Finite Group Databasebeta

← #30648249

Subgroups — #30648249

Order 16256.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861241C211yesmaximal, characteristic, complemented
2C2×SD16×D127C211yesmaximal, characteristic, complemented
3#22550444C211yesmaximal, characteristic
4#22550352C211yesmaximal, characteristic, complemented
5#22550101C211yesmaximal, characteristic, complemented
6#22550100C211yesmaximal, characteristic, complemented
7#22550095C211yesmaximal, characteristic, complemented
8#22550077C211yesmaximal, characteristic, complemented
9#22550076C211yesmaximal, characteristic, complemented
10#22550075C211yesmaximal, characteristic, complemented
11#22550074C211yesmaximal, characteristic, complemented
12#22549901C211yesmaximal, characteristic, complemented
13#22549879C211yesmaximal, characteristic, complemented
14#22549871C211yesmaximal, characteristic, complemented
15#464×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19#16861241—24no—
20C2×#14523238C2211yescharacteristic, complemented
21C2×#14523237C2211yescharacteristic
22C2×#14523237C2211yescharacteristic, complemented
23C2×#14523236C2211yescharacteristic
24C2×#14523235C2211yescharacteristic
25C2×#14523235C2211yescharacteristic, complemented
26SD16×D127—24no—
27C2×#14523231C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic
29#16861206C2211yescharacteristic
30#16861205C2211yescharacteristic
31#16861198C2211yescharacteristic
32#16861182—21no—
33#16861182—21no—
34#16861179—21no—
35#16861179—21no—
36Q8×Dic127C2211yescharacteristic, complemented
37C4×Dic254—21no—
38C4×Dic254C2211yescharacteristic, complemented
39#16861174—21no—
40#16861174C2211yescharacteristic, complemented
41#16861167—21no—
42#16861167C2211yescharacteristic
43#16861166—21no—
44#16861166C2211yescharacteristic
45C4⋊(C127⋊C8)C2211yescharacteristic, complemented
46Dic127⋊C8C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861133C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic, complemented
51#16861130C2211yescharacteristic
52#16861129C2211yescharacteristic
53#16861129C2211yescharacteristic, complemented
54C2×Q16×C127C2211yescharacteristic, complemented
55C2×SD16×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic
57C4×Q8×C127C2211yescharacteristic, complemented
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic
61C23×D127—21no—
62#14523249—22no—
63#14523249—22no—
64C2×Dic254—21no—
65C2×Dic254C2311yescharacteristic
66Q8×D127—22no—
67Q8×D127D414yescomplemented
68C2×D508—21no—
69C2×D508—21no—
70C2×D508—21no—
71C2×D508C2311yescharacteristic
72C2×D508C2311yescharacteristic
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76D4×D127—22no—
77D4×D127—22no—
78C2×C4×D127—21no—
79C2×C4×D127—21no—
80C2×C4×D127C2311yescharacteristic
81#14523238—22no—
82#14523237—22no—
83#14523237—22no—
84#14523236—22no—
85#14523235—22no—
86#14523235—22no—
87#14523231—22no—
88C2×C127⋊C8C2311yescharacteristic
89C2×C127⋊C8C2311yescharacteristic
90C8×D127—22no—
91C4⋊Dic127—21no—
92C4⋊Dic127C2311yescharacteristic
93Dic127⋊C4—21no—
94Dic127⋊C4—21no—
95Dic127⋊C4D411yescharacteristic
96Dic127⋊C4D411yescharacteristic
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4D411yescharacteristic
101(C2×D127)⋊C4D411yescharacteristic
102C4×Dic127—21no—
103C4×Dic127—21no—
104C4×Dic127—21no—
105C4×Dic127—21no—
106C4×Dic127C2311yescharacteristic
107C2×Q8×C127C2311yescharacteristic
108C2×Q8×C127C2311yescharacteristic
109C2×D4×C127C2311yescharacteristic
