Finite Group Databasebeta

← #72245523

Subgroups — #72245523

Order 2016.

292 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 69 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
1D12×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
2#66677⋊C3C211yesmaximal, characteristic, complemented
3D168⋊C3C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×#204C211yesmaximal, characteristic, complemented
5C3×#1131⋊C3C211yesmaximal, characteristic, complemented
6#66568⋊C3C211yesmaximal, characteristic, complemented
7#66562⋊C3C211yesmaximal, characteristic, complemented
8#10620811C311yesmaximal, characteristic, complemented
9#6477⋊C3—31nomaximal
10C2×D3×AGL(1,𝔽7)—21no—
11C4⋊(C3⋊AGL(1,𝔽7))C2211yescharacteristic
12C4⋊(C3⋊AGL(1,𝔽7))C2211yescharacteristic, complemented
13ΓL(1,𝔽8)×D12C2211yescharacteristic
14ΓL(1,𝔽8)×Dic6C2211yescharacteristic, complemented
15C3×C4×AGL(1,𝔽7)C2211yescharacteristic
16#4247⋊C3—21no—
17#4246⋊C3—21no—
18(C3⋊AGL(1,𝔽7))⋊C4—21no—
19C3×C8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
20C3×ΓL(1,𝔽8)⋊C8C2211yescharacteristic, complemented
21D7×D12C2×C311yescharacteristic, complemented
22#66677C2×C311yescharacteristic, complemented
23#66659—31no—
24D4×AGL(1,𝔽7)—31no—
25D168C2×C311yescharacteristic, complemented
26C7×#204C2×C311yescharacteristic, complemented
27C3×#1131C2×C311yescharacteristic, complemented
28SD16×ΓL(1,𝔽8)—31no—
29#66568C2×C311yescharacteristic, complemented
30#66562C2×C311yescharacteristic, complemented
31#1143⋊C3—31no—
32#1141⋊C3—31no—
33D56⋊C3—31no—
34#1131⋊C3—21no—
35#1131⋊C3D311yescharacteristic, complemented
36C3×#908—71nomaximal, Hall
37C2×C3⋊AGL(1,𝔽7)—21no—
38C2×C3⋊AGL(1,𝔽7)—21no—
39C2×D3×ΓL(1,𝔽8)—21no—
40C2×C3×AGL(1,𝔽7)D411yescharacteristic
41D3×AGL(1,𝔽7)—42no—
42C3×C4×ΓL(1,𝔽8)C2311yescharacteristic
43Dic3×ΓL(1,𝔽8)—21no—
44C3×ΓL(1,𝔽8)⋊C4D411yescharacteristic
45#6477—31noHall
46C2×D3×D7—21no—
47C22×AGL(1,𝔽7)—61no—
48D84C22×C311yescharacteristic
49D84C22×C311yescharacteristic, complemented
50C7×D12C22×C311yescharacteristic
51C7×Dic6C22×C311yescharacteristic, complemented
52C3×C4×D7C22×C311yescharacteristic
53Q8×ΓL(1,𝔽8)—31no—
54D4×ΓL(1,𝔽8)—31no—
55#4247—21no—
56#4246—21no—
57D21⋊C4—21no—
58(C22⋊D7)⋊C3—61no—
59C4⋊AGL(1,𝔽7)—31no—
60C4⋊AGL(1,𝔽7)—31no—
61C4⋊AGL(1,𝔽7)—61no—
62C4×AGL(1,𝔽7)—21no—
63C4×AGL(1,𝔽7)—61no—
64C4×AGL(1,𝔽7)C2×D311yescharacteristic
65C3×C7×C8C22×C311yescharacteristic, complemented
66C3×C7⋊C8C22×C311yescharacteristic, complemented
67C8×ΓL(1,𝔽8)—21no—
68C8×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
69ΓL(1,𝔽8)⋊C8—21no—
70ΓL(1,𝔽8)⋊C8C2×D311yescharacteristic, complemented
71C3×#235—71no—
72C2×C3×D12—71no—
73C32×M16—71no—
74C3×D24—71no—
75C3×D24—71no—
76C3×#204—71no—
77C3×#204—141no—
78C2×C3×ΓL(1,𝔽8)C2×D411yescharacteristic
79C3⋊AGL(1,𝔽7)—41no—
80C3⋊AGL(1,𝔽7)—41no—
