Finite Group Databasebeta

← #80395

Subgroups — #80395

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#5717C211yesmaximal, characteristic
2#5579C211yesmaximal, characteristic, complemented
3C3×#460C211yesmaximal, characteristic, complemented
4#5171C212yesmaximal, complemented
5#5169C212yesmaximal, complemented
6#5070C211yesmaximal, characteristic, complemented
7C4×#215C211yesmaximal, characteristic, complemented
8#4842C211yesmaximal, characteristic, complemented
9#4834C211yesmaximal, characteristic, complemented
10#4777C212yesmaximal, complemented
11D3×#102C212yesmaximal, complemented
12#3295—31nomaximal, Sylow
13C2×D3×D4—21no—
14C2×D3×D4C2212yes—
15C2×C3×SD16C2212yescomplemented
16C3×#127C2211yescharacteristic
17C3×C4×Q8C2211yescharacteristic, complemented
18#946C2212yescomplemented
19C2×#215C2211yescharacteristic
20#938—21no—
21#938C2212yes—
22#937—22no—
23C2×D3×C8C2212yescomplemented
24#896—22no—
25D3×C4⋊C4C2212yescomplemented
26C4×D12—21no—
27C4×D12C2211yescharacteristic
28C4×D12C2211yescharacteristic
29#870—21no—
30#869C2211yescharacteristic, complemented
31C3×C4⋊C8C2211yescharacteristic, complemented
32C3×#102C2212yescomplemented
33#835C2211yescharacteristic, complemented
34#832C2212yescomplemented
35#832C2212yescomplemented
36#821C2212yes—
37(C2×D3)⋊C8C2212yescomplemented
38#817C2212yescomplemented
39#809C2211yescharacteristic
40#808C2211yescharacteristic, complemented
41C4×C3⋊C8C2211yescharacteristic, complemented
42#554—31no—
43D42—31no—
44#486—31no—
45#481—32no—
46#476—31no—
47#465—32no—
48#460D311yescharacteristic, complemented
49#450—32no—
50C4×SD16—31no—
51C8×D4—31no—
52C2×#102—32no—
53C23×D3—22no—
54C2×C3×Q8C2311yescharacteristic
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×Dic6—21no—
69C3×SD16—42no—
70C2×C3×C8C2312yes—
71C3×C4⋊C4—21no—
72C3×C4⋊C4C2311yescharacteristic
73C3×C42—21no—
74C3×C42C2311yescharacteristic
75#217—22no—
76#215—22no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—21no—
79(C2×D3)⋊C4D412yes—
80C4⋊Dic3—21no—
81C4⋊Dic3D411yescharacteristic
82C4⋊Dic3C2311yescharacteristic
83C4⋊Dic3C2312yes—
84Dic3⋊C4—22no—
85Dic3⋊C4—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88D3×C8—42no—
89C22×D4—32no—
90C22×D4—61no—
91C2×SD16—31no—
92C2×SD16C2×D312yescomplemented
93C22×C8—32no—
94C4⋊Q8—31no—
95#127C2×D311yescharacteristic
96#125—61no—
97C22⋊Q8—32no—
98C22⋊Q8—62no—
99#121—32no—
100#121—61no—
101#120—62no—
102C4×Q8C2×D311yescharacteristic, complemented
103C4×D4—31no—
104C4×D4—31no—
105C4×D4—61no—
106C2×C4⋊C4—32no—
107#106—32no—
108#106—31no—
109C4⋊C8C2×D311yescharacteristic, complemented
110#103—31no—
111#102—32no—
112#102—32no—
113#102—32no—
114#102—31no—
115#102C2×D312yescomplemented
116C22⋊C8—32no—
117C4×C8—31no—
118C23×C3—22no—
119C22×D3—21no—
120C22×D3—44no—
121C22×D3—44no—
122C22×D3—82no—
123C22×D3C2×D412yes—
124C3×Q8—21no—
125C3×Q8—21no—
126C3×D4—22no—
127C3×D4—22no—
128C3×D4—42no—
129C3×D4—81no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2411yescharacteristic
