Finite Group Databasebeta

← #81016

Subgroups — #81016

Order 384.

332 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 113 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#5728C211yesmaximal, characteristic, complemented
2#5580C211yesmaximal, characteristic
3C3×#475C211yesmaximal, characteristic, complemented
4#5087C211yesmaximal, characteristic, complemented
5#5021C211yesmaximal, characteristic, complemented
6#4866C212yesmaximal
7#4858C212yesmaximal, complemented
8#4842C211yesmaximal, characteristic, complemented
9#4833C211yesmaximal, characteristic, complemented
10#4815C212yesmaximal, complemented
11#4800C212yesmaximal, complemented
12#3233—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4×Q8C2211yescharacteristic
15#947C2211yescharacteristic
16#946—21no—
17#946C2211yescharacteristic
18Q8×Dic3—21no—
19#898—21no—
20#897—21no—
21#895C2211yescharacteristic
22#894—21no—
23#892C2212yes—
24#890C2212yes—
25#890C2211yescharacteristic
26#887—21no—
27C4×D12—21no—
28C4×D12C2211yescharacteristic
29#872—21no—
30C42×D3C2211yescharacteristic
31#869C2211yescharacteristic
32C3×#106C2212yes—
33C3×C4⋊C8C2211yescharacteristic
34C3×#103C2212yes—
35#835C2212yes—
36#821C2212yes—
37(C2×D3)⋊C8C2212yes—
38#817C2212yes—
39C8×Dic3C2212yes—
40#809C2212yes—
41#807C2212yes—
42#807C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#533—31no—
45#529—31no—
46#499—32no—
47#488—32no—
48#485—32no—
49#482—32no—
50#477—31no—
51#476—31no—
52#475D311yescharacteristic, complemented
53#474—31no—
54#431—31no—
55C2×C3×Q8—21no—
56C2×C3×Q8C2311yescharacteristic
57C2×D12—21no—
58C2×D12C2311yescharacteristic
59C2×C4×D3—21no—
60C2×C4×D3D411yescharacteristic
61C2×C4×D3D411yescharacteristic
62C2×C4×D3C2311yescharacteristic
63C2×Dic6—21no—
64C2×C3×C8C2312yes—
65C3×C4⋊C4—21no—
66C3×C4⋊C4—21no—
67C3×C4⋊C4C2311yescharacteristic
68C3×C4⋊C4C2312yes—
69C3×C42—21no—
70C3×C42C2311yescharacteristic
71(C2×D3)⋊C4—21no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—21no—
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—21no—
76(C2×D3)⋊C4—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3—21no—
79C4⋊Dic3—22no—
80C4⋊Dic3C2311yescharacteristic
81C4⋊Dic3C2312yes—
82Dic3⋊C4—21no—
83Dic3⋊C4—21no—
84Dic3⋊C4—22no—
85C4×Dic3—22no—
86C4×Dic3D411yescharacteristic
87C4×Dic3D411yescharacteristic
88C4×Dic3C2311yescharacteristic
89C2×C3⋊C8C2312yes—
90C4⋊Q8—31no—
91C4⋊Q8C2×D311yescharacteristic
92#126—61no—
93#125—31no—
94#125—32no—
95#124—31no—
96#123—61no—
97C22⋊Q8—31no—
98C22⋊Q8—61no—
99C22⋊Q8—61no—
100#121—31no—
101C4×Q8—61no—
102C4×Q8—61no—
103C4×Q8C2×D311yescharacteristic
104C4×D4—31no—
105C4×D4—61no—
106#117—32no—
107#117—61no—
108C2×C42—31no—
109#107—32no—
110#107—32no—
111#106—32no—
112#106C2×D312yes—
113C4⋊C8—31no—
114C4⋊C8C2×D311yescharacteristic
115#103—32no—
116#103C2×D312yes—
117#102—32no—
118#102—32no—
119C22⋊C8—32no—
120C4×C8—32no—
121C22×D3—21no—
122C22×D3C2×D411yescharacteristic
123C3×Q8—21no—
