Finite Group Databasebeta

← D31×#456 #16855050

Subgroups — D31×#456 #16855050

Order 3968.

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

normal subgroups only

This group has 708 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
2#137×D31C212yesmaximal, complemented
3C4⋊Q8×D31C211yesmaximal, characteristic, complemented
4#14521234C211yesmaximal, characteristic, complemented
5#14521233C212yesmaximal, complemented
6#14521232C211yesmaximal, characteristic, complemented
7#14521231C211yesmaximal, characteristic, complemented
8#14521230C212yesmaximal, complemented
9#104×D31C212yesmaximal, complemented
10#101×D31C211yesmaximal, characteristic, complemented
11C31×#456C211yesmaximal, characteristic, complemented
12C31×2−1+4C2211yescharacteristic
13#11758567—22no—
14#11758567—22no—
15#11758567C2212yes—
16#41×D31—22no—
17#41×D31—22no—
18#41×D31C2212yes—
19#11758564—22no—
20#11758564C2211yescharacteristic
21C2×Q8×D31—22no—
22C2×Q8×D31C2211yescharacteristic
23C2×Q8×D31C2211yescharacteristic
24C31×#137C2212yescomplemented
25C31×C4⋊Q8C2211yescharacteristic, complemented
26#11758528C2212yescomplemented
27#11758521C2211yescharacteristic, complemented
28#11758521C2211yescharacteristic, complemented
29#11758519C2212yescomplemented
30#11758495—22no—
31Q16×D31—22no—
32#11758492—22no—
33SD16×D31—22no—
34#11758486C2212yescomplemented
35M16×D31C2212yes—
36C4⋊C4×D31—22no—
37#11758465—22no—
38C42×D31C2211yescharacteristic
39#11758446C2211yescharacteristic, complemented
40C31×#104C2212yescomplemented
41C31×#101C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758424C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758412C2212yescomplemented
46#11758393C2212yescomplemented
47#41×C31—22no—
48#41×C31—22no—
49#41×C31C2312yes—
50C2×Q8×C31—22no—
51C2×Q8×C31C2311yescharacteristic
52C2×Q8×C31C2311yescharacteristic
53#92499—22no—
54#92499—22no—
55#92499D412yes—
56Q8×D31—22no—
57Q8×D31—24no—
58Q8×D31—22no—
59Q8×D31—22no—
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×D31D412yes—
74#92495—24no—
75#92495—24no—
76#92495—22no—
77#92495—22no—
78#92495—22no—
79#92495C2312yes—
80C2×C4×D31—24no—
81C2×C4×D31—21no—
82C2×C4×D31—22no—
83C2×C4×D31—21no—
84C2×C4×D31D412yes—
85C2×C4×D31C2311yescharacteristic
86C2×Dic62—24no—
87C2×Dic62—22no—
88C2×Dic62—21no—
89C2×Dic62D412yes—
90C2×Dic62C2311yescharacteristic
91C2×Dic62C2311yescharacteristic
92Q16×C31—22no—
93SD16×C31—22no—
94M16×C31C2312yes—
95C4⋊C4×C31—22no—
96C42×C31C2311yescharacteristic
97#92482—22no—
98#92482—22no—
99#92481—22no—
100#92480—22no—
101C4⋊Dic31—22no—
102Dic31⋊C4—22no—
103Dic31⋊C4—22no—
104C4×Dic31—21no—
105C4×Dic31C2311yescharacteristic
106#92474C2312yes—
107Dic124—22no—
108#92470—22no—
109#92469—22no—
110C8×D31—22no—
111C22×D31—22no—
112C22×D31C2×D411yescharacteristic
113Q8×C31—22no—
114Q8×C31—22no—
