Finite Group Databasebeta

← #72229682

Subgroups — #72229682

Order 2688.

920 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 209 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
1#90453⋊C3C214yesmaximal, complemented
2ΓL(1,𝔽8)×#492C211yesmaximal, characteristic, complemented
3#89949⋊C3C214yesmaximal, complemented
4#89867⋊C3C212yesmaximal, complemented
5#89849⋊C3C212yesmaximal, complemented
6#89472⋊C3C212yesmaximal, complemented
7#11739219C311yesmaximal, characteristic, complemented, Hall
8#10621516C2212yes—
9#6548⋊C3C2212yescomplemented
10C2×SD16×ΓL(1,𝔽8)C2214yescomplemented
11ΓL(1,𝔽8)×C4⋊Q8C2211yescharacteristic, complemented
12ΓL(1,𝔽8)×#127C2211yescharacteristic
13#6505⋊C3C2214yescomplemented
14#6495⋊C3C2214yes—
15#6479⋊C3—24no—
16#6478⋊C3—24no—
17#6477⋊C3—24no—
18SD16×AGL(1,𝔽7)—24no—
19#6467⋊C3C2212yes—
20#6464⋊C3C2214yescomplemented
21C4×C8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
22#6410⋊C3C2212yescomplemented
23#6407⋊C3C2212yescomplemented
24#6393⋊C3C2212yescomplemented
25#6379⋊C3C2214yescomplemented
26#90453C2×C314yescomplemented
27C7×#492C2×C311yescharacteristic, complemented
28#89949C2×C314yescomplemented
29#89867C2×C312yescomplemented
30#89849C2×C312yescomplemented
31#89472C2×C312yescomplemented
32C3×#3291—71nomaximal, Hall
33C2×Q8×ΓL(1,𝔽8)C2312yes—
34C2×D4×ΓL(1,𝔽8)—41no—
35C2×D4×ΓL(1,𝔽8)C2312yes—
36C2×(C22⋊D7)⋊C3—42no—
37#66659—24no—
38Q8×AGL(1,𝔽7)—24no—
39#1159⋊C3—24no—
40D4×AGL(1,𝔽7)—24no—
41#1157⋊C3—42no—
42#1157⋊C3C2314yes—
43SD16×ΓL(1,𝔽8)—24no—
44SD16×ΓL(1,𝔽8)—24no—
45C2×C8×ΓL(1,𝔽8)C2312yes—
46C4⋊C4×ΓL(1,𝔽8)—41no—
47C42×ΓL(1,𝔽8)C2311yescharacteristic
48#1144⋊C3—24no—
49#1143⋊C3—24no—
50#1142⋊C3—24no—
51#1141⋊C3—24no—
52#1136⋊C3C2314yes—
53Dic28⋊C3D412yescomplemented
54D56⋊C3D412yescomplemented
55#1132⋊C3D414yescomplemented
56#1131⋊C3—24no—
57C8×AGL(1,𝔽7)—24no—
58#6551C22×C312yes—
59#6548C22×C312yescomplemented
60C2×C7×SD16C22×C314yescomplemented
61C7×C4⋊Q8C22×C311yescharacteristic, complemented
62C7×#127C22×C311yescharacteristic
63#6505C22×C314yescomplemented
64#6495C22×C314yes—
65#6479—24no—
66#6478—24no—
67#6477—24no—
68D7×SD16—24no—
69#6467C22×C312yes—
70#6464C22×C314yescomplemented
71C4×C7×C8C22×C311yescharacteristic, complemented
72#6410C22×C312yescomplemented
73#6407C22×C312yescomplemented
74#6393C22×C312yescomplemented
75#6379C22×C314yescomplemented
76C3×#577—74no—
77C3×#492—71no—
78C3×#471—74no—
79C3×#456—72no—
80C3×#453—72no—
81C3×#443—72no—
82C23×ΓL(1,𝔽8)—22no—
83C22×AGL(1,𝔽7)—42no—
84Q8×ΓL(1,𝔽8)—24no—
85D4×ΓL(1,𝔽8)—24no—
86D4×ΓL(1,𝔽8)—82no—
87C2×C4×ΓL(1,𝔽8)—21no—
88C2×C4×ΓL(1,𝔽8)—22no—
89C2×C4×ΓL(1,𝔽8)C2411yescharacteristic
90(C22⋊D7)⋊C3—24no—
91(C22⋊D7)⋊C3—44no—
92C2×ΓL(1,𝔽8)⋊C4—42no—
93C4⋊AGL(1,𝔽7)—42no—
94C4⋊AGL(1,𝔽7)C2×D414yes—
95C4×AGL(1,𝔽7)—24no—
96C4×AGL(1,𝔽7)—44no—
97ΓL(1,𝔽8)⋊Q8—42no—
98ΓL(1,𝔽8)⋊Q8C2×D414yes—
99C8×ΓL(1,𝔽8)C2×D412yes—
100C8×ΓL(1,𝔽8)C2×D412yes—
101ΓL(1,𝔽8)⋊C8—24no—
102#3291—71noSylow
103C2×C7×Q8C23×C312yes—
104C2×C7×D4—41no—
105C2×C7×D4C23×C312yes—
