Finite Group Databasebeta

← D31×#448 #16854896

Subgroups — D31×#448 #16854896

Order 3968.

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

normal subgroups only

This group has 792 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, complemented
2C2×SD16×D31C212yesmaximal, complemented
3#121×D31C211yesmaximal, characteristic
4#14521190C211yesmaximal, characteristic, complemented
5#14521189C212yesmaximal, complemented
6#14521188C211yesmaximal, characteristic, complemented
7#14521187C211yesmaximal, characteristic, complemented
8#14521186C212yesmaximal, complemented
9#103×D31C212yesmaximal, complemented
10C22⋊C8×D31C211yesmaximal, characteristic, complemented
11C31×#448C211yesmaximal, characteristic, complemented
12C22×Q8×C31C2211yescharacteristic, complemented
13C2×Q8×D31—28no—
14C2×Q8×D31—21no—
15C2×Q8×D31—21no—
16C2×Q8×D31C2212yes—
17C2×Q8×D31C2212yescomplemented
18C2×D4×D31—21no—
19C2×D4×D31C2211yescharacteristic
20C22×C4×D31—21no—
21C22×C4×D31C2211yescharacteristic
22C22×Dic62—21no—
23C22×Dic62C2211yescharacteristic, complemented
24C2×SD16×C31C2212yescomplemented
25C31×#121C2211yescharacteristic
26C2×#92481C2212yescomplemented
27#11758516—21no—
28#11758515C2211yescharacteristic
29C2×#92480C2212yescomplemented
30#11758507C2211yescharacteristic
31SD16×D31—28no—
32C2×#92470C2212yescomplemented
33C2×C8×D31C2212yes—
34#11758472C2211yescharacteristic
35C4⋊C4×D31C2211yescharacteristic
36#11758461—21no—
37C22⋊C4×D31—21no—
38C31×#103C2212yescomplemented
39C31×C22⋊C8C2211yescharacteristic, complemented
40#11758423C2212yescomplemented
41C22⋊(C31⋊C8)C2211yescharacteristic, complemented
42(C2×D31)⋊C8C2212yescomplemented
43#11758404C2212yescomplemented
44#11758398C2212yescomplemented
45C23×D31—21no—
46C23×D31D411yescharacteristic
47C2×Q8×C31—24no—
48C2×Q8×C31—21no—
49C2×Q8×C31C2312yes—
50C2×D4×C31—21no—
51C2×D4×C31C2311yescharacteristic
52C22×C4×C31—21no—
53C22×C4×C31C2311yescharacteristic
54C2×C22⋊D31—21no—
55C2×C22⋊D31—21no—
56C2×C22⋊D31—21no—
57C22×Dic31—21no—
58C22×Dic31D411yescharacteristic
59Q8×D31—28no—
60Q8×D31—28no—
61Q8×D31—24no—
62Q8×D31—24no—
63Q8×D31—24no—
64Q8×D31D418yescomplemented
65D4×D31—24no—
66D4×D31—44no—
67C2×D124—21no—
68C2×D124C2311yescharacteristic
69C2×C4×D31—28no—
70C2×C4×D31—22no—
71C2×C4×D31—22no—
72C2×C4×D31—24no—
73C2×C4×D31—21no—
74C2×C4×D31D411yescharacteristic
75C2×C4×D31D411yescharacteristic
76C2×C4×D31C2311yescharacteristic
77C2×C4×D31C2311yescharacteristic
78C2×Dic62—28no—
79C2×Dic62—21no—
80C2×Dic62—22no—
81C2×Dic62—22no—
82C2×Dic62—24no—
83C2×Dic62C2312yes—
84SD16×C31—24no—
85C2×C8×C31C2312yes—
86C4⋊C4×C31C2311yescharacteristic
87C22⋊C4×C31—21no—
88C22⋊Dic31—21no—
89#92481—24no—
90#92480—24no—
91(C2×D31)⋊C4—21no—
92(C2×D31)⋊C4—21no—
93C4⋊Dic31C2311yescharacteristic
94Dic31⋊C4—21no—
95C2×C31⋊C8C2312yes—
96#92470—24no—
97C8×D31—24no—
98C23×C31—21no—
99C23×C31C2×D411yescharacteristic
100C22×D31—22no—
101C22×D31—22no—
102C22×D31—21no—
103C22×D31—42no—
104C22×D31—42no—
105C22×D31—42no—
106C22×D31SD1614yescomplemented
