Finite Group Databasebeta

← #16857548

Subgroups — #16857548

Order 3968.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521739C211yesmaximal, characteristic, complemented
2#14521695C211yesmaximal, characteristic
3#14521404C211yesmaximal, characteristic, complemented
4#14521380C211yesmaximal, characteristic, complemented
5#14521374C211yesmaximal, characteristic, complemented
6#14521350C211yesmaximal, characteristic, complemented
7#14521290C211yesmaximal, characteristic, complemented
8#14521278C211yesmaximal, characteristic, complemented
9#14521272C211yesmaximal, characteristic, complemented
10#14521213C211yesmaximal, characteristic, complemented
11#14521202C211yesmaximal, characteristic, complemented
12C4×#92470C211yesmaximal, characteristic, complemented
13#14521046C211yesmaximal, characteristic, complemented
14C2×#11758409C211yesmaximal, characteristic, complemented
15C8×D4×C31C211yesmaximal, characteristic, complemented
16C2×#92495C2211yescharacteristic
17C22×D124C2211yescharacteristic
18C22×C8×C31C2211yescharacteristic
19C22×C8×C31C2211yescharacteristic, complemented
20C4×D4×C31C2211yescharacteristic
21#11758514—21no—
22#11758512—21no—
23#11758507C2211yescharacteristic
24#11758503C2211yescharacteristic, complemented
25C2×C4⋊Dic31C2211yescharacteristic
26#11758501C2211yescharacteristic
27#11758501C2211yescharacteristic
28#11758501C2211yescharacteristic, complemented
29C2×Dic124C2211yescharacteristic, complemented
30C2×#92470C2211yescharacteristic
31C2×#92470C2211yescharacteristic
32#11758471—21no—
33#11758466—21no—
34#11758461—21no—
35C22⋊C4×D31—21no—
36#11758456—21no—
37#11758454—21no—
38#11758452C2211yescharacteristic
39C4×D124C2211yescharacteristic
40#11758447C2211yescharacteristic, complemented
41C4×Dic62C2211yescharacteristic, complemented
42C31×C4⋊C8C2211yescharacteristic, complemented
43C31×C22⋊C8C2211yescharacteristic
44C31×C22⋊C8C2211yescharacteristic, complemented
45C4×C8×C31C2211yescharacteristic, complemented
46#11758409C2211yescharacteristic
47#11758409C2211yescharacteristic
48#11758409C2211yescharacteristic
49#11758409C2211yescharacteristic
50#11758409C2211yescharacteristic
51#11758409C2211yescharacteristic
52#11758406C2211yescharacteristic, complemented
53#11758405C2211yescharacteristic, complemented
54#11758405C2211yescharacteristic, complemented
55#11758404C2211yescharacteristic, complemented
56#11758404C2211yescharacteristic, complemented
57#11758404C2211yescharacteristic, complemented
58#11758404C2211yescharacteristic, complemented
59C23×D31—21no—
60C2×D4×C31D411yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C22×C4×C31C2311yescharacteristic
63C2×C22⋊D31—21no—
64C2×C22⋊D31—21no—
65C22×Dic31—21no—
66#92495—22no—
67#92495—41no—
68C2×D124—22no—
69C2×D124—21no—
70C2×D124—41no—
71C2×D124C2311yescharacteristic
72C2×D124C2311yescharacteristic
73C2×C4×D31—21no—
74C2×C4×D31—21no—
75C2×Dic62C2311yescharacteristic
76C2×Dic62C2311yescharacteristic
77C2×C8×C31—41no—
78C2×C8×C31—41no—
79C2×C8×C31C2311yescharacteristic
80C2×C8×C31C2311yescharacteristic
81C2×C8×C31C2311yescharacteristic
82C2×C8×C31C2311yescharacteristic
83C4⋊C4×C31D411yescharacteristic
84C22⋊C4×C31D411yescharacteristic
85C22⋊C4×C31D411yescharacteristic
86C42×C31C2311yescharacteristic
87C22⋊Dic31—21no—
88C22⋊Dic31—21no—
89C22⋊Dic31—21no—
90C22⋊Dic31—21no—
91(C2×D31)⋊C4—21no—
92(C2×D31)⋊C4—21no—
