Finite Group Databasebeta

← #30647981

Subgroups — #30647981

Order 16256.

806 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523249C211yesmaximal, characteristic, complemented
2C2×#16861236C212yesmaximal, complemented
3#121×D127C211yesmaximal, characteristic, complemented
4#22550024C212yesmaximal, complemented
5#22550020C212yesmaximal, complemented
6#22550015C211yesmaximal, characteristic, complemented
7#22550012C211yesmaximal, characteristic, complemented
8#22550009C211yesmaximal, characteristic, complemented
9#22549876C212yesmaximal, complemented
10#22549838C211yesmaximal, characteristic, complemented
11#448×C127C211yesmaximal, characteristic, complemented
12C2×#14523249—28no—
13C2×#14523249—21no—
14C2×#14523249—21no—
15C2×#14523249C2212yes—
16C2×#14523249C2212yescomplemented
17C22×D508—21no—
18C22×D508C2211yescharacteristic, complemented
19C2×D4×D127—21no—
20C2×D4×D127C2211yescharacteristic, complemented
21C22×C4×D127—21no—
22C22×C4×D127C2211yescharacteristic
23#16861236—28no—
24C2×#14523235C2212yescomplemented
25C2×D1016C2212yescomplemented
26C2×#14523233C2212yescomplemented
27C2×#14523231C2212yescomplemented
28#16861184C2211yescharacteristic, complemented
29#16861183C2211yescharacteristic
30#16861182—21no—
31#16861181—21no—
32#16861180C2211yescharacteristic, complemented
33C4⋊C4×D127C2211yescharacteristic, complemented
34C22⋊C4×D127—21no—
35#16861133C2212yescomplemented
36#16861131C2212yescomplemented
37#16861129C2212yescomplemented
38C22⋊(C127⋊C8)C2211yescharacteristic, complemented
39(C2×D127)⋊C8C2212yescomplemented
40C22×Q8×C127C2211yescharacteristic, complemented
41C2×SD16×C127C2212yescomplemented
42#121×C127C2211yescharacteristic
43#103×C127C2212yescomplemented
44C22⋊C8×C127C2211yescharacteristic, complemented
45C23×D127—21no—
46C23×D127—21no—
47C23×D127D411yescharacteristic
48#14523249—28no—
49#14523249—28no—
50#14523249—24no—
51#14523249—24no—
52#14523249—24no—
53#14523249D418yescomplemented
54C2×D508—28no—
55C2×D508—22no—
56C2×D508—22no—
57C2×D508—24no—
58C2×D508—21no—
59C2×D508—21no—
60C2×D508C2311yescharacteristic
61C2×D508C2312yes—
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65D4×D127—24no—
66D4×D127—44no—
67C22×Dic127D411yescharacteristic
68C2×C4×D127—28no—
69C2×C4×D127—22no—
70C2×C4×D127—22no—
71C2×C4×D127—24no—
72C2×C4×D127—21no—
73C2×C4×D127D411yescharacteristic
74C2×C4×D127D411yescharacteristic
75C2×C4×D127C2311yescharacteristic
76C2×C4×D127C2311yescharacteristic
77#14523235—24no—
78D1016—24no—
79#14523233—24no—
80#14523231—24no—
81C2×C127⋊C8C2312yes—
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84C22⋊Dic127—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87C2×Q8×C127—24no—
88C2×Q8×C127—21no—
89C2×Q8×C127C2312yes—
90C2×D4×C127—21no—
91C2×D4×C127C2311yescharacteristic
92C22×C4×C127—21no—
93C22×C4×C127C2311yescharacteristic
94SD16×C127—24no—
95C2×C8×C127C2312yes—
96C4⋊C4×C127C2311yescharacteristic
97C22⋊C4×C127—21no—
98C22×D127—24no—
99C22×D127—22no—
100C22×D127—22no—
101C22×D127—22no—
102C22×D127—21no—
103C22×D127—41no—
104C22×D127—42no—
105C22×D127—42no—
106C22×D127—42no—
107C22×D127—42no—
108C22×D127—42no—
109C22×D127C2×D411yescharacteristic
