Finite Group Databasebeta

← #81644

Subgroups — #81644

Order 384.

420 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 109 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#5697C211yesmaximal, characteristic, complemented
2#5674C211yesmaximal, characteristic, complemented
3C3×#488C211yesmaximal, characteristic, complemented
4#5067C211yesmaximal, characteristic, complemented
5#5058C211yesmaximal, characteristic, complemented
6#4819C212yesmaximal, complemented
7#4810C212yesmaximal, complemented
8#4786C212yesmaximal, complemented
9#4773C212yesmaximal, complemented
10#4695C211yesmaximal, characteristic, complemented
11#4690C211yesmaximal, characteristic, complemented
12#3220—31nomaximal, Sylow
13C2×#239C2211yescharacteristic
14C2×D3×D4C2211yescharacteristic
15C3×#125C2211yescharacteristic, complemented
16C3×#124C2211yescharacteristic
17#947C2211yescharacteristic
18C2×#215C2212yescomplemented
19#940C2211yescharacteristic
20C2×#213C2212yes—
21C2×D24C2212yes—
22C2×D24C2212yescomplemented
23#895—22no—
24#895C2212yes—
25#894—22no—
26#893C2212yescomplemented
27#884—22no—
28#882—22no—
29#874C2211yescharacteristic
30#874C2211yescharacteristic, complemented
31#872C2211yescharacteristic
32C3×#103C2212yescomplemented
33C3×#102C2212yes—
34C3×C4×C8C2211yescharacteristic, complemented
35(C2×D3)⋊C8C2212yes—
36Dic3⋊C8C2212yescomplemented
37#809C2212yes—
38#809C2212yescomplemented
39#803C2211yescharacteristic, complemented
40#540—31no—
41#535—31no—
42#496—31no—
43#493—31no—
44#490—31no—
45#488D311yescharacteristic, complemented
46#463—32no—
47#459—32no—
48#449—32no—
49#447—32no—
50#433—31no—
51C23×D3—21no—
52C2×C3×Q8C2311yescharacteristic
53C2×C3×D4C2311yescharacteristic
54C2×#67—21no—
55C2×#67—22no—
56#239—42no—
57D3×D4—42no—
58C2×D12—21no—
59C2×D12—22no—
60C2×D12—22no—
61C2×D12—22no—
62C2×D12C2311yescharacteristic
63C2×D12C2311yescharacteristic
64C2×D12C2312yes—
65C2×C4×D3—21no—
66C2×C4×D3—22no—
67C2×C4×D3C2311yescharacteristic
68C2×C3×C8C2312yes—
69C3×C4⋊C4—22no—
70C3×C4⋊C4C2312yes—
71C3×C22⋊C4—22no—
72C3×C42C2311yescharacteristic
73#215—42no—
74#213—42no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4D412yes—
79Dic3⋊C4D412yes—
80C4×Dic3C2311yescharacteristic
81C2×C3⋊C8C2312yes—
82D24—42no—
83D24—42no—
84C2×#41—31no—
85C22×D4—31no—
86C2×SD16—32no—
87C2×D8—32no—
88C2×D8—32no—
89C2×D8—32no—
90#127—31no—
91#127—31no—
92#127—31no—
93#125C2×D311yescharacteristic, complemented
94#124—31no—
95#124C2×D311yescharacteristic
96#123—62no—
97#121—32no—
98#121—62no—
99#121—62no—
100#120—62no—
101C4×D4—32no—
102#117—31no—
103C4⋊C8—32no—
104#103C2×D312yescomplemented
105#102—32no—
106#102—32no—
107#102C2×D312yes—
108C22⋊C8—32no—
109#97—31no—
110C4×C8C2×D311yescharacteristic, complemented
111C23×C3—21no—
112C22×D3—22no—
113C22×D3—21no—
114C22×D3—21no—
115C22×D3—42no—
116C22×D3—82no—
117C22×D3C2×D411yescharacteristic
118C3×Q8—22no—
119C3×D4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—22no—
122C2×C3×C4C2×D412yes—
123C2×C3×C4C2411yescharacteristic
124#67—42no—
125#67—82no—
