C4×Dic6C22semidirect
31. C3×C4⋊C8C22semidirect
32. C3×C22⋊C8C22semidirect
33. C3×C4×C8C22semidirect
36. #821C22semidirect
37. #821C22semidirect
38. #821C22semidirect
39. #817C22semidirect
40. #817C22semidirect
41. #816C22semidirect
51. C8×D4D3semidirect
93. C22×C8C2×D3semidirect
109. C4⋊C8C2×D3semidirect
116. C22⋊C8C2×D3semidirect
117. C4×C8C2×D3semidirect
124. C3×D4SD16semidirect
224. D4#204semidirect
266. C3#3295semidirect

Maximal subgroups

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

subgroupindex
D4×D122
#55432
C3×C8×D42
#51142
C2×#8212
#48482
#48422
#47352
#47312
#46942
C4×#2042
#32953