Finite Group Databasebeta

← #16855014

Subgroups — #16855014

Order 3968.

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

normal subgroups only

This group has 764 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×#11758561C211yesmaximal, characteristic, complemented
2C2×#11758527C211yesmaximal, characteristic, complemented
3#14521627C211yesmaximal, characteristic, complemented
4#14521208C212yesmaximal, complemented
5#14521198C212yesmaximal, complemented
6#14521191C212yesmaximal, complemented
7#14521181C212yesmaximal, complemented
8#14521051C211yesmaximal, characteristic, complemented
9C2×#11758420C211yesmaximal, characteristic, complemented
10#14521017C211yesmaximal, characteristic, complemented
11C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic, complemented
13C22×D4×C31C2211yescharacteristic
14C22×C22⋊D31—22no—
15#11758561—24no—
16#11758561—24no—
17C2×#92497—21no—
18C2×#92497C2212yes—
19C2×D4×D31—21no—
20C2×D4×D31C2212yes—
21C2×#92495C2211yescharacteristic
22C2×#92495C2211yescharacteristic
23C31×AGL(1,ℤ/8)—24no—
24C2×SD16×C31C2212yescomplemented
25C2×D8×C31C2212yescomplemented
26C2×M16×C31C2211yescharacteristic, complemented
27#11758527—24no—
28#11758523C2212yescomplemented
29C2×#92482C2211yescharacteristic, complemented
30C2×#92481C2211yescharacteristic, complemented
31#11758516—22no—
32#11758515C2212yescomplemented
33C2×#92480C2211yescharacteristic, complemented
34C2×#92479C2211yescharacteristic, complemented
35#11758507—21no—
36C2×C4⋊Dic31C2211yescharacteristic, complemented
37#11758501—21no—
38C22×C31⋊C8C2211yescharacteristic, complemented
39#11758420C2212yescomplemented
40#11758420C2212yescomplemented
41#11758409C2212yescomplemented
42(C2×D31)⋊C8C2212yescomplemented
43(C2×D31)⋊C8C2212yescomplemented
44#11758404C2212yescomplemented
45C24×C31—21no—
46C23×D31—22no—
47#41×C31—22no—
48#41×C31—22no—
49C2×Q8×C31C2311yescharacteristic
50C2×D4×C31—21no—
51C2×D4×C31—21no—
52C2×D4×C31—22no—
53C2×D4×C31D414yescomplemented
54C2×D4×C31C2311yescharacteristic
55C2×D4×C31C2312yes—
56C22×C4×C31—21no—
57C22×C4×C31C2311yescharacteristic
58C2×C22⋊D31—216no—
59C2×C22⋊D31—21no—
60C2×C22⋊D31—22no—
61C2×C22⋊D31—22no—
62C2×C22⋊D31—22no—
63C2×C22⋊D31D412yes—
64C22×Dic31—22no—
65C22×Dic31—21no—
66#92497—28no—
67#92497—24no—
68#92497—24no—
69D4×D31—28no—
70D4×D31—24no—
71D4×D31—24no—
72#92495—22no—
73#92495—22no—
74#92495—22no—
75#92495D414yescomplemented
76C2×D124D411yescharacteristic
77C2×D124C2311yescharacteristic
78C2×C4×D31—21no—
79C2×C4×D31C2312yes—
80C2×Dic62D411yescharacteristic
81C2×Dic62C2311yescharacteristic
82SD16×C31—24no—
83D8×C31—24no—
84M16×C31—22no—
85C2×C8×C31C2312yes—
86C22⋊Dic31—22no—
87#92482—22no—
88#92481—22no—
89#92480—22no—
90#92479—22no—
91(C2×D31)⋊C4—22no—
92C4⋊Dic31—21no—
93C4⋊Dic31C2312yes—
94Dic31⋊C4—22no—
95C2×C31⋊C8—22no—
96C2×C31⋊C8C2311yescharacteristic
97C2×C31⋊C8C2311yescharacteristic
98C23×C31—22no—
99C23×C31—24no—
100C23×C31—21no—
