Finite Group Databasebeta

← #30648240

Subgroups — #30648240

Order 16256.

494 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×#16861233C211yesmaximal, characteristic
2C2×SD16×D127C211yesmaximal, characteristic, complemented
3#22550446C211yesmaximal, characteristic, complemented
4#22550348C211yesmaximal, characteristic, complemented
5#22550090C211yesmaximal, characteristic, complemented
6#22550089C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550082C211yesmaximal, characteristic, complemented
9#22550081C211yesmaximal, characteristic, complemented
10#22550080C211yesmaximal, characteristic, complemented
11#22550079C211yesmaximal, characteristic, complemented
12#22549901C211yesmaximal, characteristic, complemented
13#22549883C211yesmaximal, characteristic, complemented
14#22549869C211yesmaximal, characteristic, complemented
15#463×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19#16861233—24no—
20C2×#14523237C2211yescharacteristic
21C2×#14523237C2211yescharacteristic
22C2×#14523236C2211yescharacteristic
23C2×#14523236C2211yescharacteristic
24C2×#14523235C2211yescharacteristic
25SD16×D127—24no—
26C2×#14523233C2211yescharacteristic
27C2×#14523231C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic
29#16861209C2211yescharacteristic, complemented
30#16861207C2211yescharacteristic, complemented
31#16861196C2211yescharacteristic, complemented
32#16861191—21no—
33#16861191—21no—
34#16861186—21no—
35#16861186—21no—
36C4×D508C2211yescharacteristic
37D4×Dic127C2211yescharacteristic
38#16861171C2211yescharacteristic
39C4×C22⋊D127—21no—
40(C22⋊D127)⋊C4—21no—
41#16861167C2211yescharacteristic
42#16861166C2211yescharacteristic
43C2×(C2×D127)⋊C4—21no—
44C22⋊C4×D127—21no—
45C4⋊(C127⋊C8)C2211yescharacteristic, complemented
46Dic127⋊C8C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861134C2211yescharacteristic, complemented
50#16861133C2211yescharacteristic, complemented
51#16861132C2211yescharacteristic, complemented
52#16861132C2211yescharacteristic, complemented
53#16861130C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic
55C2×D8×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic, complemented
57C4×D4×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523247—22no—
63#14523247—22no—
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×D127D414yes—
74C22×Dic127—21no—
75C22×Dic127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127—21no—
79C2×C4×D127C2311yescharacteristic
80#14523237—22no—
81#14523237—22no—
82#14523236—22no—
83#14523236—22no—
84#14523235—22no—
85#14523233—22no—
86#14523231—22no—
87C2×C127⋊C8C2311yescharacteristic
88C2×C127⋊C8C2311yescharacteristic
89C8×D127—22no—
90C4⋊Dic127—21no—
91C4⋊Dic127—21no—
92C4⋊Dic127C2311yescharacteristic
93Dic127⋊C4—21no—
94Dic127⋊C4—21no—
95Dic127⋊C4D411yescharacteristic
96Dic127⋊C4D411yescharacteristic
97C22⋊Dic127—21no—
98C22⋊Dic127—21no—
99C22⋊Dic127—21no—
100(C2×D127)⋊C4—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4D411yescharacteristic
104(C2×D127)⋊C4D411yescharacteristic
105C4×Dic127—21no—
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—24no—
122C22×D127—81no—
123C22×D127—81no—
124C22×D127C2×D411yescharacteristic
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254—42no—
130D508—21no—
131D508C2×D412yes—
132C22⋊D127—24no—
133C22⋊D127—42no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—42no—
138C2×Dic127—81no—
139C2×Dic127C2×D411yescharacteristic
140C2×Dic127C2×D411yescharacteristic
141C2×Dic127C2×D411yescharacteristic
142C4×D127—42no—
143C4×D127—81no—
144C4×D127—81no—
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—81no—
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—22no—
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—21no—
186Dic127—21no—
187Dic127—41no—
188Dic127—41no—
189Dic127—41no—
190Dic127#12112yes—
191C2×C127—42no—
192C2×C127—81no—
193C2×C127C2×#12111yescharacteristic
194C2×C127#57711yescharacteristic
195C2×C127#57711yescharacteristic
196D127—22no—
197D127—42no—
198#3042—1271nomaximal, Sylow
199C127#304211yescharacteristic, complemented, Sylow
200C2×AGL(1,ℤ/8)—1271no—
201C22×SD16—1271no—
202#536—1271no—
203#518—1271no—
204#463—1271no—
205#463—1271no—
206#463—1271no—
207#463D12711yescharacteristic, complemented
208#461—1271no—
209#461—1271no—
210#461—1271no—
211#461—1271no—
212#424—1271no—
213#419—1271no—
214#416—1271no—
215C2×#41—1271no—