93(C2×D31)⋊C4—21no—
94(C2×D31)⋊C4—21no—
95C4⋊Dic31—21no—
96C4⋊Dic31C2311yescharacteristic
97C4⋊Dic31C2311yescharacteristic
98C4⋊Dic31C2311yescharacteristic
99C4⋊Dic31C2311yescharacteristic
100Dic31⋊C4—21no—
101Dic31⋊C4—21no—
102Dic31⋊C4—21no—
103Dic31⋊C4—21no—
104Dic31⋊C4—21no—
105Dic31⋊C4—21no—
106C4×Dic31—21no—
107C4×Dic31—21no—
108Dic124—41no—
109#92470—22no—
110#92470—41no—
111C23×C31C2×D411yescharacteristic
112C23×C31C2×D411yescharacteristic
113C22×D31—21no—
114C22×D31—21no—
115C22×D31—42no—
116C22×D31—42no—
117C22×D31—81no—
118D4×C31—22no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31C2×D411yescharacteristic
124C2×C4×C31C2×D411yescharacteristic
125C2×C4×C31C2×D411yescharacteristic
126C2×C4×C31C2×D411yescharacteristic
127C2×C4×C31C2411yescharacteristic
128C22⋊D31—42no—
129C22⋊D31—81no—
130C2×Dic31—21no—
131C2×Dic31—21no—
132C2×Dic31—21no—
133C2×Dic31—21no—
134C2×Dic31—21no—
135C2×Dic31—21no—
136C2×Dic31—42no—
137D124—21no—
138D124—21no—
139D124—21no—
140D124—42no—
141D124#4112yes—
142C4×D31—42no—
143C4×D31—81no—
144Dic62—21no—
145Dic62—21no—
146Dic62—21no—
147Dic62—21no—
148C8×C31—21no—
149C8×C31—21no—
150C8×C31—21no—
151C8×C31—21no—
152#3296—311nomaximal, Sylow
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—21no—
156C22×C31—21no—
157C22×C31—21no—
158C22×C31#13512yes—
159C22×C31C22×D411yescharacteristic
160C2×D31—22no—
161C2×D31—41no—
162C2×D31—41no—
163C2×D31—41no—
164C2×D31—41no—
165C2×D31—41no—
166C2×D31—84no—
167C4×C31—21no—
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—21no—
171C4×C31C2×#4111yescharacteristic
172C4×C31H(5,𝔽2)11yescharacteristic
173Dic31—41no—
174Dic31—41no—
175Dic31—41no—
176Dic31—41no—
177Dic31—41no—
178Dic31—41no—
179#552—311no—
180#546—311no—
181#488—311no—
182#483—311no—
183#482—311no—
184#478—311no—
185#467—311no—
186#465—311no—
187#464—311no—
188#452—311no—
189#450—311no—
190C4×SD16—311no—
191C8×D4D3111yescharacteristic, complemented
192#416—311no—
193C2×#102—311no—
194C2×C31—22no—
195C2×C31—41no—
196C2×C31#54611yescharacteristic
197C2×C31C2×#13511yescharacteristic
198C2×C31#57711yescharacteristic
199D31—42no—
200D31—81no—
201C2×#41—311no—
202C22×D4—311no—
203C2×Q16—311no—
204C2×SD16—311no—
205C2×SD16—311no—
206C22×C8C2×D3111yescharacteristic
207C22×C8C2×D3111yescharacteristic, complemented
208#126—621no—
209#125—311no—
210#125—621no—
211#124—311no—
212#123—621no—
213#123—621no—
214C22⋊Q8—311no—
215C22⋊Q8—311no—
216C22⋊Q8—311no—
217#121—311no—
218#121—621no—
219#120—621no—
220C4×Q8—311no—
221C4×D4—311no—
222C4×D4C2×D3111yescharacteristic
223#117—311no—
224#117—621no—
225C2×C4⋊C4—311no—
226C2×C22⋊C4—621no—
227#107—311no—
228#106—311no—
229#106—311no—
230C4⋊C8C2×D3111yescharacteristic, complemented
231#103—311no—
232#103—311no—
233#103—311no—
234#103—311no—
235#102—311no—
236#102—311no—
237#102—311no—
238#102—311no—
239#102—311no—
240#102—311no—
241C22⋊C8C2×D3111yescharacteristic
242C22⋊C8C2×D3111yescharacteristic, complemented
243C4×C8C2×D3111yescharacteristic, complemented
244C31#329611yescharacteristic, complemented, Sylow
245C24—621no—
246#41—622no—
247#41—1241no—
248C2×Q8—311no—
249C2×Q8—311no—
250C2×D4—311no—
251C2×D4—311no—
252C2×D4—622no—
253C2×D4—621no—
254C2×D4—621no—
255C2×D4—621no—
256C2×D4—1241no—
257C2×D4D12411yescharacteristic
258C22×C4—621no—
259C22×C4—621no—
260C22×C4—621no—
261C22×C4C22×D3111yescharacteristic
