Finite Group Databasebeta

← D3×#1141⋊C3 #72245627

Subgroups — D3×#1141⋊C3 #72245627

Order 2016.

326 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 73 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×D4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2D3×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3#66632⋊C3C211yesmaximal, characteristic, complemented
4C3×#1141⋊C3C211yesmaximal, characteristic, complemented
5#66562⋊C3C211yesmaximal, characteristic, complemented
6#66560⋊C3C211yesmaximal, characteristic, complemented
7D3×ΓL(1,𝔽8)⋊C8C211yesmaximal, characteristic, complemented
8D3×#1141C311yesmaximal, characteristic, complemented
9C2×#1141⋊C3—31nomaximal
10C22×D3×ΓL(1,𝔽8)—21no—
11C2×D3×AGL(1,𝔽7)—21no—
12C3×D4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
13C4⋊(C3⋊AGL(1,𝔽7))C2211yescharacteristic, complemented
14ΓL(1,𝔽8)×#67—21no—
15ΓL(1,𝔽8)×D12C2211yescharacteristic, complemented
16C4×D3×ΓL(1,𝔽8)C2211yescharacteristic
17C3×C4⋊AGL(1,𝔽7)C2211yescharacteristic, complemented
18#4246⋊C3—21no—
19C3⋊(ΓL(1,𝔽8)⋊C8)C2211yescharacteristic, complemented
20C3×ΓL(1,𝔽8)⋊C8C2211yescharacteristic, complemented
21D3×C7×D4C2×C311yescharacteristic, complemented
22C2×D4×ΓL(1,𝔽8)—31no—
23D3×D28C2×C311yescharacteristic, complemented
24C2×C4⋊AGL(1,𝔽7)—31no—
25#66632C2×C311yescharacteristic, complemented
26C3×#1141C2×C311yescharacteristic, complemented
27#66562C2×C311yescharacteristic, complemented
28#66560C2×C311yescharacteristic, complemented
29D3×C7⋊C8C2×C311yescharacteristic, complemented
30#1141⋊C3—21no—
31#1141⋊C3—31no—
32#1141⋊C3—31no—
33#1141⋊C3—31no—
34#1141⋊C3D311yescharacteristic, direct product factor
35C2×ΓL(1,𝔽8)⋊C8—31no—
36C3×D3×D8—71nomaximal, Hall
37C22×C3×ΓL(1,𝔽8)—21no—
38C2×C3⋊AGL(1,𝔽7)—21no—
39C2×D3×ΓL(1,𝔽8)—21no—
40C2×D3×ΓL(1,𝔽8)—42no—
41C2×D3×ΓL(1,𝔽8)D411yescharacteristic
42C2×C3×AGL(1,𝔽7)—21no—
43D3×AGL(1,𝔽7)—42no—
44C3×C4×ΓL(1,𝔽8)C2311yescharacteristic
45Dic3×ΓL(1,𝔽8)D411yescharacteristic
46C2×#1141—31noHall
47C22×D3×C7—21no—
48C23×ΓL(1,𝔽8)—61no—
49C2×D3×D7—21no—
50C22×AGL(1,𝔽7)—61no—
51C3×C7×D4C22×C311yescharacteristic, complemented
52D84C22×C311yescharacteristic, complemented
53C7×#67—21no—
54C7×D12C22×C311yescharacteristic, complemented
55C4×D3×C7C22×C311yescharacteristic
56C3×D28C22×C311yescharacteristic, complemented
