Finite Group Databasebeta

← D31×#451 #16854902

Subgroups — D31×#451 #16854902

Order 3968.

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

normal subgroups only

This group has 740 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×Q8×D31C211yesmaximal, characteristic
2C2×Q16×D31C212yesmaximal, complemented
3C22⋊Q8×D31C211yesmaximal, characteristic
4#14521207C211yesmaximal, characteristic, complemented
5#14521206C212yesmaximal, complemented
6#14521205C211yesmaximal, characteristic, complemented
7#14521204C211yesmaximal, characteristic, complemented
8#14521203C212yesmaximal, complemented
9#103×D31C212yesmaximal, complemented
10C22⋊C8×D31C211yesmaximal, characteristic
11C31×#451C211yesmaximal, characteristic, complemented
12C22×Q8×C31C2211yescharacteristic
13C2×Q8×D31—28no—
14C2×Q8×D31—21no—
15C2×Q8×D31—21no—
16C2×Q8×D31C2212yes—
17C2×Q8×D31C2211yescharacteristic
18C2×Q8×D31C2212yes—
19C22×C4×D31—21no—
20C22×C4×D31C2211yescharacteristic
21C22×Dic62—21no—
22C22×Dic62C2211yescharacteristic
23C2×Q16×C31C2212yescomplemented
24C31×C22⋊Q8C2211yescharacteristic
25#11758523C2211yescharacteristic
26C2×#92482C2212yescomplemented
27C2×#92482C2212yescomplemented
28#11758501C2211yescharacteristic
29Q16×D31—28no—
30C2×Dic124C2212yescomplemented
31C2×C8×D31C2212yes—
32#11758474C2211yescharacteristic
33#11758473—21no—
34C4⋊C4×D31—21no—
35C4⋊C4×D31C2211yescharacteristic
36C22⋊C4×D31—21no—
37#11758455—21no—
38C31×#103C2212yescomplemented
39C31×C22⋊C8C2211yescharacteristic
40#11758423C2212yescomplemented
41C22⋊(C31⋊C8)C2211yescharacteristic
42(C2×D31)⋊C8C2212yes—
43#11758404C2212yescomplemented
44#11758398C2212yescomplemented
45C23×D31D411yescharacteristic
46C2×Q8×C31—24no—
47C2×Q8×C31—21no—
48C2×Q8×C31C2311yescharacteristic
49C2×Q8×C31C2312yes—
50C22×C4×C31—21no—
51C22×C4×C31C2311yescharacteristic
52C22×Dic31—21no—
53C22×Dic31D411yescharacteristic
54Q8×D31—28no—
55Q8×D31—28no—
56Q8×D31—24no—
57Q8×D31—24no—
58Q8×D31—24no—
59Q8×D31—24no—
60Q8×D31D418yes—
61C2×C4×D31—28no—
62C2×C4×D31—22no—
63C2×C4×D31—22no—
64C2×C4×D31—24no—
65C2×C4×D31—21no—
66C2×C4×D31—21no—
67C2×C4×D31D411yescharacteristic
68C2×C4×D31D411yescharacteristic
69C2×C4×D31C2311yescharacteristic
70C2×C4×D31C2311yescharacteristic
71C2×Dic62—28no—
72C2×Dic62—21no—
73C2×Dic62—22no—
74C2×Dic62—22no—
75C2×Dic62—24no—
76C2×Dic62—21no—
77C2×Dic62C2311yescharacteristic
78C2×Dic62C2312yes—
79Q16×C31—24no—
80C2×C8×C31C2312yes—
81C4⋊C4×C31—21no—
82C4⋊C4×C31C2311yescharacteristic
83C22⋊C4×C31—21no—
84C22⋊Dic31—21no—
85#92482—24no—
86#92482—24no—
87(C2×D31)⋊C4—21no—
88(C2×D31)⋊C4—21no—
89C4⋊Dic31—21no—
90C4⋊Dic31C2311yescharacteristic
91Dic31⋊C4—21no—
92Dic31⋊C4—21no—
93Dic31⋊C4—21no—
94C2×C31⋊C8C2312yes—
95Dic124—24no—
