Finite Group Databasebeta

← #81574

Subgroups — #81574

Order 384.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#5716C211yesmaximal, characteristic
3C3×#486C211yesmaximal, characteristic, complemented
4#5078C211yesmaximal, characteristic, complemented
5#5070C211yesmaximal, characteristic, complemented
6#4779C214yesmaximal, complemented
7#4763C214yesmaximal, complemented
8#4685C211yesmaximal, characteristic, complemented
9#4680C211yesmaximal, characteristic, complemented
10#3228—31nomaximal, Sylow
11C2×#237—21no—
12C3×C4⋊Q8C2211yescharacteristic
13C3×#127C2211yescharacteristic
14#946—21no—
15Q8×Dic3—21no—
16#938—21no—
17#938C2212yes—
18#936—21no—
19D4×Dic3—21no—
20D4×Dic3C2212yes—
21#897—21no—
22#892C2212yes—
23#890C2212yes—
24#887—21no—
25C42×D3C2211yescharacteristic
26#869C2211yescharacteristic
27#869C2211yescharacteristic
28C3×#102C2214yes—
29C3×C4×C8C2211yescharacteristic
30#832C2214yes—
31(C2×D3)⋊C8C2212yes—
32#817C2214yes—
33Dic3⋊C8C2212yes—
34#807C2214yes—
35C4×C3⋊C8C2211yescharacteristic
36#533—31no—
37#532—31no—
38#500—31no—
39#498—31no—
40#486—31no—
41#486D311yescharacteristic, complemented
42#482—34no—
43#478—34no—
44#431—31no—
45C2×C3×Q8—21no—
46C2×C3×D4—21no—
47C2×C3×D4—21no—
48C2×C3×D4C2312yes—
49C2×#67—22no—
50C22×Dic3—22no—
51#237—44no—
52C2×C4×D3D411yescharacteristic
53C2×C4×D3D411yescharacteristic
54C2×C4×D3C2311yescharacteristic
55C2×Dic6—21no—
56C2×Dic6—21no—
57C2×C3×C8C2312yes—
58C3×C4⋊C4—21no—
59C3×C4⋊C4C2312yes—
60C3×C42C2311yescharacteristic
61#217—22no—
62#217—22no—
63(C2×D3)⋊C4—22no—
64C4⋊Dic3—21no—
65C4⋊Dic3—21no—
66C4⋊Dic3—22no—
67C4⋊Dic3C2312yes—
68C4⋊Dic3C2312yes—
69Dic3⋊C4—22no—
70C4×Dic3—22no—
71C4×Dic3D411yescharacteristic
72C4×Dic3D411yescharacteristic
73C4×Dic3C2311yescharacteristic
74C2×C3⋊C8C2312yes—
75C2×#41—61no—
76C4⋊Q8—31no—
77C4⋊Q8—31no—
78C4⋊Q8C2×D311yescharacteristic
79#127C2×D311yescharacteristic
80#125—32no—
81#124—61no—
82C22⋊Q8—61no—
83C22⋊Q8—61no—
84#121—32no—
85#121—61no—
86C4×Q8—61no—
87C4×Q8—61no—
88C4×D4—32no—
89C4×D4—61no—
90#117—32no—
91C2×C42—31no—
92#107—34no—
93#106—34no—
94C4⋊C8—32no—
95#102—34no—
96#102C2×D314yes—
97C22⋊C8—32no—
98C4×C8—31no—
99C4×C8C2×D311yescharacteristic
100C23×C3—22no—
101C22×D3C2×D411yescharacteristic
102C3×Q8—42no—
103C3×D4—24no—
104C3×D4—42no—
105C3×D4—42no—
106C2×C3×C4—22no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2411yescharacteristic
110#67—44no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—44no—
114C2×Dic3C2×D411yescharacteristic
115C2×Dic3C2×D411yescharacteristic
116C2×Dic3C2×D411yescharacteristic
117C4×D3—22no—
118C4×D3—22no—
119C4×D3—22no—
120Dic6—42no—
121Dic6—42no—
122C3×C8—22no—
123C3⋊C8—22no—
124#41—124no—
125C2×Q8—21no—
126C2×Q8—61no—
127C2×Q8—61no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—62no—
131C2×D4C22×D312yes—
132C22×C4—31no—
133C22×C4—31no—
134C22×C4—31no—
135C22×C4—62no—
136C2×C8—32no—
137C2×C8C22×D312yes—
138C4⋊C4—21no—
139C4⋊C4—32no—
140C4⋊C4—32no—
141C4⋊C4—62no—
142C4⋊C4—62no—
143C4⋊C4—61no—
144C4⋊C4—61no—
145C4⋊C4C22×D312yes—
146C22⋊C4—62no—
147C22⋊C4—62no—
148C22⋊C4—62no—
149C42—31no—
150C42—31no—
151C42—31no—
152C42—62no—
153C42C22×D311yescharacteristic
154C22×C3—42no—
155C22×C3—42no—
156C22×C3—42no—
157C22×C3C22×D411yescharacteristic
158C2×D3—21no—
159C2×D3—21no—
160C2×D3—21no—
161C3×C4—42no—
162C3×C4#13512yes—
163C3×C4AGL(1,ℤ/8)12yes—
164C3×C42−1+412yes—
165Dic3—21no—
166Dic3—21no—
167Dic3—21no—
168Dic3—42no—
169Dic3—42no—
170C23—22no—
171C23—31no—
172Q8—42no—
173Q8—122no—
174Q8—122no—
175D4—24no—
176D4—42no—
177D4—42no—
178D4—124no—
179C2×C4—22no—
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—124no—
189C2×C4D3×D411yescharacteristic
190C2×C4D3×D411yescharacteristic
191C2×C4C23×D311yescharacteristic
192C8—22no—
193C8—62no—
194C2×C3—82no—
195C2×C3#53611yescharacteristic
196C2×C3C2×#13511yescharacteristic
197C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
198D3—41no—
199C22—42no—
200C22—42no—
201C22—42no—
202C22—61no—
203C22—61no—
204C22—61no—
205C22C2×D3×D411yescharacteristic
206C4—42no—
207C4—61no—
208C4—61no—
209C4—61no—
210C4—122no—
211C4—122no—
212C4#91112yes—
213C4#91612yes—
214C4#101012yes—
215C3#322811yescharacteristic, complemented, Sylow
216C2—82no—
217C2—121no—
218C2#567511yescharacteristic
219C2C2×#91111yescharacteristic
220C2C2×#91611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#486C2
socleC22×C3C22×D4
prime radical#5716C2
hypercenter#486D3
hypocenterC3#3228
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. #5728C2semidirect
3. C3×#486C2semidirect
4. #5078C2semidirect
5. #5070C2semidirect
6. #4779C2semidirect
7. #4763C2semidirect
8. #4685C2semidirect
9. #4680C2semidirect
41. #486D3semidirect
215. C3#3228semidirect

Maximal subgroups

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

subgroupindex
#57282
#57162
C3×#4862
#50782
#50702
#47792
#47632
#46852
#46802
#32283