Finite Group Databasebeta

← #80910

Subgroups — #80910

Order 384.

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

normal subgroups only

This group has 418 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#5627C212yesmaximal, complemented
2C3×#470C211yesmaximal, characteristic, complemented
3#5187C212yesmaximal, complemented
4#5135C211yesmaximal, characteristic, complemented
5#5126C211yesmaximal, characteristic, complemented
6#5049C212yesmaximal, complemented
7#4866C212yesmaximal, complemented
8#4815C212yesmaximal, complemented
9#4810C212yesmaximal, complemented
10#3197—31nomaximal, Sylow
11C2×#239C2212yes—
12C22×C4×D3C2211yescharacteristic
13C2×C3×Q16C2211yescharacteristic, complemented
14C2×C3×M16C2211yescharacteristic
15C3×C22⋊Q8C2212yes—
16#946C2212yes—
17C2×#215C2212yescomplemented
18#930C2212yes—
19C2×#208C2211yescharacteristic
20C2×D24C2211yescharacteristic, complemented
21#895C2212yes—
22#894—22no—
23#893—22no—
24#892C2212yes—
25#882—22no—
26#67⋊C4—22no—
27C3×#106C2211yescharacteristic, complemented
28C3×#103C2212yescomplemented
29#835C2212yescomplemented
30#821C2212yescomplemented
31(C2×D3)⋊C8C2212yes—
32(C2×D3)⋊C8C2212yes—
33#817C2211yescharacteristic, complemented
34#809C2212yescomplemented
35#807C2212yescomplemented
36#525—32no—
37#485—32no—
38#482—32no—
39#470D311yescharacteristic, complemented
40#469—31no—
41#468—32no—
42#452—32no—
43#449—32no—
44#407—31no—
45C23×D3D411yescharacteristic
46C2×C3×Q8C2312yes—
47C22×C3×C4C2311yescharacteristic
48C2×#67—22no—
49C22×Dic3D411yescharacteristic
50#239—42no—
51#239—42no—
52C2×D12—22no—
53C2×D12—22no—
54C2×D12C2312yes—
55C2×C4×D3—22no—
56C2×C4×D3—41no—
57C2×C4×D3—41no—
58C2×C4×D3D412yes—
59C2×C4×D3C2312yes—
60C3×Q16—41no—
61C3×M16—41no—
62C2×C3×C8C2311yescharacteristic
63C2×C3×C8C2311yescharacteristic
64C3×C4⋊C4—22no—
65C3×C4⋊C4C2312yes—
66C3×C22⋊C4—22no—
67#215—42no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72C4⋊Dic3C2312yes—
73Dic3⋊C4—22no—
74C4×Dic3—22no—
75#208—41no—
76C2×C3⋊C8C2312yes—
77D24—41no—
78C2×#41—32no—
79C23×C4—31no—
80C2×Q16C2×D311yescharacteristic, complemented
81C2×SD16—32no—
82C2×D8—31no—
83C2×M16—31no—
84C2×M16C2×D311yescharacteristic
85#123—62no—
86C22⋊Q8—32no—
87C22⋊Q8C2×D312yes—
88#121—32no—
89#121—32no—
90#120—62no—
91C4×D4—62no—
92C4×D4—62no—
93#117—32no—
94#107—32no—
95#106—31no—
96#106C2×D311yescharacteristic, complemented
97#103—32no—
98#103C2×D312yescomplemented
99#102—32no—
100#102—32no—
101C22⋊C8—32no—
102C22⋊C8—32no—
103C23×C3C2×D411yescharacteristic
104C22×D3—22no—
105C22×D3—41no—
106C22×D3—42no—
107C22×D3C2×D412yes—
108C3×Q8—22no—
109C3×Q8—22no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4C2×D411yescharacteristic
115C2×C3×C4C2411yescharacteristic
116#67—44no—
117C2×Dic3—22no—
118C2×Dic3—41no—
119C2×Dic3C2×D412yes—
120D12—22no—
121D12—22no—
