Finite Group Databasebeta

← #16854979

Subgroups — #16854979

Order 3968.

784 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11758526C211yesmaximal, characteristic, complemented
2C22×#92479C211yesmaximal, characteristic, complemented
3#14521613C211yesmaximal, characteristic, complemented
4#14521194C214yesmaximal, complemented
5#14521183C214yesmaximal, complemented
6#14521054C211yesmaximal, characteristic, complemented
7#14521045C211yesmaximal, characteristic, complemented
8#14521016C211yesmaximal, characteristic, complemented
9C2×C31×H(5,𝔽2)C211yesmaximal, characteristic, complemented
10C31×H(5,𝔽2)—24no—
11C31×H(5,𝔽2)—24no—
12C2×#41×C31—21no—
13C2×#41×C31C2211yescharacteristic
14C2×#41×C31C2211yescharacteristic
15C2×#41×C31C2212yescomplemented
16C22×D4×C31—21no—
17C22×D4×C31—22no—
18C22×D4×C31C2211yescharacteristic
19C22×D4×C31C2212yescomplemented
20C22×D124C2211yescharacteristic, complemented
21#11758526—24no—
22#11758524—21no—
23C2×#92481C2212yescomplemented
24#11758517—21no—
25#11758514—22no—
26C2×#92479—24no—
27C2×#92479C2212yescomplemented
28C2×#92479C2214yescomplemented
29#11758507C2214yescomplemented
30#11758503C2211yescharacteristic, complemented
31C2×#92474C2211yescharacteristic, complemented
32C22×C31⋊C8C2211yescharacteristic, complemented
33#11758423C2212yescomplemented
34#11758420C2212yescomplemented
35#11758420C2214yescomplemented
36C22⋊(C31⋊C8)C2214yescomplemented
37C24×C31—21no—
38C24×C31—22no—
39C23×D31—21no—
40#41×C31—28no—
41#41×C31—24no—
42#41×C31—24no—
43#41×C31—22no—
44#41×C31—22no—
45#41×C31—22no—
46#41×C31D414yescomplemented
47C2×Q8×C31D411yescharacteristic
48C2×Q8×C31C2311yescharacteristic
49C2×D4×C31—216no—
50C2×D4×C31—28no—
51C2×D4×C31—21no—
52C2×D4×C31—22no—
53C2×D4×C31—24no—
54C2×D4×C31—24no—
55C2×D4×C31—21no—
56C2×D4×C31—22no—
57C2×D4×C31—22no—
58C2×D4×C31D412yes—
59C2×D4×C31D411yescharacteristic
60C2×D4×C31D414yescomplemented
61C2×D4×C31C2312yes—
62C2×D4×C31C2311yescharacteristic
63C22×C4×C31—21no—
64C22×C4×C31—22no—
65C22×C4×C31C2312yes—
66C22×C4×C31C2311yescharacteristic
67C2×C22⋊D31—24no—
68C2×D124—22no—
69C2×D124—22no—
70C2×D124—21no—
71C2×D124C2312yes—
72C22⋊Dic31—21no—
73#92481—24no—
74#92479—28no—
75#92479—24no—
76(C2×D31)⋊C4—24no—
77C4⋊Dic31C2312yes—
78C4×Dic31—21no—
79#92474—22no—
80C2×C31⋊C8—22no—
81C2×C31⋊C8C2312yes—
82C2×C31⋊C8C2312yes—
83C23×C31—28no—
84C23×C31—21no—
85C23×C31—22no—
86C23×C31—24no—
87C23×C31—82no—
88C23×C31—84no—
89C23×C31C2×D412yes—
90C23×C31C2×D411yescharacteristic
91C22×D31—22no—
92C22×D31—42no—
93C22×D31—82no—
94Q8×C31—22no—
95Q8×C31—21no—
96Q8×C31C2×D412yes—
97D4×C31—28no—
98D4×C31—28no—
99D4×C31—24no—
100D4×C31—24no—
101D4×C31—22no—
102D4×C31—24no—
103D4×C31—22no—
104D4×C31—21no—
105D4×C31C2×D412yes—
106D4×C31C2×D414yes—
107C2×C4×C31—28no—
108C2×C4×C31—21no—
109C2×C4×C31—24no—
110C2×C4×C31—22no—
111C2×C4×C31—22no—
112C2×C4×C31C2×D412yes—
113C2×C4×C31C2×D411yescharacteristic
114C2×C4×C31C2×D412yes—
115C2×C4×C31C2×D412yes—
116C2×C4×C31C2411yescharacteristic
117C22⋊D31—84no—
118C2×Dic31—22no—
119D124—22no—
120D124—22no—
121D124—42no—
122C31⋊C8—22no—
123C31⋊C8—22no—
124#3008—311nomaximal, Sylow
125C22×C31—22no—
126C22×C31—22no—
127C22×C31—24no—
128C22×C31—22no—
129C22×C31—24no—
130C22×C31—44no—
131C22×C31—42no—
132C22×C31—84no—
133C22×C31—88no—
134C22×C31#12012yes—
135C22×C31C22×D411yescharacteristic
136C2×D31—42no—
137C2×D31—44no—
138C2×D31—84no—
139C4×C31—22no—
140C4×C31—22no—
141C4×C31#12012yes—
142C4×C31C22×D411yescharacteristic
143C4×C31C22×D411yescharacteristic
144Dic31—42no—
145C2×H(5,𝔽2)D3111yescharacteristic, complemented
146C2×AGL(1,ℤ/8)—311no—
