Finite Group Databasebeta

← #16854565

Subgroups — #16854565

Order 3968.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D31C211yesmaximal, characteristic
2#14521525C211yesmaximal, characteristic, complemented
3#14521521C211yesmaximal, characteristic
4#14521152C211yesmaximal, characteristic, complemented
5#14521149C211yesmaximal, characteristic, complemented
6#14521147C211yesmaximal, characteristic, complemented
7#14521146C211yesmaximal, characteristic, complemented
8C4×#92470C211yesmaximal, characteristic, complemented
9SD16×Dic31C211yesmaximal, characteristic, complemented
10#14521136C211yesmaximal, characteristic, complemented
11#14521084C211yesmaximal, characteristic, complemented
12#14521056C211yesmaximal, characteristic, complemented
13#14521052C211yesmaximal, characteristic
14#14521003C211yesmaximal, characteristic, complemented
15C4×SD16×C31C211yesmaximal, characteristic, complemented
16C2×Q8×D31C2211yescharacteristic
17C2×D4×D31C2211yescharacteristic
18C2×SD16×C31C2211yescharacteristic
19C4×Q8×C31C2211yescharacteristic, complemented
20C4×D4×C31C2211yescharacteristic
21Q8×Dic31C2211yescharacteristic, complemented
22C2×#92481C2211yescharacteristic
23D4×Dic31C2211yescharacteristic
24C2×#92480C2211yescharacteristic
25C4×C22⋊D31—21no—
26C2×(C2×D31)⋊C4—21no—
27SD16×D31C418yescomplemented
28C2×#92470C2211yescharacteristic
29C2×#92469C2211yescharacteristic
30C2×C8×D31C2211yescharacteristic
31#11758470C2211yescharacteristic
32#11758469—21no—
33#11758469C2211yescharacteristic
34#11758469C2211yescharacteristic
35#11758464—21no—
36#11758464C2211yescharacteristic, complemented
37(C22⋊D31)⋊C4—21no—
38C22⋊C4×D31—21no—
39C4×D124C2211yescharacteristic
40#11758449—21no—
41#11758449C2211yescharacteristic
42C4×Dic62—21no—
43C4×Dic62C2211yescharacteristic, complemented
44C31×#106C2211yescharacteristic, complemented
45C31×#103C2211yescharacteristic, complemented
46C31×#102C2211yescharacteristic
47C4×C8×C31C2211yescharacteristic, complemented
48#11758423C2211yescharacteristic, complemented
49#11758420C2211yescharacteristic
50#11758409C2211yescharacteristic
51#11758405C2211yescharacteristic, complemented
52#11758404C2211yescharacteristic, complemented
53#11758403C2211yescharacteristic, complemented
54C8×Dic31C2211yescharacteristic, complemented
55#11758398C2211yescharacteristic, complemented
56#11758397C2211yescharacteristic
57#11758395C2211yescharacteristic, complemented
58#11758395C2211yescharacteristic, complemented
59#11758391C2211yescharacteristic, complemented
60C23×D31—21no—
61C2×Q8×C31C2311yescharacteristic
62C2×D4×C31C2311yescharacteristic
63C22×C4×C31—21no—
64C2×C22⋊D31—21no—
65C22×Dic31—21no—
66Q8×D31—22no—
67Q8×D31C2×C414yescomplemented
68D4×D31—22no—
69D4×D31C2×C414yes—
70C2×D124C2311yescharacteristic
71C2×C4×D31—21no—
72C2×C4×D31—21no—
73C2×C4×D31—21no—
74C2×C4×D31C2311yescharacteristic
75C2×Dic62—21no—
76C2×Dic62C2311yescharacteristic
77SD16×C31C2×C414yescomplemented
78C2×C8×C31C2311yescharacteristic
79C2×C8×C31C2311yescharacteristic
80C4⋊C4×C31—21no—
81C4⋊C4×C31C2311yescharacteristic
82C4⋊C4×C31C2311yescharacteristic
83C22⋊C4×C31—21no—
84C42×C31—21no—
85C42×C31C2311yescharacteristic
86C22⋊Dic31—21no—
87#92481C2×C414yescomplemented
88#92480C2×C414yescomplemented
89(C2×D31)⋊C4—21no—
90(C2×D31)⋊C4—21no—
91(C2×D31)⋊C4—21no—
92(C2×D31)⋊C4—21no—
93(C2×D31)⋊C4D411yescharacteristic
94(C2×D31)⋊C4D411yescharacteristic
95C4⋊Dic31—21no—
