Finite Group Databasebeta

← D31×#450 #16854890

Subgroups — D31×#450 #16854890

Order 3968.

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

normal subgroups only

This group has 996 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
2C2×SD16×D31C212yesmaximal, complemented
3C22⋊Q8×D31C211yesmaximal, characteristic, complemented
4#14521202C211yesmaximal, characteristic, complemented
5#14521201C212yesmaximal, complemented
6#14521200C211yesmaximal, characteristic, complemented
7#14521199C211yesmaximal, characteristic, complemented
8#14521198C212yesmaximal, complemented
9#102×D31C212yesmaximal, complemented
10C22⋊C8×D31C211yesmaximal, characteristic, complemented
11C31×#450C211yesmaximal, characteristic, complemented
12C24×D31—21no—
13C22×D4×C31C2211yescharacteristic
14C22×C22⋊D31—21no—
15C2×Q8×D31C2211yescharacteristic, complemented
16C2×D4×D31—28no—
17C2×D4×D31—21no—
18C2×D4×D31—21no—
19C2×D4×D31C2212yes—
20C2×D4×D31C2212yes—
21C22×D124C2211yescharacteristic
22C22×C4×D31C2211yescharacteristic
23C2×SD16×C31C2212yescomplemented
24C31×C22⋊Q8C2211yescharacteristic, complemented
25#11758523C2211yescharacteristic, complemented
26C2×#92481C2212yescomplemented
27C2×#92480C2212yescomplemented
28#11758501C2211yescharacteristic, complemented
29SD16×D31—28no—
30C2×#92470C2212yescomplemented
31C2×C8×D31C2212yescomplemented
32#11758474C2211yescharacteristic, complemented
33#11758473—21no—
34C4⋊C4×D31—21no—
35C4⋊C4×D31C2211yescharacteristic, complemented
36C22⋊C4×D31—21no—
37#11758455—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—44no—
51C23×D31—44no—
52C23×D31D411yescharacteristic
53C2×Q8×C31C2311yescharacteristic, complemented
54C2×D4×C31—24no—
55C2×D4×C31—21no—
56C2×D4×C31C2312yes—
57C22×C4×C31C2311yescharacteristic
58C2×C22⋊D31—28no—
59C2×C22⋊D31—22no—
60C2×C22⋊D31—22no—
61C22×Dic31D411yescharacteristic
62Q8×D31—24no—
63D4×D31—28no—
64D4×D31—28no—
65D4×D31—24no—
66D4×D31—24no—
67D4×D31—24no—
68D4×D31D418yes—
69C2×D124—21no—
70C2×D124—24no—
71C2×D124C2312yes—
72C2×C4×D31—24no—
73C2×C4×D31—21no—
74C2×C4×D31—21no—
75C2×C4×D31D411yescharacteristic
76C2×C4×D31D411yescharacteristic
77C2×C4×D31C2311yescharacteristic
78C2×C4×D31C2311yescharacteristic
79C2×Dic62—21no—
80C2×Dic62C2311yescharacteristic, complemented
81SD16×C31—24no—
82C2×C8×C31C2312yescomplemented
83C4⋊C4×C31—21no—
84C4⋊C4×C31C2311yescharacteristic, complemented
85C22⋊C4×C31—21no—
86C22⋊Dic31—21no—
87#92481—24no—
88#92480—24no—
89(C2×D31)⋊C4—21no—
90(C2×D31)⋊C4—21no—
91C4⋊Dic31—21no—
92C4⋊Dic31C2311yescharacteristic, complemented
93Dic31⋊C4—21no—
94Dic31⋊C4—21no—
95Dic31⋊C4—21no—
96C2×C31⋊C8C2312yescomplemented
97#92470—24no—
98C8×D31—24no—
99C23×C31—24no—
100C23×C31—22no—
101C23×C31—42no—
102C23×C31—42no—
103C23×C31C2×D411yescharacteristic
104C22×D31—22no—
105C22×D31—22no—
106C22×D31—24no—
107C22×D31—28no—
108C22×D31—22no—
109C22×D31—28no—
110C22×D31—44no—
111C22×D31—432no—
112C22×D31—42no—
113C22×D31—42no—
114C22×D31—44no—
115C22×D31—44no—
116C22×D31—44no—
117C22×D31SD1614yescomplemented
118C22×D31C2×D411yescharacteristic
