Finite Group Databasebeta

← #72245629

Subgroups — #72245629

Order 2016.

294 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 67 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
2#66669⋊C3C211yesmaximal, characteristic, complemented
3#66633⋊C3C211yesmaximal, characteristic, complemented
4C3×#1141⋊C3C211yesmaximal, characteristic, complemented
5#66567⋊C3C211yesmaximal, characteristic, complemented
6#66560⋊C3C211yesmaximal, characteristic, complemented
7#66558⋊C3C211yesmaximal, characteristic, complemented
8#10621056C311yesmaximal, characteristic, complemented
9#6495⋊C3—31nomaximal
10C22×D3×ΓL(1,𝔽8)—21no—
11C3×D4×ΓL(1,𝔽8)C2211yescharacteristic
12Dic42⋊C3C2211yescharacteristic, complemented
13ΓL(1,𝔽8)×#67—21no—
14ΓL(1,𝔽8)×D12C2211yescharacteristic
15C4×D3×ΓL(1,𝔽8)C2211yescharacteristic
16C3×C4⋊AGL(1,𝔽7)C2211yescharacteristic, complemented
17#4245⋊C3—21no—
18Dic3×AGL(1,𝔽7)—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—
23#66669C2×C311yescharacteristic, complemented
24#1157⋊C3—31no—
25#66633C2×C311yescharacteristic, complemented
26C3×#1141C2×C311yescharacteristic, complemented
27#66567C2×C311yescharacteristic, complemented
28#66560C2×C311yescharacteristic, complemented
29#66558C2×C311yescharacteristic, complemented
30#1142⋊C3—31no—
31#1142⋊C3—31no—
32#1141⋊C3—21no—
33#1141⋊C3—31no—
34#1141⋊C3D311yescharacteristic, complemented
35#1136⋊C3—31no—
36C3×#911—71nomaximal, Hall
37C22×C3×ΓL(1,𝔽8)—21no—
38C2×D3×ΓL(1,𝔽8)—21no—
39C2×D3×ΓL(1,𝔽8)—42no—
40C2×D3×ΓL(1,𝔽8)D411yescharacteristic
41C2×C3×AGL(1,𝔽7)—21no—
42C3×C4×ΓL(1,𝔽8)C2311yescharacteristic
43C3⋊(ΓL(1,𝔽8)⋊C4)—21no—
44Dic3×ΓL(1,𝔽8)D411yescharacteristic
45#6495—31noHall
46C22×D3×C7—21no—
47C23×ΓL(1,𝔽8)—61no—
48C3×C7×D4C22×C311yescharacteristic
49Dic42C22×C311yescharacteristic, complemented
50C7×#67—21no—
51C7×D12C22×C311yescharacteristic
52C4×D3×C7C22×C311yescharacteristic
53C3×D28C22×C311yescharacteristic, complemented
54D4×ΓL(1,𝔽8)—21no—
55D4×ΓL(1,𝔽8)—31no—
56D4×ΓL(1,𝔽8)—61no—
57D4×ΓL(1,𝔽8)C2×D311yescharacteristic
58C2×C4×ΓL(1,𝔽8)—31no—
59#4245—21no—
60Dic3×D7—21no—
61(C22⋊D7)⋊C3—61no—
62C4⋊AGL(1,𝔽7)—21no—
63C4⋊AGL(1,𝔽7)C2×D311yescharacteristic, complemented
64C4×AGL(1,𝔽7)—61no—
65ΓL(1,𝔽8)⋊Q8—31no—
66C3⋊(C7⋊C8)C22×C311yescharacteristic, complemented
67C3×C7⋊C8C22×C311yescharacteristic, complemented
68ΓL(1,𝔽8)⋊C8—21no—
69ΓL(1,𝔽8)⋊C8—31no—
70ΓL(1,𝔽8)⋊C8C2×D311yescharacteristic, complemented
71C3×#237—71no—
72C3×D3×D4—141no—
73C32×D8—71no—
74C3×#214—71no—
75C3×#213—71no—
76C3×#204—71no—
77C3×ΓL(1,𝔽25)—71no—
78C2×C3×ΓL(1,𝔽8)—41no—
79C2×C3×ΓL(1,𝔽8)C2×D411yescharacteristic
80D3×ΓL(1,𝔽8)—21no—
