Finite Group Databasebeta

← #81982

Subgroups — #81982

Order 384.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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#5691C211yesmaximal, characteristic
3C3×#501C211yesmaximal, characteristic, complemented
4#5089C211yesmaximal, characteristic, complemented
5#5064C211yesmaximal, characteristic, complemented
6#4882C212yesmaximal
7#4875C212yesmaximal, complemented
8#4870C212yesmaximal, complemented
9#4862C212yesmaximal, complemented
10#4714C211yesmaximal, characteristic, complemented
11#4705C211yesmaximal, characteristic, complemented
12#3238—31nomaximal, Sylow
13C2×D3×Q8—21no—
14C3×C4⋊Q8C2211yescharacteristic
15C3×#125C2211yescharacteristic
16#944—21no—
17#898—21no—
18#897C2211yescharacteristic
19#895C2211yescharacteristic
20#894—21no—
21#893—21no—
22#893C2211yescharacteristic
23#892—21no—
24D3×C4⋊C4—21no—
25D3×C4⋊C4C2212yes—
26#888—21no—
27#888C2212yes—
28#887C2211yescharacteristic
29#875C2211yescharacteristic
30C42×D3C2211yescharacteristic
31#869C2211yescharacteristic
32C3×#107C2212yes—
33C3×#106C2212yes—
34C3×#97C2211yescharacteristic
35#821C2212yes—
36(C2×D3)⋊C8C2212yes—
37#816C2212yes—
38Dic3⋊C8C2212yes—
39#810C2212yes—
40#809C2212yes—
41#808C2212yes—
42#807C2212yes—
43#803C2211yescharacteristic
44C2×C4⋊Q8—31no—
45#529—31no—
46#501—31no—
47#501D311yescharacteristic, complemented
48#489—31no—
49#489—31no—
50#483—32no—
51#480—32no—
52#477—32no—
53#476—32no—
54#432—31no—
55C2×C3×Q8—21no—
56D3×Q8—44no—
57C2×D12—21no—
58C2×D12C2311yescharacteristic
59C2×C4×D3—22no—
60C2×C4×D3—21no—
61C2×C4×D3D411yescharacteristic
62C2×C4×D3D411yescharacteristic
63C2×C4×D3C2311yescharacteristic
64C2×Dic6—21no—
65C2×Dic6—22no—
66C2×Dic6C2311yescharacteristic
67C2×C3×C8C2312yes—
68C3×C4⋊C4—21no—
69C3×C4⋊C4—21no—
70C3×C4⋊C4—21no—
71C3×C4⋊C4C2311yescharacteristic
72C3×C4⋊C4C2311yescharacteristic
73C3×C4⋊C4C2312yes—
74C3×C42C2311yescharacteristic
75(C2×D3)⋊C4—21no—
76(C2×D3)⋊C4—21no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—21no—
79(C2×D3)⋊C4—21no—
80C4⋊Dic3—21no—
81C4⋊Dic3—21no—
82C4⋊Dic3C2312yes—
83Dic3⋊C4—21no—
84Dic3⋊C4—22no—
85Dic3⋊C4—22no—
86C4×Dic3—21no—
87C4×Dic3D411yescharacteristic
88C4×Dic3D411yescharacteristic
89C4×Dic3C2311yescharacteristic
90C2×C3⋊C8C2312yes—
91C22×Q8—61no—
92C4⋊Q8—32no—
93C4⋊Q8—31no—
94C4⋊Q8—61no—
95C4⋊Q8—61no—
96C4⋊Q8C2×D311yescharacteristic
97#126—61no—
98#125C2×D311yescharacteristic
99#124—31no—
100#123—61no—
101C22⋊Q8—31no—
102#121—31no—
103C4×Q8—31no—
104C4×D4—31no—
105C4×D4—61no—
106#117—61no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—61no—
109C2×C42—31no—
110#107—32no—
111#107C2×D312yes—
112#106—32no—
113#106C2×D312yes—
114C4⋊C8—32no—
115#103—32no—
116#103—32no—
117#102—32no—
118#102—32no—
119C22⋊C8—32no—
120#97—31no—
121#97C2×D311yescharacteristic
122C22×D3—21no—
123C22×D3C2×D411yescharacteristic
