Finite Group Databasebeta

← #81598

Subgroups — #81598

Order 384.

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

normal subgroups only

This group has 400 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#5728C211yesmaximal, characteristic, complemented
2#5725C211yesmaximal, characteristic
3C3×#487C211yesmaximal, characteristic, complemented
4#5088C211yesmaximal, characteristic, complemented
5#5079C211yesmaximal, characteristic, complemented
6#4804C214yesmaximal, complemented
7#4800C214yesmaximal, complemented
8#4694C211yesmaximal, characteristic, complemented
9C4×D3×C8C211yesmaximal, characteristic, complemented
10#3131—31nomaximal, Sylow
11C2×#239C2212yes—
12C3×C4⋊Q8C2211yescharacteristic
13C3×C4⋊Q8C2211yescharacteristic
14#947C2212yes—
15#946—21no—
16Q8×Dic3—21no—
17C2×D3×C8C2212yes—
18#897—21no—
19#895—21no—
20#895—21no—
21#893—21no—
22#893—21no—
23#892C2212yes—
24#890C2212yes—
25#887—21no—
26#874C2211yescharacteristic
27C42×D3C2211yescharacteristic
28#869C2211yescharacteristic
29C3×#103C2214yes—
30C3×C4×C8C2211yescharacteristic
31#835C2214yes—
32#821C2214yes—
33C8×Dic3C2212yes—
34#809C2214yes—
35C4×C3⋊C8C2211yescharacteristic
36#533—31no—
37#532—31no—
38#488—34no—
39#487—31no—
40#487D311yescharacteristic, complemented
41#486—31no—
42#486—31no—
43#416—34no—
44C2×C4×C8—31no—
45C2×C3×Q8—21no—
46C2×C3×Q8C2312yes—
47#239—28no—
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(C2×D3)⋊C4—22no—
64(C2×D3)⋊C4—22no—
65(C2×D3)⋊C4—22no—
66C4⋊Dic3—21no—
67C4⋊Dic3—22no—
68C4⋊Dic3C2312yes—
69Dic3⋊C4—22no—
70C4×Dic3—22no—
71C4×Dic3D411yescharacteristic
72C4×Dic3D411yescharacteristic
73C4×Dic3C2311yescharacteristic
74C2×C3⋊C8C2312yes—
75D3×C8—24no—
76C2×#41—32no—
77C22×C8—32no—
78C4⋊Q8—31no—
79C4⋊Q8C2×D311yescharacteristic
80C4⋊Q8C2×D311yescharacteristic
81#127—31no—
82#125—32no—
83#124—32no—
84C22⋊Q8—61no—
85C22⋊Q8—61no—
86#121—61no—
87#121—61no—
88C4×Q8—61no—
89C4×Q8—61no—
90C4×D4—61no—
91C4×D4—61no—
92#117—32no—
93C2×C42—31no—
94#103—34no—
95#103C2×D314yes—
96#102—34no—
97#102—34no—
98C4×C8—32no—
99C4×C8—31no—
100C4×C8C2×D311yescharacteristic
101C22×D3—22no—
102C22×D3C2×D411yescharacteristic
103C3×Q8—24no—
104C3×Q8—42no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2411yescharacteristic
110C2×Dic3—22no—
111C2×Dic3C2×D411yescharacteristic
112C2×Dic3C2×D411yescharacteristic
113C2×Dic3C2×D411yescharacteristic
114D12—24no—
115D12—44no—
116D12—42no—
117D12—42no—
118C4×D3—22no—
119C4×D3—22no—
120C4×D3—44no—
121C4×D3#4114yes—
122Dic6—42no—
123C3×C8—22no—
124C3⋊C8—22no—
125#41—68no—
126C2×Q8—21no—
127C2×Q8—61no—
128C2×Q8C22×D312yes—
129C2×D4—32no—
130C2×D4—62no—
131C2×D4—61no—
132C2×D4—61no—
133C22×C4—31no—
134C22×C4—31no—
135C22×C4—31no—
136C22×C4—62no—
137C2×C8—32no—
138C2×C8—64no—
139C2×C8C22×D312yes—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—32no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—61no—
147C4⋊C4C22×D312yes—
148C22⋊C4—62no—
149C22⋊C4—62no—
150C22⋊C4—62no—
151C42—31no—
152C42—31no—
153C42—31no—
154C42—62no—
155C42C22×D311yescharacteristic
156C22×C3C22×D411yescharacteristic
157C2×D3—21no—
158C2×D3—21no—
159C2×D3—42no—
160C2×D3—42no—
161C2×D3—42no—
162C2×D3#12412yes—
163C3×C4—42no—
164C3×C4—42no—
165C3×C4#13512yes—
166C3×C4#13512yes—
167C3×C4C2×#4112yes—
168Dic3—21no—
169Dic3—21no—
170Dic3—42no—
171Dic3#12412yes—
172C23—31no—
173C23—62no—
174Q8—24no—
175Q8—42no—
176Q8—122no—
177D4—64no—
178D4—124no—
179D4—122no—
180D4—122no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—31no—
184C2×C4—34no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—124no—
191C2×C4D3×D411yescharacteristic
192C2×C4D3×D411yescharacteristic
193C2×C4C23×D311yescharacteristic
194C8—22no—
195C8—62no—
196C2×C3C2×#12411yescharacteristic
197C2×C3C2×#13511yescharacteristic
198C2×C3C2×#13511yescharacteristic
199D3—22no—
200D3—82no—
201C22—32no—
202C22—61no—
203C22—61no—
204C22—122no—
205C22—122no—
206C22—122no—
207C22C2×D3×D411yescharacteristic
208C4—32no—
209C4—42no—
210C4—42no—
211C4—61no—
212C4—61no—
213C4—122no—
214C4#91612yes—
215C4#91912yes—
216C4D3×#4112yes—
217C3#313111yescharacteristic, complemented, Sylow
218C2—62no—
219C2—242no—
220C2D3×#12411yescharacteristic
221C2C2×#91611yescharacteristic
222C2C2×#91911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#487C2
socleC22×C3C22×D4
prime radical#5725C2
hypercenter#487D3
hypocenterC3#3131
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. #5088C2semidirect
5. #5079C2semidirect
6. #4804C2semidirect
7. #4800C2semidirect
8. #4694C2semidirect
9. C4×D3×C8C2semidirect
40. #487D3semidirect
217. C3#3131semidirect

Maximal subgroups

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

subgroupindex
#57282
#57252
C3×#4872
#50882
#50792
#48042
#48002
#46942
C4×D3×C82
#31313