Finite Group Databasebeta

← #16854962

Subgroups — #16854962

Order 3968.

806 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#92499C211yesmaximal, characteristic, complemented
2C2×#11758491C212yesmaximal, complemented
3#121×D31C211yesmaximal, characteristic, complemented
4#14521197C212yesmaximal, complemented
5#14521193C212yesmaximal, complemented
6#14521188C211yesmaximal, characteristic, complemented
7#14521185C211yesmaximal, characteristic, complemented
8#14521182C211yesmaximal, characteristic, complemented
9#14521049C212yesmaximal, complemented
10#14521011C211yesmaximal, characteristic, complemented
11C31×#448C211yesmaximal, characteristic, complemented
12C22×Q8×C31C2211yescharacteristic, complemented
13C2×#92499—28no—
14C2×#92499—21no—
15C2×#92499—21no—
16C2×#92499C2212yes—
17C2×#92499C2212yescomplemented
18C2×D4×D31—21no—
19C2×D4×D31C2211yescharacteristic, complemented
20C22×D124—21no—
21C22×D124C2211yescharacteristic, complemented
22C22×C4×D31—21no—
23C22×C4×D31C2211yescharacteristic
24C2×SD16×C31C2212yescomplemented
25C31×#121C2211yescharacteristic
26C2×#92481C2212yescomplemented
27#11758516—21no—
28#11758515C2211yescharacteristic, complemented
29C2×#92479C2212yescomplemented
30#11758507C2211yescharacteristic
31#11758491—28no—
32C2×D248C2212yescomplemented
33C2×#92469C2212yescomplemented
34#11758472C2211yescharacteristic, complemented
35C4⋊C4×D31C2211yescharacteristic, complemented
36#11758461—21no—
37C22⋊C4×D31—21no—
38C31×#103C2212yescomplemented
39C31×C22⋊C8C2211yescharacteristic, complemented
40#11758423C2212yescomplemented
41C22⋊(C31⋊C8)C2211yescharacteristic, complemented
42#11758409C2212yescomplemented
43(C2×D31)⋊C8C2212yescomplemented
44#11758397C2212yescomplemented
45C23×D31—21no—
46C23×D31—21no—
47C23×D31D411yescharacteristic
48C2×Q8×C31—24no—
49C2×Q8×C31—21no—
50C2×Q8×C31C2312yes—
51C2×D4×C31—21no—
52C2×D4×C31C2311yescharacteristic
53C22×C4×C31—21no—
54C22×C4×C31C2311yescharacteristic
55C2×C22⋊D31—21no—
56C2×C22⋊D31—21no—
57C2×C22⋊D31—21no—
58C22×Dic31D411yescharacteristic
59#92499—28no—
60#92499—28no—
61#92499—24no—
62#92499—24no—
63#92499—24no—
64#92499D418yescomplemented
65D4×D31—24no—
66D4×D31—44no—
67C2×D124—28no—
68C2×D124—22no—
69C2×D124—22no—
70C2×D124—21no—
71C2×D124—24no—
72C2×D124—21no—
73C2×D124C2311yescharacteristic
74C2×D124C2312yes—
75C2×C4×D31—28no—
76C2×C4×D31—22no—
77C2×C4×D31—22no—
78C2×C4×D31—24no—
79C2×C4×D31—21no—
80C2×C4×D31D411yescharacteristic
81C2×C4×D31D411yescharacteristic
82C2×C4×D31C2311yescharacteristic
83C2×C4×D31C2311yescharacteristic
84SD16×C31—24no—
85C2×C8×C31C2312yes—
86C4⋊C4×C31C2311yescharacteristic
87C22⋊C4×C31—21no—
88C22⋊Dic31—21no—
89#92481—24no—
90#92479—24no—
91(C2×D31)⋊C4—21no—
92(C2×D31)⋊C4—21no—
93C4⋊Dic31C2311yescharacteristic
94Dic31⋊C4—21no—
95C2×C31⋊C8C2312yes—
96D248—24no—
97#92469—24no—
98C23×C31—21no—
99C23×C31C2×D411yescharacteristic
100C22×D31—24no—
101C22×D31—22no—
102C22×D31—22no—
103C22×D31—22no—
104C22×D31—21no—
105C22×D31—41no—
106C22×D31—42no—
107C22×D31—42no—
108C22×D31—42no—
109C22×D31—42no—
110C22×D31—42no—
111C22×D31C2×D411yescharacteristic
