Finite Group Databasebeta

← D31×#453 #16855032

Subgroups — D31×#453 #16855032

Order 3968.

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

normal subgroups only

This group has 844 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
1H(5,𝔽2)×D31C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D31C212yesmaximal, complemented
3#127×D31C211yesmaximal, characteristic, complemented
4#14521219C211yesmaximal, characteristic, complemented
5#14521218C212yesmaximal, complemented
6#14521217C211yesmaximal, characteristic, complemented
7#14521216C211yesmaximal, characteristic, complemented
8#14521215C212yesmaximal, complemented
9#104×D31C212yesmaximal, complemented
10#100×D31C211yesmaximal, characteristic, complemented
11C31×#453C211yesmaximal, characteristic, complemented
12C31×H(5,𝔽2)C2211yescharacteristic, complemented
13#11758566—22no—
14#11758566C2212yescomplemented
15#41×D31—22no—
16#41×D31C2212yes—
17#11758561—22no—
18#11758561—22no—
19#11758561C2211yescharacteristic, complemented
20C2×D4×D31—22no—
21C2×D4×D31—22no—
22C2×D4×D31—22no—
23C2×D4×D31C2211yescharacteristic
24C2×D4×D31C2211yescharacteristic, complemented
25C31×AGL(1,ℤ/8)C2212yescomplemented
26C31×#127C2211yescharacteristic, complemented
27#11758526C2212yescomplemented
28#11758517—22no—
29#11758517C2211yescharacteristic, complemented
30#11758517C2211yescharacteristic, complemented
31#11758509C2212yescomplemented
32#11758491—22no—
33SD16×D31—22no—
34#11758488—22no—
35D8×D31—22no—
36#11758485C2212yescomplemented
37M16×D31C2212yescomplemented
38#11758451C2211yescharacteristic, complemented
39C42×D31C2211yescharacteristic, complemented
40C31×#104C2212yescomplemented
41C31×#100C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758421C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758411C2212yescomplemented
46#11758393C2212yescomplemented
47C23×D31—22no—
48C23×D31—21no—
49C23×D31D412yes—
50#41×C31—22no—
51#41×C31C2312yes—
52C2×D4×C31—22no—
53C2×D4×C31—22no—
54C2×D4×C31—22no—
55C2×D4×C31C2311yescharacteristic
56C2×D4×C31C2311yescharacteristic, complemented
57C2×C22⋊D31—22no—
58C2×C22⋊D31—24no—
59C2×C22⋊D31—22no—
60C2×C22⋊D31—22no—
61C2×C22⋊D31—22no—
62C2×C22⋊D31—21no—
63C2×C22⋊D31D412yes—
64#92499—22no—
65#92499D412yescomplemented
66Q8×D31D412yescomplemented
67#92497—24no—
68#92497—22no—
69#92497—22no—
70#92497D412yescomplemented
71D4×D31—24no—
72D4×D31—24no—
73D4×D31—24no—
74D4×D31—22no—
75D4×D31—24no—
76D4×D31—22no—
77D4×D31—22no—
78D4×D31—22no—
79D4×D31—22no—
80D4×D31—22no—
81D4×D31—22no—
82D4×D31—48no—
83D4×D31D412yescomplemented
84#92495—24no—
85#92495—22no—
86#92495—22no—
87#92495C2312yes—
88C2×D124—22no—
89C2×D124—22no—
90C2×D124—22no—
91C2×D124C2311yescharacteristic
92C2×D124C2311yescharacteristic, complemented
93C2×C4×D31—22no—
94C2×C4×D31—21no—
95C2×C4×D31—22no—
96C2×C4×D31C2311yescharacteristic
97SD16×C31—22no—
98D8×C31—22no—
99M16×C31C2312yescomplemented
100C42×C31C2311yescharacteristic, complemented
101#92481—22no—
102#92480—22no—
103#92479—22no—
104#92479—22no—
105C4×Dic31—21no—
106C4×Dic31C2311yescharacteristic, complemented
107#92474C2312yescomplemented
108D248—22no—
109#92470—22no—
110#92469—22no—
111C8×D31—22no—
112C23×C31—22no—
113C23×C31—21no—
114C23×C31C2×D412yes—
115C22×D31—24no—
116C22×D31—22no—
117C22×D31—22no—
118C22×D31—22no—
119C22×D31—22no—
120C22×D31—21no—
121C22×D31—44no—
122C22×D31—44no—
123C22×D31—41no—
124C22×D31—41no—
125C22×D31—84no—
126C22×D31C2×D411yescharacteristic
127C22×D31C2×D412yes—
