Finite Group Databasebeta

← #72244151

Subgroups — #72244151

Order 2016.

574 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C23×C3⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
2C2×C3×(C22⋊D7)⋊C3C211yesmaximal, characteristic, complemented
3C2×#4246⋊C3C212yesmaximal, complemented
4ΓL(1,𝔽8)×#217C211yesmaximal, characteristic, complemented
5#66575⋊C3C212yesmaximal, complemented
6#10621134C311yesmaximal, characteristic, complemented
7#6445⋊C3—31nomaximal
8C23×C3×ΓL(1,𝔽8)C2211yescharacteristic
9C22×C3⋊AGL(1,𝔽7)—22no—
10C22×C3⋊AGL(1,𝔽7)—24no—
11C22×C3⋊AGL(1,𝔽7)C2212yes—
12C22×C3×AGL(1,𝔽7)C2211yescharacteristic, complemented
13C2×Dic3×ΓL(1,𝔽8)C2212yescomplemented
14C3×(C22⋊D7)⋊C3—22no—
15C2×C3×ΓL(1,𝔽8)⋊C4C2211yescharacteristic, complemented
16#4246⋊C3—24no—
17C23×D21C2×C311yescharacteristic, complemented
18C23×AGL(1,𝔽7)—31no—
19C2×C3×C22⋊D7C2×C311yescharacteristic, complemented
20C2×#4246C2×C312yescomplemented
21C2×(C22⋊D7)⋊C3—21no—
22C2×(C22⋊D7)⋊C3D311yescharacteristic, complemented
23C2×C4⋊AGL(1,𝔽7)—32no—
24C7×#217C2×C311yescharacteristic, complemented
25C22⋊C4×ΓL(1,𝔽8)—31no—
26#66575C2×C312yescomplemented
27(C2×AGL(1,𝔽7))⋊C4—32no—
28C3×#937—71nomaximal, Hall
29C22×C3×ΓL(1,𝔽8)—22no—
30C22×C3×ΓL(1,𝔽8)D412yescomplemented
31C22×C3×ΓL(1,𝔽8)C2311yescharacteristic
32C2×C3⋊AGL(1,𝔽7)—28no—
33C2×C3⋊AGL(1,𝔽7)—22no—
34C2×C3⋊AGL(1,𝔽7)—22no—
35C2×C3⋊AGL(1,𝔽7)D414yescomplemented
36C2×C3×AGL(1,𝔽7)—21no—
37C2×C3×AGL(1,𝔽7)—22no—
38Dic3×ΓL(1,𝔽8)—22no—
39C3×ΓL(1,𝔽8)⋊C4—21no—
40#6445—31noHall
41C23×C3×C7C22×C311yescharacteristic
42C22×D21—22no—
43C22×D21—24no—
44C22×D21C22×C312yes—
45C22×C3×D7C22×C311yescharacteristic, complemented
46C23×ΓL(1,𝔽8)—21no—
47C23×ΓL(1,𝔽8)C2×D311yescharacteristic
48C22×AGL(1,𝔽7)—21no—
49C22×AGL(1,𝔽7)—32no—
50C22×AGL(1,𝔽7)—64no—
51C22×AGL(1,𝔽7)—62no—
52C22×AGL(1,𝔽7)C2×D311yescharacteristic, complemented
53C2×C7×Dic3C22×C312yescomplemented
54C3×C22⋊D7—22no—
55C2×C3×Dic7C22×C311yescharacteristic, complemented
56C2×C4×ΓL(1,𝔽8)—32no—
57#4246—24no—
58(C22⋊D7)⋊C3—22no—
59(C22⋊D7)⋊C3—22no—
60(C22⋊D7)⋊C3—22no—
61C2×ΓL(1,𝔽8)⋊C4—21no—
62C2×ΓL(1,𝔽8)⋊C4C2×D311yescharacteristic, complemented
63C4⋊AGL(1,𝔽7)—64no—
64C23×C3×D3—71no—
65C2×C32×D4—71no—
66C2×C3×#67—72no—
67C3×#217—141no—
68C3×(C2×D3)⋊C4—72no—
69C2×C3×ΓL(1,𝔽8)—22no—
70C2×C3×ΓL(1,𝔽8)C2×D412yes—
71C2×C3×ΓL(1,𝔽8)C2×D411yescharacteristic
72C3⋊AGL(1,𝔽7)—24no—
73C3×AGL(1,𝔽7)—41no—
74C23×D7—31no—
75C2×C22⋊D7C3×D311yescharacteristic, complemented
76C2×D28—32no—
77C7×C22⋊C4—31no—
78(C2×D7)⋊C4—32no—
79C3×#120—211no—
80#937—71no—
81C22×C3×C7—22no—
82C22×C3×C7C3×D412yescomplemented
83C22×C3×C7C23×C311yescharacteristic
84C2×D21—28no—
85C2×D21—22no—
86C2×D21—22no—
87C2×D21C3×D414yescomplemented
88C2×C3×D7—22no—
89C2×C3×D7—21no—
90C22×ΓL(1,𝔽8)—22no—
91C22×ΓL(1,𝔽8)—21no—
92C22×ΓL(1,𝔽8)—22no—
