Finite Group Databasebeta

← #80356

Subgroups — #80356

Order 384.

460 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 115 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, complemented
2#5563C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5159C212yesmaximal, complemented
5#5156C212yesmaximal, complemented
6#5070C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic, complemented
8#4842C211yesmaximal, characteristic, complemented
9#4838C211yesmaximal, characteristic, complemented
10#4779C212yesmaximal, complemented
11#4770C212yesmaximal, complemented
12#3325—31nomaximal, Sylow
13C2×D3×D4—21no—
14C2×D3×D4C2212yes—
15C2×C3×D8C2212yescomplemented
16C3×#127C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic, complemented
18#938—21no—
19#938C2212yes—
20#938C2212yescomplemented
21#937—22no—
22C2×#213C2211yescharacteristic
23#930—21no—
24#926—21no—
25C2×ΓL(1,𝔽25)C2212yescomplemented
26#892C2212yescomplemented
27#885—22no—
28C4×D12C2211yescharacteristic
29C4×D12C2211yescharacteristic
30#869C2211yescharacteristic, complemented
31C3×C4⋊C8C2211yescharacteristic, complemented
32C3×#102C2212yescomplemented
33#832C2212yescomplemented
34#832C2211yescharacteristic, complemented
35#832C2212yescomplemented
36#821C2212yes—
37(C2×D3)⋊C8C2212yescomplemented
38#817C2212yescomplemented
39#809C2211yescharacteristic
40#807C2211yescharacteristic, complemented
41C4×C3⋊C8C2211yescharacteristic, complemented
42#555—31no—
43D42—31no—
44#486—31no—
45#482—32no—
46#476—31no—
47#469—32no—
48#459D311yescharacteristic, complemented
49#447—32no—
50C4×D8—31no—
51#436—31no—
52#418—32no—
53C23×D3—22no—
54C2×C3×D4—21no—
55C2×C3×D4—21no—
56C2×C3×D4C2312yes—
57C2×C3×D4C2311yescharacteristic
58C22×C3×C4—21no—
59C2×#67—22no—
60C2×#67—22no—
61C2×#67—22no—
62D3×D4—24no—
63D3×D4—42no—
64D3×D4—44no—
65C2×D12D411yescharacteristic
66C2×D12C2311yescharacteristic
67C2×C4×D3—21no—
68C2×C4×D3C2312yes—
69C2×Dic6—21no—
70C3×D8—42no—
71C2×C3×C8C2312yes—
72C3×C4⋊C4C2311yescharacteristic
73C3×C22⋊C4—21no—
74C3×C42C2311yescharacteristic
75#217—22no—
76#217—22no—
77#213—22no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—21no—
80(C2×D3)⋊C4D412yes—
81C4⋊Dic3—21no—
82C4⋊Dic3D411yescharacteristic
83C4⋊Dic3C2311yescharacteristic
84C4⋊Dic3C2312yes—
85C4×Dic3—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88ΓL(1,𝔽25)—42no—
89C22×D4—32no—
90C22×D4—61no—
91C2×D8—31no—
92C2×D8C2×D312yescomplemented
93C2×M16—32no—
94C4⋊Q8—31no—
95#127C2×D311yescharacteristic
96#124—62no—
97#121—32no—
98#121—32no—
99#121—61no—
100#121—61no—
101#120—62no—
102C4×D4—31no—
103C4×D4—31no—
104C4×D4C2×D311yescharacteristic, complemented
105#117—32no—
106#117—61no—
107#107—31no—
108#106—32no—
109C4⋊C8C2×D311yescharacteristic, complemented
110#102—32no—
111#102—31no—
112#102—32no—
113#102—32no—
114#102—31no—
115#102C2×D312yescomplemented
116C22⋊C8—32no—
117C4×C8—31no—
118C23×C3—21no—
119C23×C3—22no—
120C22×D3—21no—
121C22×D3—44no—
122C22×D3—44no—
123C22×D3—82no—
124C22×D3C2×D412yes—
125C3×D4—21no—
126C3×D4—22no—
127C3×D4—21no—
128C3×D4—22no—
129C3×D4—42no—
130C3×D4—81no—
131C2×C3×C4—21no—
132C2×C3×C4—42no—
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2411yescharacteristic
136#67—44no—
137#67—44no—
138#67—82no—
139C2×Dic3—22no—
140C2×Dic3—21no—
141C2×Dic3C2×D412yes—
142D12—22no—
143D12—21no—
144D12#4112yes—
145C4×D3—22no—
146C4×D3—22no—