128Q8×C31C2×D412yescomplemented
129D4×C31—24no—
130D4×C31—22no—
131D4×C31—22no—
132D4×C31—22no—
133D4×C31—44no—
134D4×C31C2×D412yescomplemented
135C2×C4×C31—22no—
136C2×C4×C31—21no—
137C2×C4×C31—22no—
138C2×C4×C31C2411yescharacteristic
139C22⋊D31—24no—
140C22⋊D31—24no—
141C22⋊D31—22no—
142C22⋊D31—22no—
143C22⋊D31—24no—
144C22⋊D31—22no—
145C22⋊D31—44no—
146C22⋊D31—44no—
147C22⋊D31—42no—
148C2×Dic31—22no—
149C2×Dic31—21no—
150C2×Dic31—22no—
151C2×Dic31C2×D411yescharacteristic
152D124—24no—
153D124—22no—
154D124—22no—
155D124—22no—
156D124—44no—
157D124C2×D412yescomplemented
158C4×D31—24no—
159C4×D31—24no—
160C4×D31—22no—
161C4×D31—22no—
162C4×D31—22no—
163C4×D31C2×D412yes—
164C4×D31C2×D412yes—
165Dic62—22no—
166Dic62C2×D412yescomplemented
167C8×C31—22no—
168C31⋊C8—22no—
169C2×#453—311nomaximal, Sylow
170C22×C31—22no—
171C22×C31—22no—
172C22×C31—42no—
173C22×C31—42no—
174C22×C31—41no—
175C22×C31—82no—
176C22×C31C22×D411yescharacteristic
177C2×D31—22no—
178C2×D31—22no—
179C2×D31—22no—
180C2×D31—44no—
181C2×D31—44no—
182C2×D31—42no—
183C2×D31—42no—
184C2×D31—41no—
185C2×D31—82no—
186C2×D31—82no—
187C2×D31—84no—
188C2×D31—82no—
189C2×D31—82no—
190C2×D31#12011yescharacteristic
191C2×D31#12011yescharacteristic
192C4×C31—22no—
193C4×C31—22no—
194C4×C31C22×D412yes—
195Dic31—22no—
196Dic31—22no—
197Dic31#12012yes—
198C2×H(5,𝔽2)—311no—
199C2×AGL(1,ℤ/8)—312no—
200C2×#127—311no—
201#453—311no—
202#453—312no—
203#453—311no—
204#453—312no—
205#453—311no—
206#453D3111yescharacteristic, direct product factor
207C2×#104—312no—
208C2×#100—311no—
209C2×C31—21no—
210C2×C31—42no—
211C2×C31—42no—
212C2×C31—81no—
213C2×C31C2×#12011yescharacteristic
214D31—21no—
215D31—42no—
216D31—42no—
217D31—81no—
218D31#45312yesdirect product factor
219H(5,𝔽2)—312no—
220H(5,𝔽2)—311no—
221H(5,𝔽2)—622no—
222H(5,𝔽2)—622no—
223H(5,𝔽2)—622no—
224H(5,𝔽2)C2×D3111yescharacteristic, complemented
225C2×#41—312no—
226C2×#41—622no—
227C22×D4—311no—
228C22×D4—311no—
229C22×D4—622no—
230C22×D4—622no—
231C22×D4—622no—
232AGL(1,ℤ/8)—312no—
233AGL(1,ℤ/8)—312no—
234AGL(1,ℤ/8)—312no—
235AGL(1,ℤ/8)—622no—
236AGL(1,ℤ/8)—622no—
237AGL(1,ℤ/8)C2×D3112yescomplemented
238C2×SD16—622no—
239C2×D8—622no—
240C2×M16—312no—
241#127—311no—
242#127—311no—
243#127—311no—
244#127—622no—
245#127C2×D3111yescharacteristic, complemented
246C2×C42—311no—
247#104—312no—
248#104—312no—
249#104—312no—
250#104C2×D3112yescomplemented
251#100—311no—
252#100—312no—
253#100C2×D3111yescharacteristic, complemented
254C31C2×#45311yescharacteristic, complemented, Sylow
255C24—312no—
256C24—621no—
257C24—622no—
258#41—22no—
259#41—312no—
260#41—312no—
261#41—312no—
262#41—622no—
263#41—622no—
264#41—624no—
265#41—622no—
266#41—622no—
267#41—624no—
268#41—622no—
269#41C22×D3112yes—
270C2×Q8—312no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—312no—
275C2×D4—312no—
276C2×D4—311no—
277C2×D4—311no—
278C2×D4—622no—
279C2×D4—622no—
280C2×D4—622no—
281C2×D4—622no—
282C2×D4—622no—
283C2×D4—622no—
284C2×D4—622no—
285C2×D4—621no—
286C2×D4—622no—
287C2×D4—622no—
288C2×D4—624no—
289C2×D4—624no—
290C2×D4—624no—
291C2×D4—624no—
292C2×D4—624no—
293C2×D4—622no—
294C2×D4—622no—
295C2×D4—622no—
296C2×D4—622no—
297C2×D4—622no—
298C2×D4—1248no—
299C2×D4C22×D3111yescharacteristic
300C2×D4C22×D3111yescharacteristic, complemented
301C22×C4—311no—
302C22×C4—622no—
303C22×C4—621no—
304C22×C4—622no—
305SD16—22no—
306SD16—622no—
307SD16—622no—
308SD16—622no—
309D8—22no—
310D8—622no—
311D8—622no—
312D8—622no—
313M16—312no—
