Finite Group Databasebeta

← D3×ΓL(1,𝔽81) #14314493

Subgroups — D3×ΓL(1,𝔽81) #14314493

Order 1920.

304 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3995C211yesmaximal, characteristic
2D3×#3993C211yesmaximal, characteristic
3D3×#3930C211yesmaximal, characteristic
4#11746718C211yesmaximal, characteristic, complemented
5C3×ΓL(1,𝔽81)C211yesmaximal, characteristic, complemented
6#11746475C211yesmaximal, characteristic, complemented
7#11746474C211yesmaximal, characteristic, complemented
8C2×ΓL(1,𝔽81)—31nomaximal, Hall
9D3×C4⋊AGL(1,𝔽5)—21no—
10D3×#700—21no—
11D3×C8×D5C2211yescharacteristic
12#91224C2211yescharacteristic
13#91222C2211yescharacteristic
14C3×#3995C2211yescharacteristic
15C3×#3993C2211yescharacteristic
16#91161—21no—
17#91159—21no—
18#91082C411yescharacteristic, complemented
19C5×D3×C16C411yescharacteristic, complemented
20C3×#3930C2211yescharacteristic
21#90936C411yescharacteristic, complemented
22#90934C2211yescharacteristic
23D3×#672C411yescharacteristic, complemented
24D3×#365—51nomaximal, Hall
25C2×#3995—31no—
26C2×#3993—31no—
27C2×#3930—31no—
28ΓL(1,𝔽81)—31no—
29ΓL(1,𝔽81)—31no—
30ΓL(1,𝔽81)—31no—
31ΓL(1,𝔽81)D311yescharacteristic, direct product factor
32C2×D3×AGL(1,𝔽5)—41no—
33C4×D3×D5D411yescharacteristic
34C4⋊ΓL(1,𝔽16)—21no—
35#6758—21no—
36C3×C4⋊AGL(1,𝔽5)—21no—
37C3×#700—21no—
38#6740—41no—
39D3×#170—41no—
40Dic3⋊AGL(1,𝔽5)—41no—
41C8×D15C2×C411yescharacteristic
42C5×D3×C8C2×C411yescharacteristic
43C3×C8×D5C2311yescharacteristic
44#6648Q811yescharacteristic
45D3×#168Q811yescharacteristic
46D5×C3⋊C8D411yescharacteristic
47C3×C5×C16C2×C411yescharacteristic, complemented
48C3⋊#672C2×C411yescharacteristic, complemented
49C3×#672C2×C411yescharacteristic, complemented
50C5×C3⋊C16C2×C411yescharacteristic, complemented
51D3×M32—51no—
52D3×#108—51no—
53D3×#106—51no—
54C3×#365—51no—
55#4513—51no—
56#4494—51no—
57#4487—51no—
58C2×C4⋊AGL(1,𝔽5)—61no—
59C2×#700—61no—
60C2×C8×D5—31no—
61#3995—31no—
62#3995—61no—
63#3995C2×D311yescharacteristic
64#3993—31no—
65#3993—61no—
66#3993C2×D311yescharacteristic
67C2×C5×C16—31no—
68C2×#672—31no—
69#3930—31no—
70#3930—31no—
71#3930—31no—
72#3930C2×D311yescharacteristic
73C2×#365—151noSylow
74C2×D3×D5SD1611yescharacteristic
75C2×ΓL(1,𝔽16)—41no—
76C2×C3×AGL(1,𝔽5)—41no—
77D3×AGL(1,𝔽5)—82no—
78C4×D15C4⋊C411yescharacteristic
79C4×C5×D3C4⋊C411yescharacteristic
80C3×C4×D5C2×D411yescharacteristic
81#1200SD1611yescharacteristic
82D3×Dic5SD1611yescharacteristic
83D5×Dic3SD1611yescharacteristic
84C3⋊#170—41no—
85C3×#170—41no—
86C3×C5×C8C22×C411yescharacteristic
87C3⋊#168C4⋊C411yescharacteristic
88C3×#168C2×Q811yescharacteristic