101C23×C31—81no—
102C23×C31—81no—
103C23×C31C2×D411yescharacteristic
104C22×D31—21no—
105C22×D31—28no—
106C22×D31—82no—
107C22×D31—82no—
108C22×D31C2×D412yes—
109Q8×C31—21no—
110Q8×C31—21no—
111D4×C31—24no—
112D4×C31—22no—
113D4×C31—21no—
114D4×C31—21no—
115D4×C31—42no—
116D4×C31C2×D414yes—
117C2×C4×C31—21no—
118C2×C4×C31—42no—
119C2×C4×C31C2×D411yescharacteristic
120C2×C4×C31C2×D412yes—
121C2×C4×C31C2×D412yes—
122C2×C4×C31C2411yescharacteristic
123C22⋊D31—28no—
124C22⋊D31—28no—
125C22⋊D31—24no—
126C22⋊D31—24no—
127C22⋊D31—82no—
128C2×Dic31—28no—
129C2×Dic31—21no—
130C2×Dic31—22no—
131C2×Dic31—81no—
132C2×Dic31C2×D412yes—
133D124—22no—
134D124—21no—
135D124C2×D412yes—
136C4×D31—24no—
137C4×D31—22no—
138C4×D31—22no—
139Dic62—22no—
140Dic62—21no—
141Dic62C2×D412yes—
142C8×C31—22no—
143C31⋊C8—21no—
144C31⋊C8—21no—
145#3012—311nomaximal, Sylow
146C22×C31—21no—
147C22×C31—21no—
148C22×C31—24no—
149C22×C31—42no—
150C22×C31—42no—
151C22×C31—41no—
152C22×C31—41no—
153C22×C31—41no—
154C22×C31—84no—
155C22×C31#12012yes—
156C22×C31C22×D411yescharacteristic
157C2×D31—22no—
158C2×D31—22no—
159C2×D31—22no—
160C2×D31—22no—
161C2×D31—41no—
162C2×D31—41no—
163C2×D31—88no—
164C4×C31—41no—
165C4×C31#12012yes—
166C4×C31C22×D411yescharacteristic
167C4×C31C22×D411yescharacteristic
168Dic31—22no—
169Dic31—22no—
170Dic31—42no—
171C2×H(5,𝔽2)—311no—
172C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
173C2×#135—311no—
174#537—311no—
175#452—312no—
176#450—312no—
177#449—312no—
178#447—312no—
179#417—311no—
180C2×#102—311no—
181#408—311no—
182C2×C31—22no—
183C2×C31—44no—
184C2×C31—81no—
185C2×C31C2×#12011yescharacteristic
186C2×C31#57611yescharacteristic
187C2×C31#57711yescharacteristic
188D31—42no—
189D31—42no—
190H(5,𝔽2)—624no—
191H(5,𝔽2)—624no—
192C2×#41—311no—
193C2×#41—312no—
194C2×#41—311no—
195C2×#41—621no—
196C2×#41C2×D3111yescharacteristic, complemented
197C22×D4—312no—
198C22×D4—621no—
199C22×D4—622no—
200C22×D4C2×D3111yescharacteristic
201AGL(1,ℤ/8)—24no—
202#135—624no—
203C2×Q16—311no—
204C2×SD16—311no—
205C2×SD16—311no—
206C2×SD16C2×D3112yescomplemented
207C2×D8—311no—
208C2×D8C2×D3112yescomplemented
209C2×M16C2×D3111yescharacteristic, complemented
210C22×C8—311no—
211C22⋊Q8—312no—
212C22⋊Q8—621no—
213#121—312no—
214#121—621no—
215#121—622no—
216C2×C4⋊C4—311no—
217#103—312no—
218#102—312no—
219#102—312no—
220#102—312no—
221C22⋊C8—312no—
222C22⋊C8—312no—
223C31#301211yescharacteristic, complemented, Sylow
224C24—21no—
225C24—622no—
226#41—22no—
227#41—22no—
228#41—314no—
229#41—624no—
230#41—628no—
231#41—624no—
232#41—622no—
233#41—622no—
234#41—622no—
235C2×Q8—311no—
236C2×Q8—311no—
237C2×Q8C22×D3111yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—312no—
242C2×D4—311no—
243C2×D4—311no—
244C2×D4—622no—
245C2×D4—624no—
246C2×D4—628no—
247C2×D4—622no—
248C2×D4—6216no—
249C2×D4—621no—
250C2×D4—624no—
251C2×D4—622no—
252C2×D4C22⋊D3114yescomplemented
253C2×D4C22×D3111yescharacteristic