124C3×Q8—42no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2411yescharacteristic
131C2×Dic3—21no—
132C2×Dic3—22no—
133C2×Dic3C2×D411yescharacteristic
134C2×Dic3C2×D411yescharacteristic
135C2×Dic3C2×D411yescharacteristic
136D12—22no—
137D12—42no—
138C4×D3—22no—
139C4×D3—22no—
140C4×D3—22no—
141C4×D3—44no—
142C4×D3—42no—
143Dic6—22no—
144Dic6—44no—
145Dic6—42no—
146C3×C8—22no—
147C3⋊C8—22no—
148C2×Q8—21no—
149C2×Q8—31no—
150C2×Q8—62no—
151C2×Q8—61no—
152C2×Q8—124no—
153C2×D4—31no—
154C2×D4—61no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—61no—
159C22×C4—62no—
160C2×C8—32no—
161C2×C8C22×D312yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—32no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—61no—
169C4⋊C4—61no—
170C4⋊C4—61no—
171C4⋊C4C22×D312yes—
172C4⋊C4C22×D311yescharacteristic
173C4⋊C4C22×D311yescharacteristic
174C22⋊C4—61no—
175C22⋊C4—61no—
176C22⋊C4—61no—
177C22⋊C4—61no—
178C22⋊C4—61no—
179C42—31no—
180C42—31no—
181C42—31no—
182C42—61no—
183C42C22×D311yescharacteristic
184C22×C3C22×D411yescharacteristic
185C2×D3—21no—
186C2×D3—21no—
187C2×D3—41no—
188C2×D3—41no—
189C2×D3—41no—
190C2×D3AGL(1,ℤ/8)12yes—
191C3×C4—21no—
192C3×C4—21no—
193C3×C4—42no—
194C3×C4—41no—
195C3×C4—41no—
196C3×C42−1+412yes—
197Dic3—21no—
198Dic3—21no—
199Dic3—41no—
200Dic3—42no—
201Dic3#13712yes—
202C23—31no—
203C23—61no—
204Q8—42no—
205Q8—62no—
206Q8—122no—
207Q8—124no—
208D4—62no—
209D4—122no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—61no—
220C2×C4—62no—
221C2×C4—122no—
222C2×C4—124no—
223C2×C4D3×D411yescharacteristic
224C2×C4D3×D411yescharacteristic
225C2×C4C23×D311yescharacteristic
226C8—22no—
227C8—62no—
228C2×C3#53611yescharacteristic
229C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
230C2×C3C2×#13711yescharacteristic
231D3—22no—
232D3—81no—
233C22—32no—
234C22—61no—
235C22—61no—
236C22—121no—
237C22—121no—
238C22—121no—
239C22C2×D3×D411yescharacteristic
240C4—21no—
241C4—21no—
242C4—32no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—61no—
247C4—61no—
248C4—121no—
249C4—122no—
250C4#100712yes—
251C3#323811yescharacteristic, complemented, Sylow
252C2—62no—
253C2—241no—
254C2#572411yescharacteristic
255C2D3×AGL(1,ℤ/8)11yescharacteristic
256C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#501C2
socleC22×C3C22×D4
prime radical#895C22
hypercenter#501D3
hypocenterC3#3238
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. D3×C4⋊Q8C2semidirect
3. C3×#501C2semidirect
4. #5089C2semidirect
5. #5064C2semidirect
7. #4875C2semidirect
8. #4870C2semidirect
9. #4862C2semidirect
10. #4714C2semidirect
11. #4705C2semidirect
47. #501D3semidirect
251. C3#3238semidirect

Maximal subgroups

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

subgroupindex
D3×C4⋊Q82
#56912
C3×#5012
#50892
#50642
#48822
#48752
#48702
#48622
#47142
#47052
#32383