110Q16×C127—22no—
111SD16×C127—22no—
112C2×C8×C127C2311yescharacteristic
113C2×C8×C127C2311yescharacteristic
114C4⋊C4×C127—21no—
115C4⋊C4×C127C2311yescharacteristic
116C22⋊C4×C127—21no—
117C22⋊C4×C127—21no—
118C42×C127—21no—
119C42×C127C2311yescharacteristic
120C22×D127—21no—
121C22×D127—21no—
122C22×D127—42no—
123C22×D127—81no—
124C22×D127—81no—
125C22×D127C2×D411yescharacteristic
126Dic254—21no—
127Dic254—24no—
128Dic254C2×D412yes—
129D508—21no—
130D508—21no—
131D508—21no—
132D508—21no—
133D508—42no—
134D508—81no—
135D508—81no—
136C22⋊D127—42no—
137C22⋊D127—81no—
138C22⋊D127—81no—
139C2×Dic127—21no—
140C2×Dic127—21no—
141C2×Dic127C2×D411yescharacteristic
142C2×Dic127C2×D411yescharacteristic
143C2×Dic127C2×D411yescharacteristic
144C4×D127—24no—
145C4×D127—42no—
146C4×D127C2×D412yes—
147C4×D127#4112yes—
148C127⋊C8—21no—
149C127⋊C8—21no—
150C23×C127—21no—
151Q8×C127—21no—
152Q8×C127—21no—
153Q8×C127—21no—
154Q8×C127C2×D412yes—
155D4×C127—21no—
156D4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127C2×D411yescharacteristic
161C2×C4×C127C2×D411yescharacteristic
162C2×C4×C127C2411yescharacteristic
163C8×C127—21no—
164C8×C127—21no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127C22×D411yescharacteristic
169C2×D127—21no—
170C2×D127—21no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—41no—
176C2×D127—41no—
177C2×D127—84no—
178C2×D127#12112yes—
179C4×C127—22no—
180C4×C127—21no—
181C4×C127—21no—
182C4×C127—41no—
183C4×C127—41no—
184C4×C127C22×D411yescharacteristic
185C4×C127C2×#4111yescharacteristic
186Dic127—22no—
187Dic127—21no—
188Dic127—21no—
189Dic127—41no—
190Dic127#12112yes—
191C2×C127—81no—
192C2×C127C2×#12111yescharacteristic
193C2×C127#57711yescharacteristic
194C2×C127#57711yescharacteristic
195D127—22no—
196D127—81no—
197D127—81no—
198#3043—1271nomaximal, Sylow
199C127#304311yescharacteristic, complemented, Sylow
200C2×#137—1271no—
201C22×SD16—1271no—
202#535—1271no—
203#519—1271no—
204#464—1271no—
205#464—1271no—
206#464—1271no—
207#464D12711yescharacteristic, complemented
208#460—1271no—
209#460—1271no—
210#460—1271no—
211#460—1271no—
212#424—1271no—
213#418—1271no—
214#416—1271no—
215C2×#41—1271no—
216C22×Q8—1271no—
217C22×D4—1271no—
218#137—2544no—
219C2×Q16—1271no—
220C2×Q16C2×D12711yescharacteristic, complemented
221C2×SD16—1271no—
222C2×SD16—1271no—
223C2×SD16—1271no—
224C2×SD16—1271no—
225C2×SD16—1271no—
226C2×SD16—2544no—
227C2×SD16C2×D12711yescharacteristic
228C2×M16—1271no—
229C22×C8—1271no—
230#127—1271no—
231#127—1271no—
232#124—1271no—
233#124C2×D12711yescharacteristic
234#121—2541no—
235#121—2541no—
236#120—2541no—
237#120—2541no—
238C4×Q8—1271no—
239C4×Q8—1271no—
240C4×Q8—1271no—
241C4×Q8—2541no—
242C4×Q8—2541no—
243C4×Q8C2×D12711yescharacteristic, complemented
244#117—1271no—
245#117—1271no—
246#117—2541no—
247#117—2541no—
248C4⋊C8—1271no—
249C4⋊C8—1271no—
250C4⋊C8—1271no—
251C4⋊C8C2×D12711yescharacteristic, complemented
252#103—1271no—
253#103C2×D12711yescharacteristic, complemented
254#102—1271no—
255#102—1271no—
256#102—1271no—
257#102—1271no—
258#102—1271no—
259#102C2×D12711yescharacteristic
260C24—2541no—
261#41—2542no—
262#41—2542no—
263C2×Q8—1274no—
264C2×Q8—1271no—
265C2×Q8—2541no—
266C2×Q8—2542no—
267C2×Q8C22×D12711yescharacteristic
268C2×Q8C22×D12711yescharacteristic
269C2×D4—1271no—
270C2×D4—1271no—
271C2×D4—2541no—
272C2×D4—2542no—
273C2×D4—2542no—
274C2×D4—2541no—
275C2×D4—2541no—
276C2×D4—2541no—
277C2×D4—2541no—
