Finite Group Databasebeta

← #30648219

Subgroups — #30648219

Order 16256.

500 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×#16861238C211yesmaximal, characteristic, complemented
2C2×#16861233C211yesmaximal, characteristic
3#124×D127C211yesmaximal, characteristic, complemented
4C4×#14523247C211yesmaximal, characteristic, complemented
5#22550101C211yesmaximal, characteristic, complemented
6#22550099C211yesmaximal, characteristic, complemented
7#22550097C211yesmaximal, characteristic, complemented
8#22550095C211yesmaximal, characteristic, complemented
9#22550093C211yesmaximal, characteristic, complemented
10#22550091C211yesmaximal, characteristic, complemented
11#22550089C211yesmaximal, characteristic, complemented
12#22549898C211yesmaximal, characteristic, complemented
13#22549883C211yesmaximal, characteristic, complemented
14#22549879C211yesmaximal, characteristic, complemented
15#463×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19#16861238—24no—
20#16861233—24no—
21C2×Dic508C2211yescharacteristic, complemented
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523237C2211yescharacteristic
24C2×#14523236C2211yescharacteristic
25C2×#14523236C2211yescharacteristic, complemented
26C2×#14523233C2211yescharacteristic
27C2×#14523231C2211yescharacteristic
28C2×#14523231C2211yescharacteristic
29#16861198C2211yescharacteristic
30#16861197C2211yescharacteristic
31#16861196C2211yescharacteristic
32#16861195—21no—
33#16861195—21no—
34C4×Dic254C2211yescharacteristic, complemented
35#16861174C2211yescharacteristic, complemented
36D4×Dic127C2211yescharacteristic, complemented
37C4×C22⋊D127—21no—
38(C22⋊D127)⋊C4—21no—
39#16861167C2211yescharacteristic
40#16861165—21no—
41C22⋊C4×D127—21no—
42C22⋊C4×D127—21no—
43C2×C4×Dic127—21no—
44C42×D127C2211yescharacteristic
45C4⋊(C127⋊C8)C2211yescharacteristic, complemented
46Dic127⋊C8C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861133C2211yescharacteristic, complemented
50#16861132C2211yescharacteristic, complemented
51#16861131C2211yescharacteristic
52#16861130C2211yescharacteristic
53#16861129C2211yescharacteristic
54C2×SD16×C127C2211yescharacteristic, complemented
55C2×D8×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic
57C4×D4×C127C2211yescharacteristic, complemented
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic
61C23×D127—21no—
62#14523247—22no—
63#14523247D414yes—
64C2×Dic254—21no—
65C2×Dic254C2311yescharacteristic
66C2×Dic254C2311yescharacteristic
67Q8×D127—22no—
68Q8×D127—22no—
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72D4×D127—22no—
73D4×D127—22no—
74C22×Dic127—21no—
75C22×Dic127—21no—
76C2×C4×D127—21no—
77C2×C4×D127D411yescharacteristic
78C2×C4×D127D411yescharacteristic
79C2×C4×D127C2311yescharacteristic
80Dic508—22no—
81#14523238—22no—
82#14523237—22no—
83#14523236—22no—
84#14523236—22no—
85#14523233—22no—
86#14523231—22no—
87#14523231—22no—
88C2×C127⋊C8C2311yescharacteristic
89C2×C127⋊C8C2311yescharacteristic
90C4⋊Dic127C2311yescharacteristic
91Dic127⋊C4—21no—
92Dic127⋊C4—21no—
93C22⋊Dic127—21no—
94C22⋊Dic127—21no—
95C22⋊Dic127—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4—21no—
101C4×Dic127—21no—
102C4×Dic127—21no—
103C4×Dic127—21no—
104C4×Dic127D411yescharacteristic
105C4×Dic127D411yescharacteristic
106C4×Dic127C2311yescharacteristic
107C2×Q8×C127C2311yescharacteristic
108C2×D4×C127C2311yescharacteristic
109C2×D4×C127C2311yescharacteristic
