Finite Group Databasebeta

← #16857731

Subgroups — #16857731

Order 3968.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521737C211yesmaximal, characteristic, complemented
2#14521684C211yesmaximal, characteristic
3#14521371C211yesmaximal, characteristic, complemented
4#14521361C211yesmaximal, characteristic, complemented
5#14521346C211yesmaximal, characteristic, complemented
6#14521294C211yesmaximal, characteristic, complemented
7#14521277C211yesmaximal, characteristic, complemented
8#14521246C211yesmaximal, characteristic, complemented
9#14521201C211yesmaximal, characteristic, complemented
10#14521201C211yesmaximal, characteristic, complemented
11#14521151C211yesmaximal, characteristic, complemented
12#14521122C211yesmaximal, characteristic, complemented
13#14521052C211yesmaximal, characteristic, complemented
14#102×D31C211yesmaximal, characteristic, complemented
15C31×#490C211yesmaximal, characteristic, complemented
16C2×D4×D31—21no—
17C2×D4×D31C2211yescharacteristic
18C2×D4×D31C2211yescharacteristic
19C31×#127C2211yescharacteristic
20C31×#125C2211yescharacteristic, complemented
21#11758515—21no—
22#11758515C2211yescharacteristic
23#11758515C2211yescharacteristic
24#11758514—21no—
25#11758514—21no—
26C2×#92480C2211yescharacteristic, complemented
27C2×#92480C2211yescharacteristic, complemented
28C2×#92470C2211yescharacteristic
29C2×#92469C2211yescharacteristic, complemented
30C2×C8×D31C2211yescharacteristic, complemented
31#11758475—21no—
32#11758474C2211yescharacteristic, complemented
33#11758474C2211yescharacteristic, complemented
34#11758471—21no—
35#11758469—21no—
36#11758469C2211yescharacteristic, complemented
37C4⋊C4×D31C2211yescharacteristic, complemented
38#11758467—21no—
39C4×D124C2211yescharacteristic
40C4×D124C2211yescharacteristic
41C4×Dic62C2211yescharacteristic, complemented
42C31×#102C2211yescharacteristic, complemented
43C31×#102C2211yescharacteristic, complemented
44C31×#102C2211yescharacteristic, complemented
45C31×#102C2211yescharacteristic, complemented
46C31×#97C2211yescharacteristic, complemented
47#11758420C2211yescharacteristic, complemented
48#11758420C2211yescharacteristic, complemented
49#11758409C2211yescharacteristic
50(C2×D31)⋊C8C2211yescharacteristic, complemented
51(C2×D31)⋊C8C2211yescharacteristic, complemented
52#11758406C2211yescharacteristic, complemented
53#11758404C2211yescharacteristic, complemented
54#11758397C2211yescharacteristic
55#11758397C2211yescharacteristic
56#11758396C2211yescharacteristic, complemented
57#11758395C2211yescharacteristic, complemented
58C4⋊(C31⋊C8)C2211yescharacteristic, complemented
59C23×D31—21no—
60C23×D31—21no—
61C2×D4×C31—21no—
62C2×D4×C31—21no—
63C2×D4×C31C2311yescharacteristic
64C2×D4×C31C2311yescharacteristic
65C2×C22⋊D31—21no—
66C2×C22⋊D31—21no—
67C2×C22⋊D31—21no—
68C2×C22⋊D31—21no—
69D4×D31—22no—
70D4×D31—22no—
71D4×D31—41no—
72D4×D31—41no—
73D4×D31—44no—
74C2×D124D411yescharacteristic
75C2×D124C2311yescharacteristic
76C2×C4×D31—21no—
77C2×C4×D31—21no—
78C2×C4×D31C2311yescharacteristic
79C2×C4×D31C2311yescharacteristic
80C2×Dic62C2311yescharacteristic
81C2×C8×C31C2311yescharacteristic
82C2×C8×C31C2311yescharacteristic
83C4⋊C4×C31—21no—
84C4⋊C4×C31—21no—
85C4⋊C4×C31C2311yescharacteristic
86C4⋊C4×C31C2311yescharacteristic
87C42×C31C2311yescharacteristic
88C22⋊Dic31—21no—
89C22⋊Dic31—21no—
90#92480—41no—
91#92480—41no—
92(C2×D31)⋊C4—21no—
93(C2×D31)⋊C4—21no—
94(C2×D31)⋊C4—21no—
95(C2×D31)⋊C4—21no—
96(C2×D31)⋊C4D411yescharacteristic
97(C2×D31)⋊C4D411yescharacteristic
