Finite Group Databasebeta

← #16854538

Subgroups — #16854538

Order 3968.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758491C211yesmaximal, characteristic
2#14521531C211yesmaximal, characteristic, complemented
3#14521521C211yesmaximal, characteristic, complemented
4#14521160C211yesmaximal, characteristic, complemented
5#14521159C211yesmaximal, characteristic, complemented
6#14521150C211yesmaximal, characteristic, complemented
7#14521147C211yesmaximal, characteristic, complemented
8#14521134C211yesmaximal, characteristic, complemented
9C4×D248C211yesmaximal, characteristic, complemented
10#14521129C211yesmaximal, characteristic, complemented
11#14521088C211yesmaximal, characteristic, complemented
12#14521044C211yesmaximal, characteristic, complemented
13#102×D31C211yesmaximal, characteristic, complemented
14#14521003C211yesmaximal, characteristic, complemented
15C4×SD16×C31C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic
17C2×D4×D31C2211yescharacteristic
18C2×SD16×C31C2211yescharacteristic
19C4×Q8×C31C2211yescharacteristic, complemented
20C4×D4×C31C2211yescharacteristic
21Q8×Dic31C2211yescharacteristic, complemented
22C2×#92481C2211yescharacteristic
23D4×Dic31C2211yescharacteristic
24C2×#92479C2211yescharacteristic
25C4×C22⋊D31—21no—
26C2×(C2×D31)⋊C4—21no—
27#11758491C418yescomplemented
28C2×D248C2211yescharacteristic
29C2×#92469C2211yescharacteristic
30C2×C8×D31C2211yescharacteristic
31#11758470—21no—
32#11758470C2211yescharacteristic
33#11758470C2211yescharacteristic, complemented
34#11758469C2211yescharacteristic
35C4⋊C4×D31—21no—
36C4⋊C4×D31C2211yescharacteristic
37(C22⋊D31)⋊C4—21no—
38C22⋊C4×D31—21no—
39C4×D124—21no—
40C4×D124C2211yescharacteristic
41C4×D124C2211yescharacteristic, complemented
42#11758449—21no—
43#11758449C2211yescharacteristic
44C31×#106C2211yescharacteristic, complemented
45C31×#103C2211yescharacteristic, complemented
46C31×#102C2211yescharacteristic
47C4×C8×C31C2211yescharacteristic, complemented
48#11758423C2211yescharacteristic, complemented
49#11758420C2211yescharacteristic
50#11758409C2211yescharacteristic
51#11758409C2211yescharacteristic, complemented
52#11758406C2211yescharacteristic, complemented
53#11758403C2211yescharacteristic, complemented
54C8×Dic31C2211yescharacteristic, complemented
55#11758397C2211yescharacteristic
56#11758397C2211yescharacteristic, complemented
57#11758396C2211yescharacteristic, 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—
66#92499—22no—
67#92499C2×C414yescomplemented
68D4×D31—22no—
69D4×D31C2×C414yescomplemented
70C2×D124—21no—
71C2×D124C2311yescharacteristic
72C2×D124C2311yescharacteristic
73C2×C4×D31—21no—
74C2×C4×D31—21no—
75C2×C4×D31—21no—
76C2×C4×D31—21no—
77C2×C4×D31—21no—
78C2×C4×D31—21no—
79C2×C4×D31C2311yescharacteristic
80SD16×C31C2×C414yescomplemented
81C2×C8×C31C2311yescharacteristic
82C2×C8×C31C2311yescharacteristic
83C4⋊C4×C31—21no—
84C4⋊C4×C31C2311yescharacteristic
85C4⋊C4×C31C2311yescharacteristic
86C22⋊C4×C31—21no—
87C42×C31—21no—
88C42×C31C2311yescharacteristic
89C22⋊Dic31—21no—
90#92481C2×C414yescomplemented
91#92479C2×C414yescomplemented
92(C2×D31)⋊C4—21no—
93(C2×D31)⋊C4—21no—
94(C2×D31)⋊C4—21no—
95(C2×D31)⋊C4—21no—
96(C2×D31)⋊C4—21no—
97(C2×D31)⋊C4D411yescharacteristic
98(C2×D31)⋊C4D411yescharacteristic
99C4⋊Dic31—21no—
100C4⋊Dic31C2311yescharacteristic
101C4⋊Dic31C2311yescharacteristic
102Dic31⋊C4—21no—
103Dic31⋊C4D411yescharacteristic