216C22×Q8—1271no—
217C22×D4—1271no—
218AGL(1,ℤ/8)—2544no—
219C2×SD16—1271no—
220C2×SD16—1271no—
221C2×SD16—1271no—
222C2×SD16—1271no—
223C2×SD16—1271no—
224C2×SD16—2544no—
225C2×SD16C2×D12711yescharacteristic
226C2×D8—1271no—
227C2×D8C2×D12711yescharacteristic
228C2×M16—1271no—
229C22×C8—1271no—
230C4⋊Q8—1271no—
231C4⋊Q8—1271no—
232#124—1271no—
233#124C2×D12711yescharacteristic, complemented
234#123—2541no—
235#123—2541no—
236C22⋊Q8—2541no—
237C22⋊Q8—2541no—
238C4×D4—1271no—
239C4×D4—1271no—
240C4×D4—1271no—
241C4×D4—2541no—
242C4×D4—2541no—
243C4×D4C2×D12711yescharacteristic
244#117—1271no—
245#117—1271no—
246C2×C22⋊C4—2541no—
247C2×C22⋊C4—2541no—
248C4⋊C8—1271no—
249C4⋊C8—1271no—
250C4⋊C8—1271no—
251C4⋊C8C2×D12711yescharacteristic, complemented
252#103—1271no—
253#103—1271no—
254#103—1271no—
255#103—1271no—
256#103—1271no—
257#103C2×D12711yescharacteristic, complemented
258#102—1271no—
259#102C2×D12711yescharacteristic, complemented
260C24—2541no—
261#41—2542no—
262#41—2542no—
263C2×Q8—1271no—
264C2×Q8—1271no—
265C2×Q8—2541no—
266C2×Q8—2542no—
267C2×Q8—2542no—
268C2×Q8C22×D12711yescharacteristic
269C2×D4—1274no—
270C2×D4—1271no—
271C2×D4—2542no—
272C2×D4—2541no—
273C2×D4—2541no—
274C2×D4C22×D12711yescharacteristic
275C2×D4C22×D12711yescharacteristic
276C22×C4—21no—
277C22×C4—1271no—
278C22×C4—2541no—
279C22×C4—2541no—
280C22×C4—2541no—
281C22×C4—2541no—
282C22×C4—2541no—
283SD16—22no—
284SD16—2542no—
285SD16—2542no—
286SD16—2542no—
287SD16—2542no—
288SD16—2542no—
289D8—22no—
290D8—2542no—
291M16—2542no—
292C2×C8—1271no—
293C2×C8—1271no—
294C2×C8—2542no—
295C2×C8C22×D12711yescharacteristic
296C2×C8C22×D12711yescharacteristic
297C4⋊C4—1271no—
298C4⋊C4—1271no—
299C4⋊C4—1271no—
300C4⋊C4—2541no—
301C4⋊C4—2541no—
302C4⋊C4—2541no—
303C4⋊C4—2541no—
304C4⋊C4C22×D12711yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—1271no—
309C22⋊C4—1271no—
310C22⋊C4—2541no—
311C22⋊C4—2541no—
312C22⋊C4—2541no—
313C22⋊C4—2541no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C42—1271no—
317C42—2541no—
318C42C22×D12711yescharacteristic
319C23—21no—
320C23—21no—
321C23—1271no—
322C23—10161no—
323C23—10161no—
324C23—2544no—
325C23—2541no—
326Q8—21no—
327Q8—21no—
328Q8—5082no—
329Q8—2541no—
330Q8—2541no—
331Q8—2541no—
332Q8—2541no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—1272no—
337D4—5082no—
338D4—2544no—
339D4—2541no—
340D4D4×D12712yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—81no—
344C2×C4—1271no—
345C2×C4—1272no—
346C2×C4—1272no—
347C2×C4—1271no—
348C2×C4—1271no—
349C2×C4—10161no—
350C2×C4—10161no—
351C2×C4—10161no—
352C2×C4—5082no—
353C2×C4—5082no—
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—2542no—
375C22—2541no—
376C22—2541no—
377C22C2×D4×D12711yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—1272no—
383C4—5081no—
384C4—5081no—
385C4—5081no—
386C4—2541no—
387C4—2541no—
388C4C2×D4×D12711yescharacteristic
389C4#41×D12711yescharacteristic
390C2—42no—
391C2—81no—
392C2—5082no—
393C2—2542no—
394C2#121×D12711yescharacteristic
395C2#2255068311yescharacteristic
396C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#463×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861233C2
hypercenter#463D127
hypocenterC127#3042
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
2. C2×SD16×D127C2semidirect
3. #22550446C2semidirect
4. #22550348C2semidirect
5. #22550090C2semidirect
6. #22550089C2semidirect
7. #22550088C2semidirect
8. #22550082C2semidirect
9. #22550081C2semidirect
10. #22550080C2semidirect
11. #22550079C2semidirect
12. #22549901C2semidirect
13. #22549883C2semidirect
14. #22549869C2semidirect
15. #463×C127C2semidirect
29. #16861209C22semidirect
30. #16861207C22semidirect
31. #16861196C22semidirect
45. C4⋊(C127⋊C8)C22semidirect
46. Dic127⋊C8C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861134C22semidirect
49. #16861134C22semidirect
50. #16861133C22semidirect
51. #16861132C22semidirect
52. #16861132C22semidirect
53. #16861130C22semidirect
56. #124×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
199. C127#3042semidirect
207. #463D127semidirect
233. #124C2×D127semidirect
251. C4⋊C8C2×D127semidirect
257. #103C2×D127semidirect
259. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
C2×SD16×D1272
#225504462
#225503482
#225500902
#225500892
#225500882
#225500822
#225500812
#225500802
#225500792
#225499012
#225498832
#225498692
#463×C1272
#3042127