Finite Group Databasebeta

← #81041

Subgroups — #81041

Order 384.

348 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 115 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#5721C211yesmaximal, characteristic, complemented
2#5585C211yesmaximal, characteristic
3C3×#475C211yesmaximal, characteristic, complemented
4#5088C211yesmaximal, characteristic
5C4×#216C211yesmaximal, characteristic, complemented
6#4871C212yesmaximal, complemented
7#4852C212yesmaximal, complemented
8#4848C211yesmaximal, characteristic
9#4830C211yesmaximal, characteristic, complemented
10#4800C212yesmaximal, complemented
11#4798C212yesmaximal, complemented
12#3365—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4×Q8C2211yescharacteristic
15#947C2212yes—
16Q8×Dic3—21no—
17#944—21no—
18C2×#216C2211yescharacteristic
19C2×#204C2212yes—
20#894—22no—
21#893C2212yes—
22#890C2212yes—
23#889—22no—
24#887C2212yes—
25#875—21no—
26#874C2211yescharacteristic
27C4×D12—21no—
28C4×Dic6C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×#106C2212yes—
31C3×C4⋊C8C2211yescharacteristic
32C3×#103C2212yes—
33#835C2211yescharacteristic
34#821C2212yes—
35Dic3⋊C8C2212yes—
36C8×Dic3C2212yes—
37#810C2211yescharacteristic
38#809C2212yes—
39#809C2212yes—
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#559—31no—
43#557—31no—
44#488—32no—
45#486—31no—
46#478—32no—
47#475D311yescharacteristic, complemented
48#464—32no—
49#460—31no—
50C8×Q8—31no—
51C4×Q16—31no—
52C4×SD16—32no—
53C2×C3×Q8—21no—
54C2×C3×Q8C2311yescharacteristic
55C2×D12—21no—
56C2×D12—21no—
57C2×D12C2312yes—
58C2×C4×D3—22no—
59C2×Dic6D411yescharacteristic
60C2×Dic6C2311yescharacteristic
61C2×C3×C8C2312yes—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—21no—
64C3×C4⋊C4C2311yescharacteristic
65C3×C4⋊C4C2312yes—
66C3×C42—21no—
67C3×C42C2311yescharacteristic
68#216—22no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72C4⋊Dic3—22no—
73C4⋊Dic3D411yescharacteristic
74C4⋊Dic3C2311yescharacteristic
75Dic3⋊C4—21no—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4D412yes—
79C4×Dic3—21no—
80C4×Dic3—22no—
81C4×Dic3C2312yes—
82C2×C3⋊C8C2311yescharacteristic
83C2×C3⋊C8C2311yescharacteristic
84#204—24no—
85C2×Q16—31no—
86C2×SD16—32no—
87C4⋊Q8—61no—
88C4⋊Q8C2×D311yescharacteristic
89#127—31no—
90#125—32no—
91#125—62no—
92#124—32no—
93#124—61no—
94#123—62no—
95C4×Q8—31no—
96C4×Q8—32no—
97C4×Q8—31no—
98C4×Q8—61no—
99C4×Q8C2×D311yescharacteristic
100C4×D4—32no—
101C4×D4—61no—
102#107—31no—
103#106C2×D312yes—
104C4⋊C8—32no—
105C4⋊C8C2×D311yescharacteristic
106#103—31no—
107#103—31no—
108#103C2×D312yes—
109#102—32no—
110#102—32no—
111#102—32no—
112C4×C8—32no—
113C4×C8—31no—
114C22×D3—22no—
115C3×Q8—21no—
116C3×Q8—21no—
117C3×Q8—42no—
118C2×C3×C4—22no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2411yescharacteristic
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3C2×D412yes—
127C2×Dic3C2×D412yes—
128D12—22no—
129D12—22no—
130D12—42no—
131D12—81no—
132C4×D3—82no—
133Dic6—22no—
134Dic6—21no—
135Dic6#4112yes—
136C3×C8—22no—
137C3⋊C8—21no—
138C3⋊C8—21no—
139C2×Q8—21no—
140C2×Q8—31no—
141C2×Q8—31no—
142C2×Q8C22×D311yescharacteristic
143C2×D4—32no—
144C2×D4—61no—
145C2×D4—61no—
146C22×C4—62no—
147Q16—62no—
148SD16—64no—
149C2×C8—31no—
150C2×C8—31no—
151C2×C8C22×D312yes—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—31no—
155C4⋊C4—32no—
156C4⋊C4—31no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—61no—
161C4⋊C4C22×D312yes—
162C4⋊C4C22×D311yescharacteristic
163C22⋊C4—62no—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C42—21no—
167C42—32no—
168C42—62no—
169C42—61no—
170C42C22×D311yescharacteristic
171C22×C3C22×D411yescharacteristic
172C2×D3—42no—
173C2×D3—42no—
174C2×D3—42no—
175C3×C4—21no—
176C3×C4—41no—
177C3×C4—42no—
178C3×C4—41no—
179C3×C4#13512yes—
180C3×C4C2×#4111yescharacteristic
181C3×C42−1+411yescharacteristic
182Dic3—22no—
183Dic3—22no—
184Dic3—22no—
185Dic3—41no—
186C23—62no—
187Q8—21no—
188Q8—21no—
189Q8—32no—
190Q8—42no—
191Q8—62no—
192Q8—61no—
193D4—62no—
194D4—62no—
195D4—122no—
196D4—241no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—32no—
201C2×C4—32no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—242no—
205C2×C4D3×D411yescharacteristic
206C2×C4D3×D411yescharacteristic
207C2×C4C23×D311yescharacteristic
208C8—22no—
209C8—61no—
210C8—61no—
211C2×C3#54811yescharacteristic
212C2×C3C2×#13511yescharacteristic
213C2×C3#57711yescharacteristic
214D3—82no—
215C22—122no—
216C22—122no—
217C22—122no—
218C22C2×D3×D411yescharacteristic
219C4—21no—
220C4—41no—
221C4—42no—
222C4—41no—
223C4—62no—
224C4—62no—
225C4—62no—
226C4—121no—
227C4#91612yes—
228C4#100711yescharacteristic
229C4D3×#4111yescharacteristic
230C3#336511yescharacteristic, complemented, Sylow
231C2—242no—
232C2#563311yescharacteristic
233C2C2×#91611yescharacteristic
234C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#475C2
socleC22×C3C22×D4
prime radical#874C22
hypercenter#475D3
hypocenterC3#3365
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. #5721C2semidirect
3. C3×#475C2semidirect
5. C4×#216C2semidirect
6. #4871C2semidirect
7. #4852C2semidirect
9. #4830C2semidirect
10. #4800C2semidirect
11. #4798C2semidirect
47. #475D3semidirect
230. C3#3365semidirect

Maximal subgroups

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

subgroupindex
#57212
#55852
C3×#4752
#50882
C4×#2162
#48712
#48522
#48482
#48302
#48002
#47982
#33653