Finite Group Databasebeta

← #30648033

Subgroups — #30648033

Order 16256.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#16861254C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550454C211yesmaximal, characteristic, complemented
4#22550035C212yesmaximal, complemented
5#22550025C212yesmaximal, complemented
6#22550018C212yesmaximal, complemented
7#22550008C212yesmaximal, complemented
8#22549878C211yesmaximal, characteristic, complemented
9C2×#16861130C211yesmaximal, characteristic, complemented
10#22549844C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12#16861254—24no—
13#16861254—24no—
14C2×#14523248C2211yescharacteristic
15C2×#14523248C2211yescharacteristic
16C2×#14523247—21no—
17C2×#14523247C2212yes—
18C22×C22⋊D127—22no—
19C2×D4×D127—21no—
20C2×D4×D127C2212yes—
21#16861229—24no—
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523236C2211yescharacteristic, complemented
24C2×#14523235C2211yescharacteristic, complemented
25C2×#14523233C2211yescharacteristic, complemented
26C22×C127⋊C8C2211yescharacteristic, complemented
27#16861188—21no—
28#16861185C2212yescomplemented
29#16861183—21no—
30#16861181—22no—
31#16861180C2212yescomplemented
32C2×C4⋊Dic127C2211yescharacteristic, complemented
33#16861134C2212yescomplemented
34#16861131C2212yescomplemented
35#16861130C2212yescomplemented
36#16861130C2212yescomplemented
37(C2×D127)⋊C8C2212yescomplemented
38(C2×D127)⋊C8C2212yescomplemented
39C2×#41×C127C2211yescharacteristic, complemented
40C22×D4×C127C2211yescharacteristic
41AGL(1,ℤ/8)×C127—24no—
42C2×SD16×C127C2212yescomplemented
43C2×D8×C127C2212yescomplemented
44C2×M16×C127C2211yescharacteristic, complemented
45C23×D127—22no—
46#14523248—22no—
47#14523248—22no—
48#14523248—22no—
49#14523248D414yescomplemented
50#14523247—28no—
51#14523247—24no—
52#14523247—24no—
53C2×Dic254D411yescharacteristic
54C2×Dic254C2311yescharacteristic
55C2×D508D411yescharacteristic
56C2×D508C2311yescharacteristic
57C2×C22⋊D127—216no—
58C2×C22⋊D127—22no—
59C2×C22⋊D127—22no—
60C2×C22⋊D127—22no—
61C2×C22⋊D127—21no—
62C2×C22⋊D127D412yes—
63D4×D127—28no—
64D4×D127—24no—
65D4×D127—24no—
66C22×Dic127—22no—
67C22×Dic127—21no—
68C2×C4×D127—21no—
69C2×C4×D127C2312yes—
70#14523238—22no—
71#14523236—22no—
72#14523235—22no—
73#14523233—22no—
74C2×C127⋊C8—22no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C4⋊Dic127—21no—
78C4⋊Dic127C2312yes—
79Dic127⋊C4—22no—
80C22⋊Dic127—22no—
81(C2×D127)⋊C4—22no—
82C24×C127—21no—
83#41×C127—22no—
84#41×C127—22no—
85C2×Q8×C127C2311yescharacteristic
86C2×D4×C127—21no—
87C2×D4×C127—21no—
88C2×D4×C127—22no—
89C2×D4×C127D414yescomplemented
90C2×D4×C127C2311yescharacteristic
91C2×D4×C127C2312yes—
92C22×C4×C127—21no—
93C22×C4×C127C2311yescharacteristic
94SD16×C127—24no—
95D8×C127—24no—
96M16×C127—22no—
97C2×C8×C127C2312yes—
98C22×D127—21no—
99C22×D127—28no—
100C22×D127—82no—
101C22×D127—82no—
102C22×D127C2×D412yes—
103Dic254—22no—
104Dic254—21no—
105Dic254C2×D412yes—
106D508—22no—
107D508—21no—
108D508C2×D412yes—