119C22×D31C2×D411yescharacteristic
120Q8×C31—22no—
121D4×C31—24no—
122D4×C31—22no—
123D4×C31C2×D414yes—
124C2×C4×C31—22no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31C2×D411yescharacteristic
128C2×C4×C31C2411yescharacteristic
129C22⋊D31—28no—
130C22⋊D31—28no—
131C2×Dic31—22no—
132C2×Dic31—21no—
133C2×Dic31—21no—
134C2×Dic31C2×D411yescharacteristic
135C2×Dic31C2×D411yescharacteristic
136D124—24no—
137D124—22no—
138D124C2×D414yes—
139C4×D31—22no—
140C4×D31—22no—
141C4×D31—22no—
142C4×D31—42no—
143C4×D31—42no—
144C4×D31C2×D414yes—
145Dic62—22no—
146Dic62—42no—
147C8×C31—22no—
148C31⋊C8—22no—
149C2×#450—311nomaximal, Sylow
150C22×C31—21no—
151C22×C31—21no—
152C22×C31—24no—
153C22×C31—48no—
154C22×C31—42no—
155C22×C31—42no—
156C22×C31C2×SD1612yescomplemented
157C22×C31C22×D411yescharacteristic
158C2×D31—21no—
159C2×D31—21no—
160C2×D31—28no—
161C2×D31—24no—
162C2×D31—416no—
163C2×D31—48no—
164C2×D31—42no—
165C2×D31—42no—
166C2×D31—416no—
167C2×D31#12012yes—
168C2×D31C2×SD1612yes—
169C2×D31C2×SD1612yescomplemented
170C2×D31AGL(1,ℤ/8)12yes—
171C4×C31—21no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31C22×D412yes—
175Dic31—21no—
176Dic31—41no—
177Dic31—41no—
178Dic31#12012yes—
179C23×D4—311no—
180C22×SD16—312no—
181C2×C22⋊Q8—311no—
182#450—311no—
183#450—312no—
184#450—311no—
185#450—312no—
186#450—311no—
187#450D3111yescharacteristic, direct product factor
188C2×#102—312no—
189C2×C22⋊C8—311no—
190C2×C31—22no—
191C2×C31—44no—
192C2×C31C2×#12011yescharacteristic
193C2×C31C22×SD1611yescharacteristic
194C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
195D31—22no—
196D31—44no—
197D31#45014yesdirect product factor
198C25—621no—
199C22×Q8—311no—
200C22×D4—312no—
201C22×D4—312no—
202C22×D4—311no—
203C22×D4—621no—
204C22×D4—621no—
205C22×D4—621no—
206C22×D4—628no—
207C22×D4C2×D3111yescharacteristic
208C23×C4—311no—
209C2×SD16—312no—
210C2×SD16—312no—
211C2×SD16—312no—
212C2×SD16—628no—
213C2×SD16C2×D3112yescomplemented
214C22×C8—312no—
215C22⋊Q8—311no—
216C22⋊Q8—311no—
217C22⋊Q8—311no—
218C22⋊Q8—621no—
219C22⋊Q8—621no—
220C22⋊Q8C2×D3111yescharacteristic, complemented
221C2×C4⋊C4—311no—
222C2×C4⋊C4—621no—
223C2×C22⋊C4—621no—
224#102—312no—
225#102—312no—
226#102—312no—
227#102C2×D3112yescomplemented
228C22⋊C8—311no—
229C22⋊C8—312no—
230C22⋊C8C2×D3111yescharacteristic, complemented
231C31C2×#45011yescharacteristic, complemented, Sylow
232C24—21no—
233C24—311no—
234C24—622no—
235C24—622no—
236C24—628no—
237C24—621no—
238C24—1244no—
239C24—1244no—
240C2×Q8—311no—
241C2×Q8—621no—
242C2×Q8—624no—
243C2×Q8C22×D3111yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—318no—
247C2×D4—312no—
248C2×D4—624no—
249C2×D4—624no—
250C2×D4—622no—
251C2×D4—622no—
252C2×D4—624no—
253C2×D4—628no—
254C2×D4—628no—
255C2×D4—628no—
256C2×D4—624no—
257C2×D4—621no—
258C2×D4C22×D3112yes—
259C22×C4—311no—
260C22×C4—311no—
261C22×C4—311no—
262C22×C4—311no—
263C22×C4—311no—
264C22×C4—624no—
265C22×C4—621no—
266C22×C4—621no—
267C22×C4C22×D3111yescharacteristic
268SD16—24no—
269SD16—624no—
270SD16—624no—
271SD16—624no—
272C2×C8—312no—