89C5×C3⋊C8C4⋊C411yescharacteristic
90D3×M16—101no—
91C2×D3×C8—51no—
92D3×C4⋊C4—101no—
93C3×M32—51no—
94C3×#108—51no—
95C3×#106—51no—
96#822—101no—
97#819—51no—
98#817—51no—
99#812—51no—
100#808—101no—
101#798—51no—
102#798—51no—
103D3×C16—51no—
104D3×C16AGL(1,𝔽5)11yescharacteristic, complemented
105C22×AGL(1,𝔽5)—121no—
106C2×C4×D5—31no—
107C2×#170—121no—
108C4⋊AGL(1,𝔽5)—21no—
109C4⋊AGL(1,𝔽5)—61no—
110C4⋊AGL(1,𝔽5)—121no—
111#700—21no—
112#700—61no—
113#700—121no—
114C2×C5×C8—31no—
115C2×#168—31no—
116C8×D5—31no—
117C8×D5—31no—
118C8×D5—31no—
119C8×D5C22×D311yescharacteristic
120C5×C16—31no—
121C5×C16C4×D311yescharacteristic, complemented
122#672—31no—
123#672C4×D311yescharacteristic, complemented
124C2×M32—151no—
125C2×#108—151no—
126C2×#106—151no—
127#365—51no—
128#365—151no—
129#365—151no—
130#365—151no—
131C2×D15#10611yescharacteristic
132C2×C5×D3#10611yescharacteristic
133C2×C3×D5C2×SD1611yescharacteristic
134D3×D5—22no—
135ΓL(1,𝔽16)—81no—
136C3×AGL(1,𝔽5)—81no—
137C3×C4×C5C2×C4⋊C411yescharacteristic
138Dic15#10611yescharacteristic
139C3×Dic5C2×SD1611yescharacteristic
140C5×Dic3#10611yescharacteristic
141C2×C4×D3—51no—
142C2×C4×D3—201no—
143C3×M16—101no—
144C2×C3×C8—51no—
145C3×C4⋊C4—101no—
146C4⋊Dic3—101no—
147Dic3⋊C4—201no—
148#208—101no—
149C2×C3⋊C8—51no—
150ΓL(1,𝔽25)—201no—
151D3×C8—51no—
152D3×C8—51no—
153D3×C8—51no—
154D3×C8—201no—
155D3×C8C2×AGL(1,𝔽5)11yescharacteristic
156C3×C16—51no—
157C3×C16C2×AGL(1,𝔽5)11yescharacteristic, complemented
158C3⋊C16—51no—
159C3⋊C16C2×AGL(1,𝔽5)11yescharacteristic, complemented
160C22×D5—31no—
161C2×AGL(1,𝔽5)—41no—
162C2×AGL(1,𝔽5)—121no—
163C2×AGL(1,𝔽5)—242no—
164C2×C4×C5—31no—
165C2×Dic5—31no—
166C4×D5—31no—
167C4×D5—31no—
168C4×D5—31no—
169C4×D5D3×D411yescharacteristic
170#170—41no—
171#170—121no—
172C5×C8—31no—
173C5×C8C2×C4×D311yescharacteristic
174#168—31no—
175#168D3×Q811yescharacteristic
176C2×M16—301no—
177C22×C8—151no—
178C2×C4⋊C4—301no—
179M32—51no—
180M32—151no—
181M32—151no—
182M32—151no—
183C2×C16—31no—
184C2×C16—151no—
185#108—51no—
186#108—151no—
187#108—301no—
188#106—51no—
189#106—151no—
190#106—301no—
191C2×C3×C5C2×#10611yescharacteristic
192D15—21no—
193C3×D5—21no—
194C5×D3#36512yescomplemented
195C22×D3—51no—
196C2×C3×C4—51no—
197C2×C3×C4—201no—
198C2×Dic3—51no—
199C2×Dic3—201no—
200C4×D3—51no—
201C4×D3—51no—
202C4×D3—51no—
203C4×D3—402no—
204C4×D3C4⋊AGL(1,𝔽5)11yescharacteristic
205C3×C8—51no—
206C3×C8—201no—
207C3×C8C22×AGL(1,𝔽5)11yescharacteristic
208C3⋊C8—51no—
209C3⋊C8—201no—
210C3⋊C8C4⋊AGL(1,𝔽5)11yescharacteristic
211C22×C5—31no—
212C2×D5—31no—
213C2×D5—62no—
214C2×D5D3×SD1611yescharacteristic
215AGL(1,𝔽5)—81no—
216AGL(1,𝔽5)—241no—
