Finite Group Databasebeta

← #81607

Subgroups — #81607

Order 384.

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

normal subgroups only

This group has 360 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
2D3×C4⋊Q8C211yesmaximal, characteristic
3C3×#487C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5#5079C211yesmaximal, characteristic, complemented
6#4805C214yesmaximal
7#4791C214yesmaximal, complemented
8#4685C211yesmaximal, characteristic
9#4680C211yesmaximal, characteristic, complemented
10#3229—31nomaximal, Sylow
11C2×D3×Q8—21no—
12C3×C4⋊Q8C2211yescharacteristic
13C3×C4⋊Q8C2211yescharacteristic
14#946C2212yes—
15Q8×Dic3C2212yes—
16#944—21no—
17#897—21no—
18#892—21no—
19D3×C4⋊C4—21no—
20D3×C4⋊C4C2212yes—
21#890—21no—
22#888—21no—
23#888C2212yes—
24#887—21no—
25C42×D3C2211yescharacteristic
26#869C2211yescharacteristic
27#869C2211yescharacteristic
28C3×#103C2214yes—
29C3×C4×C8C2211yescharacteristic
30#835C2214yes—
31(C2×D3)⋊C8C2212yes—
32#817C2214yes—
33Dic3⋊C8C2212yes—
34#808C2214yes—
35C4×C3⋊C8C2211yescharacteristic
36#533—31no—
37C2×C4⋊Q8—31no—
38#498—31no—
39#498—31no—
40#487—31no—
41#487D311yescharacteristic, complemented
42#483—34no—
43#475—34no—
44#431—31no—
45C2×C3×Q8—21no—
46C2×C3×Q8C2312yes—
47D3×Q8—44no—
48C2×C4×D3—22no—
49C2×C4×D3D411yescharacteristic
50C2×C4×D3D411yescharacteristic
51C2×C4×D3C2311yescharacteristic
52C2×Dic6—21no—
53C2×Dic6—21no—
54C2×Dic6—22no—
55C2×C3×C8C2312yes—
56C3×C4⋊C4—21no—
57C3×C4⋊C4—21no—
58C3×C4⋊C4—21no—
59C3×C4⋊C4C2312yes—
60C3×C42C2311yescharacteristic
61(C2×D3)⋊C4—22no—
62C4⋊Dic3—21no—
63C4⋊Dic3—21no—
64C4⋊Dic3—22no—
65C4⋊Dic3C2312yes—
66C4⋊Dic3C2312yes—
67Dic3⋊C4—22no—
68Dic3⋊C4—22no—
69Dic3⋊C4—22no—
70C4×Dic3—22no—
71C4×Dic3D411yescharacteristic
72C4×Dic3D411yescharacteristic
73C4×Dic3C2311yescharacteristic
74C2×C3⋊C8C2312yes—
75C22×Q8—61no—
76C4⋊Q8—32no—
77C4⋊Q8—31no—
78C4⋊Q8—31no—
79C4⋊Q8—61no—
80C4⋊Q8—61no—
81C4⋊Q8C2×D311yescharacteristic
82C4⋊Q8C2×D311yescharacteristic
83#125—61no—
84C22⋊Q8—32no—
85C22⋊Q8—61no—
86C4×Q8—32no—
87C4×Q8—61no—
88#117—61no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—61no—
91C2×C42—31no—
92#106—34no—
93#106—34no—
94C4⋊C8—32no—
95#103—34no—
96#103C2×D314yes—
97C22⋊C8—32no—
98C4×C8—31no—
99C4×C8C2×D311yescharacteristic
100C22×D3C2×D411yescharacteristic
101C3×Q8—24no—
102C3×Q8—42no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4C2×D411yescharacteristic
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2411yescharacteristic
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3C2×D411yescharacteristic
111C2×Dic3C2×D411yescharacteristic
112C2×Dic3C2×D411yescharacteristic
113C4×D3—22no—
114C4×D3—22no—
115C4×D3—44no—
116C4×D3SD1614yes—
117Dic6—42no—
118Dic6—42no—
119Dic6—44no—
120C3×C8—22no—
121C3⋊C8—22no—
122C2×Q8—21no—
123C2×Q8—62no—
124C2×Q8—61no—
125C2×Q8—61no—
126C2×Q8—124no—
127C2×Q8C22×D312yes—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—31no—
131C22×C4—62no—
132C2×C8—32no—
133C2×C8C22×D312yes—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—32no—
138C4⋊C4—32no—
139C4⋊C4—62no—
140C4⋊C4—62no—
141C4⋊C4—62no—
142C4⋊C4—62no—
143C4⋊C4—61no—
144C4⋊C4—61no—
145C4⋊C4C22×D312yes—
146C22⋊C4—62no—
147C42—31no—
148C42—31no—
149C42—31no—
150C42—62no—
151C42C22×D311yescharacteristic
152C22×C3C22×D411yescharacteristic
153C2×D3—21no—
154C2×D3—21no—
155C2×D3C2×SD1612yes—
156C3×C4—42no—
157C3×C4—42no—
158C3×C4C2×SD1612yes—
159C3×C4#13712yes—
160C3×C42−1+412yes—
161Dic3—21no—
162Dic3—21no—
163Dic3—42no—
164Dic3—42no—
165Dic3C2×SD1612yes—
166C23—31no—
167Q8—24no—
168Q8—42no—
169Q8—124no—
170Q8—122no—
171Q8—122no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—31no—
175C2×C4—34no—
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—124no—
183C2×C4D3×D411yescharacteristic
184C2×C4D3×D411yescharacteristic
185C2×C4C23×D311yescharacteristic
186C8—22no—
187C8—62no—
188C2×C3#53611yescharacteristic
189C2×C3C22×SD1611yescharacteristic
190C2×C3C2×#13711yescharacteristic
191D3—22no—
192C22—32no—
193C22—61no—
194C22—61no—
195C22C2×D3×D411yescharacteristic
196C4—32no—
197C4—42no—
198C4—42no—
199C4—61no—
200C4—61no—
201C4—122no—
202C4—122no—
203C4D3×SD1612yes—
204C4#91812yes—
205C4#101012yes—
206C3#322911yescharacteristic, complemented, Sylow
207C2—62no—
208C2#567511yescharacteristic
209C2C2×D3×SD1611yescharacteristic
210C2C2×#91811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#487C2
socleC22×C3C22×D4
prime radicalC22×D3C2×D4
hypercenter#487D3
hypocenterC3#3229
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
3. C3×#487C2semidirect
4. #5080C2semidirect
5. #5079C2semidirect
7. #4791C2semidirect
9. #4680C2semidirect
41. #487D3semidirect
206. C3#3229semidirect

Maximal subgroups

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

subgroupindex
#57282
D3×C4⋊Q82
C3×#4872
#50802
#50792
#48052
#47912
#46852
#46802
#32293