Finite Group Databasebeta

← #79219

Subgroups — #79219

Order 384.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#5525C211yesmaximal, characteristic
2#5504C211yesmaximal, characteristic, complemented
3C3×#428C211yesmaximal, characteristic, complemented
4#5131C211yesmaximal, characteristic, complemented
5#5129C211yesmaximal, characteristic, complemented
6#4998C211yesmaximal, characteristic, complemented
7C2×#809C211yesmaximal, characteristic, complemented
8#4885C212yesmaximal, complemented
9#4882C212yesmaximal, complemented
10#4866C212yesmaximal, complemented
11#4863C212yesmaximal, complemented
12#3096—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C22×D12C2211yescharacteristic
15C2×C3×M16C2211yescharacteristic
16C3×#117C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C4×#67—22no—
19#926C2211yescharacteristic
20C22×C3⋊C8C2211yescharacteristic
21#895C2212yes—
22#895C2212yes—
23#893—21no—
24#892C2212yescomplemented
25D3×C4⋊C4C2212yescomplemented
26#887—21no—
27#882—22no—
28#875—21no—
29#874—21no—
30C3×#107C2212yescomplemented
31C3×#106C2212yescomplemented
32#821C2212yes—
33#821C2212yes—
34(C2×D3)⋊C8C2212yescomplemented
35(C2×D3)⋊C8C2212yescomplemented
36#809C2212yes—
37#809C2212yes—
38#808C2211yescharacteristic, complemented
39#808C2211yescharacteristic, complemented
40#807C2211yescharacteristic, complemented
41#807C2211yescharacteristic, complemented
42#535—31no—
43#520—31no—
44#482—32no—
45#482—32no—
46#481—32no—
47#480—32no—
48#428D311yescharacteristic, complemented
49#427—31no—
50#418—31no—
51C2×#102—31no—
52#408—31no—
53C23×D3—21no—
54C22×C3×C4—21no—
55C22×C3×C4C2311yescharacteristic
56C2×#67D412yes—
57#235—22no—
58#235—22no—
59C2×D12—22no—
60C2×D12—22no—
61C2×D12—21no—
62C2×D12—22no—
63C2×D12—22no—
64C2×D12D411yescharacteristic
65C2×D12C2312yes—
66C2×C4×D3—22no—
67C2×C4×D3C2312yes—
68C2×Dic6D411yescharacteristic
69C3×M16—22no—
70C2×C3×C8C2312yes—
71C3×C4⋊C4—21no—
72C3×C4⋊C4C2311yescharacteristic
73C3×C4⋊C4C2311yescharacteristic
74C3×C4⋊C4C2312yes—
75C3×C22⋊C4—21no—
76C3×C42—21no—
77#217—21no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2311yescharacteristic
83Dic3⋊C4—22no—
84C4×Dic3—21no—
85C4×Dic3—22no—
86C2×C3⋊C8—22no—
87C2×C3⋊C8C2311yescharacteristic
88C2×C3⋊C8C2311yescharacteristic
89C2×#41—31no—
90C22×D4—31no—
91C2×M16C2×D311yescharacteristic
92C22×C8—31no—
93#127—61no—
94#124—61no—
95#121—32no—
96#121—32no—
97#120—62no—
98C4×Q8—61no—
99C4×D4—61no—
100C4×D4—62no—
101#117—31no—
102#117—32no—
103#117C2×D311yescharacteristic
104C2×C4⋊C4—32no—
105C2×C4⋊C4C2×D311yescharacteristic
106#107—31no—
107#107—31no—
108#107C2×D312yescomplemented
109#106—31no—
110#106—31no—
111#106C2×D312yescomplemented
112#102—32no—
113#102—32no—
114#102—32no—
115#102—32no—
116C22⋊C8—32no—
117C22⋊C8—32no—
118C23×C3C2×D411yescharacteristic
119C22×D3—22no—
120C22×D3—42no—
121C22×D3—81no—
122C22×D3—81no—
123C22×D3C2×D412yes—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—81no—
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4#4112yes—
130C2×C3×C4#4112yes—
131C2×C3×C4C2411yescharacteristic
132#67—24no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3C2×D412yes—
136D12—22no—
