Finite Group Databasebeta

← #30648221

Subgroups — #30648221

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×#16861241C211yesmaximal, characteristic, complemented
2C2×#16861236C211yesmaximal, characteristic
3#124×D127C211yesmaximal, characteristic, complemented
4C4×#14523249C211yesmaximal, characteristic, complemented
5#22550100C211yesmaximal, characteristic, complemented
6#22550098C211yesmaximal, characteristic, complemented
7#22550096C211yesmaximal, characteristic, complemented
8#22550094C211yesmaximal, characteristic, complemented
9#22550092C211yesmaximal, characteristic, complemented
10#22550090C211yesmaximal, characteristic, complemented
11#22550088C211yesmaximal, characteristic, complemented
12#22549898C211yesmaximal, characteristic, complemented
13#22549883C211yesmaximal, characteristic, complemented
14#22549879C211yesmaximal, characteristic, complemented
15#464×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19#16861241—24no—
20#16861236—24no—
21C2×#14523238C2211yescharacteristic, complemented
22C2×#14523237C2211yescharacteristic, complemented
23C2×#14523235C2211yescharacteristic
24C2×#14523235C2211yescharacteristic, complemented
25C2×D1016C2211yescharacteristic
26C2×#14523233C2211yescharacteristic
27C2×#14523231C2211yescharacteristic
28C2×#14523231C2211yescharacteristic
29#16861198C2211yescharacteristic
30#16861197C2211yescharacteristic
31#16861196C2211yescharacteristic
32#16861195—21no—
33#16861195—21no—
34Q8×Dic127C2211yescharacteristic, complemented
35C4×D508—21no—
36C4×D508C2211yescharacteristic, complemented
37#16861171—21no—
38#16861171C2211yescharacteristic, complemented
39#16861167—21no—
40#16861167C2211yescharacteristic
41C22⋊C4×D127—21no—
42C22⋊C4×D127—21no—
43C42×D127—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×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#14523249D414yes—
64C2×Dic254—21no—
65C2×Dic254C2311yescharacteristic
66Q8×D127—22no—
67Q8×D127—22no—
68C2×D508—21no—
69C2×D508C2311yescharacteristic
70C2×D508C2311yescharacteristic
71C2×C22⋊D127—21no—
72D4×D127—22no—
73D4×D127—22no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127D411yescharacteristic
79C2×C4×D127D411yescharacteristic
80C2×C4×D127C2311yescharacteristic
81#14523238—22no—
82#14523237—22no—
83#14523235—22no—
84#14523235—22no—
85D1016—22no—
86#14523233—22no—
87#14523231—22no—
88#14523231—22no—
89C2×C127⋊C8C2311yescharacteristic
90C2×C127⋊C8C2311yescharacteristic
91C4⋊Dic127—21no—
92C4⋊Dic127C2311yescharacteristic
93C22⋊Dic127—21no—
94C22⋊Dic127—21no—
95(C2×D127)⋊C4—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—
101(C2×D127)⋊C4—21no—
102C4×Dic127—21no—
103C4×Dic127—21no—
104C4×Dic127D411yescharacteristic
105C4×Dic127D411yescharacteristic
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—42no—
124C22×D127—42no—
125C22×D127C2×D411yescharacteristic
126Dic254—21no—
127Dic254—21no—
128Dic254—42no—
129D508—24no—
130D508—21no—
131D508—21no—
132D508—21no—
133D508C2×D412yes—
134C22⋊D127—42no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127C2×D411yescharacteristic
138C2×Dic127C2×D411yescharacteristic
139C2×Dic127C2×D411yescharacteristic
140C4×D127—24no—
141C4×D127—41no—
142C4×D127—41no—
143C4×D127—42no—
144C4×D127—42no—
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—22no—
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—21no—
187Dic127—21no—
188Dic127—41no—
189Dic127—41no—
190Dic127#12112yes—
191C2×C127—81no—
192C2×C127C2×#12111yescharacteristic
193C2×C127#57511yescharacteristic
194C2×C127#57511yescharacteristic
195D127—22no—
196D127—42no—
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#464D12711yescharacteristic, complemented
