Finite Group Databasebeta

← #81572

Subgroups — #81572

Order 384.

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

normal subgroups only

This group has 404 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#5080C211yesmaximal, characteristic, complemented
5#5070C211yesmaximal, characteristic, complemented
6#4777C214yesmaximal, complemented
7#4761C214yesmaximal, complemented
8#4685C211yesmaximal, characteristic, complemented
9#4680C211yesmaximal, characteristic, complemented
10#3227—31nomaximal, Sylow
11C2×D3×Q8—21no—
12C2×#237—21no—
13C3×C4⋊Q8C2211yescharacteristic
14C3×#127C2211yescharacteristic
15#944—21no—
16#938—21no—
17#938C2212yes—
18#936—21no—
19D4×Dic3—21no—
20D4×Dic3C2212yes—
21D3×C4⋊C4—21no—
22D3×C4⋊C4C2212yes—
23#888—21no—
24#888C2212yes—
25C42×D3C2211yescharacteristic
26#869C2211yescharacteristic
27#869C2211yescharacteristic
28C3×#102C2214yes—
29C3×C4×C8C2211yescharacteristic
30#832C2214yes—
31(C2×D3)⋊C8C2212yes—
32#817C2214yes—
33Dic3⋊C8C2212yes—
34#808C2214yes—
35C4×C3⋊C8C2211yescharacteristic
36#532—31no—
37C2×C4⋊Q8—31no—
38#498—31no—
39#498—31no—
40#486—31no—
41#486D311yescharacteristic, complemented
42#481—34no—
43#476—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—
51D3×Q8—44no—
52#237—44no—
53C2×C4×D3—22no—
54C2×C4×D3D411yescharacteristic
55C2×C4×D3D411yescharacteristic
56C2×C4×D3C2311yescharacteristic
57C2×Dic6—21no—
58C2×Dic6—21no—
59C2×Dic6—22no—
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⋊Dic3C2312yes—
69C4⋊Dic3C2312yes—
70Dic3⋊C4—22no—
71Dic3⋊C4—22no—
72C4×Dic3D411yescharacteristic
73C4×Dic3D411yescharacteristic
74C4×Dic3C2311yescharacteristic
75C2×C3⋊C8C2312yes—
76C2×#41—61no—
77C22×Q8—61no—
78C4⋊Q8—32no—
79C4⋊Q8—31no—
80C4⋊Q8—31no—
81C4⋊Q8—61no—
82C4⋊Q8—61no—
83C4⋊Q8C2×D311yescharacteristic
84#127C2×D311yescharacteristic
85#124—61no—
86#121—32no—
87#121—61no—
88C4×D4—32no—
89C4×D4—61no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—61no—
92C2×C42—31no—
93#106—34no—
94#106—34no—
95C4⋊C8—32no—
96#102—34no—
97#102C2×D314yes—
98C22⋊C8—32no—
99C4×C8—31no—
100C4×C8C2×D311yescharacteristic
101C23×C3—22no—
102C22×D3C2×D411yescharacteristic
103C3×Q8—42no—
104C3×D4—24no—
105C3×D4—42no—
106C3×D4—42no—
107C2×C3×C4—22no—
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2411yescharacteristic
111#67—44no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—44no—
115C2×Dic3C2×D411yescharacteristic
116C2×Dic3C2×D411yescharacteristic
117C2×Dic3C2×D411yescharacteristic
118C4×D3—22no—
119C4×D3—22no—
120C4×D3—44no—
121C4×D3SD1614yes—
122Dic6—44no—
123Dic6—42no—
124Dic6—42no—
125C3×C8—22no—
126C3⋊C8—22no—
127#41—124no—
128C2×Q8—21no—
129C2×Q8—62no—
130C2×Q8—61no—
131C2×Q8—61no—
132C2×Q8—124no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—62no—
136C2×D4C22×D312yes—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—62no—
141C22×C4—62no—
142C2×C8—32no—
143C2×C8C22×D312yes—
144C4⋊C4—21no—
145C4⋊C4—32no—
146C4⋊C4—32no—
147C4⋊C4—62no—
148C4⋊C4—62no—
149C4⋊C4—61no—
150C4⋊C4—61no—
151C4⋊C4C22×D312yes—
152C22⋊C4—62no—
153C22⋊C4—62no—
154C42—31no—
155C42—31no—
156C42—31no—
157C42C22×D311yescharacteristic
158C22×C3—42no—
159C22×C3—42no—
160C22×C3—42no—
161C22×C3C22×D411yescharacteristic
162C2×D3—21no—
163C2×D3—21no—
164C2×D3C2×SD1612yes—
165C3×C4—42no—
166C3×C4C2×SD1612yes—
167C3×C4AGL(1,ℤ/8)12yes—
168C3×C42−1+412yes—
169Dic3—21no—
170Dic3—21no—
171Dic3—42no—
172Dic3—42no—
173Dic3C2×SD1612yes—
174C23—22no—
175C23—31no—
176Q8—42no—
177Q8—124no—
178Q8—122no—
179Q8—122no—
180D4—24no—
181D4—42no—
182D4—42no—
183D4—124no—
184C2×C4—22no—
185C2×C4—31no—
186C2×C4—34no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—124no—
194C2×C4—124no—
195C2×C4D3×D411yescharacteristic
196C2×C4D3×D411yescharacteristic
197C2×C4C23×D311yescharacteristic
198C8—22no—
199C8—62no—
200C2×C3—82no—
201C2×C3#53611yescharacteristic
202C2×C3C22×SD1611yescharacteristic
203C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
204D3—22no—
205C22—32no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—61no—
210C22—61no—
211C22C2×D3×D411yescharacteristic
212C4—32no—
213C4—42no—
214C4—61no—
215C4—61no—
216C4—122no—
217C4—122no—
218C4#91112yes—
219C4D3×SD1612yes—
220C4#101012yes—
221C3#322711yescharacteristic, complemented, Sylow
222C2—62no—
223C2—82no—
224C2#567511yescharacteristic
225C2C2×#91111yescharacteristic
226C2C2×D3×SD1611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#486C2
socleC22×C3C22×D4
prime radical#5716C2
hypercenter#486D3
hypocenterC3#3227
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. #5080C2semidirect
5. #5070C2semidirect
6. #4777C2semidirect
7. #4761C2semidirect
8. #4685C2semidirect
9. #4680C2semidirect
41. #486D3semidirect
221. C3#3227semidirect

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
#50802
#50702
#47772
#47612
#46852
#46802
#32273