Finite Group Databasebeta

← #16857734

Subgroups — #16857734

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#14521739C211yesmaximal, characteristic, complemented
2D4×D124C211yesmaximal, characteristic
3#14521414C211yesmaximal, characteristic, complemented
4#14521374C211yesmaximal, characteristic, complemented
5#14521363C211yesmaximal, characteristic, complemented
6#14521350C211yesmaximal, characteristic, complemented
7#14521295C211yesmaximal, characteristic, complemented
8#14521278C211yesmaximal, characteristic, complemented
9#14521250C211yesmaximal, characteristic, complemented
10#14521202C211yesmaximal, characteristic, complemented
11#14521202C211yesmaximal, characteristic, complemented
12#14521151C211yesmaximal, characteristic, complemented
13#14521055C211yesmaximal, characteristic, complemented
14C2×#11758409C211yesmaximal, characteristic, complemented
15C31×#435C211yesmaximal, characteristic, complemented
16C2×D4×D31—21no—
17C22×D124C2211yescharacteristic
18C22×D124C2211yescharacteristic
19C2×M16×C31C2211yescharacteristic, complemented
20C22×C8×C31C2211yescharacteristic, complemented
21C4×D4×C31C2211yescharacteristic
22#11758512—21no—
23#11758507C2211yescharacteristic
24#11758507C2211yescharacteristic
25#11758503C2211yescharacteristic, complemented
26C2×C4⋊Dic31C2211yescharacteristic, complemented
27#11758501C2211yescharacteristic, complemented
28#11758501C2211yescharacteristic, complemented
29C2×#92470C2211yescharacteristic
30C2×#92470C2211yescharacteristic, complemented
31C2×#92470C2211yescharacteristic, complemented
32#11758472—21no—
33#11758466—21no—
34#11758459—21no—
35#11758459—21no—
36#11758456—21no—
37#11758454—21no—
38#11758451C2211yescharacteristic
39C4×D124C2211yescharacteristic
40#11758447C2211yescharacteristic, complemented
41C4×Dic62C2211yescharacteristic, complemented
42C31×C4⋊C8C2211yescharacteristic, complemented
43C31×C22⋊C8C2211yescharacteristic, complemented
44C31×C22⋊C8C2211yescharacteristic, complemented
45C31×#97C2211yescharacteristic, complemented
46#11758409C2211yescharacteristic
47#11758409C2211yescharacteristic
48#11758409C2211yescharacteristic
49#11758409C2211yescharacteristic, complemented
50#11758409C2211yescharacteristic, complemented
51#11758409C2211yescharacteristic, complemented
52#11758409C2211yescharacteristic, complemented
53#11758409C2211yescharacteristic, complemented
54#11758409C2211yescharacteristic, complemented
55#11758406C2211yescharacteristic, complemented
56#11758406C2211yescharacteristic, complemented
57#11758405C2211yescharacteristic, complemented
58#11758404C2211yescharacteristic, complemented
59C23×D31—21no—
60C23×D31—21no—
61C2×D4×C31D411yescharacteristic
62C22×C4×C31C2311yescharacteristic
63C22×C4×C31C2311yescharacteristic
64C2×C22⋊D31—21no—
65C2×C22⋊D31—21no—
66C22×Dic31—21no—
67D4×D31—44no—
68C2×D124—22no—
69C2×D124—22no—
70C2×D124—21no—
71C2×D124—21no—
72C2×D124—21no—
73C2×D124—21no—
74C2×D124—41no—
75C2×D124—41no—
76C2×D124C2311yescharacteristic
77C2×D124C2311yescharacteristic
78C2×D124C2311yescharacteristic
79C2×C4×D31—21no—
80C2×Dic62C2311yescharacteristic
81M16×C31—41no—
82C2×C8×C31—41no—
83C2×C8×C31C2311yescharacteristic
84C2×C8×C31C2311yescharacteristic
85C2×C8×C31C2311yescharacteristic
86C2×C8×C31C2311yescharacteristic
87C4⋊C4×C31D411yescharacteristic
88C22⋊C4×C31D411yescharacteristic
89C22⋊C4×C31D411yescharacteristic
90C42×C31C2311yescharacteristic
91C22⋊Dic31—21no—
92C22⋊Dic31—21no—
93C22⋊Dic31—21no—
94(C2×D31)⋊C4—21no—