81D3×ΓL(1,𝔽8)—41no—
82C3×AGL(1,𝔽7)—21no—
83#1161—31no—
84D4×D7—31no—
85C7×SD16—31no—
86#1143—31no—
87#1141—31no—
88D56—31no—
89#1131C3×D311yescharacteristic, complemented
90C3×AGL(1,ℤ/8)—211no—
91#908—71no—
92C2×D21—21no—
93C2×D21—21no—
94C2×D3×C7—21no—
95C2×C3×D7C3×D411yescharacteristic
96C22×ΓL(1,𝔽8)—61no—
97D3×D7—42no—
98C2×AGL(1,𝔽7)—21no—
99C2×AGL(1,𝔽7)—61no—
100C2×AGL(1,𝔽7)—61no—
101C2×AGL(1,𝔽7)—122no—
102C2×AGL(1,𝔽7)D1211yescharacteristic
103C3×C4×C7C23×C311yescharacteristic
104C3×Dic7C3×D411yescharacteristic
105C7×Dic3—21no—
106C4×ΓL(1,𝔽8)—21no—
107C4×ΓL(1,𝔽8)—61no—
108C4×ΓL(1,𝔽8)C22×D311yescharacteristic
109ΓL(1,𝔽8)⋊C4—21no—
110ΓL(1,𝔽8)⋊C4D1211yescharacteristic
111C22×C3×D3—141no—
112C2×C32×C4—71no—
113C3×#67—141no—
114C3×D12—71no—
115C3×D12—71no—
116C3×D12—141no—
117C3×D12—141no—
118C3×C4×D3—141no—
119C3×Dic6—141no—
120C32×C8—71no—
121C32×C8—141no—
122C3×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
123C22×D7—61no—
124C7×Q8—31no—
125C7×D4—31no—
126C22⋊D7—61no—
127D28—31no—
128D28—31no—
129D28—61no—
130C4×D7—61no—
131C4×D7C2×C3×D311yescharacteristic
132C7×C8C2×C3×D311yescharacteristic, complemented
133C7⋊C8C2×C3×D311yescharacteristic, complemented
134C3×#41—211no—
135C2×C3×D4—211no—
136#235—71no—
137C2×D12—71no—
138C3×SD16—211no—
139C3×SD16—421no—
140C3×D8—211no—
141C3×D8—211no—
142C3×M16—71no—
143C3×M16—71no—
144C3×M16—141no—
145D24—71no—
146D24—71no—
147#204—71no—
148#204AGL(1,𝔽7)11yescharacteristic, complemented
149C2×C3×C7C2×C3×D411yescharacteristic
150D21—41no—
151D21—41no—
152C3×D7—21no—
153D3×C7—41no—
154C2×ΓL(1,𝔽8)—21no—
155C2×ΓL(1,𝔽8)—121no—
156C2×ΓL(1,𝔽8)C2×D1211yescharacteristic
157AGL(1,𝔽7)—21no—
158AGL(1,𝔽7)—41no—
159AGL(1,𝔽7)—121no—
160AGL(1,𝔽7)—121no—
161C22×C32—71no—
162C2×C3×D3—141no—
163C2×C3×D3—141no—
164C2×C3×D3—281no—
165C2×C3×D3—282no—
166C32×C4—71no—
167C32×C4—141no—
168C3×Dic3—281no—
169AGL(1,ℤ/8)—211noSylow
170C22×C7—61no—
171C2×D7—61no—
172C2×D7—61no—
173C2×D7—122no—
174C2×D7C3×D1211yescharacteristic
175C4×C7—61no—
176C4×C7C22×C3×D311yescharacteristic
177Dic7C3×D1211yescharacteristic
178C23×C3—421no—
179C22×D3—141no—
180C3×Q8—421no—
181C3×D4—211no—
182C3×D4—211no—
183C3×D4—421no—
184C3×D4—421no—
185C3×D4—421no—
186C2×C3×C4—71no—
187C2×C3×C4—71no—
188C2×C3×C4—141no—
189C2×C3×C4—421no—
190#67—141no—
191D12—71no—
192D12—71no—
193D12—141no—
194D12C2×AGL(1,𝔽7)11yescharacteristic
195C4×D3—141no—
196Dic6C2×AGL(1,𝔽7)11yescharacteristic, complemented
197C3×C8—71no—
198C3×C8—71no—
199C3×C8—141no—
200C3×C8—141no—
201C3×C8—281no—
202C3×C8C2×AGL(1,𝔽7)11yescharacteristic, complemented
203C3×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented
204ΓL(1,𝔽8)—21no—
205ΓL(1,𝔽8)#90811yescharacteristic, complemented
206C2×C32—141no—
207C2×C32—141no—
208C3×D3—281no—
209C3×D3—281no—
210C3×D3—561no—
211#41—211no—
212C2×D4—211no—