109C22⋊D127—28no—
110C22⋊D127—28no—
111C22⋊D127—24no—
112C22⋊D127—24no—
113C22⋊D127—82no—
114C2×Dic127—28no—
115C2×Dic127—21no—
116C2×Dic127—22no—
117C2×Dic127—81no—
118C2×Dic127C2×D412yes—
119C4×D127—24no—
120C4×D127—22no—
121C4×D127—22no—
122C127⋊C8—21no—
123C127⋊C8—21no—
124C23×C127—22no—
125C23×C127—24no—
126C23×C127—21no—
127C23×C127—81no—
128C23×C127—81no—
129C23×C127C2×D411yescharacteristic
130Q8×C127—21no—
131Q8×C127—21no—
132D4×C127—24no—
133D4×C127—22no—
134D4×C127—21no—
135D4×C127—21no—
136D4×C127—42no—
137D4×C127C2×D414yes—
138C2×C4×C127—21no—
139C2×C4×C127—42no—
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D412yes—
142C2×C4×C127C2×D412yes—
143C2×C4×C127C2411yescharacteristic
144C8×C127—22no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—24no—
148C22×C127—42no—
149C22×C127—42no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—84no—
154C22×C127#12012yes—
155C22×C127C22×D411yescharacteristic
156C2×D127—22no—
157C2×D127—22no—
158C2×D127—22no—
159C2×D127—22no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—88no—
163C4×C127—41no—
164C4×C127#12012yes—
165C4×C127C22×D411yescharacteristic
166C4×C127C22×D411yescharacteristic
167Dic127—22no—
168Dic127—22no—
169Dic127—42no—
170C2×C127—22no—
171C2×C127—44no—
172C2×C127—81no—
173C2×C127C2×#12011yescharacteristic
174C2×C127#57611yescharacteristic
175C2×C127#57711yescharacteristic
176D127—42no—
177D127—42no—
178#3012—1271nomaximal, Sylow
179C127#301211yescharacteristic, complemented, Sylow
180C2×H(5,𝔽2)—1271no—
181C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
182C2×#135—1271no—
183#537—1271no—
184#452—1272no—
185#450—1272no—
186#449—1272no—
187#447—1272no—
188#417—1271no—
189C2×#102—1271no—
190#408—1271no—
191H(5,𝔽2)—2544no—
192H(5,𝔽2)—2544no—
193C2×#41—1271no—
194C2×#41—1272no—
195C2×#41—1271no—
196C2×#41—2541no—
197C2×#41C2×D12711yescharacteristic, complemented
198C22×D4—1272no—
199C22×D4—2541no—
200C22×D4—2542no—
201C22×D4C2×D12711yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—2544no—
204C2×Q16—1271no—
205C2×SD16—1271no—
206C2×SD16—1271no—
207C2×SD16C2×D12712yescomplemented
208C2×D8—1271no—
209C2×D8C2×D12712yescomplemented
210C2×M16C2×D12711yescharacteristic, complemented
211C22×C8—1271no—
212C22⋊Q8—1272no—
213C22⋊Q8—2541no—
214#121—1272no—
215#121—2541no—
216#121—2542no—
217C2×C4⋊C4—1271no—
218#103—1272no—
219#102—1272no—
220#102—1272no—
221#102—1272no—
222C22⋊C8—1272no—
223C22⋊C8—1272no—
224C24—21no—
225C24—2542no—
226#41—22no—
227#41—22no—
228#41—1274no—
229#41—2544no—
230#41—2548no—
231#41—2544no—
232#41—2542no—
233#41—2542no—
234#41—2542no—
235C2×Q8—1271no—
236C2×Q8—1271no—
237C2×Q8C22×D12711yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—1272no—
242C2×D4—1271no—
243C2×D4—1271no—
244C2×D4—2542no—
245C2×D4—2544no—
246C2×D4—2548no—
247C2×D4—2542no—
248C2×D4—2542no—
249C2×D4—25416no—
250C2×D4—2541no—
251C2×D4—2544no—
252C2×D4C22⋊D12714yescomplemented
253C2×D4C22×D12711yescharacteristic
254C2×D4C22×D12712yes—