135#67—44no—
136#67—44no—
137C2×Dic3—22no—
138C2×Dic3—21no—
139C2×Dic3C2×D412yes—
140D12—21no—
141D12SD1614yescomplemented
142D12#4112yes—
143C4×D3—22no—
144C4×D3—22no—
145C4×D3—42no—
146C4×D3—82no—
147Dic6—81no—
148C3×C8—22no—
149C3⋊C8—21no—
150C3⋊C8—21no—
151C24—62no—
152C2×Q8—61no—
153C2×Q8C22×D311yescharacteristic
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—42no—
168SD16—62no—
169C2×C8—31no—
170C2×C8—31no—
171C2×C8—122no—
172C2×C8C22×D312yes—
173C4⋊C4—21no—
174C4⋊C4—31no—
175C4⋊C4—32no—
176C4⋊C4—31no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—61no—
180C4⋊C4C22×D311yescharacteristic
181C22⋊C4—32no—
182C22⋊C4—62no—
183C22⋊C4—62no—
184C22⋊C4—61no—
185C42—21no—
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—41no—
200C3×C4—41no—
201C3×C4C2×SD1612yes—
202C3×C4C2×#4111yescharacteristic
203C3×C4H(5,𝔽2)11yescharacteristic
204Dic3—22no—
205Dic3—41no—
206Dic3—42no—
207C23—22no—
208C23—32no—
209C23—61no—
210C23—124no—
211C23—124no—
212C23—242no—
213Q8—21no—
214Q8—21no—
215Q8—241no—
216D4—22no—
217D4—22no—
218D4—34no—
219D4—32no—
220D4—42no—
221D4—61no—
222D4—81no—
223D4—124no—
224D4—124no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—32no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—61no—
231C2×C4—62no—
232C2×C4—122no—
233C2×C4—242no—
234C2×C4D3×D411yescharacteristic
235C2×C4D3×D411yescharacteristic
236C2×C4C23×D311yescharacteristic
237C8—22no—
238C8—61no—
239C8—61no—
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—41no—
259C4—41no—
260C4—62no—
261C4—121no—
262C4—122no—
263C4D3×SD1612yes—
264C4#100411yescharacteristic
265C4D3×#4111yescharacteristic
266C3#329511yescharacteristic, complemented, Sylow
267C2—64no—
268C2—82no—
269C2—122no—
270C2#561211yescharacteristic
271C2C2×D3×SD1611yescharacteristic
272C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#460C2
socleC22×C3C22×D4
prime radical#5717C2
hypercenter#460D3
hypocenterC3#3295
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #5579C2semidirect
3. C3×#460C2semidirect
4. #5171C2semidirect
5. #5169C2semidirect
6. #5070C2semidirect
7. C4×#215C2semidirect
8. #4842C2semidirect
9. #4834C2semidirect
10. #4777C2semidirect
11. D3×#102C2semidirect
15. C2×C3×SD16C22semidirect
17. C3×C4×Q8C22semidirect
18. #946C22semidirect
23. C2×D3×C8C22semidirect
25. D3×C4⋊C4C22semidirect
30. #869C22semidirect
31. C3×C4⋊C8C22semidirect
32. C3×#102C22semidirect
33. #835C22semidirect
34. #832C22semidirect
35. #832C22semidirect
37. (C2×D3)⋊C8C22semidirect
38. #817C22semidirect
40. #808C22semidirect
41. C4×C3⋊C8C22semidirect
48. #460D3semidirect
92. C2×SD16C2×D3semidirect
102. C4×Q8C2×D3semidirect
109. C4⋊C8C2×D3semidirect
115. #102C2×D3semidirect
141. D12SD16semidirect
266. C3#3295semidirect

Maximal subgroups

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

subgroupindex
#57172
#55792
C3×#4602
#51712
#51692
#50702
C4×#2152
#48422
#48342
#47772
D3×#1022
#32953