93C22×ΓL(1,𝔽8)—42no—
94C22×ΓL(1,𝔽8)#6712yescomplemented
95C22×ΓL(1,𝔽8)C22×D311yescharacteristic
96C2×AGL(1,𝔽7)—21no—
97C2×AGL(1,𝔽7)—21no—
98C2×AGL(1,𝔽7)—22no—
99C2×AGL(1,𝔽7)—21no—
100C2×AGL(1,𝔽7)—34no—
101C2×AGL(1,𝔽7)—42no—
102C2×AGL(1,𝔽7)—68no—
103C2×AGL(1,𝔽7)—62no—
104C2×AGL(1,𝔽7)—62no—
105C3×Dic7—21no—
106C7×Dic3—22no—
107C4×ΓL(1,𝔽8)—62no—
108ΓL(1,𝔽8)⋊C4—21no—
109ΓL(1,𝔽8)⋊C4—21no—
110ΓL(1,𝔽8)⋊C4—21no—
111C23×C32—71no—
112C23×C32—141no—
113C22×C3×D3—72no—
114C22×C3×D3—144no—
115C22×C3×D3—142no—
116C32×D4—142no—
117C2×C32×C4—71no—
118C3×#67—144no—
119C2×C3×Dic3—142no—
120C3×ΓL(1,𝔽8)#12011yescharacteristic, complemented, Hall
121C23×C7C2×C3×D311yescharacteristic
122C22×D7—32no—
123C22×D7—64no—
124C22×D7—62no—
125C22×D7C2×C3×D311yescharacteristic, complemented
126C2×C4×C7—32no—
127C22⋊D7—22no—
128C2×Dic7C2×C3×D311yescharacteristic, complemented
129D28—64no—
130C24×C3—211no—
131C23×D3—71no—
132C2×C3×D4—71no—
133C2×C3×D4—71no—
134C2×C3×D4—141no—
135C2×C3×D4—212no—
136C2×#67—72no—
137C3×C22⋊C4—212no—
138C3×C22⋊C4—421no—
139#217AGL(1,𝔽7)11yescharacteristic, complemented
140(C2×D3)⋊C4—72no—
141C2×C3×C7—22no—
142C2×C3×C7C2×C3×D411yescharacteristic
143C2×C3×C7C2×C3×D412yes—
144D21—24no—
145C3×D7—41no—
146C2×ΓL(1,𝔽8)—22no—
147C2×ΓL(1,𝔽8)—21no—
148C2×ΓL(1,𝔽8)—22no—
149C2×ΓL(1,𝔽8)—42no—
150C2×ΓL(1,𝔽8)D3×D412yes—
151C2×ΓL(1,𝔽8)C2×#6711yescharacteristic
152AGL(1,𝔽7)—41no—
153AGL(1,𝔽7)—41no—
154AGL(1,𝔽7)—41no—
155AGL(1,𝔽7)—64no—
156C22×C32—141no—
157C22×C32—142no—
158C22×C32—142no—
159C22×C32—141no—
160C22×C32—282no—
161C2×C3×D3—74no—
162C2×C3×D3—142no—
163C2×C3×D3—142no—
164C2×C3×D3—148no—
165C32×C4—141no—
166C3×Dic3—282no—
167#120—211noSylow
168C22×C7—22no—
169C22×C7C3×#6712yescomplemented
170C22×C7C22×C3×D311yescharacteristic
171C2×D7—22no—
172C2×D7—21no—
173C2×D7—34no—
174C2×D7—68no—
175C2×D7—62no—
176C2×D7—62no—
177C4×C7—62no—
178Dic7—21no—
179C23×C3—71no—
180C23×C3—71no—
181C23×C3—141no—
182C23×C3—141no—
183C23×C3—141no—
184C23×C3—212no—
185C23×C3—422no—
186C23×C3—424no—
187C23×C3C2×AGL(1,𝔽7)11yescharacteristic
188C22×D3—72no—
189C22×D3—144no—
190C22×D3—142no—
191C3×D4—142no—
192C3×D4—142no—
193C3×D4—142no—
194C3×D4—142no—
195C3×D4—424no—
196C2×C3×C4—71no—
197C2×C3×C4—71no—
198C2×C3×C4—141no—
199C2×C3×C4—422no—
200#67—144no—
201C2×Dic3C2×AGL(1,𝔽7)12yescomplemented
202C3×C7C3×#12011yescharacteristic, complemented
203ΓL(1,𝔽8)—21no—
204ΓL(1,𝔽8)#93711yescharacteristic, complemented
205C2×C32—141no—
206C2×C32—142no—
207C2×C32—282no—
208C2×C32—281no—
209C3×D3—144no—
210C24—211no—
211C2×D4—71no—
212C2×D4—212no—
213C22⋊C4—31no—
214C22⋊C4—212no—
215C2×C7—22no—
216C2×C7C3×D3×D412yes—
217C2×C7C2×C3×#6711yescharacteristic
218D7—41no—
219D7—64no—
220C22×C3—22no—
221C22×C3—141no—
222C22×C3—141no—
223C22×C3—141no—
224C22×C3—142no—
225C22×C3—141no—
