Finite Group Databasebeta

← D31×#454 #16855044

Subgroups — D31×#454 #16855044

Order 3968.

728 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 135 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
12−1+4×D31C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D31C212yesmaximal, complemented
3#124×D31C211yesmaximal, characteristic, complemented
4#14521224C211yesmaximal, characteristic, complemented
5#14521223C212yesmaximal, complemented
6#14521222C211yesmaximal, characteristic, complemented
7#14521221C211yesmaximal, characteristic, complemented
8#14521220C212yesmaximal, complemented
9#104×D31C212yesmaximal, complemented
10#101×D31C211yesmaximal, characteristic, complemented
11C31×#454C211yesmaximal, characteristic, complemented
12C31×2−1+4C2211yescharacteristic, complemented
13#11758567—22no—
14#11758567—22no—
15#11758567C2212yescomplemented
16#41×D31—22no—
17#41×D31—22no—
18#41×D31C2212yes—
19#11758564—22no—
20#11758564C2211yescharacteristic, complemented
21C2×Q8×D31—22no—
22C2×Q8×D31C2211yescharacteristic
23C2×D4×D31C2211yescharacteristic
24C31×AGL(1,ℤ/8)C2212yescomplemented
25C31×#124C2211yescharacteristic, complemented
26#11758526C2212yescomplemented
27#11758524C2211yescharacteristic, complemented
28#11758513C2211yescharacteristic, complemented
29#11758509C2212yescomplemented
30#11758491—22no—
31SD16×D31—22no—
32#11758488—22no—
33D8×D31—22no—
34#11758485C2212yescomplemented
35M16×D31C2212yes—
36#11758462—22no—
37C22⋊C4×D31—22no—
38#11758452C2211yescharacteristic, complemented
39C42×D31C2211yescharacteristic
40C31×#104C2212yescomplemented
41C31×#101C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758424C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758412C2212yescomplemented
46#11758393C2212yescomplemented
47C23×D31—21no—
48#41×C31—22no—
49#41×C31—22no—
50#41×C31C2312yes—
51C2×Q8×C31—22no—
52C2×Q8×C31C2311yescharacteristic
53C2×D4×C31C2311yescharacteristic
54C2×C22⋊D31—21no—
55#92499—22no—
56#92499—22no—
57#92499D412yes—
58Q8×D31—22no—
59Q8×D31—24no—
60Q8×D31—22no—
61Q8×D31—22no—
62Q8×D31—22no—
63Q8×D31—22no—
64Q8×D31D412yes—
65#92497—22no—
66#92497—24no—
67#92497—22no—
68#92497—24no—
69#92497—22no—
70#92497D412yes—
71D4×D31—22no—
72D4×D31—22no—
73D4×D31—22no—
74D4×D31—22no—
75D4×D31D412yes—
76#92495—24no—
77#92495—24no—
78#92495—22no—
79#92495—22no—
80#92495—22no—
81#92495C2312yes—
82C2×D124C2311yescharacteristic
83C2×C4×D31—24no—
84C2×C4×D31—22no—
85C2×C4×D31—21no—
86C2×C4×D31D412yes—
87C2×C4×D31C2311yescharacteristic
88C2×Dic62—24no—
89C2×Dic62—22no—
90C2×Dic62D412yes—
91C2×Dic62C2311yescharacteristic
92SD16×C31—22no—
93D8×C31—22no—
94M16×C31C2312yes—
95C22⋊C4×C31—22no—
96C42×C31C2311yescharacteristic
97C22⋊Dic31—22no—
98#92481—22no—
99#92480—22no—
100#92479—22no—
101#92479—22no—
102(C2×D31)⋊C4—22no—
103(C2×D31)⋊C4—22no—
104C4×Dic31—21no—
105C4×Dic31C2311yescharacteristic
106#92474C2312yes—
107D248—22no—
108#92470—22no—
109#92469—22no—
110C8×D31—22no—
111C23×C31—21no—
112C22×D31—22no—
113C22×D31—21no—
114C22×D31—44no—
115C22×D31—41no—