104Dic31⋊C4D411yescharacteristic
105C4×Dic31—21no—
106C4×Dic31—21no—
107C4×Dic31C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110D248C2×C414yescomplemented
111#92469C2×C414yescomplemented
112C8×D31—22no—
113C23×C31—21no—
114C22×D31—21no—
115C22×D31—21no—
116C22×D31—24no—
117C22×D31—81no—
118C22×D31—81no—
119C22×D31C2×D411yescharacteristic
120Q8×C31—21no—
121Q8×C31C22×C412yes—
122D4×C31—21no—
123D4×C31C22×C412yes—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—81no—
128C2×C4×C31C2×D411yescharacteristic
129C2×C4×C31C2×D411yescharacteristic
130C2×C4×C31C2411yescharacteristic
131C22⋊D31—24no—
132C2×Dic31—21no—
133C2×Dic31—21no—
134C2×Dic31—81no—
135C2×Dic31C2×D411yescharacteristic
136C2×Dic31C2×D411yescharacteristic
137C2×Dic31C2×D411yescharacteristic
138D124—24no—
139D124—21no—
140D124—21no—
141D124C22×C412yes—
142D124C22×C412yes—
143C4×D31—24no—
144C4×D31—81no—
145C4×D31—81no—
146C4×D31—81no—
147C4×D31—81no—
148C4×D31—81no—
149C4×D31C22×C412yes—
150C4×D31#4112yes—
151C8×C31—21no—
152C8×C31C22×C412yes—
153C31⋊C8—21no—
154C31⋊C8C22×C412yes—
155#2946—311nomaximal, Sylow
156C22×C31—22no—
157C22×C31—41no—
158C22×C31—41no—
159C22×C31C22×D411yescharacteristic
160C2×D31—21no—
161C2×D31—21no—
162C2×D31—22no—
163C2×D31—22no—
164C2×D31—41no—
165C2×D31—41no—
166C2×D31—41no—
167C2×D31—41no—
168C2×D31—84no—
169C2×D31C4×D412yes—
170C4×C31—22no—
171C4×C31—21no—
172C4×C31—21no—
173C4×C31—41no—
174C4×C31—41no—
175C4×C31C23×C411yescharacteristic
176C4×C31C2×#4111yescharacteristic
177Dic31—21no—
178Dic31—21no—
179Dic31—41no—
180Dic31—41no—
181Dic31C4×D412yes—
182C2×AGL(1,ℤ/8)—311no—
183#520—311no—
184#518—311no—
185#442—311no—
186#442—311no—
187#440—311no—
188#440—311no—
189C4×SD16—311no—
190C4×SD16D3111yescharacteristic, complemented
191C4×D8—311no—
192C4×D8—311no—
193#428—311no—
194#416—311no—
195C2×#102—311no—
196#405—311no—
197C2×C31—42no—
198C2×C31C2×C4×D411yescharacteristic
199C2×C31#57611yescharacteristic
200C2×C31#57711yescharacteristic
201D31—22no—
202D31—42no—
203D31—42no—
204C2×#41—311no—
205C22×D4—311no—
206AGL(1,ℤ/8)—318no—
207C2×SD16—311no—
208C2×SD16C2×D3111yescharacteristic
209C2×D8—311no—
210C2×D8—311no—
211C2×M16—311no—
212C22×C8—311no—
213C4×Q8—311no—
214C4×Q8C2×D3111yescharacteristic, complemented
215C4×D4—311no—
216C4×D4—311no—
217C4×D4—311no—
218C4×D4—311no—
219C4×D4—311no—
220C4×D4—621no—
221C4×D4—621no—
222C4×D4—621no—
223C4×D4—621no—
224C4×D4C2×D3111yescharacteristic
225#117—311no—
226#117—311no—
227#117—621no—
228C2×C4⋊C4—311no—
229C2×C4⋊C4—621no—
230C2×C22⋊C4—621no—
231C2×C22⋊C4—621no—
232#107—311no—
233#107—311no—
234#106—311no—
235#106C2×D3111yescharacteristic, complemented
236#103—311no—
237#103C2×D3111yescharacteristic, complemented
238#102—311no—
239#102—311no—
240#102—311no—
241#102—311no—
242#102—311no—
243#102C2×D3111yescharacteristic
244#97—311no—
245#97—311no—
246C4×C8—311no—
247C4×C8C2×D3111yescharacteristic, complemented
248C31#294611yescharacteristic, complemented, Sylow
249C24—621no—
250#41—314no—
251#41—622no—
252C2×Q8C22×D3111yescharacteristic
253C2×D4—314no—
254C2×D4—311no—
255C2×D4—311no—
256C2×D4—621no—
257C2×D4—622no—
258C2×D4—621no—
259C2×D4C22×D3111yescharacteristic
260C22×C4—21no—
261C22×C4—311no—
262C22×C4—621no—
263C22×C4—621no—
264C22×C4—621no—
265C22×C4—621no—
266C22×C4—621no—
267C22×C4—621no—
268C22×C4—621no—
269SD16—314no—
270SD16C4×D3114yescomplemented
271D8—314no—
272D8—314no—
273M16—314no—
274C2×C8—311no—