95(C2×D31)⋊C4—21no—
96(C2×D31)⋊C4—21no—
97C4⋊Dic31—21no—
98C4⋊Dic31C2311yescharacteristic
99C4⋊Dic31C2311yescharacteristic
100C4⋊Dic31C2311yescharacteristic
101C4⋊Dic31C2311yescharacteristic
102Dic31⋊C4—21no—
103Dic31⋊C4—21no—
104Dic31⋊C4—21no—
105Dic31⋊C4—21no—
106Dic31⋊C4—21no—
107C4×Dic31—21no—
108C4×Dic31—21no—
109#92470—22no—
110#92470—41no—
111#92470—41no—
112C23×C31C2×D411yescharacteristic
113C23×C31C2×D411yescharacteristic
114C22×D31—21no—
115C22×D31—21no—
116C22×D31—21no—
117C22×D31—42no—
118C22×D31—42no—
119C22×D31—42no—
120C22×D31—42no—
121C22×D31—81no—
122C22×D31—81no—
123D4×C31—22no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31C2×D411yescharacteristic
129C2×C4×C31C2×D411yescharacteristic
130C2×C4×C31C2×D411yescharacteristic
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2411yescharacteristic
133C22⋊D31—42no—
134C22⋊D31—42no—
135C2×Dic31—21no—
136C2×Dic31—21no—
137C2×Dic31—21no—
138C2×Dic31—21no—
139C2×Dic31—21no—
140C2×Dic31—42no—
141D124—21no—
142D124—21no—
143D124—21no—
144D124—21no—
145D124—21no—
146D124—42no—
147D124—42no—
148D124—42no—
149D124—81no—
150D124#4112yes—
151C4×D31—42no—
152Dic62—21no—
153Dic62—21no—
154C8×C31—21no—
155C8×C31—21no—
156C8×C31—21no—
157C8×C31—21no—
158#3310—311nomaximal, Sylow
159C22×C31—21no—
160C22×C31—21no—
161C22×C31—21no—
162C22×C31—21no—
163C22×C31—21no—
164C22×C31AGL(1,ℤ/8)12yes—
165C22×C31C22×D411yescharacteristic
166C2×D31—22no—
167C2×D31—41no—
168C2×D31—41no—
169C2×D31—41no—
170C2×D31—41no—
171C2×D31—41no—
172C2×D31—41no—
173C2×D31—41no—
174C2×D31—41no—
175C2×D31—84no—
176C2×D31—84no—
177C4×C31—21no—
178C4×C31—21no—
179C4×C31—21no—
180C4×C31—21no—
181C4×C31C2×#4111yescharacteristic
182C4×C31H(5,𝔽2)11yescharacteristic
183Dic31—41no—
184Dic31—41no—
185Dic31—41no—
186Dic31—41no—
187Dic31—41no—
188#552—311no—
189D42—311no—
190#490—311no—
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#435D3111yescharacteristic, complemented
201#418—311no—
202C2×#102—311no—
203C2×C31—22no—
204C2×C31—41no—
205C2×C31#54611yescharacteristic
206C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
207C2×C31#57711yescharacteristic
208D31—42no—
209D31—81no—
210D31—81no—
211C22×D4—311no—
212C22×D4—311no—
213C22×D4—621no—
214C2×SD16—311no—
215C2×SD16—311no—
216C2×SD16—311no—
217C2×M16C2×D3111yescharacteristic, complemented
218C22×C8C2×D3111yescharacteristic, complemented
219#127—311no—
220#126—621no—
221#125—311no—
222#125—621no—
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×D4C2×D3111yescharacteristic
234#117—311no—
235#117—621no—
236C2×C4⋊C4—311no—
237#107—311no—
238#107—311no—
239#106—311no—
240C4⋊C8C2×D3111yescharacteristic, complemented
241#103—311no—
242#102—311no—
243#102—311no—
244#102—311no—
245#102—311no—
246#102—311no—
247#102—311no—
248#102—311no—
249#102—311no—
250#102—311no—
251C22⋊C8C2×D3111yescharacteristic, complemented
252C22⋊C8C2×D3111yescharacteristic, complemented
253#97C2×D3111yescharacteristic, complemented
254C31#331011yescharacteristic, complemented, Sylow
255C24—621no—
256C24—621no—
257C2×Q8—311no—
258C2×D4—311no—
259C2×D4—311no—
260C2×D4—311no—
261C2×D4—622no—
262C2×D4—622no—
263C2×D4—621no—
264C2×D4—621no—
265C2×D4—621no—
266C2×D4—621no—
267C2×D4—621no—
268C2×D4—621no—
269C2×D4—1241no—
270C2×D4—1241no—
271C2×D4—1244no—
272C2×D4D12411yescharacteristic
273C22×C4—621no—
274C22×C4—621no—
275C22×C4C22×D3111yescharacteristic
276C22×C4C22×D3111yescharacteristic
277SD16—622no—
278SD16—1241no—
279SD16—1241no—