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—44no—
157C2×C31—42no—
158C2×C31C2×#12011yescharacteristic
159C2×C31#57612yes—
160D31—82no—
161H(5,𝔽2)—24no—
162H(5,𝔽2)—24no—
163C2×#41—21no—
164C2×#41C2×D3111yescharacteristic
165C2×#41C2×D3111yescharacteristic
166C2×#41C2×D3112yescomplemented
167C22×D4—21no—
168C22×D4—22no—
169C22×D4—311no—
170C22×D4C2×D3111yescharacteristic
171C22×D4C2×D3112yescomplemented
172AGL(1,ℤ/8)—624no—
173C2×SD16—312no—
174C2×D8—314no—
175C2×D8—312no—
176C2×D8—624no—
177C2×M16—311no—
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—22no—
191C24—621no—
192#41—28no—
193#41—24no—
194#41—24no—
195#41—22no—
196#41—22no—
197#41—22no—
198#41C22⋊D3114yescomplemented
199C2×Q8C22⋊D3111yescharacteristic
200C2×Q8C22×D3111yescharacteristic
201C2×D4—216no—
202C2×D4—28no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—312no—
211C2×D4—624no—
212C2×D4—622no—
213C2×D4—622no—
214C2×D4—621no—
215C2×D4C22⋊D3112yes—
216C2×D4C22⋊D3111yescharacteristic
217C2×D4C22⋊D3114yescomplemented
218C2×D4C22×D3112yes—
219C2×D4C22×D3111yescharacteristic
220C22×C4—21no—
221C22×C4—22no—
222C22×C4C22×D3112yes—
223C22×C4C22×D3111yescharacteristic
224SD16—624no—
225D8—628no—
226D8—624no—
227M16—622no—
228C2×C8—312no—
229C2×C8—312no—
230C2×C8—622no—
231C4⋊C4—312no—
232C22⋊C4—621no—
233C22⋊C4—624no—
234C42—621no—
235C23—28no—
236C23—21no—
237C23—22no—
238C23—24no—
239C23—82no—
240C23—84no—
241C23—622no—
242C23—1242no—
243C23—2482no—
244C23C2×C22⋊D3112yes—
245C23C2×C22⋊D3111yescharacteristic
246Q8—22no—
247Q8—21no—
248Q8C2×C22⋊D3112yes—
249D4—28no—
250D4—28no—
251D4—24no—
252D4—24no—
253D4—22no—
254D4—24no—
255D4—22no—
256D4—21no—
257D4—622no—
258D4—622no—
259D4—1242no—
260D4—2484no—
261D4C2×C22⋊D3112yes—
262D4C2×C22⋊D3114yes—
263C2×C4—28no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—622no—
269C2×C4C2×C22⋊D3112yes—
270C2×C4C2×C22⋊D3111yescharacteristic
271C2×C4C2×C22⋊D3112yes—
272C2×C4C2×C22⋊D3112yes—
273C2×C4C23×D3111yescharacteristic
274C8—622no—
275C8—622no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—24no—
281C22—44no—
282C22—42no—
283C22—84no—
284C22—88no—
285C22—1242no—
286C22—1244no—
287C22—2484no—
288C22#1175853012yes—
289C22C22×C22⋊D3111yescharacteristic
290C4—22no—
291C4—22no—
292C4—1242no—
293C4#1175853012yes—
294C4C22×C22⋊D3111yescharacteristic
295C4C22×C22⋊D3111yescharacteristic
296C2—22no—
297C2—42no—
298C2—44no—
299C2—42no—
300C2—2482no—
301C2C2×#1175853011yescharacteristic
302C2#1452185212yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×H(5,𝔽2)C2
socleC22×C31C22×D4
hypercenterC2×H(5,𝔽2)D31
hypocenterC31#3008
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#11758526C2semidirect
2. C22×#92479C2semidirect
3. #14521613C2semidirect
4. #14521194C2semidirect
5. #14521183C2semidirect
6. #14521054C2semidirect
7. #14521045C2semidirect
8. #14521016C2semidirect
9. C2×C31×H(5,𝔽2)C2semidirect
15. C2×#41×C31C22semidirect
19. C22×D4×C31C22semidirect
20. C22×D124C22semidirect
23. C2×#92481C22semidirect
27. C2×#92479C22semidirect
28. C2×#92479C22semidirect
29. #11758507C22semidirect
30. #11758503C22semidirect
31. C2×#92474C22semidirect
32. C22×C31⋊C8C22semidirect
33. #11758423C22semidirect
34. #11758420C22semidirect
35. #11758420C22semidirect
36. C22⋊(C31⋊C8)C22semidirect
46. #41×C31D4semidirect
60. C2×D4×C31D4semidirect
145. C2×H(5,𝔽2)D31semidirect
166. C2×#41C2×D31semidirect
171. C22×D4C2×D31semidirect
188. C31#3008semidirect
198. #41C22⋊D31semidirect
217. C2×D4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585262
C22×#924792
#145216132
#145211942
#145211832
#145210542
#145210452
#145210162
C2×C31×H(5,𝔽2)2
#300831