96C8×D31—24no—
97C23×C31C2×D411yescharacteristic
98C22×D31—22no—
99C22×D31—22no—
100C22×D31Q1614yescomplemented
101C22×D31C2×D411yescharacteristic
102C22×D31C2×D411yescharacteristic
103Q8×C31—24no—
104Q8×C31—22no—
105Q8×C31—22no—
106Q8×C31C2×D414yes—
107C2×C4×C31—24no—
108C2×C4×C31—22no—
109C2×C4×C31—22no—
110C2×C4×C31—21no—
111C2×C4×C31—21no—
112C2×C4×C31C2×D411yescharacteristic
113C2×C4×C31C2411yescharacteristic
114C2×Dic31—24no—
115C2×Dic31—22no—
116C2×Dic31—22no—
117C2×Dic31—21no—
118C2×Dic31—21no—
119C2×Dic31C2×D411yescharacteristic
120C2×Dic31C2×D411yescharacteristic
121C4×D31—28no—
122C4×D31—28no—
123C4×D31—22no—
124C4×D31—22no—
125C4×D31—22no—
126C4×D31—42no—
127C4×D31—42no—
128C4×D31C2×D414yes—
129Dic62—28no—
130Dic62—24no—
131Dic62—22no—
132Dic62—28no—
133Dic62—22no—
134Dic62—42no—
135Dic62C2×D414yes—
136C8×C31—22no—
137C31⋊C8—22no—
138C2×#451—311nomaximal, Sylow
139C22×C31—21no—
140C22×C31—21no—
141C22×C31C2×Q1612yescomplemented
142C22×C31C22×D411yescharacteristic
143C2×D31—21no—
144C2×D31—21no—
145C2×D31—28no—
146C2×D31#12012yes—
147C2×D31C2×Q1612yes—
148C2×D31C2×Q1612yescomplemented
149C2×D31#13712yes—
150C4×C31—24no—
151C4×C31—21no—
152C4×C31—41no—
153C4×C31—41no—
154C4×C31C22×D412yes—
155Dic31—21no—
156Dic31—24no—
157Dic31—41no—
158Dic31—41no—
159Dic31#12012yes—
160C23×Q8—311no—
161C22×Q16—312no—
162C2×C22⋊Q8—311no—
163#451—311no—
164#451—312no—
165#451—311no—
166#451—312no—
167#451—311no—
168#451D3111yescharacteristic, direct product factor
169C2×#103—312no—
170C2×C22⋊C8—311no—
171C2×C31—22no—
172C2×C31C2×#12011yescharacteristic
173C2×C31C22×Q1611yescharacteristic
174C2×C31C2×#13711yescharacteristic
175D31—22no—
176D31#45114yesdirect product factor
177C22×Q8—312no—
178C22×Q8—311no—
179C22×Q8—312no—
180C22×Q8—311no—
181C22×Q8—621no—
182C22×Q8—621no—
183C22×Q8—621no—
184C22×Q8—628no—
185C22×Q8C2×D3111yescharacteristic
186C23×C4—311no—
187C23×C4—621no—
188C2×Q16—312no—
189C2×Q16—312no—
190C2×Q16—312no—
191C2×Q16—628no—
192C2×Q16C2×D3112yescomplemented
193C22×C8—312no—
194C22⋊Q8—311no—
195C22⋊Q8—311no—
196C22⋊Q8—311no—
197C22⋊Q8—621no—
198C22⋊Q8—621no—
199C22⋊Q8C2×D3111yescharacteristic
200C2×C4⋊C4—311no—
201C2×C4⋊C4—621no—
202C2×C22⋊C4—621no—
203#103—312no—
204#103—312no—
205#103—312no—
206#103C2×D3112yescomplemented
207C22⋊C8—311no—
208C22⋊C8—312no—
209C22⋊C8C2×D3111yescharacteristic
210C31C2×#45111yescharacteristic, complemented, Sylow
211C24—311no—
212C2×Q8—24no—
213C2×Q8—21no—
214C2×Q8—318no—
215C2×Q8—312no—
216C2×Q8—311no—
217C2×Q8—622no—
218C2×Q8—622no—
219C2×Q8—621no—
220C2×Q8—624no—
221C2×Q8—628no—
222C2×Q8—624no—
223C2×Q8—624no—
224C2×Q8—624no—
225C2×Q8—628no—
226C2×Q8—628no—
227C2×Q8—624no—
228C2×Q8—621no—
229C2×Q8C22×D3111yescharacteristic
