Finite Group Databasebeta

← D31×#447 #16854884

Subgroups — D31×#447 #16854884

Order 3968.

1048 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 155 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×D4×D31C211yesmaximal, characteristic, complemented
2C2×D8×D31C212yesmaximal, complemented
3#121×D31C211yesmaximal, characteristic, complemented
4#14521185C211yesmaximal, characteristic, complemented
5#14521184C212yesmaximal, complemented
6#14521183C211yesmaximal, characteristic, complemented
7#14521182C211yesmaximal, characteristic, complemented
8#14521181C212yesmaximal, complemented
9#102×D31C212yesmaximal, complemented
10C22⋊C8×D31C211yesmaximal, characteristic, complemented
11C31×#447C211yesmaximal, characteristic, complemented
12C24×D31—21no—
13C22×D4×C31C2211yescharacteristic, complemented
14C22×C22⋊D31—21no—
15C2×D4×D31—28no—
16C2×D4×D31—21no—
17C2×D4×D31—21no—
18C2×D4×D31—21no—
19C2×D4×D31C2212yes—
20C2×D4×D31C2212yescomplemented
21C2×D4×D31C2211yescharacteristic, complemented
22C22×D124C2211yescharacteristic, complemented
23C22×C4×D31C2211yescharacteristic
24C2×D8×C31C2212yescomplemented
25C31×#121C2211yescharacteristic, complemented
26#11758516—21no—
27#11758515C2211yescharacteristic, complemented
28C2×#92479C2212yescomplemented
29C2×#92479C2212yescomplemented
30#11758507C2211yescharacteristic, complemented
31D8×D31—28no—
32C2×D248C2212yescomplemented
33C2×C8×D31C2212yescomplemented
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#11758409C2212yescomplemented
43(C2×D31)⋊C8C2212yescomplemented
44#11758397C2212yescomplemented
45C24×C31—21no—
46C23×D31—28no—
47C23×D31—21no—
48C23×D31—22no—
49C23×D31—22no—
50C23×D31—21no—
51C23×D31—44no—
52C23×D31—44no—
53C23×D31D411yescharacteristic
54C2×D4×C31—24no—
55C2×D4×C31—21no—
56C2×D4×C31—21no—
57C2×D4×C31C2312yes—
58C2×D4×C31C2311yescharacteristic, complemented
59C22×C4×C31C2311yescharacteristic
60C2×C22⋊D31—28no—
61C2×C22⋊D31—21no—
62C2×C22⋊D31—21no—
63C2×C22⋊D31—22no—
64C2×C22⋊D31—22no—
65C2×C22⋊D31—21no—
66C22×Dic31D411yescharacteristic
67D4×D31—28no—
68D4×D31—28no—
69D4×D31—24no—
70D4×D31—24no—
71D4×D31—24no—
72D4×D31—24no—
73D4×D31—44no—
74D4×D31D418yescomplemented
75C2×D124—21no—
76C2×D124—24no—
77C2×D124—21no—
78C2×D124C2312yes—
79C2×D124C2311yescharacteristic, complemented
80C2×C4×D31—24no—
81C2×C4×D31—21no—
82C2×C4×D31D411yescharacteristic
83C2×C4×D31D411yescharacteristic
84C2×C4×D31C2311yescharacteristic
85C2×C4×D31C2311yescharacteristic
86D8×C31—24no—
87C2×C8×C31C2312yescomplemented
88C4⋊C4×C31C2311yescharacteristic, complemented
89C22⋊C4×C31—21no—
90C22⋊Dic31—21no—
91#92479—24no—
92#92479—24no—
93(C2×D31)⋊C4—21no—
94(C2×D31)⋊C4—21no—
95C4⋊Dic31C2311yescharacteristic, complemented
96Dic31⋊C4—21no—
97C2×C31⋊C8C2312yescomplemented
98D248—24no—
99C8×D31—24no—
100C23×C31—24no—
101C23×C31—22no—
102C23×C31—21no—
103C23×C31—42no—
104C23×C31—42no—
105C23×C31C2×D411yescharacteristic
106C22×D31—22no—
107C22×D31—22no—
108C22×D31—24no—
109C22×D31—28no—
110C22×D31—22no—
111C22×D31—28no—
112C22×D31—21no—
113C22×D31—44no—
114C22×D31—432no—
115C22×D31—42no—
116C22×D31—42no—
117C22×D31—44no—
118C22×D31—44no—
119C22×D31—44no—
120C22×D31—42no—
121C22×D31—42no—
122C22×D31—42no—
123C22×D31D814yescomplemented
124C22×D31C2×D411yescharacteristic
125C22×D31C2×D411yescharacteristic
126D4×C31—24no—
127D4×C31—22no—
128D4×C31—22no—
129D4×C31—42no—
130D4×C31C2×D414yescomplemented
