Finite Group Databasebeta

← #81601

Subgroups — #81601

Order 384.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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
2D3×C4⋊Q8C211yesmaximal, characteristic, complemented
3C3×#487C211yesmaximal, characteristic, complemented
4#5088C211yesmaximal, characteristic, complemented
5#5079C211yesmaximal, characteristic, complemented
6#4803C214yesmaximal, complemented
7#4794C214yesmaximal, complemented
8#4694C211yesmaximal, characteristic, complemented
9C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
10#3141—31nomaximal, Sylow
11C2×#239C2212yes—
12C2×D3×Q8—21no—
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4⋊Q8C2211yescharacteristic
15#947C2212yes—
16#944—21no—
17C2×ΓL(1,𝔽25)C2212yes—
18#895—21no—
19#895—21no—
20#893—21no—
21#893—21no—
22D3×C4⋊C4—21no—
23D3×C4⋊C4C2212yes—
24#888—21no—
25#888C2212yes—
26#874C2211yescharacteristic
27C42×D3C2211yescharacteristic
28#869C2211yescharacteristic
29C3×#103C2214yes—
30C3×C4×C8C2211yescharacteristic
31#835C2214yes—
32#821C2214yes—
33#815C2212yes—
34#809C2214yes—
35C4×C3⋊C8C2211yescharacteristic
36#532—31no—
37C2×C4⋊Q8—31no—
38#489—34no—
39#487—31no—
40#487D311yescharacteristic, complemented
41#486—31no—
42#486—31no—
43#417—34no—
44C4×M16—31no—
45C2×C3×Q8—21no—
46C2×C3×Q8C2312yes—
47#239—28no—
48D3×Q8—44no—
49C2×D12—22no—
50C2×D12—21no—
51C2×D12—21no—
52C2×D12C2312yes—
53C2×C4×D3—22no—
54C2×C4×D3—22no—
55C2×C4×D3D411yescharacteristic
56C2×C4×D3D411yescharacteristic
57C2×C4×D3C2311yescharacteristic
58C2×Dic6—21no—
59C2×Dic6—22no—
60C2×C3×C8C2312yes—
61C3×C4⋊C4—21no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—21no—
64C3×C4⋊C4C2312yes—
65C3×C42C2311yescharacteristic
66(C2×D3)⋊C4—22no—
67(C2×D3)⋊C4—22no—
68C4⋊Dic3—21no—
69C4⋊Dic3C2312yes—
70Dic3⋊C4—22no—
71Dic3⋊C4—22no—
72C4×Dic3D411yescharacteristic
73C4×Dic3D411yescharacteristic
74C4×Dic3C2311yescharacteristic
75C2×C3⋊C8C2312yes—
76ΓL(1,𝔽25)—24no—
77C2×#41—32no—
78C22×Q8—61no—
79C2×M16—32no—
80C4⋊Q8—32no—
81C4⋊Q8—31no—
82C4⋊Q8—61no—
83C4⋊Q8—61no—
84C4⋊Q8C2×D311yescharacteristic
85C4⋊Q8C2×D311yescharacteristic
86#127—31no—
87#124—32no—
88#121—61no—
89#121—61no—
90C4×D4—61no—
91C4×D4—61no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—61no—
94C2×C42—31no—
95#103—34no—
96#103C2×D314yes—
97#102—34no—
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—44no—
122C4×D3—44no—
123C4×D3#4114yes—
124Dic6—44no—
125Dic6—42no—
126C3×C8—22no—
127C3⋊C8—22no—
128#41—68no—
129C2×Q8—21no—
130C2×Q8—62no—
131C2×Q8—61no—
132C2×Q8—124no—
133C2×Q8C22×D312yes—
134C2×D4—32no—
135C2×D4—62no—
136C2×D4—61no—
137C2×D4—61no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—62no—
142C22×C4—62no—
143M16—64no—
144C2×C8—32no—
145C2×C8C22×D312yes—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—32no—
150C4⋊C4—62no—
151C4⋊C4—62no—
152C4⋊C4—61no—
153C4⋊C4C22×D312yes—
154C22⋊C4—62no—
155C22⋊C4—62no—
156C42—31no—
157C42—31no—
158C42—31no—
159C42C22×D311yescharacteristic
160C22×C3C22×D411yescharacteristic
161C2×D3—21no—
162C2×D3—21no—
163C2×D3—42no—
164C2×D3—42no—
165C2×D3—42no—
166C2×D3#12412yes—
167C3×C4—42no—
168C3×C4—42no—
169C3×C4AGL(1,ℤ/8)12yes—
170C3×C4#13712yes—
171C3×C4C2×#4112yes—
172Dic3—21no—
173Dic3—21no—
174Dic3—42no—
175Dic3#12412yes—
176C23—31no—
177C23—62no—
178Q8—24no—
179Q8—42no—
180Q8—124no—
181Q8—122no—
182D4—64no—
183D4—124no—
184D4—122no—
185D4—122no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—31no—
189C2×C4—34no—
190C2×C4—31no—
191C2×C4—31no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—124no—
196C2×C4—124no—
197C2×C4D3×D411yescharacteristic
198C2×C4D3×D411yescharacteristic
199C2×C4C23×D311yescharacteristic
200C8—22no—
201C8—62no—
202C2×C3C2×#12411yescharacteristic
203C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
204C2×C3C2×#13711yescharacteristic
205D3—22no—
206D3—82no—
207C22—32no—
208C22—61no—
209C22—61no—
210C22—122no—
211C22—122no—
212C22—122no—
213C22C2×D3×D411yescharacteristic
214C4—32no—
215C4—42no—
216C4—42no—
217C4—61no—
218C4—61no—
219C4—122no—
220C4#91412yes—
221C4#91812yes—
222C4D3×#4112yes—
223C3#314111yescharacteristic, complemented, Sylow
224C2—62no—
225C2—242no—
226C2D3×#12411yescharacteristic
227C2C2×#91411yescharacteristic
228C2C2×#91811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#487C2
socleC22×C3C22×D4
prime radical#5725C2
hypercenter#487D3
hypocenterC3#3141
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
2. D3×C4⋊Q8C2semidirect
3. C3×#487C2semidirect
4. #5088C2semidirect
5. #5079C2semidirect
6. #4803C2semidirect
7. #4794C2semidirect
8. #4694C2semidirect
9. C4×ΓL(1,𝔽25)C2semidirect
40. #487D3semidirect
223. C3#3141semidirect

Maximal subgroups

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

subgroupindex
#57252
D3×C4⋊Q82
C3×#4872
#50882
#50792
#48032
#47942
#46942
C4×ΓL(1,𝔽25)2
#31413