Finite Group Databasebeta

← #81565

Subgroups — #81565

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
1D3×C4⋊Q8C211yesmaximal, characteristic, complemented
2#5716C211yesmaximal, characteristic
3C3×#486C211yesmaximal, characteristic, complemented
4#5093C211yesmaximal, characteristic, complemented
5#5070C211yesmaximal, characteristic, complemented
6#4771C214yesmaximal, complemented
7#4765C214yesmaximal, complemented
8C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
9#4681C211yesmaximal, characteristic, complemented
10#3141—31nomaximal, Sylow
11C2×D3×Q8—21no—
12C2×#237C2212yes—
13C3×C4⋊Q8C2211yescharacteristic
14C3×#127C2211yescharacteristic
15#944—21no—
16#938—21no—
17#938—21no—
18#936C2212yes—
19D4×Dic3—21no—
20D4×Dic3—21no—
21C2×ΓL(1,𝔽25)C2212yes—
22D3×C4⋊C4—21no—
23D3×C4⋊C4C2212yes—
24#888—21no—
25#888C2212yes—
26C42×D3C2211yescharacteristic
27#869C2211yescharacteristic
28#869C2211yescharacteristic
29C3×#102C2214yes—
30C3×C4×C8C2211yescharacteristic
31#832C2214yes—
32#816C2214yes—
33#815C2212yes—
34#810C2214yes—
35C4×C3⋊C8C2211yescharacteristic
36#532—31no—
37C2×C4⋊Q8—31no—
38#489—34no—
39#487—31no—
40#487—31no—
41#486—31no—
42#486D311yescharacteristic, complemented
43#417—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—
51D3×Q8—44no—
52#237—28no—
53C2×C4×D3—22no—
54C2×C4×D3D411yescharacteristic
55C2×C4×D3D411yescharacteristic
56C2×C4×D3C2311yescharacteristic
57C2×Dic6—21no—
58C2×Dic6—22no—
59C2×Dic6C2312yes—
60C2×C3×C8C2312yes—
61C3×C4⋊C4—21no—
62C3×C4⋊C4C2312yes—
63C3×C42C2311yescharacteristic
64#217—22no—
65#217—22no—
66C4⋊Dic3—21no—
67C4⋊Dic3—21no—
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—31no—
83C4⋊Q8—61no—
84C4⋊Q8—61no—
85C4⋊Q8C2×D311yescharacteristic
86#127C2×D311yescharacteristic
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#103—34no—
97#102—34no—
98#102C2×D314yes—
99#97—32no—
100C4×C8—31no—
101C4×C8C2×D311yescharacteristic
102C23×C3—22no—
103C22×D3C2×D411yescharacteristic
104C3×Q8—42no—
105C3×D4—24no—
106C3×D4—42no—
107C3×D4—42no—
108C2×C3×C4—22no—
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2411yescharacteristic
112#67—44no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—44no—
116C2×Dic3C2×D411yescharacteristic
117C2×Dic3C2×D411yescharacteristic
118C2×Dic3C2×D411yescharacteristic
119C4×D3—22no—
120C4×D3—22no—
121C4×D3—44no—
122C4×D3#4114yes—
123Dic6—24no—
124Dic6—44no—
125Dic6—42no—
126C3×C8—22no—
127C3⋊C8—22no—
128#41—68no—
129C2×Q8—21no—
130C2×Q8—32no—
131C2×Q8—62no—
132C2×Q8—61no—
133C2×Q8—124no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—62no—
137C2×D4C22×D312yes—
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—32no—
148C4⋊C4—62no—
149C4⋊C4—62no—
150C4⋊C4—61no—
151C4⋊C4—61no—
152C4⋊C4—61no—
153C4⋊C4C22×D312yes—
154C22⋊C4—62no—
155C22⋊C4—62no—
156C42—31no—
157C42—31no—
158C42—31no—
159C42C22×D311yescharacteristic
160C22×C3—42no—
161C22×C3—42no—
162C22×C3—42no—
163C22×C3C22×D411yescharacteristic
164C2×D3—21no—
165C2×D3—21no—
166C2×D3#12412yes—
167C3×C4—42no—
168C3×C4AGL(1,ℤ/8)12yes—
169C3×C4#13712yes—
170C3×C4C2×#4112yes—
171Dic3—21no—
172Dic3—21no—
173Dic3—42no—
174Dic3—42no—
175Dic3#12412yes—
176C23—22no—
177C23—31no—
178Q8—42no—
179Q8—64no—
180Q8—124no—
181Q8—122no—
182D4—24no—
183D4—42no—
184D4—42no—
185D4—124no—
186C2×C4—22no—
187C2×C4—31no—
188C2×C4—34no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—62no—
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×C3—82no—
203C2×C3C2×#12411yescharacteristic
204C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
205C2×C3C2×#13711yescharacteristic
206D3—22no—
207C22—32no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—61no—
212C22—61no—
213C22C2×D3×D411yescharacteristic
214C4—32no—
215C4—42no—
216C4—61no—
217C4—61no—
218C4—122no—
219C4—122no—
220C4#91112yes—
221C4#91512yes—
222C4D3×#4112yes—
223C3#314111yescharacteristic, complemented, Sylow
224C2—62no—
225C2—82no—
226C2D3×#12411yescharacteristic
227C2C2×#91111yescharacteristic
228C2C2×#91511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#486C2
socleC22×C3C22×D4
prime radical#5716C2
hypercenter#486D3
hypocenterC3#3141
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. D3×C4⋊Q8C2semidirect
3. C3×#486C2semidirect
4. #5093C2semidirect
5. #5070C2semidirect
6. #4771C2semidirect
7. #4765C2semidirect
8. C4×ΓL(1,𝔽25)C2semidirect
9. #4681C2semidirect
42. #486D3semidirect
223. C3#3141semidirect

Maximal subgroups

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

subgroupindex
D3×C4⋊Q82
#57162
C3×#4862
#50932
#50702
#47712
#47652
C4×ΓL(1,𝔽25)2
#46812
#31413