Finite Group Databasebeta

← #80281

Subgroups — #80281

Order 384.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#5836C212yesmaximal, complemented
2#5775C212yesmaximal, complemented
3C3×#453C211yesmaximal, characteristic, complemented
4#5200C212yesmaximal, complemented
5#5077C211yesmaximal, characteristic, complemented
6#5076C211yesmaximal, characteristic, complemented
7#4826C212yesmaximal, complemented
8#4824C212yesmaximal, complemented
9#4748C212yesmaximal, complemented
10#3291—31nomaximal, Sylow
11C3×H(5,𝔽2)C2211yescharacteristic, complemented
12#1010C2212yes—
13#1004C2211yescharacteristic, complemented
14C3×AGL(1,ℤ/8)C2212yes—
15C3×#127C2211yescharacteristic
16#951—22no—
17#951C2212yescomplemented
18#950—22no—
19#950C2212yes—
20#948C2212yescomplemented
21C2×#214—22no—
22C2×#214C2212yescomplemented
23C2×#214C2212yescomplemented
24#932—22no—
25#932C2212yes—
26#915—22no—
27D3×SD16—22no—
28#912—22no—
29#911—22no—
30#909C2212yescomplemented
31#907C2212yescomplemented
32#869C2211yescharacteristic, complemented
33C3×#104C2212yescomplemented
34C3×#100C2211yescharacteristic, complemented
35#833C2211yescharacteristic, complemented
36#824C2212yescomplemented
37#822C2212yescomplemented
38#805C2212yescomplemented
39C4×C3⋊C8C2211yescharacteristic, complemented
40#577—32no—
41#577—32no—
42#492—31no—
43#471—32no—
44#471—32no—
45#456—32no—
46#453—31no—
47#453D311yescharacteristic, complemented
48#443—32no—
49C3×#41—22no—
50C3×#41C2312yes—
51C2×C3×D4—22no—
52C2×C3×D4—41no—
53C2×C3×D4—41no—
54C2×C3×D4C2311yescharacteristic
55C2×C3×D4C2311yescharacteristic
56C2×#67—41no—
57D3×Q8—22no—
58#237—22no—
59#237—22no—
60#237—42no—
61D3×D4—22no—
62#235—22no—
63#235C2312yes—
64C2×Dic6—22no—
65C2×Dic6C2312yes—
66C3×SD16—22no—
67C3×D8—22no—
68C3×M16C2312yes—
69C3×C42C2311yescharacteristic
70#216—22no—
71#216D412yescomplemented
72#215D412yescomplemented
73#214—22no—
74#214—22no—
75#214—22no—
76#214—22no—
77#214—22no—
78#214—22no—
79#214D412yescomplemented
80#213—22no—
81#213D412yescomplemented
82C4⋊Dic3—41no—
83#208—22no—
84#208C2312yes—
85C2×C3⋊C8—22no—
86C2×C3⋊C8C2312yes—
87Dic12—22no—
88#204—22no—
89ΓL(1,𝔽25)—22no—
90D3×C8—22no—
912−1+4—32no—
92H(5,𝔽2)—31no—
93H(5,𝔽2)C2×D311yescharacteristic, complemented
94#137—32no—
95#137—32no—
96#137—62no—
97#137—62no—
98AGL(1,ℤ/8)—32no—
99AGL(1,ℤ/8)—62no—
100AGL(1,ℤ/8)—62no—
101AGL(1,ℤ/8)C2×D312yes—
102#135—32no—
103#135—62no—
104#135—62no—
105C2×SD16—32no—
106C2×SD16—32no—
107C2×SD16—62no—
108C2×SD16—62no—
109#131—32no—
110#131—32no—
111C4⋊Q8—31no—
112#127C2×D311yescharacteristic
113#108—32no—
114#104—32no—
115#104C2×D312yescomplemented
116#101—32no—
117#100—31no—
118#100C2×D311yescharacteristic, complemented
119C4×C8—31no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—41no—
123C22×D3—41no—
124C3×Q8C2×D412yes—
125C3×D4—22no—
126C3×D4—22no—
127C3×D4—22no—
128C3×D4—42no—
129C3×D4—82no—
130C3×D4C2×D412yes—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—41no—
134C2×C3×C4C2411yescharacteristic
135#67—22no—
136#67—42no—
137#67—42no—
138C2×Dic3—22no—
139C2×Dic3—41no—
140C2×Dic3—42no—
141D12C2×D412yes—
