Finite Group Databasebeta

← #16854978

Subgroups — #16854978

Order 3968.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#11758566C211yesmaximal, characteristic, complemented
2C22×#92479C211yesmaximal, characteristic, complemented
3#14521616C211yesmaximal, characteristic, complemented
4#14521193C214yesmaximal, complemented
5#14521181C214yesmaximal, complemented
6#14521055C211yesmaximal, characteristic, complemented
7#14521045C211yesmaximal, characteristic, complemented
8#14521017C211yesmaximal, characteristic, complemented
9C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C22×D4×C31C2211yescharacteristic, complemented
12#11758566—24no—
13#11758566—24no—
14C2×#92499C2212yescomplemented
15C2×D4×D31—22no—
16C2×D4×D31C2212yescomplemented
17C2×#92495—21no—
18C2×#92495C2211yescharacteristic
19C22×D124—21no—
20C22×D124C2211yescharacteristic
21C31×AGL(1,ℤ/8)—24no—
22C2×SD16×C31C2212yescomplemented
23C2×D8×C31C2212yescomplemented
24C2×M16×C31C2211yescharacteristic, complemented
25#11758530—22no—
26#11758517—21no—
27#11758515C2214yescomplemented
28#11758513—21no—
29C2×#92479—24no—
30C2×#92479C2214yescomplemented
31#11758503C2211yescharacteristic, complemented
32C22×C31⋊C8C2211yescharacteristic, complemented
33#11758423C2212yescomplemented
34#11758420C2212yescomplemented
35#11758409C2214yescomplemented
36(C2×D31)⋊C8C2214yescomplemented
37C24×C31—21no—
38C23×D31—21no—
39C23×D31—22no—
40#41×C31—22no—
41#41×C31D414yescomplemented
42C2×Q8×C31C2311yescharacteristic
43C2×D4×C31—21no—
44C2×D4×C31—22no—
45C2×D4×C31—22no—
46C2×D4×C31—21no—
47C2×D4×C31C2312yes—
48C2×D4×C31C2311yescharacteristic
49C22×C4×C31—21no—
50C22×C4×C31C2311yescharacteristic
51C2×C22⋊D31—24no—
52C2×C22⋊D31—22no—
53C2×C22⋊D31D412yes—
54#92499—24no—
55#92499—24no—
56D4×D31—216no—
57D4×D31—24no—
58D4×D31—24no—
59#92495—28no—
60#92495—22no—
61#92495—22no—
62C2×D124—28no—
63C2×D124—22no—
64C2×D124—21no—
65C2×D124—22no—
66C2×D124D411yescharacteristic
67C2×D124D414yescomplemented
68C2×D124C2312yes—
69C2×C4×D31—22no—
70C2×C4×D31C2312yes—
71C2×Dic62D411yescharacteristic
72SD16×C31—24no—
73D8×C31—24no—
74M16×C31—22no—
75C2×C8×C31C2312yes—
76C22⋊Dic31—21no—
77C22⋊Dic31—24no—
78#92479—28no—
79C4⋊Dic31C2312yes—
80C4×Dic31—21no—
81C2×C31⋊C8—22no—
82C2×C31⋊C8C2312yes—
83C23×C31—21no—
84C23×C31—22no—
85C23×C31—42no—
86C23×C31—82no—
87C23×C31C2×D411yescharacteristic
88C22×D31—22no—
89C22×D31—28no—
90C22×D31—24no—
91C22×D31—84no—
92C22×D31—82no—
93C22×D31C2×D412yes—
94Q8×C31—21no—
95Q8×C31C2×D412yes—
96D4×C31—24no—
97D4×C31—21no—
98D4×C31—22no—
99D4×C31—22no—
100D4×C31—42no—
101D4×C31C2×D412yes—
102C2×C4×C31—21no—
103C2×C4×C31—24no—
104C2×C4×C31C2×D411yescharacteristic
105C2×C4×C31C2×D412yes—
106C2×C4×C31C2×D412yes—
107C2×C4×C31C2411yescharacteristic
108C22⋊D31—28no—
109C22⋊D31—24no—
110C22⋊D31—84no—
111C2×Dic31—22no—
112C2×Dic31C2×D412yes—
113D124—28no—
114D124—22no—
115D124—24no—
116D124—22no—
117D124C2×D414yes—
118C4×D31—28no—
119C4×D31—22no—
120C4×D31—22no—
121Dic62—22no—
122C8×C31—22no—
123C31⋊C8—22no—
124#3008—311nomaximal, Sylow
125C22×C31—22no—
126C22×C31—22no—
127C22×C31—42no—
128C22×C31—42no—
129C22×C31—44no—
130C22×C31—84no—
131C22×C31#12012yes—
132C22×C31C22×D411yescharacteristic
133C2×D31—22no—
134C2×D31—24no—
135C2×D31—24no—
136C2×D31—44no—
137C2×D31—84no—
138C2×D31—88no—
139C4×C31—22no—
140C4×C31#12012yes—
141C4×C31C22×D411yescharacteristic
142C4×C31C22×D411yescharacteristic
143Dic31—22no—
144Dic31—42no—
145C2×H(5,𝔽2)—311no—
146C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
147C22×D8—311no—
148#535—311no—
149#449—314no—
150#447—314no—
151#418—311no—
152#416—311no—
153#408—311no—
154C2×C31—22no—
155C2×C31—42no—
156C2×C31—82no—
157C2×C31C2×#12011yescharacteristic
158C2×C31#57612yes—