278C2×D4—2541no—
279C2×D4C22×D12711yescharacteristic
280C22×C4—1271no—
281C22×C4—2541no—
282C22×C4—2541no—
283Q16—22no—
284Q16—2542no—
285SD16—22no—
286SD16—2542no—
287SD16—2542no—
288SD16—2542no—
289SD16—2542no—
290SD16—2542no—
291M16—2542no—
292C2×C8—1271no—
293C2×C8—1271no—
294C2×C8—2542no—
295C2×C8C22×D12711yescharacteristic
296C2×C8C22×D12711yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—1271no—
299C4⋊C4—1271no—
300C4⋊C4—1271no—
301C4⋊C4—2541no—
302C4⋊C4—2541no—
303C4⋊C4—2541no—
304C4⋊C4C22×D12711yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—1271no—
308C22⋊C4—1271no—
309C22⋊C4—2541no—
310C22⋊C4—2541no—
311C22⋊C4—2541no—
312C42—21no—
313C42—1271no—
314C42—2541no—
315C42—2541no—
316C42—2541no—
317C42—2541no—
318C42C22×D12711yescharacteristic
319C23—21no—
320C23—1271no—
321C23—10161no—
322C23—10161no—
323C23—5082no—
324C23—2541no—
325C23—2541no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—1272no—
330Q8—2544no—
331Q8—2541no—
332Q8D4×D12712yes—
333D4—21no—
334D4—21no—
335D4—10161no—
336D4—10161no—
337D4—10161no—
338D4—10161no—
339D4—5082no—
340D4—5082no—
341D4—2541no—
342D4—2541no—
343D4—2541no—
344D4—2541no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—1271no—
349C2×C4—1272no—
350C2×C4—1272no—
351C2×C4—1271no—
352C2×C4—1271no—
353C2×C4—5082no—
354C2×C4—2544no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4D4×D12711yescharacteristic
358C2×C4D4×D12711yescharacteristic
359C2×C4C23×D12711yescharacteristic
360C8—21no—
361C8—21no—
362C8—2541no—
363C8—2541no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—1272no—
368C22—10164no—
369C22—5081no—
370C22—5081no—
371C22—5081no—
372C22—5081no—
373C22—5081no—
374C22—5081no—
375C22—2541no—
376C22—2541no—
377C22C2×D4×D12711yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—1272no—
384C4—5081no—
385C4—2542no—
386C4—2541no—
387C4—2541no—
388C4C2×D4×D12711yescharacteristic
389C4#41×D12711yescharacteristic
390C2—81no—
391C2—10161no—
392C2—10161no—
393C2—2542no—
394C2#121×D12711yescharacteristic
395C2#2255068311yescharacteristic
396C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#464×C127C2
socleC22×C127C22×D4
prime radical#22550444C2
hypercenter#464D127
hypocenterC127#3043
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#16861241C2semidirect
2. C2×SD16×D127C2semidirect
4. #22550352C2semidirect
5. #22550101C2semidirect
6. #22550100C2semidirect
7. #22550095C2semidirect
8. #22550077C2semidirect
9. #22550076C2semidirect
10. #22550075C2semidirect
11. #22550074C2semidirect
12. #22549901C2semidirect
13. #22549879C2semidirect
14. #22549871C2semidirect
15. #464×C127C2semidirect
20. C2×#14523238C22semidirect
22. C2×#14523237C22semidirect
25. C2×#14523235C22semidirect
36. Q8×Dic127C22semidirect
38. C4×Dic254C22semidirect
40. #16861174C22semidirect
45. C4⋊(C127⋊C8)C22semidirect
46. Dic127⋊C8C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861133C22semidirect
50. #16861131C22semidirect
53. #16861129C22semidirect
54. C2×Q16×C127C22semidirect
57. C4×Q8×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
67. Q8×D127D4semidirect
199. C127#3043semidirect
207. #464D127semidirect
220. C2×Q16C2×D127semidirect
243. C4×Q8C2×D127semidirect
251. C4⋊C8C2×D127semidirect
253. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612412
C2×SD16×D1272
#225504442
#225503522
#225501012
#225501002
#225500952
#225500772
#225500762
#225500752
#225500742
#225499012
#225498792
#225498712
#464×C1272
#3043127