115Q8×C31—22no—
116Q8×C31C2×D412yes—
117D4×C31—22no—
118D4×C31—22no—
119D4×C31C2×D412yes—
120C2×C4×C31—24no—
121C2×C4×C31—21no—
122C2×C4×C31—22no—
123C2×C4×C31—21no—
124C2×C4×C31C2×D412yes—
125C2×C4×C31C2411yescharacteristic
126C22⋊D31—24no—
127C22⋊D31—22no—
128C22⋊D31—22no—
129C22⋊D31—22no—
130C2×Dic31—24no—
131C2×Dic31—21no—
132C2×Dic31—22no—
133C2×Dic31—21no—
134C2×Dic31C2×D412yes—
135C2×Dic31C2×D411yescharacteristic
136D124—22no—
137D124—22no—
138D124C2×D412yes—
139C4×D31—24no—
140C4×D31—24no—
141C4×D31—24no—
142C4×D31—22no—
143C4×D31—22no—
144C4×D31—22no—
145C4×D31—22no—
146C4×D31—22no—
147C4×D31—41no—
148C4×D31—41no—
149C4×D31C2×D412yes—
150C4×D31C2×D412yes—
151Dic62—24no—
152Dic62—22no—
153Dic62—22no—
154Dic62—22no—
155Dic62—24no—
156Dic62—22no—
157Dic62—42no—
158Dic62C2×D412yes—
159C8×C31—22no—
160C31⋊C8—22no—
161C2×#456—311nomaximal, Sylow
162C22×C31—22no—
163C22×C31C22×D411yescharacteristic
164C2×D31—22no—
165C2×D31—22no—
166C2×D31—44no—
167C2×D31#12011yescharacteristic
168C2×D31#12011yescharacteristic
169C4×C31—22no—
170C4×C31—22no—
171C4×C31—22no—
172C4×C31—41no—
173C4×C31C22×D412yes—
174Dic31—22no—
175Dic31—22no—
176Dic31—22no—
177Dic31—41no—
178Dic31#12012yes—
179C2×2−1+4—311no—
180C2×#137—312no—
181C2×C4⋊Q8—311no—
182#456—311no—
183#456—312no—
184#456—311no—
185#456—312no—
186#456—311no—
187#456D3111yescharacteristic, direct product factor
188C2×#104—312no—
189C2×#101—311no—
190C2×C31—21no—
191C2×C31—42no—
192C2×C31C2×#12011yescharacteristic
193D31—21no—
194D31—42no—
195D31#45612yesdirect product factor
1962−1+4—312no—
1972−1+4—311no—
1982−1+4—622no—
1992−1+4—622no—
2002−1+4—622no—
2012−1+4C2×D3111yescharacteristic
202C2×#41—312no—
203C2×#41—622no—
204C2×#41—622no—
205C22×Q8—311no—
206C22×Q8—311no—
207C22×Q8—622no—
208#137—312no—
209#137—312no—
210#137—312no—
211#137—622no—
212#137—622no—
213#137C2×D3112yescomplemented
214C2×Q16—622no—
215C2×SD16—622no—
216C2×M16—312no—
217C4⋊Q8—311no—
218C4⋊Q8—311no—
219C4⋊Q8—311no—
220C4⋊Q8—622no—
221C4⋊Q8C2×D3111yescharacteristic, complemented
222C2×C4⋊C4—622no—
223C2×C42—311no—
224#104—312no—
225#104—312no—
226#104—312no—
227#104C2×D3112yescomplemented
228#101—311no—
229#101—312no—
230#101C2×D3111yescharacteristic, complemented
231C31C2×#45611yescharacteristic, complemented, Sylow
232#41—22no—
233#41—22no—
234#41—312no—
235#41—312no—
236#41—312no—
237#41—622no—
238#41—622no—
239#41—624no—
240#41—624no—
241#41—622no—
242#41—624no—
243#41—622no—
244#41—624no—
245#41—622no—
246#41—622no—
247#41—622no—
248#41—622no—
249#41C22×D3112yes—
250C2×Q8—22no—
251C2×Q8—312no—
252C2×Q8—311no—
253C2×Q8—312no—
254C2×Q8—311no—
255C2×Q8—621no—
256C2×Q8—622no—
257C2×Q8—624no—
258C2×Q8—622no—
259C2×Q8—622no—
260C2×Q8—622no—
261C2×Q8—622no—
262C2×Q8—622no—
263C2×Q8—622no—
264C2×Q8—624no—
265C2×Q8—622no—
266C2×Q8C22×D3111yescharacteristic