110C22×D127C2×D411yescharacteristic
111D508—28no—
112D508—24no—
113D508—28no—
114D508—22no—
115D508—22no—
116D508—42no—
117D508C2×D414yescomplemented
118C22⋊D127—42no—
119C22⋊D127—42no—
120C22⋊D127—42no—
121C2×Dic127—22no—
122C2×Dic127—21no—
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C4×D127—28no—
126C4×D127—22no—
127C4×D127—22no—
128C4×D127—28no—
129C4×D127—22no—
130C4×D127—42no—
131C4×D127C2×D414yes—
132C127⋊C8—22no—
133C23×C127—21no—
134C23×C127C2×D411yescharacteristic
135Q8×C127—24no—
136Q8×C127—22no—
137Q8×C127C2×D414yescomplemented
138D4×C127—22no—
139D4×C127—42no—
140C2×C4×C127—24no—
141C2×C4×C127—22no—
142C2×C4×C127—22no—
143C2×C4×C127—21no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—22no—
147C22×C127—21no—
148C22×C127—21no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127AGL(1,ℤ/8)12yescomplemented
153C22×C127C22×D411yescharacteristic
154C2×D127—21no—
155C2×D127—21no—
156C2×D127—21no—
157C2×D127—24no—
158C2×D127—21no—
159C2×D127—21no—
160C2×D127—44no—
161C2×D127—48no—
162C2×D127—42no—
163C2×D127—42no—
164C2×D127—41no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—84no—
168C2×D127#12012yes—
169C4×C127—24no—
170C4×C127—21no—
171C4×C127—41no—
172C4×C127C22×D412yes—
173Dic127—21no—
174Dic127—41no—
175Dic127#12012yes—
176C2×C127—22no—
177C2×C127—81no—
178C2×C127C2×#12011yescharacteristic
179C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C127#57511yescharacteristic
181D127—22no—
182D127—41no—
183D127—44no—
184D127—81no—
185#3006—1271nomaximal, Sylow
186C127#300611yescharacteristic, complemented, Sylow
187C22×#41—1271no—
188C2×AGL(1,ℤ/8)—1272no—
189C2×#121—1271no—
190#449—1272no—
191#449—1272no—
192#448—1271no—
193#448D12711yescharacteristic, complemented
194#447—1271no—
195#447—1271no—
196#417—1272no—
197#407—1271no—
198C2×#41—1272no—
199C2×#41—1272no—
200C2×#41—2541no—
201C2×#41—2541no—
202C2×#41—2548no—
203C22×Q8C2×D12711yescharacteristic, complemented
204C22×D4—1271no—
205C22×D4—1271no—
206C22×D4—2541no—
207C22×D4—2541no—
208C23×C4—1271no—
209C23×C4—2541no—
210AGL(1,ℤ/8)—2548no—
211C2×SD16—1272no—
212C2×SD16C2×D12712yescomplemented
213C2×D8—1272no—
214C2×D8—1272no—
215C2×M16—1272no—
216#121—1271no—
217#121—1271no—
218#121—1271no—
219#121—2541no—
220#121—2541no—
221#121C2×D12711yescharacteristic
222C2×C4⋊C4—1271no—
223C2×C22⋊C4—2541no—
224#103—1272no—
225#103C2×D12712yescomplemented
226#102—1272no—
227#102—1272no—
228C22⋊C8—1271no—
229C22⋊C8—1272no—
230C22⋊C8C2×D12711yescharacteristic, complemented
231C24—1271no—
232C24—2541no—
233C24—2541no—
234#41—1278no—
235#41—2544no—
236#41—2544no—
237#41—2548no—
238#41—2548no—
239#41—2544no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D12712yes—
243C2×D4—21no—
244C2×D4—1272no—
245C2×D4—1271no—
246C2×D4—5084no—
247C2×D4—2544no—
248C2×D4—2541no—
249C2×D4—2542no—
250C2×D4—2542no—
251C2×D4—2548no—
252C2×D4—2544no—
253C2×D4—2541no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4—2541no—
257C2×D4C22×D12711yescharacteristic
258C22×C4—21no—
259C22×C4—1271no—
260C22×C4—1271no—
261C22×C4—1271no—
262C22×C4—1271no—
263C22×C4—1271no—
264C22×C4—2544no—
265C22×C4—2542no—
266C22×C4—2542no—
267C22×C4—2541no—
268C22×C4—2548no—