116C22×D31—41no—
117C22×D31C2×D411yescharacteristic
118Q8×C31—22no—
119Q8×C31—22no—
120Q8×C31C2×D412yes—
121D4×C31—22no—
122D4×C31—22no—
123D4×C31—22no—
124D4×C31C2×D412yes—
125C2×C4×C31—24no—
126C2×C4×C31—22no—
127C2×C4×C31—21no—
128C2×C4×C31C2×D412yes—
129C2×C4×C31C2411yescharacteristic
130C22⋊D31—24no—
131C22⋊D31—22no—
132C22⋊D31—22no—
133C22⋊D31—22no—
134C22⋊D31—42no—
135C2×Dic31—24no—
136C2×Dic31—22no—
137C2×Dic31—21no—
138C2×Dic31C2×D412yes—
139C2×Dic31C2×D411yescharacteristic
140D124—22no—
141D124—22no—
142D124—22no—
143D124C2×D412yes—
144C4×D31—24no—
145C4×D31—24no—
146C4×D31—22no—
147C4×D31—22no—
148C4×D31—22no—
149C4×D31—22no—
150C4×D31—22no—
151C4×D31—42no—
152C4×D31C2×D412yes—
153C4×D31C2×D412yes—
154Dic62—24no—
155Dic62—22no—
156Dic62—22no—
157Dic62—22no—
158Dic62—24no—
159Dic62C2×D412yes—
160C8×C31—22no—
161C31⋊C8—22no—
162C2×#454—311nomaximal, Sylow
163C22×C31—22no—
164C22×C31—42no—
165C22×C31—41no—
166C22×C31C22×D411yescharacteristic
167C2×D31—22no—
168C2×D31—22no—
169C2×D31—44no—
170C2×D31—42no—
171C2×D31—41no—
172C2×D31—82no—
173C2×D31—82no—
174C2×D31#12011yescharacteristic
175C2×D31#12011yescharacteristic
176C4×C31—22no—
177C4×C31—22no—
178C4×C31—41no—
179C4×C31C22×D412yes—
180Dic31—22no—
181Dic31—22no—
182Dic31—41no—
183Dic31#12012yes—
184C2×2−1+4—311no—
185C2×AGL(1,ℤ/8)—312no—
186C2×#124—311no—
187#454—311no—
188#454—312no—
189#454—311no—
190#454—312no—
191#454—311no—
192#454D3111yescharacteristic, direct product factor
193C2×#104—312no—
194C2×#101—311no—
195C2×C31—21no—
196C2×C31—42no—
197C2×C31—81no—
198C2×C31C2×#12011yescharacteristic
199D31—21no—
200D31—42no—
201D31—81no—
202D31#45412yesdirect product factor
2032−1+4—312no—
2042−1+4—311no—
2052−1+4—622no—
2062−1+4—622no—
2072−1+4—622no—
2082−1+4C2×D3111yescharacteristic, complemented
209C2×#41—312no—
210C2×#41—622no—
211C2×#41—622no—
212C22×Q8—311no—
213C22×Q8—622no—
214C22×D4—311no—
215AGL(1,ℤ/8)—312no—
216AGL(1,ℤ/8)—312no—
217AGL(1,ℤ/8)—312no—
218AGL(1,ℤ/8)—622no—
219AGL(1,ℤ/8)—622no—
220AGL(1,ℤ/8)C2×D3112yescomplemented
221C2×SD16—622no—
222C2×D8—622no—
223C2×M16—312no—
224#124—311no—
225#124—311no—
226#124—311no—
227#124—622no—
228#124C2×D3111yescharacteristic, complemented
229C2×C22⋊C4—622no—
230C2×C42—311no—
231#104—312no—
232#104—312no—
233#104—312no—
234#104C2×D3112yescomplemented
235#101—311no—
236#101—312no—
237#101C2×D3111yescharacteristic, complemented
238C31C2×#45411yescharacteristic, complemented, Sylow
239C24—621no—
240#41—22no—
241#41—22no—
242#41—312no—
243#41—312no—
244#41—312no—
245#41—622no—
246#41—622no—
247#41—624no—
248#41—624no—
249#41—622no—
250#41—624no—
251#41—622no—
252#41—624no—
253#41—622no—
254#41—622no—
255#41—622no—
256#41—622no—
257#41C22×D3112yes—
258C2×Q8—22no—
259C2×Q8—312no—
260C2×Q8—311no—
261C2×Q8—312no—
262C2×Q8—622no—
263C2×Q8—624no—
264C2×Q8—622no—
265C2×Q8—622no—
266C2×Q8—622no—
267C2×Q8—622no—
268C2×Q8—624no—
269C2×Q8—622no—
270C2×Q8C22×D3111yescharacteristic
271C2×D4—312no—
272C2×D4—311no—
273C2×D4—622no—
274C2×D4—622no—
275C2×D4—621no—
