Finite Group Databasebeta

← #72244185

Subgroups — #72244185

Order 2016.

654 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 101 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×#4268⋊C3C211yesmaximal, characteristic, complemented
3C2×#4246⋊C3C212yesmaximal, complemented
4ΓL(1,𝔽8)×#217C211yesmaximal, characteristic, complemented
5(C2×AGL(1,𝔽7))⋊Dic3C212yesmaximal, complemented
6#10621126C311yesmaximal, characteristic, complemented
7#6445⋊C3—31nomaximal
8C23×C3×ΓL(1,𝔽8)C2211yescharacteristic
9C22×C3⋊AGL(1,𝔽7)C2211yescharacteristic, complemented
10C22×C3×AGL(1,𝔽7)—22no—
11C22×C3×AGL(1,𝔽7)—24no—
12C22×C3×AGL(1,𝔽7)C2212yes—
13#4268⋊C3—22no—
14C2×C3⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic, complemented
15C2×Dic3×ΓL(1,𝔽8)C2212yescomplemented
16#4246⋊C3—24no—
17C23×C3×D7C2×C311yescharacteristic, complemented
18C23×AGL(1,𝔽7)—21no—
19C23×AGL(1,𝔽7)D311yescharacteristic, complemented
20C2×#4268C2×C311yescharacteristic, complemented
21C2×#4246C2×C312yescomplemented
22C2×(C22⋊D7)⋊C3—31no—
23C2×C4⋊AGL(1,𝔽7)—32no—
24C7×#217C2×C311yescharacteristic, complemented
25C22⋊C4×ΓL(1,𝔽8)—31no—
26(C2×D7)⋊Dic3C2×C312yescomplemented
27(C2×AGL(1,𝔽7))⋊C4—32no—
28C3×#953—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)—22no—
33C2×C3⋊AGL(1,𝔽7)—21no—
34C2×C3×AGL(1,𝔽7)—28no—
35C2×C3×AGL(1,𝔽7)—22no—
36C2×C3×AGL(1,𝔽7)—22no—
37C2×C3×AGL(1,𝔽7)D414yescomplemented
38C3⋊(ΓL(1,𝔽8)⋊C4)—21no—
39Dic3×ΓL(1,𝔽8)—22no—
40#6445—31noHall
41C23×C3×C7C22×C311yescharacteristic
42C22×D21C22×C311yescharacteristic, complemented
43C22×C3×D7—22no—
44C22×C3×D7—24no—
45C22×C3×D7C22×C312yes—
46C23×ΓL(1,𝔽8)—21no—
47C23×ΓL(1,𝔽8)C2×D311yescharacteristic
48C22×AGL(1,𝔽7)—22no—
49C22×AGL(1,𝔽7)—24no—
50C22×AGL(1,𝔽7)—22no—
51C22×AGL(1,𝔽7)—24no—
52C22×AGL(1,𝔽7)—22no—
53C22×AGL(1,𝔽7)—24no—
54C22×AGL(1,𝔽7)—22no—
55C22×AGL(1,𝔽7)—31no—
56C22×AGL(1,𝔽7)C2×D312yes—
57#4268—22no—
58C2×Dic21C22×C311yescharacteristic, complemented
59C2×C7×Dic3C22×C312yescomplemented
60C2×C4×ΓL(1,𝔽8)—32no—
61#4246—24no—
62(C22⋊D7)⋊C3—62no—
63C2×ΓL(1,𝔽8)⋊C4—31no—
64C4⋊AGL(1,𝔽7)—64no—
65C24×C32—71no—
66C2×C3×#67—72no—
67C2×C3×#67—71no—
68C3×#217—72no—
69C3×#217—71no—
70C2×C3×ΓL(1,𝔽8)—22no—
71C2×C3×ΓL(1,𝔽8)C2×D412yes—
72C2×C3×ΓL(1,𝔽8)C2×D411yescharacteristic
73C3⋊AGL(1,𝔽7)—41no—
74C3×AGL(1,𝔽7)—24no—
75C23×D7C3×D311yescharacteristic, complemented
76C2×C22⋊D7—31no—
77C2×D28—32no—
78C7×C22⋊C4—31no—
79(C2×D7)⋊C4—32no—
80C3×#120—211no—
81#953—71no—
82C22×C3×C7—22no—
83C22×C3×C7C3×D412yescomplemented
84C22×C3×C7C23×C311yescharacteristic
85C2×D21—22no—
86C2×D21—21no—
87C2×C3×D7—28no—
88C2×C3×D7—22no—
89C2×C3×D7—22no—
90C2×C3×D7C3×D414yescomplemented
91C22×ΓL(1,𝔽8)—22no—
92C22×ΓL(1,𝔽8)—22no—
93C22×ΓL(1,𝔽8)—22no—