255C22×C4—21no—
256C22×C4—1272no—
257C22×C4—2542no—
258C22×C4—2541no—
259C22×C4—2541no—
260C22×C4C22×D12711yescharacteristic
261Q16—2542no—
262SD16—24no—
263SD16—2542no—
264SD16—2542no—
265D8—24no—
266D8—2542no—
267M16—22no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—2542no—
271C2×C8C22×D12712yes—
272C4⋊C4—1272no—
273C4⋊C4—2542no—
274C4⋊C4—2541no—
275C22⋊C4—2542no—
276C22⋊C4—2542no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—1272no—
283C23—10162no—
284C23—10162no—
285C23—2548no—
286C23—2541no—
287C23D4×D12711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—1272no—
291Q8—2541no—
292Q8—2542no—
293D4—22no—
294D4—21no—
295D4—21no—
296D4—24no—
297D4—42no—
298D4—1272no—
299D4—10162no—
300D4—2548no—
301D4—2548no—
302D4—2544no—
303D4—2544no—
304D4—2541no—
305D4—2542no—
306D4C2×C22⋊D12714yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—1272no—
310C2×C4—10161no—
311C2×C4—2542no—
312C2×C4—2548no—
313C2×C4—2541no—
314C2×C4—2542no—
315C2×C4—2542no—
316C2×C4—2544no—
317C2×C4D4×D12712yes—
318C2×C4D4×D12711yescharacteristic
319C2×C4C2×C22⋊D12712yes—
320C2×C4C23×D12711yescharacteristic
321C8—22no—
322C8—2541no—
323C8—2541no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—10168no—
334C22—5081no—
335C22—5081no—
336C22—2542no—
337C22—2542no—
338C22—2542no—
339C22—2542no—
340C22#1686117712yes—
341C22C2×D4×D12711yescharacteristic
342C4—41no—
343C4—5082no—
344C4—2542no—
345C4—2542no—
346C4#1686117712yes—
347C4C2×D4×D12711yescharacteristic
348C4C22×C22⋊D12711yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—5082no—
353C2—5082no—
354C2C2×#1686117711yescharacteristic
355C2#2255068011yescharacteristic
356C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3012
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861254C2semidirect
2. C2×#16861229C2semidirect
3. #22550454C2semidirect
4. #22550035C2semidirect
5. #22550025C2semidirect
6. #22550018C2semidirect
7. #22550008C2semidirect
8. #22549878C2semidirect
9. C2×#16861130C2semidirect
10. #22549844C2semidirect
11. C2×AGL(1,ℤ/8)×C127C2semidirect
22. C2×#14523238C22semidirect
23. C2×#14523236C22semidirect
24. C2×#14523235C22semidirect
25. C2×#14523233C22semidirect
26. C22×C127⋊C8C22semidirect
28. #16861185C22semidirect
31. #16861180C22semidirect
32. C2×C4⋊Dic127C22semidirect
33. #16861134C22semidirect
34. #16861131C22semidirect
35. #16861130C22semidirect
36. #16861130C22semidirect
37. (C2×D127)⋊C8C22semidirect
38. (C2×D127)⋊C8C22semidirect
39. C2×#41×C127C22semidirect
42. C2×SD16×C127C22semidirect
43. C2×D8×C127C22semidirect
44. C2×M16×C127C22semidirect
49. #14523248D4semidirect
89. C2×D4×C127D4semidirect
179. C127#3012semidirect
181. C2×AGL(1,ℤ/8)D127semidirect
197. C2×#41C2×D127semidirect
207. C2×SD16C2×D127semidirect
209. C2×D8C2×D127semidirect
210. C2×M16C2×D127semidirect
252. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
C2×#168612292
#225504542
#225500352
#225500252
#225500182
#225500082
#225498782
C2×#168611302
#225498442
C2×AGL(1,ℤ/8)×C1272
#3012127