96C4⋊Dic31C2311yescharacteristic
97C4⋊Dic31C2311yescharacteristic
98Dic31⋊C4—21no—
99Dic31⋊C4—21no—
100Dic31⋊C4D411yescharacteristic
101Dic31⋊C4D411yescharacteristic
102C4×Dic31—21no—
103C4×Dic31—21no—
104C4×Dic31—21no—
105C4×Dic31—21no—
106C4×Dic31—21no—
107C4×Dic31C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110#92470C2×C414yescomplemented
111#92469—22no—
112C8×D31C2×C414yescomplemented
113C23×C31—21no—
114C22×D31—21no—
115C22×D31—24no—
116C22×D31—81no—
117C22×D31—81no—
118C22×D31C2×D411yescharacteristic
119Q8×C31—21no—
120Q8×C31C22×C412yes—
121D4×C31—21no—
122D4×C31C22×C412yes—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—81no—
127C2×C4×C31C2×D411yescharacteristic
128C2×C4×C31C2×D411yescharacteristic
129C2×C4×C31C2411yescharacteristic
130C22⋊D31—24no—
131C2×Dic31—21no—
132C2×Dic31—21no—
133C2×Dic31—21no—
134C2×Dic31—81no—
135C2×Dic31C2×D411yescharacteristic
136C2×Dic31C2×D411yescharacteristic
137C2×Dic31C2×D411yescharacteristic
138D124—21no—
139D124C22×C412yes—
140C4×D31—24no—
141C4×D31—81no—
142C4×D31—81no—
143C4×D31C22×C412yes—
144C4×D31#4112yes—
145Dic62—21no—
146Dic62—24no—
147Dic62C22×C412yes—
148C8×C31—21no—
149C8×C31C22×C412yes—
150C31⋊C8—21no—
151C31⋊C8C22×C412yes—
152#2949—311nomaximal, Sylow
153C22×C31—22no—
154C22×C31—41no—
155C22×C31—41no—
156C22×C31C22×D411yescharacteristic
157C2×D31—21no—
158C2×D31—21no—
159C2×D31—22no—
160C2×D31—41no—
161C2×D31—41no—
162C2×D31—84no—
163C2×D31C4×D412yes—
164C4×C31—22no—
165C4×C31—21no—
166C4×C31—21no—
167C4×C31—41no—
168C4×C31—41no—
169C4×C31C23×C411yescharacteristic
170C4×C31C2×#4111yescharacteristic
171Dic31—22no—
172Dic31—21no—
173Dic31—21no—
174Dic31—41no—
175Dic31—41no—
176Dic31C4×D412yes—
177C22×SD16—311no—
178#519—311no—
179#518—311no—
180#440—311no—
181#440—311no—
182#440—311no—
183#440—311no—
184C4×SD16—311no—
185C4×SD16—311no—
186C4×SD16—311no—
187C4×SD16D3111yescharacteristic, complemented
188#427—311no—
189#419—311no—
190#418—311no—
191#405—311no—
192C2×C31—42no—
193C2×C31C2×C4×D411yescharacteristic
194C2×C31#57711yescharacteristic
195C2×C31#57711yescharacteristic
196D31—22no—
197D31—42no—
198C22×Q8—311no—
199C22×D4—311no—
200C2×SD16—311no—
201C2×SD16—311no—
202C2×SD16—318no—
203C2×SD16—311no—
204C2×SD16C2×D3111yescharacteristic
205C2×M16—311no—
206C22×C8—311no—
207C4×Q8—311no—
208C4×Q8—311no—
209C4×Q8—311no—
210C4×Q8—621no—
211C4×Q8—621no—
212C4×Q8C2×D3111yescharacteristic, complemented
213C4×D4—311no—
214C4×D4—311no—
215C4×D4—311no—
216C4×D4—621no—
217C4×D4—621no—
218C4×D4C2×D3111yescharacteristic
219#117—311no—
220#117—311no—
221#117—311no—
222#117—621no—
223#117—621no—
224C2×C22⋊C4—621no—
225C2×C22⋊C4—621no—
226#107—311no—
227#107—311no—
228#106—311no—
229#106C2×D3111yescharacteristic, complemented
230#103—311no—
231#103—311no—
232#103—311no—
233#103C2×D3111yescharacteristic, complemented
234#102—311no—
235#102—311no—
236#102—311no—
237#102C2×D3111yescharacteristic
238#97—311no—
239#97—311no—
240C4×C8—311no—
241C4×C8C2×D3111yescharacteristic, complemented
242C31#294911yescharacteristic, complemented, Sylow
243C24—621no—
244C2×Q8—314no—
245C2×Q8—311no—
246C2×Q8—621no—
247C2×Q8—622no—
248C2×Q8C22×D3111yescharacteristic
249C2×D4—314no—
250C2×D4—311no—
251C2×D4—622no—
252C2×D4—621no—
253C2×D4C22×D3111yescharacteristic
254C22×C4—21no—
255C22×C4—311no—
256C22×C4—621no—
257C22×C4—621no—
258C22×C4—621no—
259C22×C4—621no—
260SD16—314no—
261SD16—314no—