137D12—22no—
138D12—22no—
139D12—42no—
140D12—82no—
141D12—82no—
142C4×D3—22no—
143C4×D3—22no—
144C4×D3—82no—
145Dic6—22no—
146C3×C8—22no—
147C3⋊C8—21no—
148C3⋊C8—21no—
149C24—61no—
150#41—62no—
151#41—62no—
152C2×Q8—31no—
153C2×D4—32no—
154C2×D4—31no—
155C2×D4—32no—
156C2×D4—62no—
157C2×D4—62no—
158C2×D4—62no—
159C2×D4—61no—
160C2×D4—62no—
161C22×C4—21no—
162C22×C4—32no—
163C22×C4—62no—
164C22×C4C22×D311yescharacteristic
165M16—22no—
166C2×C8—31no—
167C2×C8—31no—
168C2×C8—62no—
169C2×C8C22×D312yes—
170C4⋊C4—21no—
171C4⋊C4—31no—
172C4⋊C4—31no—
173C4⋊C4—62no—
174C4⋊C4C22×D312yes—
175C4⋊C4C22×D311yescharacteristic
176C4⋊C4C22×D311yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—62no—
179C22⋊C4—62no—
180C22⋊C4—62no—
181C22⋊C4—61no—
182C42—21no—
183C42—62no—
184C42—61no—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3#12312yes—
188C22×C3C22×D411yescharacteristic
189C2×D3—22no—
190C2×D3—22no—
191C2×D3—22no—
192C2×D3—42no—
193C2×D3—42no—
194C2×D3—42no—
195C2×D3—84no—
196C3×C4—42no—
197C3×C4—41no—
198C3×C4—41no—
199C3×C4#12312yes—
200C3×C4C2×#4111yescharacteristic
201C3×C4C2×#4111yescharacteristic
202Dic3—22no—
203Dic3—41no—
204Dic3—41no—
205C23—32no—
206C23—62no—
207C23—122no—
208C23—241no—
209C23—241no—
210C23D3×D411yescharacteristic
211Q8—62no—
212D4—64no—
213D4—62no—
214D4—62no—
215D4—62no—
216D4—122no—
217D4—242no—
218D4—242no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—32no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—81no—
228C2×C4—242no—
229C2×C4#23512yes—
230C2×C4D3×D411yescharacteristic
231C2×C4#23912yes—
232C2×C4C23×D311yescharacteristic
233C8—22no—
234C8—61no—
235C8—61no—
236C2×C3—22no—
237C2×C3C2×#12311yescharacteristic
238C2×C3#57611yescharacteristic
239C2×C3#57711yescharacteristic
240D3—42no—
241D3—82no—
242C22—21no—
243C22—21no—
244C22—62no—
245C22—62no—
246C22—62no—
247C22—122no—
248C22—122no—
249C22—122no—
250C22—244no—
251C22#89412yes—
252C22C2×D3×D411yescharacteristic
253C4—42no—
254C4—41no—
255C4—41no—
256C4—62no—
257C4—121no—
258C4—121no—
259C4#89412yes—
260C4C2×#23511yescharacteristic
261C4C2×#23911yescharacteristic
262C3#309611yescharacteristic, complemented, Sylow
263C2—22no—
264C2—122no—
265C2—242no—
266C2C2×#89411yescharacteristic
267C2#577411yescharacteristic
268C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
prime radical#5525C2
hypercenter#428D3
hypocenterC3#3096
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #5504C2semidirect
3. C3×#428C2semidirect
4. #5131C2semidirect
5. #5129C2semidirect
6. #4998C2semidirect
7. C2×#809C2semidirect
8. #4885C2semidirect
9. #4882C2semidirect
10. #4866C2semidirect
11. #4863C2semidirect
24. #892C22semidirect
25. D3×C4⋊C4C22semidirect
30. C3×#107C22semidirect
31. C3×#106C22semidirect
34. (C2×D3)⋊C8C22semidirect
35. (C2×D3)⋊C8C22semidirect
38. #808C22semidirect
39. #808C22semidirect
40. #807C22semidirect
41. #807C22semidirect
48. #428D3semidirect
108. #107C2×D3semidirect
111. #106C2×D3semidirect
262. C3#3096semidirect

Maximal subgroups

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

subgroupindex
#55252
#55042
C3×#4282
#51312
#51292
#49982
C2×#8092
#48852
#48822
#48662
#48632
#30963