Finite Group Databasebeta

← #81560

Subgroups — #81560

Order 384.

446 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 121 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#5725C211yesmaximal, characteristic, complemented
2#5716C211yesmaximal, characteristic, complemented
3C3×#486C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5#5073C211yesmaximal, characteristic, complemented
6#4785C214yesmaximal, complemented
7#4782C214yesmaximal, complemented
8#4694C211yesmaximal, characteristic, complemented
9C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
10#3184—31nomaximal, Sylow
11C2×#239—21no—
12C2×#237—21no—
13C3×C4⋊Q8C2211yescharacteristic
14C3×#127C2211yescharacteristic
15#947—21no—
16#938—21no—
17#938C2212yes—
18#936—21no—
19D4×Dic3—21no—
20D4×Dic3C2212yes—
21C2×#213C2214yes—
22C2×ΓL(1,𝔽25)C2212yes—
23#895—21no—
24#895C2212yes—
25#893—21no—
26#893C2212yes—
27#874C2211yescharacteristic
28C42×D3C2211yescharacteristic
29#869C2211yescharacteristic
30C3×#102C2214yes—
31C3×C4×C8C2211yescharacteristic
32#821C2214yes—
33#815C2212yes—
34#808C2214yes—
35C4×C3⋊C8C2211yescharacteristic
36#532—31no—
37#532—31no—
38#498—31no—
39#493—31no—
40#486—31no—
41#486D311yescharacteristic, complemented
42#469—34no—
43#442—34no—
44C4×M16—31no—
45C2×C3×Q8—21no—
46C2×C3×D4—21no—
47C2×C3×D4—21no—
48C2×C3×D4C2312yes—
49C2×#67—22no—
50C22×Dic3—22no—
51#239—44no—
52#237—44no—
53C2×D12—22no—
54C2×D12—21no—
55C2×D12—21no—
56C2×D12C2312yes—
57C2×C4×D3—22no—
58C2×C4×D3D411yescharacteristic
59C2×C4×D3D411yescharacteristic
60C2×C4×D3C2311yescharacteristic
61C2×Dic6—21no—
62C2×C3×C8C2312yes—
63C3×C4⋊C4—21no—
64C3×C4⋊C4C2312yes—
65C3×C42C2311yescharacteristic
66#217—22no—
67#217—22no—
68#213—28no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71C4⋊Dic3—21no—
72C4⋊Dic3C2312yes—
73C4×Dic3D411yescharacteristic
74C4×Dic3D411yescharacteristic
75C4×Dic3C2311yescharacteristic
76C2×C3⋊C8C2312yes—
77ΓL(1,𝔽25)—24no—
78C2×#41—61no—
79C2×#41—61no—
80C2×D8—34no—
81C2×M16—32no—
82C4⋊Q8—31no—
83C4⋊Q8C2×D311yescharacteristic
84#127—31no—
85#127C2×D311yescharacteristic
86#124—61no—
87#124—61no—
88#121—32no—
89#121—32no—
90#121—61no—
91#121—61no—
92C4×D4—32no—
93C4×D4—32no—
94C4×D4—61no—
95C4×D4—61no—
96C2×C42—31no—
97#106—34no—
98#102—34no—
99#102C2×D314yes—
100#97—32no—
101C4×C8—31no—
102C4×C8C2×D311yescharacteristic
103C23×C3—22no—
104C22×D3—22no—
105C22×D3C2×D411yescharacteristic
106C3×Q8—42no—
107C3×D4—24no—
108C3×D4—42no—
109C3×D4—42no—
110C2×C3×C4—22no—
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4C2411yescharacteristic
114#67—44no—
115C2×Dic3—22no—
116C2×Dic3—44no—
117C2×Dic3C2×D411yescharacteristic
118C2×Dic3C2×D411yescharacteristic
119C2×Dic3C2×D411yescharacteristic
120D12—24no—
121D12—44no—
122D12—42no—
123D12—42no—
124C4×D3—22no—
125C4×D3—22no—
126C4×D3—22no—
127C4×D3—44no—
128Dic6—42no—
129C3×C8—22no—
130C3⋊C8#4114yescomplemented
131#41—124no—
132#41—124no—
133C2×Q8—21no—
134C2×Q8—61no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—32no—
138C2×D4—62no—
139C2×D4—62no—
140C2×D4—61no—
141C2×D4—61no—
142C2×D4C22×D312yes—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—62no—
147C22×C4—62no—
148D8—68no—
149M16—64no—
150C2×C8—32no—
151C2×C8C22×D312yes—
152C4⋊C4—21no—
153C4⋊C4—32no—
154C4⋊C4—61no—
155C4⋊C4C22×D312yes—
156C22⋊C4—62no—
157C22⋊C4—62no—
158C22⋊C4—62no—
159C22⋊C4—62no—
160C42—31no—
161C42—31no—
162C42—31no—
163C42C22×D311yescharacteristic
164C22×C3—42no—
165C22×C3—42no—
166C22×C3—42no—
167C22×C3C22×D411yescharacteristic
168C2×D3—21no—
169C2×D3—21no—
170C2×D3—21no—
171C2×D3—42no—
172C2×D3—42no—
173C2×D3—42no—
174C3×C4—42no—
175C3×C4AGL(1,ℤ/8)12yes—
176C3×C4AGL(1,ℤ/8)12yes—
177C3×C4C2×#4112yes—
178Dic3—21no—
179Dic3—21no—
180Dic3—21no—
181Dic3—42no—
182C23—22no—
183C23—31no—
184C23—62no—
185Q8—42no—
186Q8—122no—
187D4—24no—
188D4—42no—
189D4—42no—
190D4—64no—
191D4—124no—
192D4—124no—
193D4—122no—
194D4—122no—
195C2×C4—22no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—124no—
204C2×C4—124no—
205C2×C4D3×D411yescharacteristic
206C2×C4D3×D411yescharacteristic
207C2×C4C23×D311yescharacteristic
208C8—22no—
209C8—34no—
210C2×C3—82no—
211C2×C3#53211yescharacteristic
212C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
214D3—41no—
215D3—82no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—61no—
220C22—61no—
221C22—61no—
222C22—122no—
223C22—122no—
224C22—122no—
225C22C2×D3×D411yescharacteristic
226C4—42no—
227C4—61no—
228C4—61no—
229C4—61no—
230C4—122no—
231C4#91112yes—
232C4#91412yes—
233C4D3×#4112yes—
234C3#318411yescharacteristic, complemented, Sylow
235C2—82no—
236C2—121no—
237C2—242no—
238C2#566811yescharacteristic
239C2C2×#91111yescharacteristic
240C2C2×#91411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#486C2
socleC22×C3C22×D4
hypercenter#486D3
hypocenterC3#3184
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #5725C2semidirect
2. #5716C2semidirect
3. C3×#486C2semidirect
4. #5080C2semidirect
5. #5073C2semidirect
6. #4785C2semidirect
7. #4782C2semidirect
8. #4694C2semidirect
9. C4×ΓL(1,𝔽25)C2semidirect
41. #486D3semidirect
130. C3⋊C8#41semidirect
234. C3#3184semidirect

Maximal subgroups

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

subgroupindex
#57252
#57162
C3×#4862
#50802
#50732
#47852
#47822
#46942
C4×ΓL(1,𝔽25)2
#31843