107C22×D31C2×D411yescharacteristic
108C22×D31C2×D411yescharacteristic
109Q8×C31—24no—
110Q8×C31—22no—
111Q8×C31C2×D414yescomplemented
112D4×C31—22no—
113D4×C31—42no—
114C2×C4×C31—24no—
115C2×C4×C31—22no—
116C2×C4×C31—22no—
117C2×C4×C31—21no—
118C2×C4×C31C2×D411yescharacteristic
119C2×C4×C31C2411yescharacteristic
120C22⋊D31—42no—
121C22⋊D31—42no—
122C22⋊D31—42no—
123C2×Dic31—24no—
124C2×Dic31—22no—
125C2×Dic31—22no—
126C2×Dic31—21no—
127C2×Dic31C2×D411yescharacteristic
128C2×Dic31C2×D411yescharacteristic
129D124—22no—
130D124—42no—
131C4×D31—28no—
132C4×D31—28no—
133C4×D31—22no—
134C4×D31—22no—
135C4×D31—22no—
136C4×D31—42no—
137C4×D31C2×D414yes—
138Dic62—28no—
139Dic62—24no—
140Dic62—22no—
141Dic62—28no—
142Dic62C2×D414yescomplemented
143C8×C31—22no—
144C31⋊C8—22no—
145C2×#448—311nomaximal, Sylow
146C22×C31—21no—
147C22×C31—21no—
148C22×C31—41no—
149C22×C31—41no—
150C22×C31—41no—
151C22×C31C2×SD1612yescomplemented
152C22×C31C22×D411yescharacteristic
153C2×D31—21no—
154C2×D31—21no—
155C2×D31—28no—
156C2×D31—41no—
157C2×D31—41no—
158C2×D31—41no—
159C2×D31—84no—
160C2×D31#12012yes—
161C2×D31C2×SD1612yes—
162C2×D31C2×SD1612yescomplemented
163C2×D31#13712yes—
164C4×C31—24no—
165C4×C31—21no—
166C4×C31—41no—
167C4×C31C22×D412yes—
168Dic31—21no—
169Dic31—24no—
170Dic31—41no—
171Dic31#12012yes—
172C23×Q8—311no—
173C22×SD16—312no—
174C2×#121—311no—
175#448—311no—
176#448—312no—
177#448—311no—
178#448—312no—
179#448—311no—
180#448D3111yescharacteristic, direct product factor
181C2×#103—312no—
182C2×C22⋊C8—311no—
183C2×C31—22no—
184C2×C31—81no—
185C2×C31C2×#12011yescharacteristic
186C2×C31C22×SD1611yescharacteristic
187C2×C31C2×#13711yescharacteristic
188D31—22no—
189D31—81no—
190D31#44814yesdirect product factor
191C22×Q8—312no—
192C22×Q8—312no—
193C22×Q8—311no—
194C22×Q8—621no—
195C22×Q8—621no—
196C22×Q8—621no—
197C22×Q8—628no—
198C22×Q8C2×D3111yescharacteristic, complemented
199C22×D4—311no—
200C22×D4—621no—
201C23×C4—311no—
202C23×C4—621no—
203C2×SD16—312no—
204C2×SD16—312no—
205C2×SD16—312no—
206C2×SD16—628no—
207C2×SD16C2×D3112yescomplemented
208C22×C8—312no—
209#121—311no—
210#121—311no—
211#121—311no—
212#121—621no—
213#121—621no—
214#121C2×D3111yescharacteristic
215C2×C4⋊C4—311no—
216C2×C22⋊C4—621no—
217#103—312no—
218#103—312no—
219#103—312no—
220#103C2×D3112yescomplemented
221C22⋊C8—311no—
222C22⋊C8—312no—
223C22⋊C8C2×D3111yescharacteristic, complemented
224C31C2×#44811yescharacteristic, complemented, Sylow
225C24—311no—
226C24—621no—
227C2×Q8—24no—
228C2×Q8—21no—
229C2×Q8—318no—
230C2×Q8—312no—
231C2×Q8—622no—
232C2×Q8—622no—
233C2×Q8—624no—
234C2×Q8—628no—
235C2×Q8—624no—
236C2×Q8—624no—
237C2×Q8—628no—
238C2×Q8—628no—
239C2×Q8—624no—
240C2×Q8—621no—
241C2×Q8C22×D3112yes—
242C2×D4—21no—
243C2×D4—311no—
244C2×D4—621no—
245C2×D4—624no—
246C2×D4—621no—
247C2×D4—621no—
248C2×D4—621no—
249C2×D4—1244no—
250C2×D4C22×D3111yescharacteristic
251C22×C4—21no—
252C22×C4—311no—
253C22×C4—311no—
254C22×C4—311no—
255C22×C4—311no—
256C22×C4—311no—
257C22×C4—624no—
258C22×C4—622no—