112C22×D31C2×D411yescharacteristic
113Q8×C31—24no—
114Q8×C31—22no—
115Q8×C31C2×D414yescomplemented
116D4×C31—22no—
117D4×C31—42no—
118C2×C4×C31—24no—
119C2×C4×C31—22no—
120C2×C4×C31—22no—
121C2×C4×C31—21no—
122C2×C4×C31C2×D411yescharacteristic
123C2×C4×C31C2411yescharacteristic
124C22⋊D31—42no—
125C22⋊D31—42no—
126C22⋊D31—42no—
127C2×Dic31—22no—
128C2×Dic31—21no—
129C2×Dic31C2×D411yescharacteristic
130C2×Dic31C2×D411yescharacteristic
131D124—28no—
132D124—24no—
133D124—28no—
134D124—22no—
135D124—22no—
136D124—42no—
137D124C2×D414yescomplemented
138C4×D31—28no—
139C4×D31—22no—
140C4×D31—22no—
141C4×D31—28no—
142C4×D31—22no—
143C4×D31—42no—
144C4×D31C2×D414yes—
145C8×C31—22no—
146C31⋊C8—22no—
147#3006—311nomaximal, Sylow
148C22×C31—21no—
149C22×C31—21no—
150C22×C31—41no—
151C22×C31—41no—
152C22×C31—41no—
153C22×C31AGL(1,ℤ/8)12yescomplemented
154C22×C31C22×D411yescharacteristic
155C2×D31—21no—
156C2×D31—21no—
157C2×D31—21no—
158C2×D31—24no—
159C2×D31—21no—
160C2×D31—21no—
161C2×D31—44no—
162C2×D31—48no—
163C2×D31—42no—
164C2×D31—42no—
165C2×D31—41no—
166C2×D31—41no—
167C2×D31—41no—
168C2×D31—84no—
169C2×D31#12012yes—
170C4×C31—24no—
171C4×C31—21no—
172C4×C31—41no—
173C4×C31C22×D412yes—
174Dic31—21no—
175Dic31—41no—
176Dic31#12012yes—
177C22×#41—311no—
178C2×AGL(1,ℤ/8)—312no—
179C2×#121—311no—
180#449—312no—
181#449—312no—
182#448—311no—
183#448D3111yescharacteristic, complemented
184#447—311no—
185#447—311no—
186#417—312no—
187#407—311no—
188C2×C31—22no—
189C2×C31—81no—
190C2×C31C2×#12011yescharacteristic
191C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C31#57511yescharacteristic
193D31—22no—
194D31—41no—
195D31—44no—
196D31—81no—
197C2×#41—312no—
198C2×#41—312no—
199C2×#41—621no—
200C2×#41—621no—
201C2×#41—628no—
202C22×Q8C2×D3111yescharacteristic, complemented
203C22×D4—311no—
204C22×D4—311no—
205C22×D4—621no—
206C22×D4—621no—
207C23×C4—311no—
208C23×C4—621no—
209AGL(1,ℤ/8)—628no—
210C2×SD16—312no—
211C2×SD16C2×D3112yescomplemented
212C2×D8—312no—
213C2×D8—312no—
214C2×M16—312no—
215#121—311no—
216#121—311no—
217#121—311no—
218#121—621no—
219#121—621no—
220#121C2×D3111yescharacteristic
221C2×C4⋊C4—311no—
222C2×C22⋊C4—621no—
223#103—312no—
224#103C2×D3112yescomplemented
225#102—312no—
226#102—312no—
227C22⋊C8—311no—
228C22⋊C8—312no—
229C22⋊C8C2×D3111yescharacteristic, complemented
230C31#300611yescharacteristic, complemented, Sylow
231C24—311no—
232C24—621no—
233C24—621no—
234#41—318no—
235#41—624no—
236#41—624no—
237#41—628no—
238#41—628no—
239#41—624no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D3112yes—
243C2×D4—21no—
244C2×D4—312no—
245C2×D4—311no—
246C2×D4—624no—
247C2×D4—622no—
248C2×D4—622no—
249C2×D4—628no—
250C2×D4—621no—
251C2×D4—624no—
252C2×D4—621no—
253C2×D4—621no—
254C2×D4—621no—
255C2×D4—621no—
256C2×D4—1244no—
257C2×D4C22×D3111yescharacteristic
258C22×C4—21no—
259C22×C4—311no—
260C22×C4—311no—
261C22×C4—311no—
262C22×C4—311no—
263C22×C4—311no—
264C22×C4—624no—
265C22×C4—622no—
266C22×C4—622no—
267C22×C4—628no—
268C22×C4—621no—
269C22×C4C22×D3111yescharacteristic
270SD16—24no—