57D4×ΓL(1,𝔽8)—21no—
58D4×ΓL(1,𝔽8)—31no—
59D4×ΓL(1,𝔽8)—61no—
60D4×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
61C2×C4×ΓL(1,𝔽8)—31no—
62#4246—21no—
63C4⋊AGL(1,𝔽7)—21no—
64C4⋊AGL(1,𝔽7)—31no—
65C4⋊AGL(1,𝔽7)—61no—
66C4⋊AGL(1,𝔽7)C2×D311yescharacteristic, complemented
67C3⋊(C7⋊C8)C22×C311yescharacteristic, complemented
68C3×C7⋊C8C22×C311yescharacteristic, complemented
69ΓL(1,𝔽8)⋊C8—21no—
70ΓL(1,𝔽8)⋊C8—31no—
71ΓL(1,𝔽8)⋊C8C2×D311yescharacteristic, complemented
72C3×D3×D4—71no—
73C3×D3×D4—141no—
74C32×D8—71no—
75C3×#213—71no—
76C3×#213—71no—
77C3×D24—71no—
78C3×D3×C8—71no—
79C2×C3×ΓL(1,𝔽8)—41no—
80C2×C3×ΓL(1,𝔽8)C2×D411yescharacteristic
81C3⋊AGL(1,𝔽7)—41no—
82D3×ΓL(1,𝔽8)—41no—
83D3×ΓL(1,𝔽8)D812yescomplemented
84C3×AGL(1,𝔽7)—41no—
85C2×C7×D4—31no—
86C2×D28—31no—
87#1141—31no—
88#1141—31no—
89#1141—31no—
90#1141C3×D311yescharacteristic, complemented
91C2×C7⋊C8—31no—
92C2×C3×D8—211no—
93D3×D8—71no—
94C22×C3×C7—21no—
95C2×D21—21no—
96C2×D3×C7—21no—
97C2×D3×C7—42no—
98C2×D3×C7C3×D411yescharacteristic
99C2×C3×D7—21no—
100C22×ΓL(1,𝔽8)—21no—
101C22×ΓL(1,𝔽8)—31no—
102C22×ΓL(1,𝔽8)—41no—
103C22×ΓL(1,𝔽8)—61no—
104C22×ΓL(1,𝔽8)—122no—
105D3×D7—42no—
106C2×AGL(1,𝔽7)—21no—
107C2×AGL(1,𝔽7)—41no—
108C2×AGL(1,𝔽7)—61no—
109C2×AGL(1,𝔽7)—122no—
110C3×C4×C7C23×C311yescharacteristic
111C7×Dic3C3×D411yescharacteristic
112C4×ΓL(1,𝔽8)—21no—
113C4×ΓL(1,𝔽8)—31no—
114C4×ΓL(1,𝔽8)C22×D311yescharacteristic
115C22×C3×D3—141no—
116C22×C3×D3—141no—
117C32×D4—71no—
118C32×D4—141no—
119C3×#67—141no—
120C3×#67—141no—
121C3×D12—71no—
122C3×D12—141no—
123C3×C4×D3—141no—
124C32×C8—71no—
125C3×C3⋊C8—71no—
126C3×ΓL(1,𝔽8)C2×D811yescharacteristic, complemented, Hall
127C23×C7—61no—
128C22×D7—61no—
129C7×D4—31no—
130C7×D4—61no—
131C7×D4C2×C3×D311yescharacteristic, complemented
132C2×C4×C7—31no—
133D28—31no—
134D28—61no—
135D28C2×C3×D311yescharacteristic, complemented
136C7⋊C8—31no—
137C7⋊C8C2×C3×D311yescharacteristic, complemented
138C2×C3×D4—211no—
139C2×C3×D4—421no—
140D3×D4—71no—
141D3×D4AGL(1,𝔽7)11yescharacteristic, complemented
142C3×D8—71no—
143C3×D8—71no—
144C3×D8—141no—
145C3×D8—211no—
146C3×D8—211no—
147C3×D8—211no—
148C2×C3×C8—211no—
149#213—71no—
150#213—71no—
151D24—71no—
152D3×C8—71no—
153C2×C3×C7—41no—
154C2×C3×C7C2×C3×D411yescharacteristic
155D21—41no—
156C3×D7—41no—
157D3×C7—41no—