262C22×C4C22×D3111yescharacteristic
263Q16—1241no—
264SD16—622no—
265SD16—1241no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D3111yescharacteristic
269C2×C8C22×D3111yescharacteristic
270C2×C8C22×D3111yescharacteristic
271C2×C8C22×D3111yescharacteristic
272C4⋊C4—311no—
273C4⋊C4—311no—
274C4⋊C4—311no—
275C4⋊C4—311no—
276C4⋊C4—621no—
277C4⋊C4—621no—
278C4⋊C4—621no—
279C4⋊C4—621no—
280C4⋊C4—621no—
281C4⋊C4—621no—
282C4⋊C4—621no—
283C4⋊C4D12411yescharacteristic
284C22⋊C4—621no—
285C22⋊C4—621no—
286C22⋊C4—621no—
287C22⋊C4—621no—
288C22⋊C4—621no—
289C22⋊C4—621no—
290C22⋊C4—621no—
291C22⋊C4—621no—
292C22⋊C4D12411yescharacteristic
293C22⋊C4D12411yescharacteristic
294C42—621no—
295C42—621no—
296C42C22×D3111yescharacteristic
297C23—621no—
298C23—621no—
299C23—1242no—
300C23—1242no—
301C23—2481no—
302C23C2×D12411yescharacteristic
303C23C2×D12411yescharacteristic
304Q8—621no—
305Q8—621no—
306Q8—621no—
307Q8—621no—
308D4—22no—
309D4—312no—
310D4—621no—
311D4—621no—
312D4—621no—
313D4—1242no—
314D4—1242no—
315D4—2481no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—621no—
321C2×C4—621no—
322C2×C4—621no—
323C2×C4—621no—
324C2×C4—621no—
325C2×C4—621no—
326C2×C4—1242no—
327C2×C4—1242no—
328C2×C4—2481no—
329C2×C4C2×D12411yescharacteristic
330C2×C4C2×D12411yescharacteristic
331C2×C4C2×D12411yescharacteristic
332C2×C4C2×D12411yescharacteristic
333C2×C4C23×D3111yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—622no—
344C22—1241no—
345C22—1241no—
346C22—1241no—
347C22—1241no—
348C22—1241no—
349C22—2484no—
350C22#1175848112yes—
351C22C22×D12411yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—1241no—
357C4—1241no—
358C4—1241no—
359C4—1241no—
360C4—1241no—
361C4—1241no—
362C4#1175856111yescharacteristic
363C4#41×D3111yescharacteristic
364C2—22no—
365C2—41no—
366C2—1242no—
367C2—2481no—
368C2#1452169411yescharacteristic
369C2C2×#1175848111yescharacteristic
370C2#1452186111yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC8×D4×C31C2
socleC22×C31C22×D4
prime radical#14521695C2
hypercenterC8×D4D31
hypocenterC31#3296
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #14521739C2semidirect
3. #14521404C2semidirect
4. #14521380C2semidirect
5. #14521374C2semidirect
6. #14521350C2semidirect
7. #14521290C2semidirect
8. #14521278C2semidirect
9. #14521272C2semidirect
10. #14521213C2semidirect
11. #14521202C2semidirect
12. C4×#92470C2semidirect
13. #14521046C2semidirect
14. C2×#11758409C2semidirect
15. C8×D4×C31C2semidirect
19. C22×C8×C31C22semidirect
24. #11758503C22semidirect
28. #11758501C22semidirect
29. C2×Dic124C22semidirect
40. #11758447C22semidirect
41. C4×Dic62C22semidirect
42. C31×C4⋊C8C22semidirect
44. C31×C22⋊C8C22semidirect
45. C4×C8×C31C22semidirect
52. #11758406C22semidirect
53. #11758405C22semidirect
54. #11758405C22semidirect
55. #11758404C22semidirect
56. #11758404C22semidirect
57. #11758404C22semidirect
58. #11758404C22semidirect
191. C8×D4D31semidirect
207. C22×C8C2×D31semidirect
230. C4⋊C8C2×D31semidirect
242. C22⋊C8C2×D31semidirect
243. C4×C8C2×D31semidirect
244. C31#3296semidirect

Maximal subgroups

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

subgroupindex
#145217392
#145216952
#145214042
#145213802
#145213742
#145213502
#145212902
#145212782
#145212722
#145212132
#145212022
C4×#924702
#145210462
C2×#117584092
C8×D4×C312
#329631