159D31—42no—
160D31—44no—
161H(5,𝔽2)—624no—
162H(5,𝔽2)—624no—
163C2×#41—311no—
164C2×#41—312no—
165C2×#41—621no—
166C2×#41C2×D3111yescharacteristic
167C22×D4—312no—
168C22×D4—311no—
169C22×D4—621no—
170C22×D4—622no—
171C22×D4C2×D3111yescharacteristic, complemented
172AGL(1,ℤ/8)—24no—
173C2×SD16C2×D3112yescomplemented
174C2×D8—314no—
175C2×D8—624no—
176C2×D8C2×D3112yescomplemented
177C2×M16C2×D3111yescharacteristic, complemented
178C22×C8—311no—
179#127—621no—
180#124—621no—
181#121—314no—
182#120—622no—
183#117—311no—
184#103—312no—
185#102—312no—
186#102—314no—
187C22⋊C8—314no—
188C31#300811yescharacteristic, complemented, Sylow
189C24—21no—
190C24—622no—
191C24—621no—
192#41—22no—
193#41—628no—
194#41—624no—
195#41—624no—
196#41—622no—
197#41—622no—
198#41C22⋊D3114yescomplemented
199C2×Q8—311no—
200C2×Q8C22×D3111yescharacteristic
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—312no—
206C2×D4—314no—
207C2×D4—311no—
208C2×D4—312no—
209C2×D4—622no—
210C2×D4—628no—
211C2×D4—622no—
212C2×D4—624no—
213C2×D4—6216no—
214C2×D4—624no—
215C2×D4—624no—
216C2×D4—621no—
217C2×D4—622no—
218C2×D4C22×D3112yes—
219C2×D4C22×D3111yescharacteristic
220C22×C4—21no—
221C22×C4—312no—
222C22×C4—622no—
223C22×C4C22×D3111yescharacteristic
224SD16—24no—
225D8—24no—
226D8—628no—
227M16—22no—
228C2×C8—312no—
229C2×C8—622no—
230C2×C8C22×D3112yes—
231C4⋊C4—312no—
232C22⋊C4—624no—
233C22⋊C4—621no—
234C42—621no—
235C23—21no—
236C23—22no—
237C23—42no—
238C23—82no—
239C23—312no—
240C23—628no—
241C23—622no—
242C23—624no—
243C23—2482no—
244C23—2484no—
245C23D4×D3111yescharacteristic
246Q8—21no—
247Q8—622no—
248Q8C2×C22⋊D3112yes—
249D4—24no—
250D4—21no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—314no—
255D4—628no—
256D4—628no—
257D4—624no—
258D4—624no—
259D4—622no—
260D4—622no—
261D4—2484no—
262D4C2×C22⋊D3112yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—312no—
266C2×C4—628no—
267C2×C4—622no—
268C2×C4—622no—
269C2×C4—622no—
270C2×C4D4×D3111yescharacteristic
271C2×C4D4×D3112yes—
272C2×C4C2×C22⋊D3112yes—
273C2×C4C23×D3111yescharacteristic
274C8—22no—
275C8—622no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—624no—
283C22—622no—
284C22—624no—
285C22—1244no—
286C22—2484no—
287C22—2488no—
288C22#1175851412yes—
289C22C2×D4×D3111yescharacteristic
290C4—22no—
291C4—622no—
292C4—1242no—
293C4#1175851412yes—
294C4C2×D4×D3111yescharacteristic
295C4C22×C22⋊D3111yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—1242no—
300C2—1244no—
301C2C2×#1175851411yescharacteristic
302C2#1452185312yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×AGL(1,ℤ/8)C2
socleC22×C31C22×D4
hypercenterC2×AGL(1,ℤ/8)D31
hypocenterC31#3008
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11758566C2semidirect
2. C22×#92479C2semidirect
3. #14521616C2semidirect
4. #14521193C2semidirect
5. #14521181C2semidirect
6. #14521055C2semidirect
7. #14521045C2semidirect
8. #14521017C2semidirect
9. C2×C31×AGL(1,ℤ/8)C2semidirect
11. C22×D4×C31C22semidirect
14. C2×#92499C22semidirect
16. C2×D4×D31C22semidirect
22. C2×SD16×C31C22semidirect
23. C2×D8×C31C22semidirect
24. C2×M16×C31C22semidirect
27. #11758515C22semidirect
30. C2×#92479C22semidirect
31. #11758503C22semidirect
32. C22×C31⋊C8C22semidirect
33. #11758423C22semidirect
34. #11758420C22semidirect
35. #11758409C22semidirect
36. (C2×D31)⋊C8C22semidirect
41. #41×C31D4semidirect
67. C2×D124D4semidirect
146. C2×AGL(1,ℤ/8)D31semidirect
171. C22×D4C2×D31semidirect
173. C2×SD16C2×D31semidirect
176. C2×D8C2×D31semidirect
177. C2×M16C2×D31semidirect
188. C31#3008semidirect
198. #41C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
C22×#924792
#145216162
#145211932
#145211812
#145210552
#145210452
#145210172
C2×C31×AGL(1,ℤ/8)2
#300831