208#463—1271no—
209#463—1271no—
210#463—1271no—
211#463—1271no—
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×Q16C2×D12711yescharacteristic, complemented
222C2×SD16—1271no—
223C2×SD16—1271no—
224C2×SD16—1271no—
225C2×SD16C2×D12711yescharacteristic
226C2×D8—1271no—
227C2×D8—1271no—
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×Q8C2×D12711yescharacteristic, complemented
238C4×D4—1271no—
239C4×D4—1271no—
240C4×D4—2541no—
241C4×D4—2541no—
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—2541no—
265C2×Q8—2542no—
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×D4C22×D12711yescharacteristic
276C22×C4—1271no—
277C22×C4—1271no—
278C22×C4—1271no—
279C22×C4—2541no—
280C22×C4—2541no—
281C22×C4—2541no—
282C22×C4—2541no—
283Q16—22no—
284Q16—2542no—
285SD16—22no—
286SD16—2542no—
287SD16—2542no—
288SD16—2542no—
289D8—2542no—
290D8—2542no—
291M16—2542no—
292M16—2542no—
293C2×C8—1271no—
294C2×C8—1271no—
295C2×C8C22×D12711yescharacteristic
296C2×C8C22×D12711yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—1271no—
299C4⋊C4—2541no—
300C4⋊C4C22×D12711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—2541no—
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—21no—
313C42—1271no—
314C42—1271no—
315C42—1271no—
316C42—2541no—
317C42—2541no—
318C42C22×D12711yescharacteristic
319C23—21no—
320C23—1271no—
321C23—5082no—
322C23—5082no—
323C23—5082no—
324C23—2541no—
325C23—2541no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—5082no—
330Q8—2541no—
331Q8—2541no—
332Q8D4×D12712yes—
333D4—21no—
334D4—21no—
335D4—1272no—
336D4—5082no—
337D4—2544no—
338D4—2541no—
339D4—2541no—
340D4—2541no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—1271no—
345C2×C4—1272no—
346C2×C4—1272no—
347C2×C4—1271no—
348C2×C4—1271no—
349C2×C4—5082no—
350C2×C4—5081no—
351C2×C4—5081no—
352C2×C4—5082no—
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—2542no—
375C22—2541no—
376C22—2541no—
377C22C2×D4×D12711yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—1272no—
384C4—5081no—
385C4—5081no—
386C4—2541no—
387C4—2541no—
388C4C2×D4×D12711yescharacteristic
389C4#41×D12711yescharacteristic
390C2—81no—
391C2—10161no—
392C2—5082no—
393C2—2542no—
394C2#121×D12711yescharacteristic
395C2#2255067311yescharacteristic
396C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#464×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861236C2
hypercenter#464D127
hypocenterC127#3040
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
3. #124×D127C2semidirect
4. C4×#14523249C2semidirect
5. #22550100C2semidirect
6. #22550098C2semidirect
7. #22550096C2semidirect
8. #22550094C2semidirect
9. #22550092C2semidirect
10. #22550090C2semidirect
11. #22550088C2semidirect
12. #22549898C2semidirect
13. #22549883C2semidirect
14. #22549879C2semidirect
15. #464×C127C2semidirect
21. C2×#14523238C22semidirect
22. C2×#14523237C22semidirect
24. C2×#14523235C22semidirect
34. Q8×Dic127C22semidirect
36. C4×D508C22semidirect
38. #16861171C22semidirect
45. C4⋊(C127⋊C8)C22semidirect
46. Dic127⋊C8C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861134C22semidirect
49. #16861133C22semidirect
50. #16861132C22semidirect
54. C2×Q16×C127C22semidirect
57. C4×Q8×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
199. C127#3040semidirect
207. #464D127semidirect
221. C2×Q16C2×D127semidirect
237. C4×Q8C2×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×#168612412
C2×#168612362
#124×D1272
C4×#145232492
#225501002
#225500982
#225500962
#225500942
#225500922
#225500902
#225500882
#225498982
#225498832
#225498792
#464×C1272
#3040127