314M16—622no—
315M16C22×D3112yescomplemented
316C2×C8—622no—
317C42—311no—
318C42—621no—
319C42C22×D3111yescharacteristic, complemented
320C23—22no—
321C23—21no—
322C23—311no—
323C23—312no—
324C23—622no—
325C23—622no—
326C23—622no—
327C23—622no—
328C23—624no—
329C23—621no—
330C23—1241no—
331C23—1241no—
332C23—1244no—
333C23—1244no—
334C23—2484no—
335C23D4×D3112yes—
336Q8—312no—
337Q8—622no—
338Q8D4×D3112yescomplemented
339D4—24no—
340D4—22no—
341D4—22no—
342D4—22no—
343D4—44no—
344D4—312no—
345D4—622no—
346D4—624no—
347D4—624no—
348D4—622no—
349D4—622no—
350D4—624no—
351D4—624no—
352D4—622no—
353D4—622no—
354D4—622no—
355D4—1244no—
356D4—1242no—
357D4—1244no—
358D4—1244no—
359D4D4×D3112yescomplemented
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—311no—
364C2×C4—312no—
365C2×C4—312no—
366C2×C4—622no—
367C2×C4—622no—
368C2×C4—624no—
369C2×C4—624no—
370C2×C4—622no—
371C2×C4—622no—
372C2×C4—621no—
373C2×C4—622no—
374C2×C4C23×D3111yescharacteristic
375C8—22no—
376C8—622no—
377C22—22no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—82no—
383C22—311no—
384C22—311no—
385C22—622no—
386C22—622no—
387C22—622no—
388C22—1242no—
389C22—1242no—
390C22—1244no—
391C22—1244no—
392C22—1241no—
393C22—2482no—
394C22—2482no—
395C22—2484no—
396C22—2482no—
397C22—2482no—
398C22C2×D4×D3111yescharacteristic
399C4—22no—
400C4—22no—
401C4—312no—
402C4—622no—
403C4—622no—
404C4C2×D4×D3112yes—
405C2—21no—
406C2—42no—
407C2—42no—
408C2—81no—
409C2—312no—
410C2—621no—
411C2—1242no—
412C2—1242no—
413C2—2481no—
414C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#453C2
socleC2×C31C2×#120
hypercenter#453D31
hypocenterC31C2×#453
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. H(5,𝔽2)×D31C2semidirect
2. AGL(1,ℤ/8)×D31C2semidirect
3. #127×D31C2semidirect
4. #14521219C2semidirect
5. #14521218C2semidirect
6. #14521217C2semidirect
7. #14521216C2semidirect
8. #14521215C2semidirect
9. #104×D31C2semidirect
10. #100×D31C2semidirect
11. C31×#453C2semidirect
12. C31×H(5,𝔽2)C22semidirect
14. #11758566C22semidirect
19. #11758561C22semidirect
24. C2×D4×D31C22semidirect
25. C31×AGL(1,ℤ/8)C22semidirect
26. C31×#127C22semidirect
27. #11758526C22semidirect
29. #11758517C22semidirect
30. #11758517C22semidirect
31. #11758509C22semidirect
36. #11758485C22semidirect
37. M16×D31C22semidirect
38. #11758451C22semidirect
39. C42×D31C22semidirect
40. C31×#104C22semidirect
41. C31×#100C22semidirect
42. #11758425C22semidirect
43. #11758421C22semidirect
44. #11758413C22semidirect
45. #11758411C22semidirect
46. #11758393C22semidirect
56. C2×D4×C31C23semidirect
65. #92499D4semidirect
66. Q8×D31D4semidirect
70. #92497D4semidirect
83. D4×D31D4semidirect
92. C2×D124C23semidirect
99. M16×C31C23semidirect
100. C42×C31C23semidirect
106. C4×Dic31C23semidirect
107. #92474C23semidirect
128. Q8×C31C2×D4semidirect
134. D4×C31C2×D4semidirect
157. D124C2×D4semidirect
166. Dic62C2×D4semidirect
206. #453D31direct (N × H)
218. D31#453direct (N × H)
224. H(5,𝔽2)C2×D31semidirect
237. AGL(1,ℤ/8)C2×D31semidirect
245. #127C2×D31semidirect
250. #104C2×D31semidirect
253. #100C2×D31semidirect
254. C31C2×#453semidirect
300. C2×D4C22×D31semidirect
315. M16C22×D31semidirect
319. C42C22×D31semidirect
338. Q8D4×D31semidirect
359. D4D4×D31semidirect

Maximal subgroups

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

subgroupindex
H(5,𝔽2)×D312
AGL(1,ℤ/8)×D312
#127×D312
#145212192
#145212182
#145212172
#145212162
#145212152
#104×D312
#100×D312
C31×#4532
C2×#45331