280M16—41no—
281C2×C8—41no—
282C2×C8C22×D3111yescharacteristic
283C2×C8C22×D3111yescharacteristic
284C2×C8C22×D3111yescharacteristic
285C2×C8C22×D3111yescharacteristic
286C4⋊C4—311no—
287C4⋊C4—311no—
288C4⋊C4—311no—
289C4⋊C4—311no—
290C4⋊C4—621no—
291C4⋊C4—621no—
292C4⋊C4—621no—
293C4⋊C4—621no—
294C4⋊C4—621no—
295C4⋊C4—621no—
296C4⋊C4D12411yescharacteristic
297C22⋊C4—621no—
298C22⋊C4—621no—
299C22⋊C4—621no—
300C22⋊C4—621no—
301C22⋊C4—621no—
302C22⋊C4—621no—
303C22⋊C4D12411yescharacteristic
304C22⋊C4D12411yescharacteristic
305C42—621no—
306C42—621no—
307C42C22×D3111yescharacteristic
308C23—621no—
309C23—621no—
310C23—621no—
311C23—1242no—
312C23—1242no—
313C23—1242no—
314C23—1242no—
315C23—2481no—
316C23—2481no—
317C23C2×D12411yescharacteristic
318C23C2×D12411yescharacteristic
319Q8—621no—
320Q8—621no—
321D4—22no—
322D4—312no—
323D4—621no—
324D4—621no—
325D4—621no—
326D4—621no—
327D4—621no—
328D4—1242no—
329D4—1242no—
330D4—1242no—
331D4—1242no—
332D4—1242no—
333D4—2481no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—621no—
339C2×C4—621no—
340C2×C4—621no—
341C2×C4—621no—
342C2×C4—621no—
343C2×C4—1242no—
344C2×C4—1242no—
345C2×C4C2×D12411yescharacteristic
346C2×C4C2×D12411yescharacteristic
347C2×C4C2×D12411yescharacteristic
348C2×C4C2×D12411yescharacteristic
349C2×C4C23×D3111yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—622no—
360C22—1241no—
361C22—1241no—
362C22—1241no—
363C22—1241no—
364C22—1241no—
365C22—1241no—
366C22—1241no—
367C22—1241no—
368C22—2484no—
369C22—2484no—
370C22#1175848512yes—
371C22C22×D12411yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—1241no—
377C4—1241no—
378C4—1241no—
379C4—1241no—
380C4—1241no—
381C4#1175856111yescharacteristic
382C4#41×D3111yescharacteristic
383C2—22no—
384C2—41no—
385C2—1242no—
386C2—2481no—
387C2—2481no—
388C2#1452169411yescharacteristic
389C2C2×#1175848511yescharacteristic
390C2#1452186111yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#435C2
socleC22×C31C22×D4
prime radicalD4×D124C2
hypercenter#435D31
hypocenterC31#3310
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #14521739C2semidirect
3. #14521414C2semidirect
4. #14521374C2semidirect
5. #14521363C2semidirect
6. #14521350C2semidirect
7. #14521295C2semidirect
8. #14521278C2semidirect
9. #14521250C2semidirect
10. #14521202C2semidirect
11. #14521202C2semidirect
12. #14521151C2semidirect
13. #14521055C2semidirect
14. C2×#11758409C2semidirect
15. C31×#435C2semidirect
19. C2×M16×C31C22semidirect
20. C22×C8×C31C22semidirect
25. #11758503C22semidirect
26. C2×C4⋊Dic31C22semidirect
27. #11758501C22semidirect
28. #11758501C22semidirect
30. C2×#92470C22semidirect
31. C2×#92470C22semidirect
40. #11758447C22semidirect
41. C4×Dic62C22semidirect
42. C31×C4⋊C8C22semidirect
43. C31×C22⋊C8C22semidirect
44. C31×C22⋊C8C22semidirect
45. C31×#97C22semidirect
49. #11758409C22semidirect
50. #11758409C22semidirect
51. #11758409C22semidirect
52. #11758409C22semidirect
53. #11758409C22semidirect
54. #11758409C22semidirect
55. #11758406C22semidirect
56. #11758406C22semidirect
57. #11758405C22semidirect
58. #11758404C22semidirect
200. #435D31semidirect
217. C2×M16C2×D31semidirect
218. C22×C8C2×D31semidirect
240. C4⋊C8C2×D31semidirect
251. C22⋊C8C2×D31semidirect
252. C22⋊C8C2×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
#145217392
D4×D1242
#145214142
#145213742
#145213632
#145213502
#145212952
#145212782
#145212502
#145212022
#145211512
#145210552
C2×#117584092
C31×#4352
#331031