217C4×C5—31no—
218C4×C5D3×C4⋊C411yescharacteristic
219Dic5—31no—
220Dic5D3×SD1611yescharacteristic
221C22×C4—151no—
222C22×C4—601no—
223M16—101no—
224M16—301no—
225M16—601no—
226C2×C8—31no—
227C2×C8—51no—
228C2×C8—151no—
229C2×C8—151no—
230C2×C8—151no—
231C2×C8—151no—
232C2×C8—601no—
233C4⋊C4—101no—
234C4⋊C4—301no—
235C4⋊C4—601no—
236C16—31no—
237C16—51no—
238C16—151no—
239C16D3×AGL(1,𝔽5)11yescharacteristic, complemented
240C3×C5C2×#36511yescharacteristic, complemented, Hall
241C22×C3—51no—
242C2×D3—51no—
243C2×D3—102no—
244C2×D3#399311yescharacteristic
245C3×C4—51no—
246C3×C4—401no—
247C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
248Dic3—51no—
249Dic3—401no—
250Dic3#399311yescharacteristic
251C2×C5—32no—
252C2×C5D3×#10611yescharacteristic
253D5—21no—
254D5—61no—
255C23—151no—
256C2×C4—31no—
257C2×C4—51no—
258C2×C4—151no—
259C2×C4—151no—
260C2×C4—151no—
261C2×C4—151no—
262C2×C4—201no—
263C2×C4—601no—
264C2×C4—1202no—
265C8—31no—
266C8—51no—
267C8—151no—
268C8—201no—
269C8—601no—
270C8C2×D3×AGL(1,𝔽5)11yescharacteristic
271C2×C3—101no—
272C2×C3C2×#399311yescharacteristic
273D3—101no—
274D3ΓL(1,𝔽81)12yesdirect product factor
275C5D3×#36511yescharacteristic, complemented, Sylow
276C22—31no—
277C22—51no—
278C22—151no—
279C22—302no—
280C4—31no—
281C4—51no—
282C4—151no—
283C4—401no—
284C4—1201no—
285C4D3×C4⋊AGL(1,𝔽5)11yescharacteristic
286C3C2×ΓL(1,𝔽81)11yescharacteristic, complemented, Sylow
287C2—32no—
288C2—101no—
289C2—301no—
290C2D3×#399311yescharacteristic
subgroupG/H
centerC2D3×#3993
commutatorC3×C5×C8C22×C4
FrattiniC8C2×D3×AGL(1,𝔽5)
FittingC3×C5×C16C2×C4
socleC2×C3×C5C2×#106
prime radicalC2×D3×D5SD16
cosocleC3×C8×D5C23
hypercenterC16D3×AGL(1,𝔽5)
hypocenterC3×C5C2×#365
LCSG ▷ C3×C5×C8 ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
4. #11746718C2semidirect
5. C3×ΓL(1,𝔽81)C2semidirect
6. #11746475C2semidirect
7. #11746474C2semidirect
18. #91082C4semidirect
19. C5×D3×C16C4semidirect
21. #90936C4semidirect
23. D3×#672C4semidirect
31. ΓL(1,𝔽81)D3direct (N × H)
47. C3×C5×C16C2×C4semidirect
48. C3⋊#672C2×C4semidirect
49. C3×#672C2×C4semidirect
50. C5×C3⋊C16C2×C4semidirect
104. D3×C16AGL(1,𝔽5)semidirect
121. C5×C16C4×D3semidirect
123. #672C4×D3semidirect
157. C3×C16C2×AGL(1,𝔽5)semidirect
159. C3⋊C16C2×AGL(1,𝔽5)semidirect
194. C5×D3#365semidirect
239. C16D3×AGL(1,𝔽5)semidirect
240. C3×C5C2×#365semidirect
274. D3ΓL(1,𝔽81)direct (N × H)
275. C5D3×#365semidirect
286. C3C2×ΓL(1,𝔽81)semidirect

Maximal subgroups

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

subgroupindex
D3×#39952
D3×#39932
D3×#39302
#117467182
C3×ΓL(1,𝔽81)2
#117464752
#117464742
C2×ΓL(1,𝔽81)3
D3×#3655