106C2×C22⋊D7—42no—
107#1161—24no—
108Q8×D7—24no—
109#1159—24no—
110D4×D7—24no—
111#1157—42no—
112#1157C23×C314yes—
113C7×SD16—24no—
114C7×SD16—24no—
115C2×C7×C8C23×C312yes—
116C7×C4⋊C4—41no—
117C42×C7C23×C311yescharacteristic
118#1144—24no—
119#1143—24no—
120#1142—24no—
121#1141—24no—
122#1136C23×C314yes—
123Dic28C3×D412yescomplemented
124D56C3×D412yescomplemented
125#1132C3×D414yescomplemented
126#1131—24no—
127C8×D7—24no—
128C3×2−1+4—72no—
129C3×H(5,𝔽2)—72no—
130C3×#137—74no—
131C3×#137—144no—
132C3×AGL(1,ℤ/8)—74no—
133C3×AGL(1,ℤ/8)—144no—
134C3×#135—72no—
135C3×#135—144no—
136C2×C3×SD16—74no—
137C2×C3×SD16—144no—
138C3×#131—74no—
139C3×C4⋊Q8—71no—
140C3×#127—71no—
141C3×#108—72no—
142C3×#104—74no—
143C3×#101—72no—
144C3×#100—72no—
145C3×C4×C8—71no—
146C22×ΓL(1,𝔽8)—42no—
147C22×ΓL(1,𝔽8)—82no—
148C22×ΓL(1,𝔽8)H(5,𝔽2)11yescharacteristic
149C2×AGL(1,𝔽7)—24no—
150C2×AGL(1,𝔽7)—162no—
151C4×ΓL(1,𝔽8)—22no—
152C4×ΓL(1,𝔽8)—42no—
153C4×ΓL(1,𝔽8)C22×D412yes—
154ΓL(1,𝔽8)⋊C4—24no—
155#577—74no—
156#492AGL(1,𝔽7)11yescharacteristic, complemented
157#471—74no—
158#456—72no—
159#453—72no—
160#443—72no—
161C23×C7—22no—
162C22×D7—42no—
163C7×Q8—24no—
164C7×D4—24no—
165C7×D4—82no—
166C2×C4×C7—21no—
167C2×C4×C7—22no—
168C2×C4×C7C24×C311yescharacteristic
169C22⋊D7—24no—
170C22⋊D7—44no—
171C2×Dic7—42no—
172D28—42no—
173D28C2×C3×D414yes—
174C4×D7—24no—
175C4×D7—44no—
176Dic14—42no—
177Dic14C2×C3×D414yes—
178C7×C8C2×C3×D412yes—
179C7×C8C2×C3×D412yes—
180C7⋊C8—24no—
181C3×#41—74no—
182C3×#41—144no—
183C3×#41—144no—
184C3×#41—282no—
185C2×C3×Q8—72no—
186C2×C3×Q8—144no—
187C2×C3×D4—72no—
188C2×C3×D4—144no—
189C2×C3×D4—281no—
190C2×C3×D4—282no—
191C3×Q16—72no—
192C3×Q16—144no—
193C3×SD16—74no—
194C3×SD16—144no—
195C3×SD16—144no—
196C3×SD16—144no—
197C3×SD16—144no—
198C3×D8—72no—
199C3×D8—144no—
200C3×M16—74no—
201C3×M16—144no—
202C2×C3×C8—72no—
203C2×C3×C8—144no—
204C3×C4⋊C4—281no—
205C3×C42—71no—
206C2×ΓL(1,𝔽8)—21no—
207C2×ΓL(1,𝔽8)—82no—
208C2×ΓL(1,𝔽8)D4211yescharacteristic
209AGL(1,𝔽7)—44no—
2102−1+4—72no—
211H(5,𝔽2)—72no—
212#137—74no—
213#137—144no—
214AGL(1,ℤ/8)—74no—
215AGL(1,ℤ/8)—144no—
216#135—72no—
217#135—144no—
218C2×SD16—144no—
219C2×SD16C2×AGL(1,𝔽7)14yescomplemented
220#131—74no—
221C4⋊Q8C2×AGL(1,𝔽7)11yescharacteristic, complemented
222#127C2×AGL(1,𝔽7)11yescharacteristic
223#108—72no—
224#104—74no—
225#101—72no—
226#100—72no—
227C4×C8C2×AGL(1,𝔽7)11yescharacteristic, complemented
228C22×C7—42no—
229C22×C7—82no—
230C22×C7C3×H(5,𝔽2)11yescharacteristic
231C2×D7—24no—
232C2×D7—162no—
233C4×C7—22no—
234C4×C7—42no—
235C4×C7C22×C3×D412yes—
236Dic7—24no—
237C23×C3—142no—
238C23×C3—282no—
239C3×Q8—74no—
240C3×Q8—144no—
241C3×Q8—282no—
242C3×D4—74no—
243C3×D4—144no—
244C3×D4—144no—
245C3×D4—284no—
246C3×D4—282no—
247C3×D4—562no—
248C2×C3×C4—71no—
249C2×C3×C4—141no—
250C2×C3×C4—142no—
251C2×C3×C4—144no—
252C2×C3×C4—282no—
253C2×C3×C4—284no—
254C3×C8—72no—
255C3×C8—72no—
256C3×C8—144no—
257ΓL(1,𝔽8)#329111yescharacteristic, complemented, Hall