142C4×D3—22no—
143C4×D3—42no—
144Dic6—22no—
145Dic6—22no—
146Dic6—42no—
147Dic6C2×D412yes—
148C3×C8—22no—
149C3⋊C8—22no—
150C3⋊C8C2×D412yes—
151C3⋊C8C2×D412yes—
152#41—22no—
153#41—32no—
154#41—62no—
155#41—62no—
156#41—62no—
157#41—122no—
158#41C22×D312yes—
159C2×Q8—32no—
160C2×Q8—62no—
161C2×Q8—62no—
162C2×D4—22no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—62no—
166C2×D4—121no—
167C2×D4C22×D311yescharacteristic
168C2×D4C22×D311yescharacteristic
169Q16—32no—
170Q16—62no—
171Q16—62no—
172SD16—22no—
173SD16—32no—
174SD16—32no—
175SD16—62no—
176SD16—62no—
177SD16—62no—
178SD16—62no—
179SD16—62no—
180SD16—62no—
181SD16—62no—
182D8—22no—
183D8—32no—
184D8—62no—
185M16—32no—
186M16—62no—
187M16—62no—
188M16C22×D312yes—
189C2×C8—32no—
190C2×C8—62no—
191C2×C8—62no—
192C4⋊C4—121no—
193C42C22×D311yescharacteristic
194C22×C3—22no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—81no—
198C22×C3—81no—
199C22×C3—161no—
200C22×C3H(5,𝔽2)11yescharacteristic
201C2×D3—22no—
202C2×D3—161no—
203C3×C4—22no—
204C3×C4—22no—
205C3×C4C22×D412yes—
206Dic3—22no—
207Dic3—42no—
208C23—21no—
209C23—21no—
210C23—41no—
211C23—121no—
212Q8—32no—
213Q8—62no—
214Q8—62no—
215Q8—122no—
216Q8D3×D412yes—
217D4—22no—
218D4—22no—
219D4—22no—
220D4—32no—
221D4—42no—
222D4—62no—
223D4—82no—
224D4—122no—
225D4—122no—
226D4D3×D412yes—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—122no—
233C2×C4—121no—
234C2×C4—122no—
235C2×C4C23×D311yescharacteristic
236C8—22no—
237C8—32no—
238C8—32no—
239C8—62no—
240C2×C3—21no—
241C2×C3—42no—
242C2×C3—81no—
243C2×C3—81no—
244C2×C3D4211yescharacteristic
245D3—42no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—62no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—481no—
254C22#100411yescharacteristic
255C4—22no—
256C4—22no—
257C4—62no—
258C4—122no—
259C4C2×D3×D412yes—
260C3#329111yescharacteristic, complemented, Sylow
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—122no—
266C2#559011yescharacteristic
subgroupG/H
centerC2#5590
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#453C2
socleC2×C3D42
hypercenter#453D3
hypocenterC3#3291
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #5836C2semidirect
2. #5775C2semidirect
3. C3×#453C2semidirect
4. #5200C2semidirect
5. #5077C2semidirect
6. #5076C2semidirect
7. #4826C2semidirect
8. #4824C2semidirect
9. #4748C2semidirect
11. C3×H(5,𝔽2)C22semidirect
13. #1004C22semidirect
17. #951C22semidirect
20. #948C22semidirect
22. C2×#214C22semidirect
23. C2×#214C22semidirect
30. #909C22semidirect
31. #907C22semidirect
32. #869C22semidirect
33. C3×#104C22semidirect
34. C3×#100C22semidirect
35. #833C22semidirect
36. #824C22semidirect
37. #822C22semidirect
38. #805C22semidirect
39. C4×C3⋊C8C22semidirect
47. #453D3semidirect
71. #216D4semidirect
72. #215D4semidirect
79. #214D4semidirect
81. #213D4semidirect
93. H(5,𝔽2)C2×D3semidirect
115. #104C2×D3semidirect
118. #100C2×D3semidirect
260. C3#3291semidirect

Maximal subgroups

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

subgroupindex
#58362
#57752
C3×#4532
#52002
#50772
#50762
#48262
#48242
#47482
#32913