122D12—44no—
123D12—44no—
124C4×D3—22no—
125C4×D3—22no—
126C4×D3—42no—
127C4×D3—44no—
128C3×C8—21no—
129C3×C8—21no—
130C3⋊C8—22no—
131C24—31no—
132#41—122no—
133#41—122no—
134C2×Q8C22×D312yes—
135C2×D4—32no—
136C2×D4—62no—
137C2×D4—62no—
138C2×D4—62no—
139C22×C4—31no—
140C22×C4—32no—
141C22×C4—32no—
142C22×C4—62no—
143C22×C4—121no—
144C22×C4—121no—
145C22×C4C22×D311yescharacteristic
146Q16—41no—
147SD16—122no—
148D8—121no—
149M16—41no—
150M16—121no—
151C2×C8—32no—
152C2×C8C22×D311yescharacteristic
153C2×C8C22×D311yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—32no—
156C4⋊C4—62no—
157C4⋊C4C22×D312yes—
158C22⋊C4—22no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C42—62no—
164C22×C3—21no—
165C22×C3—22no—
166C22×C3C22×D411yescharacteristic
167C2×D3—22no—
168C2×D3—22no—
169C2×D3—22no—
170C2×D3—44no—
171C2×D3—42no—
172C2×D3—42no—
173C2×D3—42no—
174C3×C4—21no—
175C3×C4—42no—
176C3×C4—42no—
177C3×C4H(5,𝔽2)11yescharacteristic
178C3×C4H(5,𝔽2)11yescharacteristic
179Dic3—22no—
180Dic3—42no—
181C23—32no—
182C23—62no—
183C23—121no—
184C23—122no—
185C23D3×D411yescharacteristic
186Q8—22no—
187Q8—22no—
188D4—62no—
189D4—62no—
190D4—124no—
191D4—124no—
192D4—124no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—32no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—122no—
202C2×C4—124no—
203C2×C4—121no—
204C2×C4D3×D411yescharacteristic
205C2×C4C23×D311yescharacteristic
206C8—21no—
207C8—21no—
208C8—62no—
209C2×C3—41no—
210C2×C3#53411yescharacteristic
211C2×C3#57512yes—
212D3—42no—
213D3—82no—
214C22—22no—
215C22—21no—
216C22—62no—
217C22—62no—
218C22—62no—
219C22—124no—
220C22—122no—
221C22—122no—
222C22—122no—
223C22C2×D3×D411yescharacteristic
224C4—21no—
225C4—42no—
226C4—42no—
227C4—62no—
228C4—122no—
229C4#100411yescharacteristic
230C4#100911yescharacteristic
231C3#319711yescharacteristic, complemented, Sylow
232C2—41no—
233C2—122no—
234C2—242no—
235C2#560511yescharacteristic
236C2#577712yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#470C2
socleC22×C3C22×D4
hypercenter#470D3
hypocenterC3#3197
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #5627C2semidirect
2. C3×#470C2semidirect
3. #5187C2semidirect
4. #5135C2semidirect
5. #5126C2semidirect
6. #5049C2semidirect
7. #4866C2semidirect
8. #4815C2semidirect
9. #4810C2semidirect
13. C2×C3×Q16C22semidirect
17. C2×#215C22semidirect
20. C2×D24C22semidirect
27. C3×#106C22semidirect
28. C3×#103C22semidirect
29. #835C22semidirect
30. #821C22semidirect
33. #817C22semidirect
34. #809C22semidirect
35. #807C22semidirect
39. #470D3semidirect
80. C2×Q16C2×D3semidirect
96. #106C2×D3semidirect
98. #103C2×D3semidirect
231. C3#3197semidirect

Maximal subgroups

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

subgroupindex
#56272
C3×#4702
#51872
#51352
#51262
#50492
#48662
#48152
#48102
#31973