94C22×ΓL(1,𝔽8)—22no—
95C22×ΓL(1,𝔽8)—21no—
96C22×ΓL(1,𝔽8)#6712yescomplemented
97C22×ΓL(1,𝔽8)C22×D311yescharacteristic
98C2×AGL(1,𝔽7)—22no—
99C2×AGL(1,𝔽7)—22no—
100C2×AGL(1,𝔽7)—28no—
101C2×AGL(1,𝔽7)—28no—
102C2×AGL(1,𝔽7)—28no—
103C2×AGL(1,𝔽7)—22no—
104C2×AGL(1,𝔽7)—22no—
105C2×AGL(1,𝔽7)—22no—
106C2×AGL(1,𝔽7)—22no—
107C2×AGL(1,𝔽7)—24no—
108C2×AGL(1,𝔽7)—62no—
109C2×AGL(1,𝔽7)—61no—
110C2×AGL(1,𝔽7)#6714yescomplemented
111Dic21—21no—
112C7×Dic3—22no—
113C4×ΓL(1,𝔽8)—62no—
114ΓL(1,𝔽8)⋊C4—61no—
115C23×C32—72no—
116C23×C32—71no—
117C23×C32—144no—
118C23×C32—142no—
119C22×C3×D3—71no—
120C3×#67—144no—
121C3×#67—142no—
122C2×C3×Dic3—71no—
123C2×C3×Dic3—72no—
124C3×ΓL(1,𝔽8)#12011yescharacteristic, complemented, Hall
125C23×C7C2×C3×D311yescharacteristic
126C22×D7—22no—
127C22×D7—24no—
128C22×D7—31no—
129C22×D7C2×C3×D312yes—
130C2×C4×C7—32no—
131C22⋊D7—62no—
132C2×Dic7—31no—
133D28—64no—
134C24×C3—71no—
135C24×C3—71no—
136C24×C3—141no—
137C2×C3×D4—211no—
138C2×C3×D4—212no—
139C2×#67—71no—
140C2×#67—72no—
141C3×C22⋊C4—212no—
142C3×C22⋊C4—421no—
143#217—72no—
144#217AGL(1,𝔽7)11yescharacteristic, complemented
145C2×C3×C7—22no—
146C2×C3×C7C2×C3×D411yescharacteristic
147C2×C3×C7C2×C3×D412yes—
148D21—41no—
149C3×D7—24no—
150C2×ΓL(1,𝔽8)—22no—
151C2×ΓL(1,𝔽8)—22no—
152C2×ΓL(1,𝔽8)—22no—
153C2×ΓL(1,𝔽8)—21no—
154C2×ΓL(1,𝔽8)—22no—
155C2×ΓL(1,𝔽8)C2×#6711yescharacteristic
156C2×ΓL(1,𝔽8)C2×#6712yes—
157AGL(1,𝔽7)—24no—
158AGL(1,𝔽7)—24no—
159AGL(1,𝔽7)—24no—
160AGL(1,𝔽7)—121no—
161C22×C32—74no—
162C22×C32—72no—
163C22×C32—71no—
164C22×C32—142no—
165C22×C32—142no—
166C22×C32—142no—
167C22×C32—148no—
168C2×C3×D3—141no—
169C2×C3×D3—142no—
170C3×Dic3—142no—
171C3×Dic3—141no—
172#120—211noSylow
173C22×C7—22no—
174C22×C7C3×#6712yescomplemented
175C22×C7C22×C3×D311yescharacteristic
176C2×D7—28no—
177C2×D7—22no—
178C2×D7—22no—
179C2×D7—62no—
180C2×D7—61no—
181C2×D7C3×#6714yescomplemented
182C4×C7—62no—
183Dic7—61no—
184C23×C3—72no—
185C23×C3—72no—
186C23×C3—142no—
187C23×C3—144no—
188C23×C3—144no—
189C23×C3—144no—
190C23×C3—142no—
191C23×C3—142no—
192C23×C3—144no—
193C23×C3—142no—
194C23×C3—142no—
195C23×C3—141no—
196C23×C3—211no—
197C23×C3—281no—
198C23×C3C2×AGL(1,𝔽7)11yescharacteristic
199C22×D3—71no—
200C3×D4—422no—
201C3×D4—424no—
202C2×C3×C4—211no—
203C2×C3×C4—422no—
204#67—144no—
205#67—142no—
206C2×Dic3—71no—
207C2×Dic3C2×AGL(1,𝔽7)12yescomplemented
208C3×C7C3×#12011yescharacteristic, complemented
209ΓL(1,𝔽8)—21no—
210ΓL(1,𝔽8)#95311yescharacteristic, complemented
211C2×C32—71no—
212C2×C32—72no—
213C2×C32—142no—
214C2×C32—144no—
215C3×D3—281no—
216C24—71no—
217C2×D4—212no—
218C2×D4—211no—
219C22⋊C4—31no—
220C22⋊C4—212no—
221C2×C7—22no—
222C2×C7C2×C3×#6711yescharacteristic
223C2×C7C2×C3×#6712yes—