158D3×C7C3×D812yescomplemented
159C2×ΓL(1,𝔽8)—21no—
160C2×ΓL(1,𝔽8)—32no—
161C2×ΓL(1,𝔽8)—41no—
162C2×ΓL(1,𝔽8)—81no—
163C2×ΓL(1,𝔽8)—121no—
164C2×ΓL(1,𝔽8)D3×D411yescharacteristic
165AGL(1,𝔽7)—41no—
166AGL(1,𝔽7)—81no—
167AGL(1,𝔽7)—121no—
168C22×C32—141no—
169C22×C32—141no—
170C2×C3×D3—141no—
171C2×C3×D3—141no—
172C2×C3×D3—141no—
173C2×C3×D3—282no—
174C2×C3×D3—282no—
175C32×C4—141no—
176C3×Dic3—141no—
177C2×D8—211noSylow
178C22×C7—21no—
179C22×C7—31no—
180C22×C7—61no—
181C22×C7—122no—
182C2×D7—21no—
183C2×D7—61no—
184C2×D7—122no—
185C4×C7—31no—
186C4×C7C22×C3×D311yescharacteristic
187C23×C3—421no—
188C23×C3—421no—
189C22×D3—21no—
190C22×D3—141no—
191C3×D4—71no—
192C3×D4—71no—
193C3×D4—141no—
194C3×D4—141no—
195C3×D4—211no—
196C3×D4—281no—
197C3×D4—421no—
198C3×D4—421no—
199C3×D4—421no—
200C3×D4C2×AGL(1,𝔽7)11yescharacteristic, complemented
201C2×C3×C4—421no—
202#67—21no—
203#67—141no—
204D12—71no—
205D12C2×AGL(1,𝔽7)11yescharacteristic, complemented
206C4×D3C2×AGL(1,𝔽7)11yescharacteristic
207C3×C8—71no—
208C3×C8—71no—
209C3×C8—141no—
210C3×C8—211no—
211C3⋊C8—71no—
212C3×C7C2×C3×D811yescharacteristic, complemented
213ΓL(1,𝔽8)—21no—
214ΓL(1,𝔽8)D3×D811yescharacteristic, complemented
215C2×C32—141no—
216C2×C32—281no—
217C2×C32—281no—
218C3×D3—142no—
219C3×D3—281no—
220C3×D3—281no—
221C2×D4—31no—
222C2×D4—211no—
223D8—71no—
224D8—211no—
225D8—211no—
226D8—211no—
227C2×C8—211no—
228C2×C7—32no—
229C2×C7—41no—
230C2×C7—121no—
231C2×C7C3×D3×D411yescharacteristic
232D7—41no—
233D7—121no—
234C22×C3—21no—
235C22×C3—141no—
236C22×C3—141no—
237C22×C3—141no—
238C22×C3—281no—
239C22×C3—281no—
240C22×C3—421no—
241C22×C3—421no—
242C22×C3—421no—
243C22×C3—842no—
244C22×C3—842no—
245C2×D3—21no—
246C2×D3—42no—
247C2×D3—141no—
248C2×D3—282no—
249C2×D3(C22⋊D7)⋊C311yescharacteristic
250C3×C4—141no—
251C3×C4—281no—
252C3×C4—421no—
253C3×C4C22×AGL(1,𝔽7)11yescharacteristic
254Dic3(C22⋊D7)⋊C311yescharacteristic
255C32—141noSylow
256C23—61no—
257C23—421no—
258D4—31no—
259D4—61no—
260D4—71no—
261D4—211no—
262D4—421no—
263D4D3×AGL(1,𝔽7)11yescharacteristic, complemented
264C2×C4—31no—
265C8—71no—
266C8—211no—
267C7C3×D3×D811yescharacteristic, complemented, Sylow
268C2×C3—41no—
269C2×C3—141no—
270C2×C3—281no—
271C2×C3—281no—
272C2×C3—281no—
273C2×C3—281no—
274C2×C3—422no—
275C2×C3—561no—
276C2×C3—561no—