276C2×D4—622no—
277C2×D4—622no—
278C2×D4C22×D3111yescharacteristic
279C22×C4—311no—
280C22×C4—312no—
281C22×C4—622no—
282C22×C4—624no—
283C22×C4—621no—
284SD16—22no—
285SD16—622no—
286SD16—622no—
287SD16—622no—
288D8—22no—
289D8—622no—
290D8—622no—
291D8—622no—
292M16—312no—
293M16—622no—
294M16C22×D3112yes—
295C2×C8—622no—
296C22⋊C4—22no—
297C22⋊C4—622no—
298C22⋊C4—622no—
299C22⋊C4—622no—
300C42—311no—
301C42—621no—
302C42C22×D3111yescharacteristic
303C23—21no—
304C23—311no—
305C23—622no—
306C23—621no—
307C23—1241no—
308C23—1241no—
309C23—1244no—
310Q8—22no—
311Q8—22no—
312Q8—312no—
313Q8—622no—
314Q8—624no—
315Q8—624no—
316Q8—622no—
317Q8—622no—
318Q8D4×D3112yes—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—312no—
323D4—622no—
324D4—622no—
325D4—622no—
326D4—622no—
327D4—624no—
328D4—622no—
329D4—622no—
330D4—1242no—
331D4D4×D3112yes—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—311no—
336C2×C4—312no—
337C2×C4—312no—
338C2×C4—312no—
339C2×C4—622no—
340C2×C4—622no—
341C2×C4—622no—
342C2×C4—622no—
343C2×C4—624no—
344C2×C4—624no—
345C2×C4—622no—
346C2×C4—624no—
347C2×C4—622no—
348C2×C4—621no—
349C2×C4—1242no—
350C2×C4D4×D3112yes—
351C2×C4C23×D3111yescharacteristic
352C8—22no—
353C8—622no—
354C22—22no—
355C22—42no—
356C22—41no—
357C22—311no—
358C22—311no—
359C22—622no—
360C22—622no—
361C22—1242no—
362C22—1244no—
363C22—1241no—
364C22—2482no—
365C22—2482no—
366C22C2×D4×D3111yescharacteristic
367C4—22no—
368C4—22no—
369C4—41no—
370C4—312no—
371C4—622no—
372C4—622no—
373C4—1241no—
374C4C2×D4×D3112yes—
375C2—21no—
376C2—42no—
377C2—81no—
378C2—312no—
379C2—621no—
380C2—1242no—
381C2—2481no—
382C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#454C2
socleC2×C31C2×#120
hypercenter#454D31
hypocenterC31C2×#454
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. 2−1+4×D31C2semidirect
2. AGL(1,ℤ/8)×D31C2semidirect
3. #124×D31C2semidirect
4. #14521224C2semidirect
5. #14521223C2semidirect
6. #14521222C2semidirect
7. #14521221C2semidirect
8. #14521220C2semidirect
9. #104×D31C2semidirect
10. #101×D31C2semidirect
11. C31×#454C2semidirect
12. C31×2−1+4C22semidirect
15. #11758567C22semidirect
20. #11758564C22semidirect
24. C31×AGL(1,ℤ/8)C22semidirect
25. C31×#124C22semidirect
26. #11758526C22semidirect
27. #11758524C22semidirect
28. #11758513C22semidirect
29. #11758509C22semidirect
34. #11758485C22semidirect
38. #11758452C22semidirect
40. C31×#104C22semidirect
41. C31×#101C22semidirect
42. #11758425C22semidirect
43. #11758424C22semidirect
44. #11758413C22semidirect
45. #11758412C22semidirect
46. #11758393C22semidirect
192. #454D31direct (N × H)
202. D31#454direct (N × H)
208. 2−1+4C2×D31semidirect
220. AGL(1,ℤ/8)C2×D31semidirect
228. #124C2×D31semidirect
234. #104C2×D31semidirect
237. #101C2×D31semidirect
238. C31C2×#454semidirect

Maximal subgroups

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

subgroupindex
2−1+4×D312
AGL(1,ℤ/8)×D312
#124×D312
#145212242
#145212232
#145212222
#145212212
#145212202
#104×D312
#101×D312
C31×#4542
C2×#45431