131C2×C4×C31—22no—
132C2×C4×C31—21no—
133C2×C4×C31C2×D411yescharacteristic
134C2×C4×C31C2411yescharacteristic
135C22⋊D31—28no—
136C22⋊D31—28no—
137C22⋊D31—42no—
138C22⋊D31—42no—
139C22⋊D31—42no—
140C2×Dic31—22no—
141C2×Dic31—21no—
142C2×Dic31C2×D411yescharacteristic
143C2×Dic31C2×D411yescharacteristic
144D124—24no—
145D124—22no—
146D124—22no—
147D124—42no—
148D124C2×D414yescomplemented
149C4×D31—22no—
150C4×D31—22no—
151C4×D31—22no—
152C4×D31—42no—
153C4×D31C2×D414yes—
154C8×C31—22no—
155C31⋊C8—22no—
156C2×#447—311nomaximal, Sylow
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—24no—
160C22×C31—48no—
161C22×C31—42no—
162C22×C31—42no—
163C22×C31—41no—
164C22×C31—41no—
165C22×C31—41no—
166C22×C31C2×D812yescomplemented
167C22×C31C22×D411yescharacteristic
168C2×D31—21no—
169C2×D31—21no—
170C2×D31—28no—
171C2×D31—24no—
172C2×D31—416no—
173C2×D31—48no—
174C2×D31—42no—
175C2×D31—42no—
176C2×D31—416no—
177C2×D31—41no—
178C2×D31—41no—
179C2×D31—41no—
180C2×D31—84no—
181C2×D31#12012yes—
182C2×D31C2×D812yes—
183C2×D31C2×D812yescomplemented
184C2×D31AGL(1,ℤ/8)12yes—
185C4×C31—21no—
186C4×C31—41no—
187C4×C31C22×D412yes—
188Dic31—21no—
189Dic31—41no—
190Dic31#12012yes—
191C23×D4—311no—
192C22×D8—312no—
193C2×#121—311no—
194#447—311no—
195#447—312no—
196#447—311no—
197#447—312no—
198#447—311no—
199#447D3111yescharacteristic, direct product factor
200C2×#102—312no—
201C2×C22⋊C8—311no—
202C2×C31—22no—
203C2×C31—44no—
204C2×C31—81no—
205C2×C31C2×#12011yescharacteristic
206C2×C31C22×D811yescharacteristic
207C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
208D31—22no—
209D31—44no—
210D31—81no—
211D31#44714yesdirect product factor
212C25—621no—
213C22×D4—312no—
214C22×D4—312no—
215C22×D4—311no—
216C22×D4—311no—
217C22×D4—621no—
218C22×D4—621no—
219C22×D4—621no—
220C22×D4—628no—
221C22×D4—621no—
222C22×D4C2×D3111yescharacteristic, complemented
223C23×C4—311no—
224C2×D8—312no—
225C2×D8—312no—
226C2×D8—312no—
227C2×D8—628no—
228C2×D8C2×D3112yescomplemented
229C22×C8—312no—
230#121—311no—
231#121—311no—
232#121—311no—
233#121—621no—
234#121—621no—
235#121C2×D3111yescharacteristic, complemented
236C2×C4⋊C4—311no—
237C2×C22⋊C4—621no—
238#102—312no—
239#102—312no—
240#102—312no—
241#102C2×D3112yescomplemented
242C22⋊C8—311no—
243C22⋊C8—312no—
244C22⋊C8C2×D3111yescharacteristic, complemented
245C31C2×#44711yescharacteristic, complemented, Sylow
246C24—21no—
247C24—311no—
248C24—622no—
249C24—622no—
250C24—621no—
251C24—628no—
252C24—621no—
253C24—1244no—
254C24—1244no—
255C2×D4—24no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—318no—
259C2×D4—312no—
260C2×D4—311no—
261C2×D4—624no—
262C2×D4—621no—
263C2×D4—624no—
264C2×D4—624no—
265C2×D4—622no—
266C2×D4—622no—
267C2×D4—621no—
268C2×D4—624no—
269C2×D4—628no—
270C2×D4—628no—
271C2×D4—628no—
272C2×D4—624no—
273C2×D4—621no—
274C2×D4—621no—
275C2×D4—621no—
276C2×D4—1244no—
277C2×D4C22×D3112yes—
278C2×D4C22×D3111yescharacteristic, complemented
279C22×C4—311no—
280C22×C4—311no—
281C22×C4—311no—
282C22×C4—311no—
283C22×C4—311no—
284C22×C4—624no—
285C22×C4—621no—
286C22×C4C22×D3111yescharacteristic
287D8—24no—
288D8—624no—
289D8—624no—
290D8—624no—
291C2×C8—312no—
292C2×C8—624no—
293C2×C8C22×D3112yescomplemented
294C4⋊C4—311no—
295C4⋊C4—621no—
296C4⋊C4C22×D3111yescharacteristic, complemented
297C22⋊C4—21no—
298C22⋊C4—621no—
299C22⋊C4—621no—
300C22⋊C4—621no—
301C23—24no—
302C23—22no—