110C22×C4×C127—21no—
111SD16×C127—22no—
112D8×C127—22no—
113C2×C8×C127C2311yescharacteristic
114C2×C8×C127C2311yescharacteristic
115C4⋊C4×C127C2311yescharacteristic
116C22⋊C4×C127—21no—
117C22⋊C4×C127—21no—
118C22⋊C4×C127—21no—
119C42×C127C2311yescharacteristic
120C22×D127—21no—
121C22×D127—42no—
122C22×D127—42no—
123C22×D127—42no—
124C22×D127C2×D411yescharacteristic
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—42no—
129Dic254C2×D412yes—
130D508—21no—
131D508—21no—
132C22⋊D127—24no—
133C22⋊D127—42no—
134C2×Dic127—21no—
135C2×Dic127—24no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—42no—
139C2×Dic127C2×D411yescharacteristic
140C2×Dic127C2×D411yescharacteristic
141C2×Dic127C2×D411yescharacteristic
142C4×D127—41no—
143C4×D127—41no—
144C4×D127—42no—
145C4×D127C2×D412yes—
146C4×D127#4112yes—
147C127⋊C8—21no—
148C127⋊C8—21no—
149C23×C127—21no—
150C23×C127—21no—
151Q8×C127—21no—
152Q8×C127—21no—
153D4×C127—21no—
154D4×C127—21no—
155D4×C127—21no—
156D4×C127C2×D412yes—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—42no—
160C2×C4×C127C2×D411yescharacteristic
161C2×C4×C127C2×D411yescharacteristic
162C2×C4×C127C2411yescharacteristic
163C8×C127—21no—
164C8×C127—21no—
165C22×C127—22no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127C22×D411yescharacteristic
172C2×D127—21no—
173C2×D127—21no—
174C2×D127—41no—
175C2×D127—41no—
176C2×D127—41no—
177C2×D127—84no—
178C2×D127#12112yes—
179C4×C127—21no—
180C4×C127—21no—
181C4×C127—41no—
182C4×C127—41no—
183C4×C127C22×D411yescharacteristic
184C4×C127C2×#4111yescharacteristic
185Dic127—22no—
186Dic127—21no—
187Dic127—21no—
188Dic127—41no—
189Dic127—41no—
190Dic127#12112yes—
191C2×C127—42no—
192C2×C127—81no—
193C2×C127C2×#12111yescharacteristic
194C2×C127#57511yescharacteristic
195C2×C127#57511yescharacteristic
196D127—22no—
197D127—81no—
198#3040—1271nomaximal, Sylow
199C127#304011yescharacteristic, complemented, Sylow
200C2×#137—1271no—
201C2×AGL(1,ℤ/8)—1271no—
202C2×#124—1271no—
203C4×#41—1271no—
204#464—1271no—
205#464—1271no—
206#464—1271no—
207#464—1271no—
208#463—1271no—
209#463—1271no—
210#463—1271no—
211#463D12711yescharacteristic, complemented
212#423—1271no—
213#419—1271no—
214#418—1271no—
215C2×#41—1271no—
216C22×Q8—1271no—
217C22×D4—1271no—
218#137—2544no—
219AGL(1,ℤ/8)—2544no—
220C2×Q16—1271no—
221C2×Q16—1271no—
222C2×SD16—1271no—
223C2×SD16—1271no—
224C2×SD16—1271no—
225C2×SD16C2×D12711yescharacteristic, complemented
226C2×D8—1271no—
227C2×D8C2×D12711yescharacteristic
228C2×M16—1271no—
229C2×M16—1271no—
230#124—1271no—
231#124—1271no—
232#124—1271no—
233#124—2541no—
234#124—2541no—
235#124C2×D12711yescharacteristic
236C4×Q8—1271no—
237C4×Q8—1271no—
238C4×D4—1271no—
239C4×D4—2541no—
240C4×D4—2541no—
241C4×D4C2×D12711yescharacteristic, complemented
242#117—1271no—
243#117—2541no—
244C2×C22⋊C4—2541no—
245C2×C22⋊C4—2541no—
246C2×C42—1271no—
247C2×C42—2541no—
248C4⋊C8—1271no—
249C4⋊C8—1271no—
250C4⋊C8—1271no—
251C4⋊C8C2×D12711yescharacteristic, complemented
252#103—1271no—
253#103—1271no—
254#103—1271no—
255#103C2×D12711yescharacteristic, complemented
256#102—1271no—
257#102—1271no—
258#102—1271no—
259#102C2×D12711yescharacteristic
260C24—2541no—
261#41—1274no—
262#41—2542no—
263C2×Q8—1271no—
264C2×Q8—1271no—
265C2×Q8—2541no—
266C2×Q8—2542no—
267C2×Q8—2542no—
268C2×Q8C22×D12711yescharacteristic
269C2×D4—1271no—
270C2×D4—2542no—
271C2×D4—2542no—
272C2×D4—2541no—
273C2×D4—2541no—