98C4⋊Dic31—21no—
99C4⋊Dic31—21no—
100C4⋊Dic31D411yescharacteristic
101C4⋊Dic31C2311yescharacteristic
102C4⋊Dic31C2311yescharacteristic
103Dic31⋊C4—21no—
104Dic31⋊C4—21no—
105C4×Dic31—21no—
106C4×Dic31—21no—
107C2×C31⋊C8C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109#92470—22no—
110#92469—41no—
111C8×D31—41no—
112C23×C31—21no—
113C23×C31—21no—
114C22×D31—21no—
115C22×D31—42no—
116C22×D31—42no—
117C22×D31—42no—
118C22×D31—42no—
119C22×D31—81no—
120C22×D31—81no—
121C22×D31C2×D411yescharacteristic
122C22×D31C2×D411yescharacteristic
123D4×C31—21no—
124D4×C31—21no—
125D4×C31—21no—
126D4×C31—21no—
127D4×C31—42no—
128D4×C31—81no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2×D411yescharacteristic
133C2×C4×C31C2411yescharacteristic
134C22⋊D31—42no—
135C22⋊D31—42no—
136C22⋊D31—42no—
137C22⋊D31—42no—
138C2×Dic31—21no—
139C2×Dic31—21no—
140C2×Dic31—21no—
141C2×Dic31C2×D411yescharacteristic
142C2×Dic31C2×D411yescharacteristic
143D124—22no—
144D124—21no—
145D124#4112yes—
146C4×D31—21no—
147C4×D31—21no—
148C4×D31—21no—
149C4×D31—21no—
150C4×D31—42no—
151C4×D31—42no—
152Dic62—21no—
153Dic62—21no—
154C8×C31—21no—
155C8×C31—21no—
156C31⋊C8—21no—
157C31⋊C8—21no—
158#3310—311nomaximal, Sylow
159C22×C31—41no—
160C22×C31—41no—
161C22×C31—41no—
162C22×C31—41no—
163C22×C31—41no—
164C22×C31—41no—
165C22×C31C22×D411yescharacteristic
166C2×D31—21no—
167C2×D31—21no—
168C2×D31—21no—
169C2×D31—22no—
170C2×D31—21no—
171C2×D31—21no—
172C2×D31—41no—
173C2×D31—41no—
174C2×D31—84no—
175C2×D31—84no—
176C2×D31AGL(1,ℤ/8)12yes—
177C4×C31—21no—
178C4×C31—21no—
179C4×C31—41no—
180C4×C31—41no—
181C4×C31C2×#4111yescharacteristic
182C4×C31H(5,𝔽2)11yescharacteristic
183Dic31—21no—
184Dic31—21no—
185Dic31—41no—
186Dic31—41no—
187Dic31—41no—
188#552—311no—
189D42—311no—
190#490D3111yescharacteristic, complemented
191#482—311no—
192#480—311no—
193#478—311no—
194#468—311no—
195#465—311no—
196#460—311no—
197#450—311no—
198#450—311no—
199#440—311no—
200#435—311no—
201#418—311no—
202C2×#102—311no—
203C2×C31—81no—
204C2×C31—81no—
205C2×C31#54611yescharacteristic
206C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
207C2×C31#57711yescharacteristic
208D31—22no—
209D31—41no—
210D31—42no—
211C22×D4—311no—
212C22×D4—311no—
213C22×D4—621no—
214C2×SD16—311no—
215C2×SD16—311no—
216C2×SD16—311no—
217C2×M16—311no—
218C22×C8—311no—
219#127C2×D3111yescharacteristic
220#126—621no—
221#125—621no—
222#125C2×D3111yescharacteristic, complemented
223#123—621no—
224C22⋊Q8—311no—
225C22⋊Q8—311no—
226#121—311no—
227#121—311no—
228#121—621no—
229#120—621no—
230#120—621no—
231C4×Q8—311no—
232C4×D4—311no—
233C4×D4—311no—
234#117—311no—
235#117—621no—
236C2×C4⋊C4—311no—
237#107—311no—
238#107—311no—
239#106—311no—
240C4⋊C8—311no—
241#103—311no—
242#102—311no—
243#102—311no—
244#102—311no—
245#102—311no—
246#102—311no—
247#102C2×D3111yescharacteristic, complemented
248#102C2×D3111yescharacteristic, complemented
249#102C2×D3111yescharacteristic, complemented
250#102C2×D3111yescharacteristic, complemented
251C22⋊C8—311no—
252C22⋊C8—311no—
253#97C2×D3111yescharacteristic, complemented
254C31#331011yescharacteristic, complemented, Sylow
255C24—621no—
256C24—621no—
257C2×Q8—311no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—311no—
261C2×D4—311no—
262C2×D4—621no—
263C2×D4—621no—
264C2×D4—622no—
265C2×D4—622no—
266C2×D4—621no—
267C2×D4—621no—
268C2×D4—1241no—
269C2×D4—1241no—
270C2×D4—1244no—
271C2×D4C22×D3111yescharacteristic
272C2×D4C22×D3111yescharacteristic
273C22×C4—311no—
274C22×C4—311no—
275C22×C4—621no—