258#41—74no—
259#41—144no—
260#41—144no—
261#41—282no—
262C2×Q8—144no—
263C2×Q8C22×AGL(1,𝔽7)12yes—
264C2×D4—41no—
265C2×D4—144no—
266C2×D4—282no—
267C2×D4C22×AGL(1,𝔽7)12yes—
268Q16—72no—
269Q16—144no—
270SD16—24no—
271SD16—24no—
272SD16—74no—
273SD16—144no—
274SD16—144no—
275D8—72no—
276D8—144no—
277M16—74no—
278M16—144no—
279C2×C8—144no—
280C2×C8C22×AGL(1,𝔽7)12yes—
281C4⋊C4—41no—
282C42C22×AGL(1,𝔽7)11yescharacteristic
283C2×C7—21no—
284C2×C7—82no—
285C2×C7C3×D4211yescharacteristic
286D7—44no—
287C22×C3—71no—
288C22×C3—144no—
289C22×C3—282no—
290C22×C3—562no—
291C22×C3—1122no—
292C3×C4—72no—
293C3×C4—142no—
294C3×C4—144no—
295C3×C4—282no—
296C23—22no—
297C23—282no—
298Q8—24no—
299Q8—74no—
300Q8—282no—
301D4—24no—
302D4—74no—
303D4—82no—
304D4—144no—
305D4—282no—
306D4—284no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—144no—
310C2×C4—284no—
311C2×C4—282no—
312C2×C4C23×AGL(1,𝔽7)11yescharacteristic
313C8—144no—
314C8D4×AGL(1,𝔽7)12yes—
315C8D4×AGL(1,𝔽7)12yes—
316C7C3×#329111yescharacteristic, complemented, Sylow
317C2×C3—71no—
318C2×C3—141no—
319C2×C3—284no—
320C2×C3—562no—
321C22—42no—
322C22—82no—
323C22—144no—
324C22—1122no—
325C22#6548⋊C311yescharacteristic
326C4—22no—
327C4—42no—
328C4—144no—
329C4C2×D4×AGL(1,𝔽7)12yes—
330C3—71noSylow
331C2—21no—
332C2—82no—
333C2—284no—
334C2#90408⋊C311yescharacteristic
subgroupG/H
centerC2#90408⋊C3
commutator, cosocleC2×C4×C7C24×C3
FrattiniC2×C4C23×AGL(1,𝔽7)
FittingC7×#492C2×C3
socleC2×C7C3×D42
hypercenter#492AGL(1,𝔽7)
hypocenterC7C3×#3291
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #90453⋊C3C2semidirect
2. ΓL(1,𝔽8)×#492C2semidirect
3. #89949⋊C3C2semidirect
4. #89867⋊C3C2semidirect
5. #89849⋊C3C2semidirect
6. #89472⋊C3C2semidirect
7. #11739219C3semidirect
9. #6548⋊C3C22semidirect
10. C2×SD16×ΓL(1,𝔽8)C22semidirect
11. ΓL(1,𝔽8)×C4⋊Q8C22semidirect
13. #6505⋊C3C22semidirect
20. #6464⋊C3C22semidirect
21. C4×C8×ΓL(1,𝔽8)C22semidirect
22. #6410⋊C3C22semidirect
23. #6407⋊C3C22semidirect
24. #6393⋊C3C22semidirect
25. #6379⋊C3C22semidirect
26. #90453C2×C3semidirect
27. C7×#492C2×C3semidirect
28. #89949C2×C3semidirect
29. #89867C2×C3semidirect
30. #89849C2×C3semidirect
31. #89472C2×C3semidirect
53. Dic28⋊C3D4semidirect
54. D56⋊C3D4semidirect
55. #1132⋊C3D4semidirect
59. #6548C22×C3semidirect
60. C2×C7×SD16C22×C3semidirect
61. C7×C4⋊Q8C22×C3semidirect
63. #6505C22×C3semidirect
70. #6464C22×C3semidirect
71. C4×C7×C8C22×C3semidirect
72. #6410C22×C3semidirect
73. #6407C22×C3semidirect
74. #6393C22×C3semidirect
75. #6379C22×C3semidirect
123. Dic28C3×D4semidirect
124. D56C3×D4semidirect
125. #1132C3×D4semidirect
156. #492AGL(1,𝔽7)semidirect
219. C2×SD16C2×AGL(1,𝔽7)semidirect
221. C4⋊Q8C2×AGL(1,𝔽7)semidirect
227. C4×C8C2×AGL(1,𝔽7)semidirect
257. ΓL(1,𝔽8)#3291semidirect
316. C7C3×#3291semidirect

Maximal subgroups

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

subgroupindex
#90453⋊C32
ΓL(1,𝔽8)×#4922
#89949⋊C32
#89867⋊C32
#89849⋊C32
#89472⋊C32
#117392193
C3×#32917