81D3×ΓL(1,𝔽8)—41no—
82C3×AGL(1,𝔽7)—41no—
83C2×C7×D4—31no—
84#1157—31no—
85#1142—31no—
86#1142—31no—
87#1141—31no—
88#1141C3×D311yescharacteristic, complemented
89#1136—31no—
90C3×AGL(1,ℤ/8)—211no—
91#911—71no—
92C22×C3×C7—21no—
93C2×D3×C7—21no—
94C2×D3×C7—42no—
95C2×D3×C7C3×D411yescharacteristic
96C2×C3×D7—21no—
97C22×ΓL(1,𝔽8)—21no—
98C22×ΓL(1,𝔽8)—31no—
99C22×ΓL(1,𝔽8)—41no—
100C22×ΓL(1,𝔽8)—61no—
101C22×ΓL(1,𝔽8)—122no—
102C2×AGL(1,𝔽7)—21no—
103C2×AGL(1,𝔽7)—41no—
104C3×C4×C7C23×C311yescharacteristic
105Dic21—21no—
106C7×Dic3C3×D411yescharacteristic
107C4×ΓL(1,𝔽8)—21no—
108C4×ΓL(1,𝔽8)—31no—
109C4×ΓL(1,𝔽8)C22×D311yescharacteristic
110ΓL(1,𝔽8)⋊C4—61no—
111C22×C3×D3—141no—
112C32×D4—71no—
113C32×D4—141no—
114C3×#67—141no—
115C3×#67—141no—
116C2×C3×Dic3—141no—
117C3×D12—141no—
118C3×C4×D3—141no—
119C3×Dic6—71no—
120C32×C8—71no—
121C3×C3⋊C8—71no—
122C3×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
123C23×C7—61no—
124C7×D4—31no—
125C7×D4—61no—
126C7×D4C2×C3×D311yescharacteristic
127C2×C4×C7—31no—
128C22⋊D7—61no—
129D28C2×C3×D311yescharacteristic, complemented
130C4×D7—61no—
131Dic14—31no—
132C7⋊C8—31no—
133C7⋊C8C2×C3×D311yescharacteristic, complemented
134C3×#41—211no—
135C2×C3×D4—421no—
136#237—71no—
137D3×D4AGL(1,𝔽7)11yescharacteristic, complemented
138C3×SD16—211no—
139C3×SD16—211no—
140C3×D8—71no—
141C3×D8—71no—
142C3×D8—141no—
143C3×D8—211no—
144C3×M16—211no—
145#214—71no—
146#213—71no—
147#204—71no—
148ΓL(1,𝔽25)—71no—
149C2×C3×C7—41no—
150C2×C3×C7C2×C3×D411yescharacteristic
151C3×D7—41no—
152D3×C7—21no—
153D3×C7—41no—
154C2×ΓL(1,𝔽8)—21no—
155C2×ΓL(1,𝔽8)—41no—
156C2×ΓL(1,𝔽8)—61no—
157C2×ΓL(1,𝔽8)—81no—
158C2×ΓL(1,𝔽8)—121no—
159C2×ΓL(1,𝔽8)D3×D411yescharacteristic
160AGL(1,𝔽7)—41no—
161AGL(1,𝔽7)—81no—
162C22×C32—141no—
163C22×C32—141no—
164C2×C3×D3—141no—
165C2×C3×D3—141no—
166C2×C3×D3—282no—
167C32×C4—141no—
168C3×Dic3—141no—
169C3×Dic3—141no—
170AGL(1,ℤ/8)—211noSylow
171C22×C7—21no—
172C22×C7—31no—
173C22×C7—61no—
174C22×C7—122no—
175C2×D7—21no—
176C4×C7—31no—
177C4×C7C22×C3×D311yescharacteristic
178Dic7—61no—
179C23×C3—421no—
180C22×D3—21no—
181C3×Q8—211no—
182C3×D4—71no—
183C3×D4—71no—
184C3×D4—141no—
185C3×D4—141no—
186C3×D4—141no—
187C3×D4—421no—
188C3×D4—421no—
189C3×D4—421no—
190C3×D4C2×AGL(1,𝔽7)11yescharacteristic
191C2×C3×C4—421no—
192C2×C3×C4—421no—
193#67—21no—
194#67—141no—
195C2×Dic3—141no—
196D12C2×AGL(1,𝔽7)11yescharacteristic
197C4×D3C2×AGL(1,𝔽7)11yescharacteristic
198Dic6—71no—
199C3×C8—71no—
200C3×C8—71no—
201C3×C8—141no—
202C3×C8—211no—
203C3⋊C8—71no—
204C3×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented
205ΓL(1,𝔽8)—21no—
