Finite Group Databasebeta

← #81577

Subgroups — #81577

Order 384.

462 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#5730C211yesmaximal, characteristic, complemented
2#5717C211yesmaximal, characteristic
3C3×#486C211yesmaximal, characteristic, complemented
4#5084C211yesmaximal, characteristic, complemented
5#5075C211yesmaximal, characteristic, complemented
6#4778C212yesmaximal, complemented
7#4777C212yesmaximal, complemented
8#4776C212yesmaximal, complemented
9D3×#102C212yesmaximal, complemented
10C4×#204C211yesmaximal, characteristic, complemented
11C8×D12C211yesmaximal, characteristic, complemented
12#3295—31nomaximal, Sylow
13C2×D3×D4—21no—
14C2×D3×D4C2212yes—
15C3×C4⋊Q8C2211yescharacteristic, complemented
16C3×#127C2211yescharacteristic
17#946—22no—
18#938—21no—
19#938C2212yes—
20#937—22no—
21C2×#214C2212yescomplemented
22C2×#204C2211yescharacteristic
23C2×D3×C8C2212yescomplemented
24#897C2212yescomplemented
25#893—21no—
26D3×C4⋊C4C2212yescomplemented
27#890—21no—
28C4×D12C2211yescharacteristic
29C4×D12C2211yescharacteristic
30C4×Dic6C2211yescharacteristic, complemented
31C3×#102C2212yescomplemented
32C3×#102C2212yescomplemented
33C3×C4×C8C2211yescharacteristic, complemented
34#832C2212yescomplemented
35#821C2211yescharacteristic
36(C2×D3)⋊C8C2212yescomplemented
37#817C2211yescharacteristic, complemented
38#816C2211yescharacteristic, complemented
39#809C2212yes—
40#808C2212yescomplemented
41C4⋊(C3⋊C8)C2211yescharacteristic, complemented
42#554—31no—
43D42—31no—
44#486D311yescharacteristic, complemented
45#481—32no—
46#476—31no—
47#465—32no—
48#460—31no—
49#450—32no—
50C4×SD16—31no—
51C8×D4—31no—
52C2×#102—32no—
53C23×D3—22no—
54C2×C3×Q8—21no—
55C2×C3×D4—21no—
56C2×C3×D4—21no—
57C2×C3×D4C2312yes—
58C2×#67—22no—
59C2×#67—22no—
60D3×D4—24no—
61D3×D4—42no—
62D3×D4—44no—
63C2×D12D411yescharacteristic
64C2×D12C2311yescharacteristic
65C2×C4×D3—21no—
66C2×C4×D3—22no—
67C2×C4×D3C2312yes—
68C2×Dic6C2311yescharacteristic
69C2×C3×C8C2311yescharacteristic
70C2×C3×C8C2311yescharacteristic
71C3×C4⋊C4—21no—
72C3×C4⋊C4C2312yes—
73C3×C42C2311yescharacteristic
74#217—22no—
75#214—42no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4D412yes—
79C4⋊Dic3—22no—
80C4⋊Dic3D411yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2311yescharacteristic
83Dic3⋊C4—21no—
84Dic3⋊C4—22no—
85C4×Dic3—21no—
86C2×C3⋊C8C2312yes—
87#204—22no—
88D3×C8—42no—
89C22×D4—32no—
90C22×D4—61no—
91C2×SD16—32no—
92C2×SD16—31no—
93C22×C8—32no—
94C4⋊Q8C2×D311yescharacteristic, complemented
95#127C2×D311yescharacteristic
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—31no—
105C4×D4—61no—
106C2×C4⋊C4—32no—
107#106—31no—
108#106—32no—
109C4⋊C8—31no—
110#103—31no—
111#102—32no—
112#102—31no—
113#102—32no—
114#102C2×D312yescomplemented
115#102C2×D312yescomplemented
116C22⋊C8—32no—
117C4×C8C2×D311yescharacteristic, complemented
118C23×C3—22no—
119C22×D3—21no—
120C22×D3—44no—
121C22×D3—44no—
122C22×D3—82no—
123C22×D3C2×D412yes—
124C3×Q8—81no—
125C3×D4—22no—
126C3×D4—22no—
127C3×D4—42no—
128C3×D4—81no—
129C2×C3×C4—22no—
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2×D411yescharacteristic
132C2×C3×C4C2411yescharacteristic
133#67—44no—
134#67—44no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138C2×Dic3C2×D412yes—
139D12—21no—
140D12SD1614yescomplemented
141D12#4112yes—
142C4×D3—22no—
143C4×D3—22no—
144C4×D3—42no—
145C4×D3—82no—