147C4×D3—42no—
148Dic6—81no—
149C3×C8—22no—
150C3⋊C8—21no—
151C3⋊C8—21no—
152C24—62no—
153C2×Q8—61no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—31no—
157C2×D4—31no—
158C2×D4—62no—
159C2×D4—64no—
160C2×D4—62no—
161C2×D4—62no—
162C2×D4—122no—
163C2×D4—124no—
164C2×D4C22×D312yes—
165C2×D4C22×D311yescharacteristic
166C22×C4—21no—
167C22×C4—32no—
168C22×C4—61no—
169D8—42no—
170D8—62no—
171M16—122no—
172C2×C8—31no—
173C2×C8—31no—
174C2×C8C22×D312yes—
175C4⋊C4—31no—
176C4⋊C4—32no—
177C4⋊C4—31no—
178C4⋊C4—61no—
179C4⋊C4C22×D311yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—32no—
182C22⋊C4—62no—
183C22⋊C4—62no—
184C22⋊C4—62no—
185C22⋊C4—61no—
186C42—62no—
187C42C22×D311yescharacteristic
188C22×C3—42no—
189C22×C3—41no—
190C22×C3—42no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3—42no—
194C22×C3C22×D411yescharacteristic
195C2×D3—22no—
196C2×D3—22no—
197C2×D3—22no—
198C2×D3—22no—
199C2×D3—41no—
200C2×D3—41no—
201C2×D3—88no—
202C3×C4—21no—
203C3×C4—41no—
204C3×C4AGL(1,ℤ/8)12yes—
205C3×C4C2×#4111yescharacteristic
206C3×C4H(5,𝔽2)11yescharacteristic
207Dic3—22no—
208Dic3—41no—
209Dic3—42no—
210C23—21no—
211C23—22no—
212C23—32no—
213C23—61no—
214C23—124no—
215C23—124no—
216C23—242no—
217Q8—241no—
218D4—21no—
219D4—22no—
220D4—21no—
221D4—22no—
222D4—32no—
223D4—42no—
224D4—62no—
225D4—61no—
226D4—81no—
227D4—124no—
228D4—124no—
229D4—242no—
230C2×C4—21no—
231C2×C4—32no—
232C2×C4—42no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—61no—
237C2×C4—122no—
238C2×C4D3×D411yescharacteristic
239C2×C4D3×D411yescharacteristic
240C2×C4C23×D311yescharacteristic
241C8—22no—
242C8—61no—
243C8—61no—
244C2×C3—82no—
245C2×C3—81no—
246C2×C3#54611yescharacteristic
247C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
248C2×C3#57611yescharacteristic
249D3—42no—
250D3—42no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—41no—
255C22—42no—
256C22—41no—
257C22—62no—
258C22—62no—
259C22—62no—
260C22—62no—
261C22—121no—
262C22—121no—
263C22—248no—
264C22C2×D3×D411yescharacteristic
265C4—21no—
266C4—41no—
267C4—62no—
268C4—121no—
269C4—122no—
270C4#91112yes—
271C4#100411yescharacteristic
272C4D3×#4111yescharacteristic
273C3#332511yescharacteristic, complemented, Sylow
274C2—82no—
275C2—81no—
276C2—122no—
277C2—122no—
278C2#561211yescharacteristic
279C2C2×#91111yescharacteristic
280C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3325
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5717C2semidirect
2. #5563C2semidirect
3. C3×#459C2semidirect
4. #5159C2semidirect
5. #5156C2semidirect
6. #5070C2semidirect
7. C4×#213C2semidirect
8. #4842C2semidirect
9. #4838C2semidirect
10. #4779C2semidirect
11. #4770C2semidirect
15. C2×C3×D8C22semidirect
17. C3×C4×D4C22semidirect
20. #938C22semidirect
25. C2×ΓL(1,𝔽25)C22semidirect
26. #892C22semidirect
30. #869C22semidirect
31. C3×C4⋊C8C22semidirect
32. C3×#102C22semidirect
33. #832C22semidirect
34. #832C22semidirect
35. #832C22semidirect
37. (C2×D3)⋊C8C22semidirect
38. #817C22semidirect
40. #807C22semidirect
41. C4×C3⋊C8C22semidirect
48. #459D3semidirect
92. C2×D8C2×D3semidirect
104. C4×D4C2×D3semidirect
109. C4⋊C8C2×D3semidirect
115. #102C2×D3semidirect
273. C3#3325semidirect

Maximal subgroups

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

subgroupindex
#57172
#55632
C3×#4592
#51592
#51562
#50702
C4×#2132
#48422
#48382
#47792
#47702
#33253