206ΓL(1,𝔽8)#91111yescharacteristic, complemented
207C2×C32—141no—
208C2×C32—281no—
209C2×C32—281no—
210C3×D3—141no—
211C3×D3—281no—
212#41—211no—
213C2×D4—31no—
214SD16—211no—
215SD16—211no—
216D8—71no—
217D8—211no—
218M16—211no—
219C2×C7—41no—
220C2×C7—61no—
221C2×C7—121no—
222C2×C7C3×D3×D411yescharacteristic
223D7—41no—
224C22×C3—21no—
225C22×C3—141no—
226C22×C3—141no—
227C22×C3—141no—
228C22×C3—281no—
229C22×C3—281no—
230C22×C3—421no—
231C22×C3—421no—
232C22×C3—842no—
233C2×D3—21no—
234C2×D3—42no—
235C2×D3(C22⋊D7)⋊C311yescharacteristic
236C3×C4—141no—
237C3×C4—141no—
238C3×C4—421no—
239C3×C4—421no—
240C3×C4C22×AGL(1,𝔽7)11yescharacteristic
241Dic3—141no—
242Dic3(C22⋊D7)⋊C311yescharacteristic
243C32—141noSylow
244C23—61no—
245Q8—211no—
246D4—31no—
247D4—61no—
248D4—71no—
249D4—421no—
250D4D3×AGL(1,𝔽7)11yescharacteristic
251C2×C4—31no—
252C2×C4—421no—
253C8—71no—
254C8—211no—
255C7C3×#91111yescharacteristic, complemented, Sylow
256C2×C3—41no—
257C2×C3—141no—
258C2×C3—141no—
259C2×C3—281no—
260C2×C3—281no—
261C2×C3—281no—
262C2×C3—421no—
263C2×C3—561no—
264C2×C3—561no—
265C2×C3—841no—
266C2×C3C2×(C22⋊D7)⋊C311yescharacteristic
267D3—21no—
268D3—41no—
269C22—21no—
270C22—31no—
271C22—61no—
272C22—122no—
273C22—141no—
274C4—31no—
275C4—421no—
276C4C2×D3×AGL(1,𝔽7)11yescharacteristic
277C3—141no—
278C3—141no—
279C3#6495⋊C311yescharacteristic, complemented
280C2—41no—
281C2—61no—
282C2—121no—
283C2—281no—
284C2D3×(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×C7C3×AGL(1,ℤ/8)
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. #66669⋊C3C2semidirect
3. #66633⋊C3C2semidirect
4. C3×#1141⋊C3C2semidirect
5. #66567⋊C3C2semidirect
6. #66560⋊C3C2semidirect
7. #66558⋊C3C2semidirect
8. #10621056C3semidirect
12. Dic42⋊C3C22semidirect
16. 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. #66669C2×C3semidirect
25. #66633C2×C3semidirect
26. C3×#1141C2×C3semidirect
27. #66567C2×C3semidirect
28. #66560C2×C3semidirect
29. #66558C2×C3semidirect
34. #1141⋊C3D3semidirect
49. Dic42C22×C3semidirect
53. C3×D28C22×C3semidirect
63. C4⋊AGL(1,𝔽7)C2×D3semidirect
66. C3⋊(C7⋊C8)C22×C3semidirect
67. C3×C7⋊C8C22×C3semidirect
70. ΓL(1,𝔽8)⋊C8C2×D3semidirect
88. #1141C3×D3semidirect
122. C3×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
129. D28C2×C3×D3semidirect
133. C7⋊C8C2×C3×D3semidirect
137. D3×D4AGL(1,𝔽7)semidirect
204. C3×C7C3×AGL(1,ℤ/8)semidirect
206. ΓL(1,𝔽8)#911semidirect
255. C7C3×#911semidirect
279. C3#6495⋊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
#66669⋊C32
#66633⋊C32
C3×#1141⋊C32
#66567⋊C32
#66560⋊C32
#66558⋊C32
#106210563
#6495⋊C33
C3×#9117