274C2×D4C22×D12711yescharacteristic
275C2×D4C22×D12711yescharacteristic
276C22×C4—21no—
277C22×C4—1271no—
278C22×C4—1271no—
279C22×C4—1271no—
280C22×C4—2541no—
281C22×C4—2541no—
282C22×C4—2541no—
283Q16—2542no—
284Q16—2542no—
285SD16—22no—
286SD16—2542no—
287SD16—2542no—
288SD16—2542no—
289D8—22no—
290D8—2542no—
291M16—2542no—
292M16—2542no—
293C2×C8—1271no—
294C2×C8—1271no—
295C2×C8C22×D12711yescharacteristic
296C2×C8C22×D12711yescharacteristic
297C4⋊C4—1271no—
298C4⋊C4—2541no—
299C4⋊C4—2541no—
300C4⋊C4C22×D12711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—2541no—
305C22⋊C4—2541no—
306C22⋊C4—2541no—
307C22⋊C4—2541no—
308C22⋊C4—2541no—
309C22⋊C4—2541no—
310C22⋊C4—2541no—
311C22⋊C4—2541no—
312C42—1271no—
313C42—1271no—
314C42—1271no—
315C42—2541no—
316C42—2541no—
317C42—2541no—
318C42C22×D12711yescharacteristic
319C23—21no—
320C23—21no—
321C23—1271no—
322C23—5082no—
323C23—5082no—
324C23—5082no—
325C23—2541no—
326Q8—21no—
327Q8—21no—
328Q8—1272no—
329Q8—5082no—
330Q8—2541no—
331Q8—2541no—
332Q8—2541no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—5082no—
337D4—2544no—
338D4—2541no—
339D4—2541no—
340D4D4×D12712yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—1271no—
345C2×C4—1272no—
346C2×C4—1272no—
347C2×C4—1271no—
348C2×C4—1271no—
349C2×C4—5081no—
350C2×C4—5081no—
351C2×C4—5082no—
352C2×C4—5082no—
353C2×C4—2544no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4D4×D12711yescharacteristic
358C2×C4D4×D12711yescharacteristic
359C2×C4C23×D12711yescharacteristic
360C8—21no—
361C8—21no—
362C8—2541no—
363C8—2541no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—1272no—
371C22—10164no—
372C22—5081no—
373C22—5081no—
374C22—5081no—
375C22—2541no—
376C22—2541no—
377C22C2×D4×D12711yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—1272no—
383C4—5081no—
384C4—5081no—
385C4—2542no—
386C4—2541no—
387C4—2541no—
388C4C2×D4×D12711yescharacteristic
389C4#41×D12711yescharacteristic
390C2—42no—
391C2—81no—
392C2—10161no—
393C2—2542no—
394C2#121×D12711yescharacteristic
395C2#2255067211yescharacteristic
396C2#2255067311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#463×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861233C2
hypercenter#463D127
hypocenterC127#3040
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16861238C2semidirect
3. #124×D127C2semidirect
4. C4×#14523247C2semidirect
5. #22550101C2semidirect
6. #22550099C2semidirect
7. #22550097C2semidirect
8. #22550095C2semidirect
9. #22550093C2semidirect
10. #22550091C2semidirect
11. #22550089C2semidirect
12. #22549898C2semidirect
13. #22549883C2semidirect
14. #22549879C2semidirect
15. #463×C127C2semidirect
21. C2×Dic508C22semidirect
22. C2×#14523238C22semidirect
25. C2×#14523236C22semidirect
34. C4×Dic254C22semidirect
35. #16861174C22semidirect
36. D4×Dic127C22semidirect
45. C4⋊(C127⋊C8)C22semidirect
46. Dic127⋊C8C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861134C22semidirect
49. #16861133C22semidirect
50. #16861132C22semidirect
54. C2×SD16×C127C22semidirect
57. C4×D4×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
199. C127#3040semidirect
211. #463D127semidirect
225. C2×SD16C2×D127semidirect
241. C4×D4C2×D127semidirect
251. C4⋊C8C2×D127semidirect
255. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
C2×#168612332
#124×D1272
C4×#145232472
#225501012
#225500992
#225500972
#225500952
#225500932
#225500912
#225500892
#225498982
#225498832
#225498792
#463×C1272
#3040127