303C23—21no—
304C23—42no—
305C23—42no—
306C23—311no—
307C23—314no—
308C23—311no—
309C23—622no—
310C23—622no—
311C23—628no—
312C23—624no—
313C23—628no—
314C23—622no—
315C23—621no—
316C23—1242no—
317C23—1244no—
318C23—1244no—
319C23—1244no—
320C23—1242no—
321C23—1242no—
322C23—1242no—
323C23—12432no—
324C23—1244no—
325C23—1242no—
326C23D4×D3111yescharacteristic
327D4—24no—
328D4—22no—
329D4—22no—
330D4—42no—
331D4—314no—
332D4—624no—
333D4—628no—
334D4—628no—
335D4—622no—
336D4—622no—
337D4—1242no—
338D4—1242no—
339D4—1242no—
340D4—1242no—
341D4D4×D3114yescomplemented
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—311no—
345C2×C4—314no—
346C2×C4—311no—
347C2×C4—622no—
348C2×C4—622no—
349C2×C4—622no—
350C2×C4—622no—
351C2×C4—621no—
352C2×C4—1242no—
353C2×C4D4×D3111yescharacteristic
354C2×C4C23×D3111yescharacteristic
355C8—22no—
356C8—622no—
357C22—21no—
358C22—21no—
359C22—24no—
360C22—48no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—312no—
367C22—312no—
368C22—312no—
369C22—312no—
370C22—628no—
371C22—621no—
372C22—621no—
373C22—624no—
374C22—1248no—
375C22—12416no—
376C22—12416no—
377C22—1242no—
378C22—1242no—
379C22—1241no—
380C22—1241no—
381C22—1241no—
382C22—2484no—
383C22D8×D3112yescomplemented
384C22C2×D4×D3111yescharacteristic
385C4—21no—
386C4—41no—
387C4—312no—
388C4—621no—
389C4—1241no—
390C4C2×D4×D3112yes—
391C2—22no—
392C2—44no—
393C2—81no—
394C2—314no—
395C2—622no—
396C2—1244no—
397C2—2481no—
398C2#120×D3111yescharacteristic
399C2C2×D8×D3111yescharacteristic
400C2AGL(1,ℤ/8)×D3111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#447C2
socleC22×C31C22×D4
hypercenter#447D31
hypocenterC31C2×#447
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×D4×D31C2semidirect
2. C2×D8×D31C2semidirect
3. #121×D31C2semidirect
4. #14521185C2semidirect
5. #14521184C2semidirect
6. #14521183C2semidirect
7. #14521182C2semidirect
8. #14521181C2semidirect
9. #102×D31C2semidirect
10. C22⋊C8×D31C2semidirect
11. C31×#447C2semidirect
13. C22×D4×C31C22semidirect
20. C2×D4×D31C22semidirect
21. C2×D4×D31C22semidirect
22. C22×D124C22semidirect
24. C2×D8×C31C22semidirect
25. C31×#121C22semidirect
27. #11758515C22semidirect
28. C2×#92479C22semidirect
29. C2×#92479C22semidirect
30. #11758507C22semidirect
32. C2×D248C22semidirect
33. C2×C8×D31C22semidirect
34. #11758472C22semidirect
35. C4⋊C4×D31C22semidirect
38. C31×#102C22semidirect
39. C31×C22⋊C8C22semidirect
40. #11758420C22semidirect
41. C22⋊(C31⋊C8)C22semidirect
42. #11758409C22semidirect
43. (C2×D31)⋊C8C22semidirect
44. #11758397C22semidirect
58. C2×D4×C31C23semidirect
74. D4×D31D4semidirect
79. C2×D124C23semidirect
87. C2×C8×C31C23semidirect
88. C4⋊C4×C31C23semidirect
95. C4⋊Dic31C23semidirect
97. C2×C31⋊C8C23semidirect
123. C22×D31D8semidirect
130. D4×C31C2×D4semidirect
148. D124C2×D4semidirect
166. C22×C31C2×D8semidirect
183. C2×D31C2×D8semidirect
199. #447D31direct (N × H)
211. D31#447direct (N × H)
222. C22×D4C2×D31semidirect
228. C2×D8C2×D31semidirect
235. #121C2×D31semidirect
241. #102C2×D31semidirect
244. C22⋊C8C2×D31semidirect
245. C31C2×#447semidirect
278. C2×D4C22×D31semidirect
293. C2×C8C22×D31semidirect
296. C4⋊C4C22×D31semidirect
341. D4D4×D31semidirect
383. C22D8×D31semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D312
C2×D8×D312
#121×D312
#145211852
#145211842
#145211832
#145211822
#145211812
#102×D312
C22⋊C8×D312
C31×#4472
C2×#44731