126C2×Dic3C2×D412yes—
127C2×Dic3C2×D411yescharacteristic
128D12—22no—
129D12—22no—
130D12—22no—
131D12—22no—
132D12—42no—
133D12—82no—
134D12—82no—
135D12—82no—
136C4×D3—22no—
137C4×D3—42no—
138C4×D3—82no—
139C3×C8—22no—
140C3⋊C8—22no—
141C24—61no—
142#41—122no—
143C2×Q8C22×D311yescharacteristic
144C2×D4—31no—
145C2×D4—31no—
146C2×D4—32no—
147C2×D4—61no—
148C2×D4—62no—
149C2×D4—61no—
150C2×D4—62no—
151C2×D4—62no—
152C2×D4—62no—
153C2×D4—122no—
154C2×D4C22×D311yescharacteristic
155C22×C4—31no—
156C22×C4—61no—
157C22×C4—62no—
158SD16—122no—
159D8—122no—
160D8—122no—
161D8—122no—
162C2×C8—32no—
163C2×C8C22×D312yes—
164C4⋊C4—22no—
165C4⋊C4—32no—
166C4⋊C4C22×D312yes—
167C22⋊C4—22no—
168C22⋊C4—32no—
169C22⋊C4—62no—
170C22⋊C4—62no—
171C22⋊C4—62no—
172C42—31no—
173C42C22×D311yescharacteristic
174C22×C3—41no—
175C22×C3—42no—
176C22×C3C22×D411yescharacteristic
177C2×D3—21no—
178C2×D3—22no—
179C2×D3—41no—
180C2×D3—41no—
181C2×D3—42no—
182C2×D3—42no—
183C2×D3—42no—
184C2×D3—42no—
185C2×D3—42no—
186C2×D3—84no—
187C3×C4—22no—
188C3×C4—41no—
189C3×C4—42no—
190C3×C4H(5,𝔽2)12yes—
191Dic3—21no—
192Dic3—22no—
193C23—21no—
194C23—31no—
195C23—62no—
196C23—61no—
197C23—61no—
198C23—122no—
199C23—242no—
200Q8—22no—
201D4—22no—
202D4—62no—
203D4—62no—
204D4—62no—
205D4—62no—
206D4—122no—
207D4—122no—
208D4—242no—
209D4—242no—
210D4—242no—
211D4—242no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—32no—
215C2×C4—31no—
216C2×C4—62no—
217C2×C4—122no—
218C2×C4—242no—
219C2×C4D3×D412yes—
220C2×C4C23×D311yescharacteristic
221C8—22no—
222C8—62no—
223C2×C3—81no—
224C2×C3#53511yescharacteristic
225C2×C3#57612yes—
226D3—41no—
227D3—81no—
228D3—82no—
229D3—81no—
230C22—41no—
231C22—42no—
232C22—61no—
233C22—62no—
234C22—121no—
235C22—121no—
236C22—122no—
237C22—122no—
238C22—122no—
239C22—122no—
240C22—122no—
241C22—244no—
242C22C2×D3×D411yescharacteristic
243C4—22no—
244C4—41no—
245C4—42no—
246C4—61no—
247C4—62no—
248C4#100912yes—
249C3#322011yescharacteristic, complemented, Sylow
250C2—81no—
251C2—121no—
252C2—241no—
253C2—241no—
254C2—242no—
255C2#567411yescharacteristic
256C2#576912yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#488C2
socleC22×C3C22×D4
hypercenter#488D3
hypocenterC3#3220
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #5697C2semidirect
2. #5674C2semidirect
3. C3×#488C2semidirect
4. #5067C2semidirect
5. #5058C2semidirect
6. #4819C2semidirect
7. #4810C2semidirect
8. #4786C2semidirect
9. #4773C2semidirect
10. #4695C2semidirect
11. #4690C2semidirect
15. C3×#125C22semidirect
18. C2×#215C22semidirect
22. C2×D24C22semidirect
26. #893C22semidirect
30. #874C22semidirect
32. C3×#103C22semidirect
34. C3×C4×C8C22semidirect
36. Dic3⋊C8C22semidirect
38. #809C22semidirect
39. #803C22semidirect
45. #488D3semidirect
93. #125C2×D3semidirect
104. #103C2×D3semidirect
110. C4×C8C2×D3semidirect
249. C3#3220semidirect

Maximal subgroups

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

subgroupindex
#56972
#56742
C3×#4882
#50672
#50582
#48192
#48102
#47862
#47732
#46952
#46902
#32203