273C2×C8—624no—
274C2×C8C22×D3112yescomplemented
275C4⋊C4—21no—
276C4⋊C4—311no—
277C4⋊C4—621no—
278C4⋊C4—621no—
279C4⋊C4—621no—
280C4⋊C4—621no—
281C4⋊C4C22×D3111yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—621no—
284C22⋊C4—621no—
285C22⋊C4—621no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—311no—
291C23—314no—
292C23—311no—
293C23—622no—
294C23—622no—
295C23—628no—
296C23—624no—
297C23—628no—
298C23—622no—
299C23—1242no—
300C23—1244no—
301C23—1244no—
302C23—1244no—
303C23—12432no—
304C23—1244no—
305C23—1242no—
306C23D4×D3111yescharacteristic
307Q8—22no—
308Q8—622no—
309Q8—1242no—
310D4—24no—
311D4—22no—
312D4—314no—
313D4—624no—
314D4—628no—
315D4—628no—
316D4—622no—
317D4D4×D3114yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—311no—
322C2×C4—314no—
323C2×C4—311no—
324C2×C4—622no—
325C2×C4—622no—
326C2×C4—622no—
327C2×C4—622no—
328C2×C4—621no—
329C2×C4—621no—
330C2×C4—1242no—
331C2×C4—1242no—
332C2×C4D4×D3111yescharacteristic
333C2×C4C23×D3111yescharacteristic
334C8—22no—
335C8—622no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—312no—
343C22—312no—
344C22—312no—
345C22—312no—
346C22—628no—
347C22—621no—
348C22—621no—
349C22—624no—
350C22—1248no—
351C22—12416no—
352C22—12416no—
353C22—1242no—
354C22—1242no—
355C22SD16×D3112yescomplemented
356C22C2×D4×D3111yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—312no—
361C4—621no—
362C4—1241no—
363C4—1241no—
364C4C2×D4×D3112yes—
365C2—22no—
366C2—44no—
367C2—314no—
368C2—622no—
369C2—1244no—
370C2#120×D3111yescharacteristic
371C2C2×SD16×D3111yescharacteristic
372C2AGL(1,ℤ/8)×D3111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#450C2
socleC22×C31C22×D4
prime radicalC22×D4×D31C2
hypercenter#450D31
hypocenterC31C2×#450
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D31C2semidirect
3. C22⋊Q8×D31C2semidirect
4. #14521202C2semidirect
5. #14521201C2semidirect
6. #14521200C2semidirect
7. #14521199C2semidirect
8. #14521198C2semidirect
9. #102×D31C2semidirect
10. C22⋊C8×D31C2semidirect
11. C31×#450C2semidirect
15. C2×Q8×D31C22semidirect
23. C2×SD16×C31C22semidirect
24. C31×C22⋊Q8C22semidirect
25. #11758523C22semidirect
26. C2×#92481C22semidirect
27. C2×#92480C22semidirect
28. #11758501C22semidirect
30. C2×#92470C22semidirect
31. C2×C8×D31C22semidirect
32. #11758474C22semidirect
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
53. C2×Q8×C31C23semidirect
80. C2×Dic62C23semidirect
82. C2×C8×C31C23semidirect
84. C4⋊C4×C31C23semidirect
92. C4⋊Dic31C23semidirect
96. C2×C31⋊C8C23semidirect
117. C22×D31SD16semidirect
156. C22×C31C2×SD16semidirect
169. C2×D31C2×SD16semidirect
187. #450D31direct (N × H)
197. D31#450direct (N × H)
213. C2×SD16C2×D31semidirect
220. C22⋊Q8C2×D31semidirect
227. #102C2×D31semidirect
230. C22⋊C8C2×D31semidirect
231. C31C2×#450semidirect
243. C2×Q8C22×D31semidirect
274. C2×C8C22×D31semidirect
281. C4⋊C4C22×D31semidirect
355. C22SD16×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×SD16×D312
C22⋊Q8×D312
#145212022
#145212012
#145212002
#145211992
#145211982
#102×D312
C22⋊C8×D312
C31×#4502
C2×#45031