269C22×C4C22×D12711yescharacteristic
270SD16—24no—
271SD16—2544no—
272D8—2544no—
273D8—2544no—
274M16—2544no—
275C2×C8—1272no—
276C2×C8C22×D12712yes—
277C4⋊C4—1271no—
278C4⋊C4—2541no—
279C4⋊C4C22×D12711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C23—21no—
285C23—1271no—
286C23—1271no—
287C23—5082no—
288C23—5081no—
289C23—5082no—
290C23—5082no—
291C23—5082no—
292C23—5082no—
293C23—2542no—
294C23—2544no—
295C23—2541no—
296C23—2542no—
297C23—2542no—
298C23D4×D12711yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D12714yescomplemented
302D4—22no—
303D4—42no—
304D4—1274no—
305D4—5082no—
306D4—5082no—
307D4—5082no—
308D4—5082no—
309D4—2548no—
310D4—2548no—
311D4—2544no—
312D4—2542no—
313D4—2542no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—1271no—
319C2×C4—1274no—
320C2×C4—1271no—
321C2×C4—5082no—
322C2×C4—2542no—
323C2×C4—2548no—
324C2×C4—2542no—
325C2×C4—2548no—
326C2×C4—2542no—
327C2×C4—2541no—
328C2×C4—2542no—
329C2×C4D4×D12711yescharacteristic
330C2×C4C23×D12711yescharacteristic
331C8—22no—
332C8—2542no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—1272no—
339C22—10164no—
340C22—5088no—
341C22—5084no—
342C22—5081no—
343C22—5081no—
344C22—5081no—
345C22—5082no—
346C22—5082no—
347C22—2544no—
348C22—2541no—
349C22—2541no—
350C22—2541no—
351C22—2541no—
352C22—2541no—
353C22#1686123612yescomplemented
354C22C2×D4×D12711yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—1272no—
359C4—5081no—
360C4—2541no—
361C4C2×D4×D12712yes—
362C2—22no—
363C2—81no—
364C2—10161no—
365C2—5084no—
366C2—5081no—
367C2—2542no—
368C2#120×D12711yescharacteristic
369C2C2×#1686123611yescharacteristic
370C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#448×C127C2
socleC22×C127C22×D4
hypercenter#448D127
hypocenterC127#3006
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#14523249C2semidirect
2. C2×#16861236C2semidirect
3. #121×D127C2semidirect
4. #22550024C2semidirect
5. #22550020C2semidirect
6. #22550015C2semidirect
7. #22550012C2semidirect
8. #22550009C2semidirect
9. #22549876C2semidirect
10. #22549838C2semidirect
11. #448×C127C2semidirect
16. C2×#14523249C22semidirect
18. C22×D508C22semidirect
20. C2×D4×D127C22semidirect
24. C2×#14523235C22semidirect
25. C2×D1016C22semidirect
26. C2×#14523233C22semidirect
27. C2×#14523231C22semidirect
28. #16861184C22semidirect
32. #16861180C22semidirect
33. C4⋊C4×D127C22semidirect
35. #16861133C22semidirect
36. #16861131C22semidirect
37. #16861129C22semidirect
38. C22⋊(C127⋊C8)C22semidirect
39. (C2×D127)⋊C8C22semidirect
40. C22×Q8×C127C22semidirect
41. C2×SD16×C127C22semidirect
43. #103×C127C22semidirect
44. C22⋊C8×C127C22semidirect
53. #14523249D4semidirect
117. D508C2×D4semidirect
137. Q8×C127C2×D4semidirect
152. C22×C127AGL(1,ℤ/8)semidirect
186. C127#3006semidirect
193. #448D127semidirect
203. C22×Q8C2×D127semidirect
212. C2×SD16C2×D127semidirect
225. #103C2×D127semidirect
230. C22⋊C8C2×D127semidirect
301. Q8D4×D127semidirect
353. C22#16861236semidirect

Maximal subgroups

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

subgroupindex
C22×#145232492
C2×#168612362
#121×D1272
#225500242
#225500202
#225500152
#225500122
#225500092
#225498762
#225498382
#448×C1272
#3006127