275C2×C8—311no—
276C2×C8—622no—
277C2×C8C22×D3111yescharacteristic
278C2×C8C22×D3111yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—311no—
281C4⋊C4—311no—
282C4⋊C4—311no—
283C4⋊C4—311no—
284C4⋊C4—621no—
285C4⋊C4—621no—
286C4⋊C4C22×D3111yescharacteristic
287C4⋊C4C22×D3111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—311no—
290C22⋊C4—311no—
291C22⋊C4—621no—
292C22⋊C4—621no—
293C22⋊C4—621no—
294C22⋊C4—621no—
295C22⋊C4—621no—
296C22⋊C4—621no—
297C42—21no—
298C42—311no—
299C42—621no—
300C42—621no—
301C42C22×D3111yescharacteristic
302C23—21no—
303C23—311no—
304C23—624no—
305C23—621no—
306C23—621no—
307C23—2481no—
308C23—2481no—
309Q8—21no—
310Q8C2×C4×D3112yes—
311D4—21no—
312D4—312no—
313D4—312no—
314D4—624no—
315D4—624no—
316D4—621no—
317D4—621no—
318D4C2×C4×D3112yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—311no—
324C2×C4—312no—
325C2×C4—312no—
326C2×C4—311no—
327C2×C4—311no—
328C2×C4—624no—
329C2×C4—621no—
330C2×C4—621no—
331C2×C4—2481no—
332C2×C4—2481no—
333C2×C4—2481no—
334C2×C4—2481no—
335C2×C4—2481no—
336C2×C4—2481no—
337C2×C4D4×D3111yescharacteristic
338C2×C4D4×D3111yescharacteristic
339C2×C4C23×D3111yescharacteristic
340C8—21no—
341C8—312no—
342C8—621no—
343C8C2×C4×D3112yes—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—312no—
348C22—621no—
349C22—621no—
350C22—622no—
351C22—622no—
352C22—1241no—
353C22—1241no—
354C22—1241no—
355C22—1241no—
356C22—2484no—
357C22C2×D4×D3111yescharacteristic
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—312no—
364C4—621no—
365C4—621no—
366C4—1241no—
367C4—1241no—
368C4C22×C4×D3111yescharacteristic
369C4#41×D3111yescharacteristic
370C2—42no—
371C2—622no—
372C2—1242no—
373C2—1242no—
374C2C4×D4×D3111yescharacteristic
375C2#1452185111yescharacteristic
376C2#1452185511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutatorC4×C31C23×C4
FrattiniC2×C4C23×D31
FittingC4×SD16×C31C2
socleC22×C31C22×D4
prime radicalC2×#11758491C2
cosocleC2×C4×C31C24
hypercenterC4×SD16D31
hypocenterC31#2946
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. #14521531C2semidirect
3. #14521521C2semidirect
4. #14521160C2semidirect
5. #14521159C2semidirect
6. #14521150C2semidirect
7. #14521147C2semidirect
8. #14521134C2semidirect
9. C4×D248C2semidirect
10. #14521129C2semidirect
11. #14521088C2semidirect
12. #14521044C2semidirect
13. #102×D31C2semidirect
14. #14521003C2semidirect
15. C4×SD16×C31C2semidirect
19. C4×Q8×C31C22semidirect
21. Q8×Dic31C22semidirect
27. #11758491C4semidirect
33. #11758470C22semidirect
41. C4×D124C22semidirect
44. C31×#106C22semidirect
45. C31×#103C22semidirect
47. C4×C8×C31C22semidirect
48. #11758423C22semidirect
51. #11758409C22semidirect
52. #11758406C22semidirect
53. #11758403C22semidirect
54. C8×Dic31C22semidirect
56. #11758397C22semidirect
57. #11758396C22semidirect
58. #11758395C22semidirect
59. #11758391C22semidirect
67. #92499C2×C4semidirect
69. D4×D31C2×C4semidirect
80. SD16×C31C2×C4semidirect
90. #92481C2×C4semidirect
91. #92479C2×C4semidirect
110. D248C2×C4semidirect
111. #92469C2×C4semidirect
190. C4×SD16D31semidirect
214. C4×Q8C2×D31semidirect
235. #106C2×D31semidirect
237. #103C2×D31semidirect
247. C4×C8C2×D31semidirect
248. C31#2946semidirect
270. SD16C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584912
#145215312
#145215212
#145211602
#145211592
#145211502
#145211472
#145211342
C4×D2482
#145211292
#145210882
#145210442
#102×D312
#145210032
C4×SD16×C312
#294631