Finite Group Databasebeta

← #16854958

Subgroups — #16854958

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×#92497C211yesmaximal, characteristic, complemented
2C2×#11758488C212yesmaximal, complemented
3#121×D31C211yesmaximal, characteristic, complemented
4#14521195C212yesmaximal, complemented
5#14521191C212yesmaximal, complemented
6#14521190C211yesmaximal, characteristic, complemented
7#14521187C211yesmaximal, characteristic, complemented
8#14521183C211yesmaximal, characteristic, complemented
9#14521047C212yesmaximal, complemented
10#14521011C211yesmaximal, characteristic, complemented
11C31×#447C211yesmaximal, characteristic, complemented
12C22×D4×C31C2211yescharacteristic, complemented
13C22×C22⋊D31—21no—
14C23×Dic31—21no—
15C2×#92497—28no—
16C2×#92497—21no—
17C2×#92497—21no—
18C2×#92497C2212yes—
19C2×#92497C2212yescomplemented
20C2×D4×D31—21no—
21C2×D4×D31C2211yescharacteristic, complemented
22C22×C4×D31C2211yescharacteristic
23C22×Dic62C2211yescharacteristic, complemented
24C2×D8×C31C2212yescomplemented
25C31×#121C2211yescharacteristic
26#11758516—21no—
27#11758515C2211yescharacteristic, complemented
28C2×#92480C2212yescomplemented
29C2×#92479C2212yescomplemented
30#11758507C2211yescharacteristic
31#11758488—28no—
32C2×#92470C2212yescomplemented
33C2×#92469C2212yescomplemented
34#11758472C2211yescharacteristic, complemented
35C4⋊C4×D31C2211yescharacteristic, complemented
36#11758461—21no—
37C22⋊C4×D31—21no—
38C31×#102C2212yescomplemented
39C31×C22⋊C8C2211yescharacteristic, complemented
40#11758420C2212yescomplemented
41C22⋊(C31⋊C8)C2211yescharacteristic, complemented
42(C2×D31)⋊C8C2212yescomplemented
43#11758404C2212yescomplemented
44#11758398C2212yescomplemented
45C24×C31—21no—
46C23×D31—21no—
47C23×D31D411yescharacteristic
48C2×D4×C31—24no—
49C2×D4×C31—21no—
50C2×D4×C31—21no—
51C2×D4×C31C2311yescharacteristic
52C2×D4×C31C2312yes—
53C22×C4×C31C2311yescharacteristic
54C2×C22⋊D31—28no—
55C2×C22⋊D31—22no—
56C2×C22⋊D31—22no—
57C2×C22⋊D31—21no—
58C2×C22⋊D31—21no—
59C2×C22⋊D31—21no—
60C22×Dic31—28no—
61C22×Dic31—21no—
62C22×Dic31—22no—
63C22×Dic31—22no—
64C22×Dic31D411yescharacteristic
65#92497—28no—
66#92497—28no—
67#92497—24no—
68#92497—24no—
69#92497—24no—
70#92497D418yescomplemented
71D4×D31—24no—
72D4×D31—44no—
73C2×D124—21no—
74C2×D124C2311yescharacteristic
75C2×C4×D31—24no—
76C2×C4×D31—21no—
77C2×C4×D31D411yescharacteristic
78C2×C4×D31D411yescharacteristic
79C2×C4×D31C2311yescharacteristic
80C2×C4×D31C2311yescharacteristic
81C2×Dic62—21no—
82C2×Dic62—24no—
83C2×Dic62C2312yes—
84D8×C31—24no—
85C2×C8×C31C2312yes—
86C4⋊C4×C31C2311yescharacteristic
87C22⋊C4×C31—21no—
88C22⋊Dic31—21no—
89#92480—24no—
90#92479—24no—
91(C2×D31)⋊C4—21no—
92(C2×D31)⋊C4—21no—
93C4⋊Dic31C2311yescharacteristic
94Dic31⋊C4—21no—
95C2×C31⋊C8C2312yes—
96#92470—24no—
97#92469—24no—
98C23×C31—24no—
99C23×C31—22no—
100C23×C31—21no—
101C23×C31—42no—
102C23×C31—42no—
