Finite Group Databasebeta

← #81595

Subgroups — #81595

Order 384.

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

normal subgroups only

This group has 402 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#5728C211yesmaximal, characteristic, complemented
2#5725C211yesmaximal, characteristic
3C3×#487C211yesmaximal, characteristic, complemented
4#5085C211yesmaximal, characteristic, complemented
5#5078C211yesmaximal, characteristic, complemented
6#4817C214yesmaximal, complemented
7#4812C214yesmaximal, complemented
8#4694C211yesmaximal, characteristic, complemented
9C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
10#3182—31nomaximal, Sylow
11C2×#239—21no—
12C3×C4⋊Q8C2211yescharacteristic
13C3×C4⋊Q8C2211yescharacteristic
14#947—21no—
15#946C2212yes—
16Q8×Dic3C2212yes—
17C2×#215C2214yes—
18C2×ΓL(1,𝔽25)C2212yes—
19#897—21no—
20#895—21no—
21#895C2212yes—
22#893—21no—
23#893C2212yes—
24#892—21no—
25#890—21no—
26#887—21no—
27#874C2211yescharacteristic
28C42×D3C2211yescharacteristic
29#869C2211yescharacteristic
30C3×#103C2214yes—
31C3×C4×C8C2211yescharacteristic
32#821C2214yes—
33#815C2212yes—
34#807C2214yes—
35C4×C3⋊C8C2211yescharacteristic
36#533—31no—
37#532—31no—
38#500—31no—
39#492—31no—
40#487D311yescharacteristic, complemented
41#486—31no—
42#468—34no—
43#440—34no—
44C4×M16—31no—
45C2×C3×Q8—21no—
46C2×C3×Q8C2312yes—
47#239—44no—
48C2×D12—22no—
49C2×D12—21no—
50C2×D12—21no—
51C2×D12C2312yes—
52C2×C4×D3—22no—
53C2×C4×D3D411yescharacteristic
54C2×C4×D3D411yescharacteristic
55C2×C4×D3C2311yescharacteristic
56C2×Dic6—21no—
57C2×C3×C8C2312yes—
58C3×C4⋊C4—21no—
59C3×C4⋊C4—21no—
60C3×C4⋊C4—21no—
61C3×C4⋊C4C2312yes—
62C3×C42C2311yescharacteristic
63#215—28no—
64(C2×D3)⋊C4—22no—
65(C2×D3)⋊C4—22no—
66(C2×D3)⋊C4—22no—
67C4⋊Dic3—21no—
68C4⋊Dic3—22no—
69C4⋊Dic3C2312yes—
70Dic3⋊C4—22no—
71C4×Dic3—22no—
72C4×Dic3D411yescharacteristic
73C4×Dic3D411yescharacteristic
74C4×Dic3C2311yescharacteristic
75C2×C3⋊C8C2312yes—
76ΓL(1,𝔽25)—24no—
77C2×#41—61no—
78C2×SD16—34no—
79C2×M16—32no—
80C4⋊Q8—31no—
81C4⋊Q8C2×D311yescharacteristic
82C4⋊Q8C2×D311yescharacteristic
83#127—31no—
84#125—61no—
85#124—61no—
86C22⋊Q8—32no—
87C22⋊Q8—61no—
88#121—32no—
89#121—61no—
90C4×Q8—32no—
91C4×Q8—61no—
92C4×D4—32no—
93C4×D4—61no—
94#117—61no—
95C2×C42—31no—
96#107—34no—
97#103C2×D314yes—
98#102—34no—
99#97—32no—
100C4×C8—31no—
101C4×C8C2×D311yescharacteristic
102C22×D3—22no—
103C22×D3C2×D411yescharacteristic
104C3×Q8—24no—
105C3×Q8—42no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2411yescharacteristic
111C2×Dic3—22no—
112C2×Dic3C2×D411yescharacteristic
113C2×Dic3C2×D411yescharacteristic
114C2×Dic3C2×D411yescharacteristic
115D12—24no—
116D12—44no—
117D12—42no—
118D12—42no—
119C4×D3—22no—
120C4×D3—22no—
121C4×D3—22no—
122C4×D3—44no—
123Dic6—42no—
124C3×C8—22no—
125C3⋊C8#4114yescomplemented
126#41—124no—
127C2×Q8—21no—
128C2×Q8—61no—
129C2×Q8C22×D312yes—
130C2×D4—32no—
131C2×D4—62no—
132C2×D4—61no—
133C2×D4—61no—
134C22×C4—31no—
135C22×C4—31no—
136C22×C4—31no—
137C22×C4—62no—
138SD16—68no—
139M16—64no—
140C2×C8—32no—
141C2×C8C22×D312yes—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—32no—
146C4⋊C4—62no—
147C4⋊C4—62no—
148C4⋊C4—61no—
149C4⋊C4C22×D312yes—
150C22⋊C4—62no—
151C22⋊C4—62no—
152C22⋊C4—62no—
153C42—31no—
154C42—31no—
155C42—31no—
156C42—62no—
157C42C22×D311yescharacteristic
158C22×C3C22×D411yescharacteristic
159C2×D3—21no—
160C2×D3—21no—
161C2×D3—21no—
162C2×D3—42no—
163C2×D3—42no—
164C2×D3—42no—
165C3×C4—42no—
166C3×C4—42no—
167C3×C4AGL(1,ℤ/8)12yes—
168C3×C4#13712yes—
169C3×C4C2×#4112yes—
170Dic3—21no—
171Dic3—21no—
172Dic3—21no—
173Dic3—42no—
174C23—31no—
175C23—62no—
176Q8—24no—
177Q8—42no—
178Q8—122no—
179D4—64no—
180D4—124no—
181D4—122no—
182D4—122no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—124no—
193C2×C4D3×D411yescharacteristic
194C2×C4D3×D411yescharacteristic
195C2×C4C23×D311yescharacteristic
196C8—22no—
197C8—34no—
198C2×C3#53211yescharacteristic
199C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
200C2×C3C2×#13711yescharacteristic
201D3—41no—
202D3—82no—
203C22—61no—
204C22—61no—
205C22—61no—
206C22—122no—
207C22—122no—
208C22—122no—
209C22C2×D3×D411yescharacteristic
210C4—42no—
211C4—42no—
212C4—61no—
213C4—61no—
214C4—61no—
215C4—122no—
216C4#91412yes—
217C4#91812yes—
218C4D3×#4112yes—
219C3#318211yescharacteristic, complemented, Sylow
220C2—121no—
221C2—242no—
222C2#566811yescharacteristic
223C2C2×#91411yescharacteristic
224C2C2×#91811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#487C2
socleC22×C3C22×D4
prime radical#5725C2
hypercenter#487D3
hypocenterC3#3182
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #5728C2semidirect
3. C3×#487C2semidirect
4. #5085C2semidirect
5. #5078C2semidirect
6. #4817C2semidirect
7. #4812C2semidirect
8. #4694C2semidirect
9. C4×ΓL(1,𝔽25)C2semidirect
40. #487D3semidirect
125. C3⋊C8#41semidirect
219. C3#3182semidirect

Maximal subgroups

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

subgroupindex
#57282
#57252
C3×#4872
#50852
#50782
#48172
#48122
#46942
C4×ΓL(1,𝔽25)2
#31823