Finite Group Databasebeta

← #81719

Subgroups — #81719

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#5715C211yesmaximal, characteristic, complemented
2#5697C211yesmaximal, characteristic, complemented
3C3×#490C211yesmaximal, characteristic, complemented
4#5073C211yesmaximal, characteristic, complemented
5#5068C211yesmaximal, characteristic, complemented
6#4787C212yesmaximal, complemented
7#4786C212yesmaximal, complemented
8#4781C212yesmaximal, complemented
9#4773C212yesmaximal, complemented
10#4712C211yesmaximal, characteristic, complemented
11#4709C211yesmaximal, characteristic, complemented
12#3220—31nomaximal, Sylow
13C2×#237C2211yescharacteristic
14C2×D3×D4C2211yescharacteristic
15C3×#127C2211yescharacteristic
16C3×#125C2211yescharacteristic, complemented
17#940C2211yescharacteristic
18#939—22no—
19#937—22no—
20#936C2211yescharacteristic
21C2×#213C2212yes—
22C2×#213C2212yescomplemented
23C2×D24C2212yes—
24C2×#204C2212yescomplemented
25#895—22no—
26#895C2212yes—
27#894—22no—
28#893C2212yescomplemented
29#875C2211yescharacteristic
30#874C2211yescharacteristic, complemented
31#872C2211yescharacteristic
32C3×#102C2212yes—
33C3×#102C2212yescomplemented
34C3×#97C2211yescharacteristic, complemented
35(C2×D3)⋊C8C2212yes—
36Dic3⋊C8C2212yescomplemented
37#810C2212yescomplemented
38#809C2212yes—
39C4×C3⋊C8C2211yescharacteristic, complemented
40#540—31no—
41#535—31no—
42#496—31no—
43#493—31no—
44#490D311yescharacteristic, complemented
45#488—31no—
46#463—32no—
47#459—32no—
48#449—32no—
49#447—32no—
50#433—31no—
51C23×D3—21no—
52C2×C3×D4—22no—
53C2×C3×D4C2311yescharacteristic
54C2×C3×D4C2311yescharacteristic
55C2×#67—21no—
56C2×#67—21no—
57C2×#67—22no—
58C22×Dic3—21no—
59#237—42no—
60D3×D4—42no—
61C2×D12—22no—
62C2×D12—22no—
63C2×D12C2311yescharacteristic
64C2×D12C2312yes—
65C2×C4×D3—22no—
66C2×C4×D3C2311yescharacteristic
67C2×Dic6C2311yescharacteristic
68C2×C3×C8C2312yes—
69C3×C4⋊C4—22no—
70C3×C4⋊C4C2312yes—
71C3×C42C2311yescharacteristic
72#217—22no—
73#213—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—
83#204—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#127C2×D311yescharacteristic
93#125C2×D311yescharacteristic, complemented
94#124—31no—
95#124—31no—
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#103—32no—
105#102—32no—
106#102C2×D312yes—
107#102C2×D312yescomplemented
108C22⋊C8—32no—
109#97C2×D311yescharacteristic, complemented
110C4×C8—31no—
111C23×C3—21no—
112C23×C3—21no—
113C22×D3—22no—
114C22×D3—21no—
115C22×D3—42no—
116C22×D3—82no—
117C22×D3C2×D411yescharacteristic
118C3×D4—22no—
119C3×D4—22no—
120C3×D4—82no—
121C2×C3×C4—22no—
122C2×C3×C4C2×D412yes—
123C2×C3×C4C2411yescharacteristic
124#67—42no—
125#67—42no—
126#67—82no—
127C2×Dic3—21no—
128C2×Dic3—42no—
129C2×Dic3C2×D412yes—
130C2×Dic3C2×D411yescharacteristic
131D12—22no—
132D12—22no—
133D12—22no—
134D12—82no—
135D12—82no—
136C4×D3—22no—
137C4×D3—82no—
138Dic6—22no—
139C3×C8—22no—
140C3⋊C8—22no—
141C24—61no—
142#41—122no—
143C2×Q8—31no—
144C2×D4—22no—
145C2×D4—31no—
146C2×D4—32no—
147C2×D4—62no—
148C2×D4—61no—
149C2×D4—62no—
150C2×D4—61no—
151C2×D4—62no—
152C2×D4—122no—
153C2×D4C22×D311yescharacteristic
154C2×D4C22×D311yescharacteristic
155C22×C4—31no—
156C22×C4—62no—
157C22×C4—61no—
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—32no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C22⋊C4—62no—
171C22⋊C4—62no—
172C42—31no—
173C42C22×D311yescharacteristic
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—42no—
177C22×C3—42no—
178C22×C3C22×D411yescharacteristic
179C2×D3—21no—
180C2×D3—22no—
181C2×D3—41no—
182C2×D3—42no—
183C2×D3—42no—
184C2×D3—42no—
185C2×D3—42no—
186C2×D3—84no—
187C3×C4—22no—
188C3×C4—42no—
189C3×C4H(5,𝔽2)12yes—
190Dic3—21no—
191Dic3—22no—
192Dic3—41no—
193C23—21no—
194C23—21no—
195C23—31no—
196C23—62no—
197C23—61no—
198C23—122no—
199C23—242no—
200Q8—62no—
201D4—22no—
202D4—22no—
203D4—62no—
204D4—62no—
205D4—62no—
206D4—82no—
207D4—122no—
208D4—122no—
209D4—242no—
210D4—242no—
211D4—242no—
212C2×C4—22no—
213C2×C4—32no—
214C2×C4—31no—
215C2×C4—62no—
216C2×C4—61no—
217C2×C4—122no—
218C2×C4—242no—
219C2×C4D3×D412yes—
220C2×C4C23×D311yescharacteristic
221C8—22no—
222C8—62no—
223C2×C3—81no—
224C2×C3—81no—
225C2×C3#53511yescharacteristic
226C2×C3#57612yes—
227D3—41no—
228D3—82no—
229D3—81no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—42no—
234C22—61no—
235C22—62no—
236C22—121no—
237C22—122no—
238C22—122no—
239C22—122no—
240C22—122no—
241C22—244no—
242C22C2×D3×D411yescharacteristic
243C4—22no—
244C4—42no—
245C4—61no—
246C4—62no—
247C4—121no—
248C4#100912yes—
249C3#322011yescharacteristic, complemented, Sylow
250C2—81no—
251C2—81no—
252C2—121no—
253C2—241no—
254C2—242no—
255C2#567411yescharacteristic
256C2#577412yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#490C2
socleC22×C3C22×D4
hypercenter#490D3
hypocenterC3#3220
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #5715C2semidirect
2. #5697C2semidirect
3. C3×#490C2semidirect
4. #5073C2semidirect
5. #5068C2semidirect
6. #4787C2semidirect
7. #4786C2semidirect
8. #4781C2semidirect
9. #4773C2semidirect
10. #4712C2semidirect
11. #4709C2semidirect
16. C3×#125C22semidirect
22. C2×#213C22semidirect
24. C2×#204C22semidirect
28. #893C22semidirect
30. #874C22semidirect
33. C3×#102C22semidirect
34. C3×#97C22semidirect
36. Dic3⋊C8C22semidirect
37. #810C22semidirect
39. C4×C3⋊C8C22semidirect
44. #490D3semidirect
93. #125C2×D3semidirect
107. #102C2×D3semidirect
109. #97C2×D3semidirect
249. C3#3220semidirect

Maximal subgroups

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

subgroupindex
#57152
#56972
C3×#4902
#50732
#50682
#47872
#47862
#47812
#47732
#47122
#47092
#32203