277C2×C3—841no—
278C2×C3—841no—
279C2×C3C2×(C22⋊D7)⋊C311yescharacteristic
280D3—41no—
281D3—281no—
282D3#1141⋊C312yesdirect product factor
283C22—21no—
284C22—31no—
285C22—61no—
286C22—122no—
287C22—141no—
288C22—421no—
289C22—842no—
290C4—31no—
291C4C2×D3×AGL(1,𝔽7)11yescharacteristic
292C3—141no—
293C3—281no—
294C3C2×#1141⋊C311yescharacteristic, complemented
295C2—32no—
296C2—41no—
297C2—121no—
298C2—281no—
299C2—841no—
300C2D3×(C22⋊D7)⋊C311yescharacteristic
subgroupG/H
centerC2D3×(C22⋊D7)⋊C3
commutator, cosocleC3×C4×C7C23×C3
FrattiniC4C2×D3×AGL(1,𝔽7)
FittingC3×C7×D4C22×C3
socleC2×C3×C7C2×C3×D4
hypercenterD4D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C3×D8
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. D3×D4×ΓL(1,𝔽8)C2semidirect
2. D3×C4⋊AGL(1,𝔽7)C2semidirect
3. #66632⋊C3C2semidirect
4. C3×#1141⋊C3C2semidirect
5. #66562⋊C3C2semidirect
6. #66560⋊C3C2semidirect
7. D3×ΓL(1,𝔽8)⋊C8C2semidirect
8. D3×#1141C3semidirect
12. C3×D4×ΓL(1,𝔽8)C22semidirect
13. C4⋊(C3⋊AGL(1,𝔽7))C22semidirect
15. ΓL(1,𝔽8)×D12C22semidirect
17. C3×C4⋊AGL(1,𝔽7)C22semidirect
19. C3⋊(ΓL(1,𝔽8)⋊C8)C22semidirect
20. C3×ΓL(1,𝔽8)⋊C8C22semidirect
21. D3×C7×D4C2×C3semidirect
23. D3×D28C2×C3semidirect
25. #66632C2×C3semidirect
26. C3×#1141C2×C3semidirect
27. #66562C2×C3semidirect
28. #66560C2×C3semidirect
29. D3×C7⋊C8C2×C3semidirect
34. #1141⋊C3D3direct (N × H)
51. C3×C7×D4C22×C3semidirect
52. D84C22×C3semidirect
54. C7×D12C22×C3semidirect
56. C3×D28C22×C3semidirect
60. D4×ΓL(1,𝔽8)C2×D3semidirect
66. C4⋊AGL(1,𝔽7)C2×D3semidirect
67. C3⋊(C7⋊C8)C22×C3semidirect
68. C3×C7⋊C8C22×C3semidirect
71. ΓL(1,𝔽8)⋊C8C2×D3semidirect
83. D3×ΓL(1,𝔽8)D8semidirect
90. #1141C3×D3semidirect
126. C3×ΓL(1,𝔽8)C2×D8semidirect
131. C7×D4C2×C3×D3semidirect
135. D28C2×C3×D3semidirect
137. C7⋊C8C2×C3×D3semidirect
141. D3×D4AGL(1,𝔽7)semidirect
158. D3×C7C3×D8semidirect
200. C3×D4C2×AGL(1,𝔽7)semidirect
205. D12C2×AGL(1,𝔽7)semidirect
212. C3×C7C2×C3×D8semidirect
214. ΓL(1,𝔽8)D3×D8semidirect
263. D4D3×AGL(1,𝔽7)semidirect
267. C7C3×D3×D8semidirect
282. D3#1141⋊C3direct (N × H)
294. C3C2×#1141⋊C3semidirect

Maximal subgroups

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

subgroupindex
D3×D4×ΓL(1,𝔽8)2
D3×C4⋊AGL(1,𝔽7)2
#66632⋊C32
C3×#1141⋊C32
#66562⋊C32
#66560⋊C32
D3×ΓL(1,𝔽8)⋊C82
D3×#11413
C2×#1141⋊C33
C3×D3×D87