230C2×Q8C22×D3112yes—
231C22×C4—21no—
232C22×C4—311no—
233C22×C4—311no—
234C22×C4—311no—
235C22×C4—311no—
236C22×C4—311no—
237C22×C4—624no—
238C22×C4—622no—
239C22×C4—622no—
240C22×C4—628no—
241C22×C4—621no—
242C22×C4—621no—
243C22×C4—621no—
244C22×C4C22×D3111yescharacteristic
245Q16—24no—
246Q16—624no—
247Q16—624no—
248Q16—624no—
249C2×C8—312no—
250C2×C8—624no—
251C2×C8C22×D3112yes—
252C4⋊C4—21no—
253C4⋊C4—311no—
254C4⋊C4—621no—
255C4⋊C4—621no—
256C4⋊C4—621no—
257C4⋊C4—621no—
258C4⋊C4C22×D3111yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—621no—
261C22⋊C4—621no—
262C22⋊C4—621no—
263C23—311no—
264C23—314no—
265C23—311no—
266C23—622no—
267C23—622no—
268C23D4×D3111yescharacteristic
269Q8—24no—
270Q8—22no—
271Q8—22no—
272Q8—314no—
273Q8—628no—
274Q8—628no—
275Q8—624no—
276Q8—622no—
277Q8—622no—
278Q8—1242no—
279Q8D4×D3114yes—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—311no—
286C2×C4—314no—
287C2×C4—311no—
288C2×C4—622no—
289C2×C4—622no—
290C2×C4—628no—
291C2×C4—628no—
292C2×C4—622no—
293C2×C4—624no—
294C2×C4—622no—
295C2×C4—622no—
296C2×C4—621no—
297C2×C4—621no—
298C2×C4—1242no—
299C2×C4—1242no—
300C2×C4D4×D3111yescharacteristic
301C2×C4C23×D3111yescharacteristic
302C8—22no—
303C8—622no—
304C22—21no—
305C22—21no—
306C22—312no—
307C22—312no—
308C22—312no—
309C22—312no—
310C22—628no—
311C22—621no—
312C22—621no—
313C22Q16×D3112yescomplemented
314C22C2×D4×D3111yescharacteristic
315C4—24no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—312no—
320C4—621no—
321C4—624no—
322C4—1241no—
323C4—1241no—
324C4C2×D4×D3112yes—
325C2—22no—
326C2—314no—
327C2—622no—
328C2#120×D3111yescharacteristic
329C2C2×Q16×D3111yescharacteristic
330C2#137×D3111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#451C2
socleC22×C31C22×D4
prime radicalC23×D31D4
hypercenter#451D31
hypocenterC31C2×#451
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q16×D31C2semidirect
4. #14521207C2semidirect
5. #14521206C2semidirect
6. #14521205C2semidirect
7. #14521204C2semidirect
8. #14521203C2semidirect
9. #103×D31C2semidirect
11. C31×#451C2semidirect
23. C2×Q16×C31C22semidirect
26. C2×#92482C22semidirect
27. C2×#92482C22semidirect
30. C2×Dic124C22semidirect
38. C31×#103C22semidirect
40. #11758423C22semidirect
43. #11758404C22semidirect
44. #11758398C22semidirect
100. C22×D31Q16semidirect
141. C22×C31C2×Q16semidirect
148. C2×D31C2×Q16semidirect
168. #451D31direct (N × H)
176. D31#451direct (N × H)
192. C2×Q16C2×D31semidirect
206. #103C2×D31semidirect
210. C31C2×#451semidirect
313. C22Q16×D31semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D312
C2×Q16×D312
C22⋊Q8×D312
#145212072
#145212062
#145212052
#145212042
#145212032
#103×D312
C22⋊C8×D312
C31×#4512
C2×#45131