146Dic6—21no—
147Dic6—21no—
148C3×C8—21no—
149C3×C8—21no—
150C3⋊C8—22no—
151C24—62no—
152C2×Q8—21no—
153C2×Q8—31no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—31no—
157C2×D4—31no—
158C2×D4—62no—
159C2×D4—64no—
160C2×D4—62no—
161C2×D4—122no—
162C2×D4—124no—
163C2×D4C22×D312yes—
164C22×C4—32no—
165C22×C4—62no—
166C22×C4—61no—
167SD16—62no—
168SD16—122no—
169C2×C8—32no—
170C2×C8—122no—
171C2×C8C22×D311yescharacteristic
172C2×C8C22×D311yescharacteristic
173C4⋊C4—21no—
174C4⋊C4—31no—
175C4⋊C4—31no—
176C4⋊C4—31no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—61no—
180C4⋊C4C22×D312yes—
181C22⋊C4—32no—
182C22⋊C4—62no—
183C22⋊C4—62no—
184C22⋊C4—61no—
185C42—61no—
186C42C22×D311yescharacteristic
187C22×C3—42no—
188C22×C3—42no—
189C22×C3—42no—
190C22×C3C22×D411yescharacteristic
191C2×D3—22no—
192C2×D3—22no—
193C2×D3—22no—
194C2×D3—41no—
195C2×D3—41no—
196C2×D3—88no—
197C2×D3C2×SD1614yes—
198C3×C4—21no—
199C3×C4—42no—
200C3×C4C2×SD1612yes—
201C3×C4C2×#4111yescharacteristic
202C3×C4H(5,𝔽2)11yescharacteristic
203Dic3—22no—
204Dic3—41no—
205Dic3—41no—
206Dic3—41no—
207C23—22no—
208C23—32no—
209C23—61no—
210C23—124no—
211C23—124no—
212C23—242no—
213Q8—61no—
214Q8—61no—
215Q8—81no—
216D4—22no—
217D4—22no—
218D4—34no—
219D4—32no—
220D4—42no—
221D4—61no—
222D4—81no—
223D4—124no—
224D4—124no—
225C2×C4—22no—
226C2×C4—32no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—61no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4—122no—
233C2×C4—242no—
234C2×C4D3×D411yescharacteristic
235C2×C4D3×D411yescharacteristic
236C2×C4C23×D311yescharacteristic
237C8—21no—
238C8—21no—
239C8—62no—
240C2×C3—82no—
241C2×C3#54611yescharacteristic
242C2×C3C22×SD1611yescharacteristic
243C2×C3#57611yescharacteristic
244D3—24no—
245D3—42no—
246C22—34no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—62no—
251C22—62no—
252C22—62no—
253C22—121no—
254C22—121no—
255C22—248no—
256C22C2×D3×D411yescharacteristic
257C4—21no—
258C4—42no—
259C4—62no—
260C4—121no—
261C4—121no—
262C4—121no—
263C4D3×SD1612yes—
264C4D3×#4111yescharacteristic
265C4#100911yescharacteristic
266C3#329511yescharacteristic, complemented, Sylow
267C2—64no—
268C2—82no—
269C2—122no—
270C2#567611yescharacteristic
271C2#575711yescharacteristic
272C2C2×D3×SD1611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#486C2
socleC22×C3C22×D4
prime radical#5717C2
hypercenter#486D3
hypocenterC3#3295
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #5730C2semidirect
3. C3×#486C2semidirect
4. #5084C2semidirect
5. #5075C2semidirect
6. #4778C2semidirect
7. #4777C2semidirect
8. #4776C2semidirect
9. D3×#102C2semidirect
10. C4×#204C2semidirect
11. C8×D12C2semidirect
15. C3×C4⋊Q8C22semidirect
21. C2×#214C22semidirect
23. C2×D3×C8C22semidirect
24. #897C22semidirect
26. D3×C4⋊C4C22semidirect
30. C4×Dic6C22semidirect
31. C3×#102C22semidirect
32. C3×#102C22semidirect
33. C3×C4×C8C22semidirect
34. #832C22semidirect
36. (C2×D3)⋊C8C22semidirect
37. #817C22semidirect
38. #816C22semidirect
40. #808C22semidirect
41. C4⋊(C3⋊C8)C22semidirect
44. #486D3semidirect
94. C4⋊Q8C2×D3semidirect
114. #102C2×D3semidirect
115. #102C2×D3semidirect
117. C4×C8C2×D3semidirect
140. D12SD16semidirect
266. C3#3295semidirect

Maximal subgroups

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

subgroupindex
#57302
#57172
C3×#4862
#50842
#50752
#47782
#47772
#47762
D3×#1022
C4×#2042
C8×D122
#32953