271SD16—624no—
272D8—624no—
273D8—624no—
274M16—624no—
275C2×C8—312no—
276C2×C8C22×D3112yes—
277C4⋊C4—311no—
278C4⋊C4—621no—
279C4⋊C4C22×D3111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—621no—
282C22⋊C4—621no—
283C22⋊C4—621no—
284C23—21no—
285C23—311no—
286C23—311no—
287C23—622no—
288C23—624no—
289C23—622no—
290C23—622no—
291C23—621no—
292C23—1241no—
293C23—1242no—
294C23—1242no—
295C23—1242no—
296C23—1242no—
297C23—1242no—
298C23D4×D3111yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D3114yescomplemented
302D4—22no—
303D4—42no—
304D4—314no—
305D4—628no—
306D4—628no—
307D4—624no—
308D4—622no—
309D4—622no—
310D4—1242no—
311D4—1242no—
312D4—1242no—
313D4—1242no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—311no—
319C2×C4—314no—
320C2×C4—311no—
321C2×C4—622no—
322C2×C4—628no—
323C2×C4—622no—
324C2×C4—628no—
325C2×C4—622no—
326C2×C4—622no—
327C2×C4—621no—
328C2×C4—1242no—
329C2×C4D4×D3111yescharacteristic
330C2×C4C23×D3111yescharacteristic
331C8—22no—
332C8—622no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—312no—
339C22—621no—
340C22—621no—
341C22—624no—
342C22—621no—
343C22—621no—
344C22—621no—
345C22—1244no—
346C22—1248no—
347C22—1242no—
348C22—1242no—
349C22—1241no—
350C22—1241no—
351C22—1241no—
352C22—2484no—
353C22#1175849112yescomplemented
354C22C2×D4×D3111yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—312no—
359C4—621no—
360C4—1241no—
361C4C2×D4×D3112yes—
362C2—22no—
363C2—81no—
364C2—622no—
365C2—1241no—
366C2—1244no—
367C2—2481no—
368C2#120×D3111yescharacteristic
369C2C2×#1175849111yescharacteristic
370C2#1452184811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#448C2
socleC22×C31C22×D4
hypercenter#448D31
hypocenterC31#3006
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#92499C2semidirect
2. C2×#11758491C2semidirect
3. #121×D31C2semidirect
4. #14521197C2semidirect
5. #14521193C2semidirect
6. #14521188C2semidirect
7. #14521185C2semidirect
8. #14521182C2semidirect
9. #14521049C2semidirect
10. #14521011C2semidirect
11. C31×#448C2semidirect
12. C22×Q8×C31C22semidirect
17. C2×#92499C22semidirect
19. C2×D4×D31C22semidirect
21. C22×D124C22semidirect
24. C2×SD16×C31C22semidirect
26. C2×#92481C22semidirect
28. #11758515C22semidirect
29. C2×#92479C22semidirect
32. C2×D248C22semidirect
33. C2×#92469C22semidirect
34. #11758472C22semidirect
35. C4⋊C4×D31C22semidirect
38. C31×#103C22semidirect
39. C31×C22⋊C8C22semidirect
40. #11758423C22semidirect
41. C22⋊(C31⋊C8)C22semidirect
42. #11758409C22semidirect
43. (C2×D31)⋊C8C22semidirect
44. #11758397C22semidirect
64. #92499D4semidirect
115. Q8×C31C2×D4semidirect
137. D124C2×D4semidirect
153. C22×C31AGL(1,ℤ/8)semidirect
183. #448D31semidirect
202. C22×Q8C2×D31semidirect
211. C2×SD16C2×D31semidirect
224. #103C2×D31semidirect
229. C22⋊C8C2×D31semidirect
230. C31#3006semidirect
301. Q8D4×D31semidirect
353. C22#11758491semidirect

Maximal subgroups

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

subgroupindex
C22×#924992
C2×#117584912
#121×D312
#145211972
#145211932
#145211882
#145211852
#145211822
#145210492
#145210112
C31×#4482
#300631