213SD16—31no—
214SD16—211no—
215D8—211no—
216D8—211no—
217M16—71no—
218C2×C7—121no—
219C2×C7C2×C3×D1211yescharacteristic
220D7—21no—
221D7—121no—
222D7—121no—
223C22×C3—71no—
224C22×C3—71no—
225C22×C3—141no—
226C22×C3—421no—
227C22×C3—421no—
228C22×C3—841no—
229C22×C3—842no—
230C2×D3—21no—
231C2×D3—141no—
232C2×D3—141no—
233C2×D3—282no—
234C3×C4—71no—
235C3×C4—71no—
236C3×C4—141no—
237C3×C4—141no—
238C3×C4—281no—
239C3×C4—841no—
240C3×C4C22×AGL(1,𝔽7)11yescharacteristic
241Dic3—21no—
242C32—141noSylow
243C23—421no—
244Q8—31no—
245D4—31no—
246D4—211no—
247D4—211no—
248D4—421no—
249D4—421no—
250C2×C4—71no—
251C2×C4—421no—
252C8—71no—
253C8D3×AGL(1,𝔽7)11yescharacteristic, complemented
254C7C3×#90811yescharacteristic, complemented, Sylow
255C2×C3—141no—
256C2×C3—141no—
257C2×C3—141no—
258C2×C3—281no—
259C2×C3—281no—
260C2×C3—841no—
261C2×C3—841no—
262C2×C3—1681no—
263C2×C3D4×AGL(1,𝔽7)11yescharacteristic
264D3—41no—
265D3—281no—
266D3—281no—
267C22—61no—
268C22—71no—
269C22—421no—
270C22—421no—
271C22—842no—
272C4—61no—
273C4—71no—
274C4C2×D3×AGL(1,𝔽7)11yescharacteristic
275C3—141no—
276C3—281no—
277C3#6477⋊C311yescharacteristic, complemented
278C2—121no—
279C2—141no—
280C2—841no—
281C2—841no—
282C2D12×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
centerC2D12×AGL(1,𝔽7)
commutator, cosocleC3×C4×C7C23×C3
FrattiniC4C2×D3×AGL(1,𝔽7)
FittingC3×C7×C8C22×C3
socleC2×C3×C7C2×C3×D4
hypercenterC8D3×AGL(1,𝔽7)
hypocenterC3×C7C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. D12×AGL(1,𝔽7)C2semidirect
2. #66677⋊C3C2semidirect
3. D168⋊C3C2semidirect
4. ΓL(1,𝔽8)×#204C2semidirect
5. C3×#1131⋊C3C2semidirect
6. #66568⋊C3C2semidirect
7. #66562⋊C3C2semidirect
8. #10620811C3semidirect
12. C4⋊(C3⋊AGL(1,𝔽7))C22semidirect
14. ΓL(1,𝔽8)×Dic6C22semidirect
19. C3×C8×ΓL(1,𝔽8)C22semidirect
20. C3×ΓL(1,𝔽8)⋊C8C22semidirect
21. D7×D12C2×C3semidirect
22. #66677C2×C3semidirect
25. D168C2×C3semidirect
26. C7×#204C2×C3semidirect
27. C3×#1131C2×C3semidirect
29. #66568C2×C3semidirect
30. #66562C2×C3semidirect
35. #1131⋊C3D3semidirect
49. D84C22×C3semidirect
51. C7×Dic6C22×C3semidirect
65. C3×C7×C8C22×C3semidirect
66. C3×C7⋊C8C22×C3semidirect
68. C8×ΓL(1,𝔽8)C2×D3semidirect
70. ΓL(1,𝔽8)⋊C8C2×D3semidirect
89. #1131C3×D3semidirect
122. C3×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
132. C7×C8C2×C3×D3semidirect
133. C7⋊C8C2×C3×D3semidirect
148. #204AGL(1,𝔽7)semidirect
196. Dic6C2×AGL(1,𝔽7)semidirect
202. C3×C8C2×AGL(1,𝔽7)semidirect
203. C3×C7C3×AGL(1,ℤ/8)semidirect
205. ΓL(1,𝔽8)#908semidirect
253. C8D3×AGL(1,𝔽7)semidirect
254. C7C3×#908semidirect
277. C3#6477⋊C3semidirect

Maximal subgroups

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

subgroupindex
D12×AGL(1,𝔽7)2
#66677⋊C32
D168⋊C32
ΓL(1,𝔽8)×#2042
C3×#1131⋊C32
#66568⋊C32
#66562⋊C32
#106208113
#6477⋊C33
C3×#9087