224D7—24no—
225D7—121no—
226C22×C3—22no—
227C22×C3—74no—
228C22×C3—74no—
229C22×C3—142no—
230C22×C3—144no—
231C22×C3—148no—
232C22×C3—142no—
233C22×C3—142no—
234C22×C3—148no—
235C22×C3—148no—
236C22×C3—148no—
237C22×C3—142no—
238C22×C3—142no—
239C22×C3—142no—
240C22×C3—142no—
241C22×C3—142no—
242C22×C3—142no—
243C22×C3—141no—
244C22×C3—142no—
245C22×C3—282no—
246C22×C3—282no—
247C22×C3—281no—
248C22×C3—282no—
249C22×C3—421no—
250C22×C3—422no—
251C22×C3C4⋊AGL(1,𝔽7)12yescomplemented
252C22×C3C22×AGL(1,𝔽7)11yescharacteristic
253C2×D3—141no—
254C2×D3—142no—
255C3×C4—421no—
256C3×C4—842no—
257Dic3—22no—
258Dic3—141no—
259C32—71noSylow
260C23—72no—
261C23—144no—
262C23—142no—
263C23—211no—
264C23D3×AGL(1,𝔽7)11yescharacteristic
265D4—424no—
266D4—422no—
267C2×C4—32no—
268C2×C4—211no—
269C7C3×#95311yescharacteristic, complemented, Sylow
270C2×C3—22no—
271C2×C3—142no—
272C2×C3—144no—
273C2×C3—144no—
274C2×C3—144no—
275C2×C3—144no—
276C2×C3—141no—
277C2×C3—142no—
278C2×C3—282no—
279C2×C3—282no—
280C2×C3—281no—
281C2×C3—282no—
282C2×C3—841no—
283C2×C3C2×C4⋊AGL(1,𝔽7)11yescharacteristic
284C2×C3D4×AGL(1,𝔽7)12yes—
285D3—281no—
286C22—22no—
287C22—74no—
288C22—148no—
289C22—142no—
290C22—142no—
291C22—421no—
292C22—422no—
293C22#4246⋊C312yescomplemented
294C22C2×D3×AGL(1,𝔽7)11yescharacteristic
295C4—62no—
296C4—421no—
297C3—141no—
298C3—281no—
299C3#6445⋊C311yescharacteristic, complemented
300C2—22no—
301C2—144no—
302C2—841no—
303C2C2×#4246⋊C311yescharacteristic
304C2#67×AGL(1,𝔽7)12yes—
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×#4268⋊C3C2semidirect
3. C2×#4246⋊C3C2semidirect
4. ΓL(1,𝔽8)×#217C2semidirect
5. (C2×AGL(1,𝔽7))⋊Dic3C2semidirect
6. #10621126C3semidirect
9. C22×C3⋊AGL(1,𝔽7)C22semidirect
14. C2×C3⋊(ΓL(1,𝔽8)⋊C4)C22semidirect
15. C2×Dic3×ΓL(1,𝔽8)C22semidirect
17. C23×C3×D7C2×C3semidirect
19. C23×AGL(1,𝔽7)D3semidirect
20. C2×#4268C2×C3semidirect
21. C2×#4246C2×C3semidirect
24. C7×#217C2×C3semidirect
26. (C2×D7)⋊Dic3C2×C3semidirect
30. C22×C3×ΓL(1,𝔽8)D4semidirect
37. C2×C3×AGL(1,𝔽7)D4semidirect
42. C22×D21C22×C3semidirect
58. C2×Dic21C22×C3semidirect
59. C2×C7×Dic3C22×C3semidirect
75. C23×D7C3×D3semidirect
83. C22×C3×C7C3×D4semidirect
90. C2×C3×D7C3×D4semidirect
96. C22×ΓL(1,𝔽8)#67semidirect
110. C2×AGL(1,𝔽7)#67semidirect
124. C3×ΓL(1,𝔽8)#120semidirect
144. #217AGL(1,𝔽7)semidirect
174. C22×C7C3×#67semidirect
181. C2×D7C3×#67semidirect
207. C2×Dic3C2×AGL(1,𝔽7)semidirect
208. C3×C7C3×#120semidirect
210. ΓL(1,𝔽8)#953semidirect
251. C22×C3C4⋊AGL(1,𝔽7)semidirect
269. C7C3×#953semidirect
293. C22#4246⋊C3semidirect
299. 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×#4268⋊C32
C2×#4246⋊C32
ΓL(1,𝔽8)×#2172
(C2×AGL(1,𝔽7))⋊Dic32
#106211263
#6445⋊C33
C3×#9537