276C22×C4—621no—
277SD16—622no—
278SD16—1241no—
279SD16—1241no—
280M16—1241no—
281C2×C8—311no—
282C2×C8—311no—
283C2×C8—1241no—
284C2×C8C22×D3111yescharacteristic
285C2×C8C22×D3111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—311no—
289C4⋊C4—311no—
290C4⋊C4—311no—
291C4⋊C4—621no—
292C4⋊C4—621no—
293C4⋊C4—621no—
294C4⋊C4—621no—
295C4⋊C4C22×D3111yescharacteristic
296C4⋊C4C22×D3111yescharacteristic
297C22⋊C4—311no—
298C22⋊C4—311no—
299C22⋊C4—621no—
300C22⋊C4—621no—
301C22⋊C4—621no—
302C22⋊C4—621no—
303C22⋊C4—621no—
304C22⋊C4—621no—
305C42—621no—
306C42—621no—
307C42C22×D3111yescharacteristic
308C23—21no—
309C23—21no—
310C23—311no—
311C23—311no—
312C23—621no—
313C23—1242no—
314C23—1242no—
315C23—1242no—
316C23—1242no—
317C23—2481no—
318C23—2481no—
319Q8—621no—
320Q8—621no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—81no—
327D4—312no—
328D4—622no—
329D4—621no—
330D4—1242no—
331D4—1242no—
332D4—1242no—
333D4—1242no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—311no—
337C2×C4—311no—
338C2×C4—621no—
339C2×C4—621no—
340C2×C4—621no—
341C2×C4—621no—
342C2×C4—621no—
343C2×C4—621no—
344C2×C4—621no—
345C2×C4—1242no—
346C2×C4—1242no—
347C2×C4D4×D3111yescharacteristic
348C2×C4D4×D3111yescharacteristic
349C2×C4C23×D3111yescharacteristic
350C8—21no—
351C8—21no—
352C8—621no—
353C8—621no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—312no—
361C22—621no—
362C22—621no—
363C22—621no—
364C22—621no—
365C22—622no—
366C22—621no—
367C22—1241no—
368C22—1241no—
369C22—2484no—
370C22—2484no—
371C22C2×D4×D3111yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—621no—
377C4—621no—
378C4—1241no—
379C4—1241no—
380C4—1241no—
381C4#41×D3111yescharacteristic
382C4#1175856611yescharacteristic
383C2—81no—
384C2—81no—
385C2—622no—
386C2—1241no—
387C2—1242no—
388C2#1452169211yescharacteristic
389C2AGL(1,ℤ/8)×D3111yescharacteristic
390C2#1452185911yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#490C2
socleC22×C31C22×D4
prime radical#14521684C2
hypercenter#490D31
hypocenterC31#3310
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #14521737C2semidirect
3. #14521371C2semidirect
4. #14521361C2semidirect
5. #14521346C2semidirect
6. #14521294C2semidirect
7. #14521277C2semidirect
8. #14521246C2semidirect
9. #14521201C2semidirect
10. #14521201C2semidirect
11. #14521151C2semidirect
12. #14521122C2semidirect
13. #14521052C2semidirect
14. #102×D31C2semidirect
15. C31×#490C2semidirect
20. C31×#125C22semidirect
26. C2×#92480C22semidirect
27. C2×#92480C22semidirect
29. C2×#92469C22semidirect
30. C2×C8×D31C22semidirect
32. #11758474C22semidirect
33. #11758474C22semidirect
36. #11758469C22semidirect
37. C4⋊C4×D31C22semidirect
41. C4×Dic62C22semidirect
42. C31×#102C22semidirect
43. C31×#102C22semidirect
44. C31×#102C22semidirect
45. C31×#102C22semidirect
46. C31×#97C22semidirect
47. #11758420C22semidirect
48. #11758420C22semidirect
50. (C2×D31)⋊C8C22semidirect
51. (C2×D31)⋊C8C22semidirect
52. #11758406C22semidirect
53. #11758404C22semidirect
56. #11758396C22semidirect
57. #11758395C22semidirect
58. C4⋊(C31⋊C8)C22semidirect
190. #490D31semidirect
222. #125C2×D31semidirect
247. #102C2×D31semidirect
248. #102C2×D31semidirect
249. #102C2×D31semidirect
250. #102C2×D31semidirect
253. #97C2×D31semidirect
254. C31#3310semidirect

Maximal subgroups

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

subgroupindex
#145217372
#145216842
#145213712
#145213612
#145213462
#145212942
#145212772
#145212462
#145212012
#145211512
#145211222
#145210522
#102×D312
C31×#4902
#331031