124C3×Q8—21no—
125C3×Q8—42no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2411yescharacteristic
132C2×Dic3—22no—
133C2×Dic3—21no—
134C2×Dic3C2×D411yescharacteristic
135C2×Dic3C2×D411yescharacteristic
136C2×Dic3C2×D411yescharacteristic
137D12—21no—
138D12—21no—
139D12—42no—
140C4×D3—22no—
141C4×D3—21no—
142C4×D3—21no—
143C4×D3—41no—
144C4×D3—42no—
145Dic6—42no—
146C3×C8—22no—
147C3⋊C8—22no—
148C2×Q8—21no—
149C2×Q8—61no—
150C2×Q8C22×D311yescharacteristic
151C2×D4—31no—
152C2×D4—61no—
153C22×C4—31no—
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—61no—
157C2×C8—32no—
158C2×C8C22×D312yes—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—32no—
162C4⋊C4—31no—
163C4⋊C4—61no—
164C4⋊C4—61no—
165C4⋊C4—61no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—61no—
169C4⋊C4C22×D312yes—
170C4⋊C4C22×D311yescharacteristic
171C22⋊C4—61no—
172C22⋊C4—61no—
173C22⋊C4—61no—
174C22⋊C4—62no—
175C22⋊C4—61no—
176C22⋊C4—61no—
177C42—21no—
178C42—31no—
179C42—31no—
180C42—31no—
181C42—62no—
182C42C22×D311yescharacteristic
183C22×C3C22×D411yescharacteristic
184C2×D3—21no—
185C2×D3—21no—
186C2×D3—21no—
187C2×D3—41no—
188C2×D3—41no—
189C2×D3—41no—
190C3×C4—21no—
191C3×C4—41no—
192C3×C4—42no—
193C3×C4—41no—
194C3×C4#13512yes—
195C3×C42−1+411yescharacteristic
196C3×C42−1+411yescharacteristic
197Dic3—21no—
198Dic3—21no—
199Dic3—21no—
200Dic3—41no—
201Dic3—42no—
202C23—31no—
203C23—61no—
204Q8—21no—
205Q8—21no—
206Q8—42no—
207Q8—122no—
208D4—61no—
209D4—61no—
210D4—122no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—62no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—62no—
221C2×C4—61no—
222C2×C4—121no—
223C2×C4—122no—
224C2×C4D3×D411yescharacteristic
225C2×C4D3×D411yescharacteristic
226C2×C4C23×D311yescharacteristic
227C8—22no—
228C8—62no—
229C2×C3#53611yescharacteristic
230C2×C3C2×#13511yescharacteristic
231C2×C3#57511yescharacteristic
232D3—41no—
233D3—81no—
234C22—61no—
235C22—61no—
236C22—61no—
237C22—121no—
238C22—121no—
239C22—121no—
240C22C2×D3×D411yescharacteristic
241C4—21no—
242C4—42no—
243C4—41no—
244C4—41no—
245C4—61no—
246C4—61no—
247C4—61no—
248C4—121no—
249C4—122no—
250C4#91612yes—
251C4#100711yescharacteristic
252C4#101011yescharacteristic
253C3#323311yescharacteristic, complemented, Sylow
254C2—121no—
255C2—241no—
256C2#562811yescharacteristic
257C2C2×#91611yescharacteristic
258C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#475C2
socleC22×C3C22×D4
prime radical#895C22
hypercenter#475D3
hypocenterC3#3233
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #5728C2semidirect
3. C3×#475C2semidirect
4. #5087C2semidirect
5. #5021C2semidirect
7. #4858C2semidirect
8. #4842C2semidirect
9. #4833C2semidirect
10. #4815C2semidirect
11. #4800C2semidirect
52. #475D3semidirect
253. C3#3233semidirect

Maximal subgroups

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

subgroupindex
#57282
#55802
C3×#4752
#50872
#50212
#48662
#48582
#48422
#48332
#48152
#48002
#32333