259C22×C4—622no—
260C22×C4—628no—
261C22×C4—621no—
262C22×C4—621no—
263C22×C4C22×D3111yescharacteristic
264SD16—24no—
265SD16—624no—
266SD16—624no—
267SD16—624no—
268C2×C8—312no—
269C2×C8—624no—
270C2×C8C22×D3112yes—
271C4⋊C4—311no—
272C4⋊C4—621no—
273C4⋊C4C22×D3111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—621no—
276C22⋊C4—621no—
277C22⋊C4—621no—
278C23—21no—
279C23—311no—
280C23—314no—
281C23—311no—
282C23—622no—
283C23—622no—
284C23—621no—
285C23—1242no—
286C23—1242no—
287C23—1242no—
288C23D4×D3111yescharacteristic
289Q8—24no—
290Q8—22no—
291Q8—314no—
292Q8—628no—
293Q8—628no—
294Q8—624no—
295Q8—622no—
296Q8D4×D3114yescomplemented
297D4—22no—
298D4—42no—
299D4—622no—
300D4—1242no—
301D4—1242no—
302D4—1242no—
303D4—1242no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—311no—
309C2×C4—314no—
310C2×C4—311no—
311C2×C4—622no—
312C2×C4—622no—
313C2×C4—628no—
314C2×C4—628no—
315C2×C4—622no—
316C2×C4—624no—
317C2×C4—622no—
318C2×C4—622no—
319C2×C4—621no—
320C2×C4—1242no—
321C2×C4D4×D3111yescharacteristic
322C2×C4C23×D3111yescharacteristic
323C8—22no—
324C8—622no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—312no—
331C22—312no—
332C22—312no—
333C22—312no—
334C22—628no—
335C22—621no—
336C22—621no—
337C22—1241no—
338C22—1241no—
339C22—1241no—
340C22—2484no—
341C22SD16×D3112yescomplemented
342C22C2×D4×D3111yescharacteristic
343C4—24no—
344C4—21no—
345C4—41no—
346C4—312no—
347C4—621no—
348C4—624no—
349C4—1241no—
350C4C2×D4×D3112yes—
351C2—22no—
352C2—81no—
353C2—314no—
354C2—622no—
355C2—2481no—
356C2#120×D3111yescharacteristic
357C2C2×SD16×D3111yescharacteristic
358C2#137×D3111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#448C2
socleC22×C31C22×D4
prime radical#121×D31C2
hypercenter#448D31
hypocenterC31C2×#448
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D31C2semidirect
2. C2×SD16×D31C2semidirect
4. #14521190C2semidirect
5. #14521189C2semidirect
6. #14521188C2semidirect
7. #14521187C2semidirect
8. #14521186C2semidirect
9. #103×D31C2semidirect
10. C22⋊C8×D31C2semidirect
11. C31×#448C2semidirect
12. C22×Q8×C31C22semidirect
17. C2×Q8×D31C22semidirect
23. C22×Dic62C22semidirect
24. C2×SD16×C31C22semidirect
26. C2×#92481C22semidirect
29. C2×#92480C22semidirect
32. C2×#92470C22semidirect
38. C31×#103C22semidirect
39. C31×C22⋊C8C22semidirect
40. #11758423C22semidirect
41. C22⋊(C31⋊C8)C22semidirect
42. (C2×D31)⋊C8C22semidirect
43. #11758404C22semidirect
44. #11758398C22semidirect
64. Q8×D31D4semidirect
106. C22×D31SD16semidirect
111. Q8×C31C2×D4semidirect
142. Dic62C2×D4semidirect
151. C22×C31C2×SD16semidirect
162. C2×D31C2×SD16semidirect
180. #448D31direct (N × H)
190. D31#448direct (N × H)
198. C22×Q8C2×D31semidirect
207. C2×SD16C2×D31semidirect
220. #103C2×D31semidirect
223. C22⋊C8C2×D31semidirect
224. C31C2×#448semidirect
296. Q8D4×D31semidirect
341. C22SD16×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×SD16×D312
#121×D312
#145211902
#145211892
#145211882
#145211872
#145211862
#103×D312
C22⋊C8×D312
C31×#4482
C2×#44831