267C2×Q8C22×D3111yescharacteristic
268C2×D4—312no—
269C2×D4—622no—
270C2×D4—622no—
271C22×C4—311no—
272C22×C4—312no—
273C22×C4—622no—
274C22×C4—621no—
275C22×C4—624no—
276C22×C4—621no—
277Q16—22no—
278Q16—622no—
279Q16—622no—
280Q16—622no—
281SD16—22no—
282SD16—622no—
283SD16—622no—
284SD16—622no—
285M16—312no—
286M16—622no—
287M16C22×D3112yes—
288C2×C8—622no—
289C4⋊C4—22no—
290C4⋊C4—622no—
291C4⋊C4—622no—
292C4⋊C4—622no—
293C42—311no—
294C42—621no—
295C42C22×D3111yescharacteristic
296C23—311no—
297C23—622no—
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8—312no—
302Q8—622no—
303Q8—624no—
304Q8—624no—
305Q8—622no—
306Q8—622no—
307Q8—622no—
308Q8—1242no—
309Q8D4×D3112yes—
310D4—22no—
311D4—22no—
312D4—312no—
313D4—622no—
314D4—622no—
315D4—622no—
316D4—622no—
317D4—624no—
318D4—622no—
319D4D4×D3112yes—
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—311no—
325C2×C4—312no—
326C2×C4—312no—
327C2×C4—312no—
328C2×C4—622no—
329C2×C4—622no—
330C2×C4—622no—
331C2×C4—622no—
332C2×C4—624no—
333C2×C4—624no—
334C2×C4—624no—
335C2×C4—622no—
336C2×C4—624no—
337C2×C4—621no—
338C2×C4—622no—
339C2×C4—621no—
340C2×C4—1241no—
341C2×C4—1241no—
342C2×C4D4×D3112yes—
343C2×C4C23×D3111yescharacteristic
344C8—22no—
345C8—622no—
346C22—22no—
347C22—311no—
348C22—311no—
349C22—622no—
350C22—622no—
351C22—1244no—
352C22C2×D4×D3111yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—312no—
358C4—622no—
359C4—622no—
360C4—622no—
361C4—1241no—
362C4C2×D4×D3112yes—
363C2—21no—
364C2—42no—
365C2—312no—
366C2—621no—
367C2—1242no—
368C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#456C2
socleC2×C31C2×#120
prime radical2−1+4×D31C2
hypercenter#456D31
hypocenterC31C2×#456
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
2. #137×D31C2semidirect
3. C4⋊Q8×D31C2semidirect
4. #14521234C2semidirect
5. #14521233C2semidirect
6. #14521232C2semidirect
7. #14521231C2semidirect
8. #14521230C2semidirect
9. #104×D31C2semidirect
10. #101×D31C2semidirect
11. C31×#456C2semidirect
24. C31×#137C22semidirect
25. C31×C4⋊Q8C22semidirect
26. #11758528C22semidirect
27. #11758521C22semidirect
28. #11758521C22semidirect
29. #11758519C22semidirect
34. #11758486C22semidirect
39. #11758446C22semidirect
40. C31×#104C22semidirect
41. C31×#101C22semidirect
42. #11758425C22semidirect
43. #11758424C22semidirect
44. #11758413C22semidirect
45. #11758412C22semidirect
46. #11758393C22semidirect
187. #456D31direct (N × H)
195. D31#456direct (N × H)
213. #137C2×D31semidirect
221. C4⋊Q8C2×D31semidirect
227. #104C2×D31semidirect
230. #101C2×D31semidirect
231. C31C2×#456semidirect

Maximal subgroups

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

subgroupindex
2−1+4×D312
#137×D312
C4⋊Q8×D312
#145212342
#145212332
#145212322
#145212312
#145212302
#104×D312
#101×D312
C31×#4562
C2×#45631