254C2×D4C22×D3112yes—
255C22×C4—21no—
256C22×C4—312no—
257C22×C4—622no—
258C22×C4—621no—
259C22×C4—621no—
260C22×C4C22×D3111yescharacteristic
261Q16—622no—
262SD16—24no—
263SD16—622no—
264SD16—622no—
265D8—24no—
266D8—622no—
267M16—22no—
268C2×C8—311no—
269C2×C8—311no—
270C2×C8—622no—
271C2×C8C22×D3112yes—
272C4⋊C4—312no—
273C4⋊C4—622no—
274C4⋊C4—621no—
275C22⋊C4—622no—
276C22⋊C4—622no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—312no—
283C23—628no—
284C23—621no—
285C23—2482no—
286C23—2482no—
287C23D4×D3111yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—312no—
291Q8—622no—
292Q8—621no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—312no—
299D4—628no—
300D4—628no—
301D4—624no—
302D4—624no—
303D4—622no—
304D4—621no—
305D4—2482no—
306D4C2×C22⋊D3114yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—312no—
310C2×C4—628no—
311C2×C4—621no—
312C2×C4—624no—
313C2×C4—622no—
314C2×C4—622no—
315C2×C4—622no—
316C2×C4—2481no—
317C2×C4D4×D3111yescharacteristic
318C2×C4D4×D3112yes—
319C2×C4C2×C22⋊D3112yes—
320C2×C4C23×D3111yescharacteristic
321C8—22no—
322C8—621no—
323C8—621no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—622no—
334C22—622no—
335C22—622no—
336C22—622no—
337C22—1241no—
338C22—1241no—
339C22—2488no—
340C22#1175851412yes—
341C22C2×D4×D3111yescharacteristic
342C4—41no—
343C4—622no—
344C4—622no—
345C4—1242no—
346C4#1175851412yes—
347C4C2×D4×D3111yescharacteristic
348C4C22×C22⋊D3111yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—1242no—
353C2—1242no—
354C2C2×#1175851411yescharacteristic
355C2#1452185311yescharacteristic
356C2#1452185911yescharacteristic
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#3012
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×#11758561C2semidirect
2. C2×#11758527C2semidirect
3. #14521627C2semidirect
4. #14521208C2semidirect
5. #14521198C2semidirect
6. #14521191C2semidirect
7. #14521181C2semidirect
8. #14521051C2semidirect
9. C2×#11758420C2semidirect
10. #14521017C2semidirect
11. C2×C31×AGL(1,ℤ/8)C2semidirect
12. C2×#41×C31C22semidirect
24. C2×SD16×C31C22semidirect
25. C2×D8×C31C22semidirect
26. C2×M16×C31C22semidirect
28. #11758523C22semidirect
29. C2×#92482C22semidirect
30. C2×#92481C22semidirect
32. #11758515C22semidirect
33. C2×#92480C22semidirect
34. C2×#92479C22semidirect
36. C2×C4⋊Dic31C22semidirect
38. C22×C31⋊C8C22semidirect
39. #11758420C22semidirect
40. #11758420C22semidirect
41. #11758409C22semidirect
42. (C2×D31)⋊C8C22semidirect
43. (C2×D31)⋊C8C22semidirect
44. #11758404C22semidirect
53. C2×D4×C31D4semidirect
75. #92495D4semidirect
172. C2×AGL(1,ℤ/8)D31semidirect
196. C2×#41C2×D31semidirect
206. C2×SD16C2×D31semidirect
208. C2×D8C2×D31semidirect
209. C2×M16C2×D31semidirect
223. C31#3012semidirect
252. C2×D4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585612
C2×#117585272
#145216272
#145212082
#145211982
#145211912
#145211812
#145210512
C2×#117584202
#145210172
C2×C31×AGL(1,ℤ/8)2
#301231