262SD16—314no—
263SD16C4×D3114yescomplemented
264M16—622no—
265C2×C8—311no—
266C2×C8—314no—
267C2×C8—311no—
268C2×C8C22×D3111yescharacteristic
269C2×C8C22×D3111yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—311no—
272C4⋊C4—311no—
273C4⋊C4—311no—
274C4⋊C4—311no—
275C4⋊C4—621no—
276C4⋊C4—621no—
277C4⋊C4—621no—
278C4⋊C4C22×D3111yescharacteristic
279C4⋊C4C22×D3111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—311no—
282C22⋊C4—311no—
283C22⋊C4—621no—
284C22⋊C4—621no—
285C22⋊C4—621no—
286C22⋊C4—621no—
287C22⋊C4—621no—
288C42—21no—
289C42—311no—
290C42—621no—
291C42—621no—
292C42—621no—
293C42—621no—
294C42—621no—
295C42C22×D3111yescharacteristic
296C23—21no—
297C23—311no—
298C23—624no—
299C23—621no—
300C23—2481no—
301C23—2481no—
302Q8—21no—
303Q8—312no—
304Q8—624no—
305Q8—621no—
306Q8C2×C4×D3112yes—
307D4—21no—
308D4—312no—
309D4—624no—
310D4—621no—
311D4C2×C4×D3112yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—311no—
317C2×C4—312no—
318C2×C4—312no—
319C2×C4—311no—
320C2×C4—311no—
321C2×C4—624no—
322C2×C4—621no—
323C2×C4—621no—
324C2×C4—621no—
325C2×C4—2481no—
326C2×C4—2481no—
327C2×C4—2481no—
328C2×C4D4×D3111yescharacteristic
329C2×C4D4×D3111yescharacteristic
330C2×C4C23×D3111yescharacteristic
331C8—21no—
332C8—312no—
333C8—621no—
334C8C2×C4×D3112yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—312no—
339C22—621no—
340C22—621no—
341C22—622no—
342C22—1241no—
343C22—1241no—
344C22—2484no—
345C22C2×D4×D3111yescharacteristic
346C4—21no—
347C4—22no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—312no—
352C4—621no—
353C4—622no—
354C4—621no—
355C4—1241no—
356C4—1241no—
357C4C22×C4×D3111yescharacteristic
358C4#41×D3111yescharacteristic
359C2—42no—
360C2—622no—
361C2—1242no—
362C2C4×D4×D3111yescharacteristic
363C2#1452185511yescharacteristic
364C2#1452185611yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutatorC4×C31C23×C4
FrattiniC2×C4C23×D31
FittingC4×SD16×C31C2
socleC22×C31C22×D4
prime radicalC2×D4×D31C22
cosocleC2×C4×C31C24
hypercenterC4×SD16D31
hypocenterC31#2949
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #14521525C2semidirect
4. #14521152C2semidirect
5. #14521149C2semidirect
6. #14521147C2semidirect
7. #14521146C2semidirect
8. C4×#92470C2semidirect
9. SD16×Dic31C2semidirect
10. #14521136C2semidirect
11. #14521084C2semidirect
12. #14521056C2semidirect
14. #14521003C2semidirect
15. C4×SD16×C31C2semidirect
19. C4×Q8×C31C22semidirect
21. Q8×Dic31C22semidirect
27. SD16×D31C4semidirect
36. #11758464C22semidirect
43. C4×Dic62C22semidirect
44. C31×#106C22semidirect
45. C31×#103C22semidirect
47. C4×C8×C31C22semidirect
48. #11758423C22semidirect
51. #11758405C22semidirect
52. #11758404C22semidirect
53. #11758403C22semidirect
54. C8×Dic31C22semidirect
55. #11758398C22semidirect
57. #11758395C22semidirect
58. #11758395C22semidirect
59. #11758391C22semidirect
67. Q8×D31C2×C4semidirect
77. SD16×C31C2×C4semidirect
87. #92481C2×C4semidirect
88. #92480C2×C4semidirect
110. #92470C2×C4semidirect
112. C8×D31C2×C4semidirect
187. C4×SD16D31semidirect
212. C4×Q8C2×D31semidirect
229. #106C2×D31semidirect
233. #103C2×D31semidirect
241. C4×C8C2×D31semidirect
242. C31#2949semidirect
263. SD16C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D312
#145215252
#145215212
#145211522
#145211492
#145211472
#145211462
C4×#924702
SD16×Dic312
#145211362
#145210842
#145210562
#145210522
#145210032
C4×SD16×C312
#294931