103C23×C31C2×D411yescharacteristic
104C22×D31—22no—
105C22×D31—22no—
106C22×D31—21no—
107C22×D31—41no—
108C22×D31—42no—
109C22×D31—42no—
110C22×D31—42no—
111C22×D31C2×D411yescharacteristic
112C22×D31C2×D411yescharacteristic
113D4×C31—24no—
114D4×C31—22no—
115D4×C31—22no—
116D4×C31—42no—
117D4×C31C2×D414yescomplemented
118C2×C4×C31—22no—
119C2×C4×C31—21no—
120C2×C4×C31C2×D411yescharacteristic
121C2×C4×C31C2411yescharacteristic
122C22⋊D31—28no—
123C22⋊D31—28no—
124C22⋊D31—42no—
125C22⋊D31—42no—
126C22⋊D31—42no—
127C2×Dic31—24no—
128C2×Dic31—28no—
129C2×Dic31—22no—
130C2×Dic31—22no—
131C2×Dic31—28no—
132C2×Dic31—21no—
133C2×Dic31C2×D411yescharacteristic
134C2×Dic31C2×D411yescharacteristic
135D124—22no—
136D124—42no—
137C4×D31—22no—
138C4×D31—22no—
139C4×D31—22no—
140C4×D31—42no—
141C4×D31C2×D414yes—
142Dic62—24no—
143Dic62—22no—
144Dic62C2×D414yescomplemented
145C8×C31—22no—
146C31⋊C8—22no—
147#3006—311nomaximal, Sylow
148C22×C31—21no—
149C22×C31—21no—
150C22×C31—24no—
151C22×C31—48no—
152C22×C31—42no—
153C22×C31—42no—
154C22×C31—41no—
155C22×C31—41no—
156C22×C31—41no—
157C22×C31AGL(1,ℤ/8)12yescomplemented
158C22×C31C22×D411yescharacteristic
159C2×D31—21no—
160C2×D31—21no—
161C2×D31—21no—
162C2×D31—21no—
163C2×D31—21no—
164C2×D31—44no—
165C2×D31—41no—
166C2×D31—41no—
167C2×D31—41no—
168C2×D31—84no—
169C2×D31#12012yes—
170C4×C31—21no—
171C4×C31—41no—
172C4×C31C22×D412yes—
173Dic31—24no—
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#448—311no—
184#447—311no—
185#447D3111yescharacteristic, complemented
186#417—312no—
187#407—311no—
188C2×C31—22no—
189C2×C31—44no—
190C2×C31—81no—
191C2×C31C2×#12011yescharacteristic
192C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
193C2×C31#57511yescharacteristic
194D31—22no—
195D31—41no—
196D31—81no—
197C2×#41—312no—
198C2×#41—312no—
199C2×#41—621no—
200C2×#41—621no—
201C2×#41—628no—
202C22×Q8—311no—
203C22×D4—311no—
204C22×D4—621no—
205C22×D4—621no—
206C22×D4C2×D3111yescharacteristic, complemented
207C23×C4—311no—
208C23×C4—621no—
209AGL(1,ℤ/8)—628no—
210C2×SD16—312no—
211C2×SD16—312no—
212C2×D8—312no—
213C2×D8C2×D3112yescomplemented
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#103—312no—
225#102—312no—
226#102C2×D3112yescomplemented
227C22⋊C8—311no—
228C22⋊C8—312no—
229C22⋊C8C2×D3111yescharacteristic, complemented
230C31#300611yescharacteristic, complemented, Sylow
231C24—21no—
232C24—311no—
233C24—621no—
234#41—318no—
235#41—624no—
236#41—624no—
237#41—628no—
238#41—628no—
239#41—624no—
240C2×Q8—312no—
241C2×Q8—624no—
242C2×Q8—621no—
243C2×D4—24no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—311no—
247C2×D4—621no—
248C2×D4—624no—
249C2×D4—621no—
250C2×D4—622no—
251C2×D4—622no—
252C2×D4—628no—
253C2×D4—621no—
254C2×D4—621no—
255C2×D4—1244no—
256C2×D4C22×D3111yescharacteristic
257C2×D4C22×D3112yes—
258C22×C4—311no—
259C22×C4—311no—
260C22×C4—311no—
261C22×C4—311no—
262C22×C4—311no—
263C22×C4—624no—
264C22×C4—621no—