226C22×C3—142no—
227C22×C3—141no—
228C22×C3—141no—
229C22×C3—142no—
230C22×C3—142no—
231C22×C3—214no—
232C22×C3—282no—
233C22×C3—282no—
234C22×C3—282no—
235C22×C3—422no—
236C22×C3—422no—
237C22×C3—428no—
238C22×C3C4⋊AGL(1,𝔽7)12yescomplemented
239C22×C3C22×AGL(1,𝔽7)11yescharacteristic
240C2×D3—74no—
241C2×D3—148no—
242C2×D3—142no—
243C2×D3—142no—
244C3×C4—141no—
245C3×C4—141no—
246C3×C4—141no—
247C3×C4—141no—
248C3×C4—842no—
249Dic3—22no—
250C32—141noSylow
251C23—71no—
252C23—212no—
253C23—424no—
254C23—422no—
255C23D3×AGL(1,𝔽7)11yescharacteristic
256D4—142no—
257D4—424no—
258C2×C4—32no—
259C2×C4—71no—
260C7C3×#93711yescharacteristic, complemented, Sylow
261C2×C3—22no—
262C2×C3—141no—
263C2×C3—141no—
264C2×C3—142no—
265C2×C3—142no—
266C2×C3—282no—
267C2×C3—282no—
268C2×C3—281no—
269C2×C3—281no—
270C2×C3—281no—
271C2×C3—281no—
272C2×C3—424no—
273C2×C3C2×C4⋊AGL(1,𝔽7)11yescharacteristic
274C2×C3D4×AGL(1,𝔽7)12yes—
275D3—144no—
276C22—22no—
277C22—141no—
278C22—142no—
279C22—214no—
280C22—428no—
281C22—422no—
282C22—422no—
283C22#4246⋊C312yescomplemented
284C22C2×D3×AGL(1,𝔽7)11yescharacteristic
285C4—62no—
286C4—141no—
287C3—141no—
288C3—141no—
289C3#6445⋊C311yescharacteristic, complemented
290C2—22no—
291C2—281no—
292C2—424no—
293C2C2×#4246⋊C311yescharacteristic
294C2#66694⋊C312yes—
subgroupG/H
center, FrattiniC22C2×D3×AGL(1,𝔽7)
commutator, socle, cosocleC22×C3×C7C23×C3
FittingC23×C3×C7C22×C3
hypercenterC23D3×AGL(1,𝔽7)
hypocenterC3×C7C3×#120
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊AGL(1,𝔽7)C2semidirect
2. C2×C3×(C22⋊D7)⋊C3C2semidirect
3. C2×#4246⋊C3C2semidirect
4. ΓL(1,𝔽8)×#217C2semidirect
5. #66575⋊C3C2semidirect
6. #10621134C3semidirect
12. C22×C3×AGL(1,𝔽7)C22semidirect
13. C2×Dic3×ΓL(1,𝔽8)C22semidirect
15. C2×C3×ΓL(1,𝔽8)⋊C4C22semidirect
17. C23×D21C2×C3semidirect
19. C2×C3×C22⋊D7C2×C3semidirect
20. C2×#4246C2×C3semidirect
22. C2×(C22⋊D7)⋊C3D3semidirect
24. C7×#217C2×C3semidirect
26. #66575C2×C3semidirect
30. C22×C3×ΓL(1,𝔽8)D4semidirect
35. C2×C3⋊AGL(1,𝔽7)D4semidirect
45. C22×C3×D7C22×C3semidirect
52. C22×AGL(1,𝔽7)C2×D3semidirect
53. C2×C7×Dic3C22×C3semidirect
55. C2×C3×Dic7C22×C3semidirect
62. C2×ΓL(1,𝔽8)⋊C4C2×D3semidirect
75. C2×C22⋊D7C3×D3semidirect
82. C22×C3×C7C3×D4semidirect
87. C2×D21C3×D4semidirect
94. C22×ΓL(1,𝔽8)#67semidirect
120. C3×ΓL(1,𝔽8)#120semidirect
125. C22×D7C2×C3×D3semidirect
128. C2×Dic7C2×C3×D3semidirect
139. #217AGL(1,𝔽7)semidirect
169. C22×C7C3×#67semidirect
201. C2×Dic3C2×AGL(1,𝔽7)semidirect
202. C3×C7C3×#120semidirect
204. ΓL(1,𝔽8)#937semidirect
238. C22×C3C4⋊AGL(1,𝔽7)semidirect
260. C7C3×#937semidirect
283. C22#4246⋊C3semidirect
289. C3#6445⋊C3semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊AGL(1,𝔽7)2
C2×C3×(C22⋊D7)⋊C32
C2×#4246⋊C32
ΓL(1,𝔽8)×#2172
#66575⋊C32
#106211343
#6445⋊C33
C3×#9377