265C22×C4—622no—
266C22×C4—622no—
267C22×C4—628no—
268C22×C4—621no—
269C22×C4C22×D3111yescharacteristic
270SD16—624no—
271SD16—624no—
272D8—24no—
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—24no—
285C23—22no—
286C23—21no—
287C23—42no—
288C23—42no—
289C23—311no—
290C23—311no—
291C23—622no—
292C23—622no—
293C23—621no—
294C23—1241no—
295C23—1242no—
296C23—1242no—
297C23—1242no—
298C23D4×D3111yescharacteristic
299Q8—314no—
300Q8—624no—
301Q8—622no—
302D4—24no—
303D4—22no—
304D4—22no—
305D4—42no—
306D4—628no—
307D4—628no—
308D4—622no—
309D4—1242no—
310D4—1242no—
311D4—1242no—
312D4—1242no—
313D4D4×D3114yescomplemented
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—311no—
317C2×C4—314no—
318C2×C4—311no—
319C2×C4—622no—
320C2×C4—622no—
321C2×C4—622no—
322C2×C4—628no—
323C2×C4—624no—
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—24no—
336C22—48no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—312no—
343C22—621no—
344C22—621no—
345C22—621no—
346C22—621no—
347C22—621no—
348C22—1244no—
349C22—1241no—
350C22—1241no—
351C22—1241no—
352C22—2484no—
353C22#1175848812yescomplemented
354C22C2×D4×D3111yescharacteristic
355C4—21no—
356C4—41no—
357C4—312no—
358C4—621no—
359C4—624no—
360C4—1241no—
361C4C2×D4×D3112yes—
362C2—22no—
363C2—44no—
364C2—81no—
365C2—622no—
366C2—1241no—
367C2—2481no—
368C2#120×D3111yescharacteristic
369C2C2×#1175848811yescharacteristic
370C2#1452184511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#447C2
socleC22×C31C22×D4
hypercenter#447D31
hypocenterC31#3006
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×#92497C2semidirect
2. C2×#11758488C2semidirect
3. #121×D31C2semidirect
4. #14521195C2semidirect
5. #14521191C2semidirect
6. #14521190C2semidirect
7. #14521187C2semidirect
8. #14521183C2semidirect
9. #14521047C2semidirect
10. #14521011C2semidirect
11. C31×#447C2semidirect
12. C22×D4×C31C22semidirect
19. C2×#92497C22semidirect
21. C2×D4×D31C22semidirect
23. C22×Dic62C22semidirect
24. C2×D8×C31C22semidirect
27. #11758515C22semidirect
28. C2×#92480C22semidirect
29. C2×#92479C22semidirect
32. C2×#92470C22semidirect
33. C2×#92469C22semidirect
34. #11758472C22semidirect
35. C4⋊C4×D31C22semidirect
38. C31×#102C22semidirect
39. C31×C22⋊C8C22semidirect
40. #11758420C22semidirect
41. C22⋊(C31⋊C8)C22semidirect
42. (C2×D31)⋊C8C22semidirect
43. #11758404C22semidirect
44. #11758398C22semidirect
70. #92497D4semidirect
117. D4×C31C2×D4semidirect
144. Dic62C2×D4semidirect
157. C22×C31AGL(1,ℤ/8)semidirect
185. #447D31semidirect
206. C22×D4C2×D31semidirect
213. C2×D8C2×D31semidirect
226. #102C2×D31semidirect
229. C22⋊C8C2×D31semidirect
230. C31#3006semidirect
313. D4D4×D31semidirect
353. C22#11758488semidirect

Maximal subgroups

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

subgroupindex
C22×#924972
C2×#117584882
#121×D312
#145211952
#145211912
#145211902
#145211872
#145211832
#145210472
#145210112
C31×#4472
#300631