Finite Group Databasebeta

← #30648036

Subgroups — #30648036

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×#16861255C211yesmaximal, characteristic, complemented
2C2×#16861237C211yesmaximal, characteristic, complemented
3C2×#16861229C211yesmaximal, characteristic, complemented
4#22550457C211yesmaximal, characteristic, complemented
5#22550041C212yesmaximal, complemented
6#22550028C212yesmaximal, complemented
7#22550024C212yesmaximal, complemented
8#22550011C212yesmaximal, complemented
9C2×#16861129C211yesmaximal, characteristic, complemented
10#22549844C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12#16861255—24no—
13#16861255—24no—
14C2×#14523249C2212yes—
15C2×#14523248—21no—
16C2×#14523248C2211yescharacteristic
17C2×#14523248C2211yescharacteristic, complemented
18C22×D508—21no—
19C22×D508C2211yescharacteristic
20C2×D4×D127—22no—
21C2×D4×D127C2212yes—
22#16861237—24no—
23#16861229—24no—
24C2×#14523238C2211yescharacteristic, complemented
25C2×#14523237C2212yescomplemented
26C2×#14523236C2211yescharacteristic, complemented
27C2×#14523235C2211yescharacteristic, complemented
28C2×D1016C2212yescomplemented
29C2×#14523233C2211yescharacteristic, complemented
30C22×C127⋊C8C2211yescharacteristic, complemented
31#16861189—21no—
32#16861189C2212yescomplemented
33#16861184—21no—
34#16861184C2212yescomplemented
35#16861183—22no—
36#16861129C2212yescomplemented
37#16861129C2212yescomplemented
38(C2×D127)⋊C8C2212yescomplemented
39(C2×D127)⋊C8C2212yescomplemented
40C2×#41×C127C2211yescharacteristic
41C2×M16×C127C2211yescharacteristic, complemented
42C2×C4⋊C4×C127C2211yescharacteristic, complemented
43#103×C127C2212yescomplemented
44#102×C127C2212yescomplemented
45C23×D127—22no—
46C23×D127—21no—
47#14523249—24no—
48#14523249—24no—
49#14523248—28no—
50#14523248—22no—
51#14523248—22no—
52#14523248—22no—
53#14523248—22no—
54C2×Dic254D411yescharacteristic
55C2×Dic254C2311yescharacteristic
56C2×D508—28no—
57C2×D508—22no—
58C2×D508—22no—
59C2×D508—21no—
60C2×D508—22no—
61C2×D508D411yescharacteristic
62C2×D508D414yescomplemented
63C2×D508C2311yescharacteristic
64C2×D508C2312yes—
65C2×C22⋊D127—22no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127D412yes—
68D4×D127—216no—
69D4×D127—24no—
70D4×D127—24no—
71C2×C4×D127—22no—
72C2×C4×D127—21no—
73C2×C4×D127C2312yes—
74#14523238—22no—
75#14523237—24no—
76#14523236—22no—
77#14523235—22no—
78D1016—24no—
79#14523233—22no—
80C2×C127⋊C8—22no—
81C2×C127⋊C8C2311yescharacteristic
82C2×C127⋊C8C2311yescharacteristic
83C4⋊Dic127—22no—
84(C2×D127)⋊C4—22no—
85(C2×D127)⋊C4—22no—
86#41×C127—22no—
87#41×C127D414yescomplemented
88C2×Q8×C127C2311yescharacteristic
89C2×D4×C127—21no—
90C2×D4×C127C2311yescharacteristic
91C22×C4×C127—21no—
92C22×C4×C127—21no—
93C22×C4×C127C2311yescharacteristic
94M16×C127—22no—
95C2×C8×C127C2312yes—
96C4⋊C4×C127—21no—
97C4⋊C4×C127C2312yes—
98C22×D127—22no—
99C22×D127—28no—
100C22×D127—24no—
101C22×D127—21no—
102C22×D127—82no—
103C22×D127—82no—
104C22×D127—81no—
105C22×D127—81no—
106C22×D127C2×D412yes—
107Dic254—22no—
108Dic254—21no—
109Dic254—21no—
110D508—28no—
111D508—22no—
112D508—24no—
113D508—22no—
114D508—21no—
115D508—21no—
116D508—82no—
117D508C2×D414yes—
118C22⋊D127—28no—
119C22⋊D127—24no—
120C22⋊D127—42no—
121C2×Dic127—21no—
122C2×Dic127C2×D412yes—
123C4×D127—28no—
124C4×D127—22no—
125C4×D127—22no—
126C4×D127—42no—
127C127⋊C8—21no—
128C127⋊C8—21no—
129C23×C127—21no—
130C23×C127C2×D411yescharacteristic
131Q8×C127—21no—
132Q8×C127C2×D412yes—
133D4×C127—24no—
134D4×C127—21no—
135D4×C127C2×D412yes—
136C2×C4×C127—21no—
137C2×C4×C127—24no—
138C2×C4×C127—22no—
139C2×C4×C127—81no—
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—22no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127#12012yes—
151C22×C127C22×D411yescharacteristic
152C2×D127—22no—
153C2×D127—24no—
154C2×D127—22no—
155C2×D127—22no—
156C2×D127—42no—
157C2×D127—42no—
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—84no—
162C2×D127—88no—
163C4×C127—22no—
164C4×C127—42no—
165C4×C127#12012yes—
166C4×C127C22×D411yescharacteristic
167C4×C127C22×D411yescharacteristic
168Dic127—22no—
169Dic127—41no—
170C2×C127—22no—
171C2×C127—42no—
172C2×C127C2×#12011yescharacteristic
173C2×C127#57611yescharacteristic
174C2×C127#57711yescharacteristic
175D127—42no—
176D127—44no—
177D127—81no—
178#3012—1271nomaximal, Sylow
179C127#301211yescharacteristic, complemented, Sylow
180C2×H(5,𝔽2)—1271no—
181C2×AGL(1,ℤ/8)—1271no—
182C2×#135—1271no—
183#537—1271no—
184#452—1272no—
185#450—1272no—
186#449—1272no—
187#447—1272no—
188#417D12711yescharacteristic, complemented
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
198C22×D4—1272no—
199C22×D4—1271no—
200C22×D4—2541no—
201C22×D4—2542no—
202AGL(1,ℤ/8)—2544no—
203#135—2544no—
204C2×Q16—1271no—
205C2×SD16—1271no—
206C2×SD16—1271no—
207C2×SD16—1272no—
208C2×D8—1271no—
209C2×D8—1272no—
210C2×M16C2×D12711yescharacteristic, complemented
211C22×C8—1271no—
212C22⋊Q8—1272no—
213C22⋊Q8—2541no—
214#121—1272no—
215#121—2541no—
216#121—2542no—
217C2×C4⋊C4C2×D12711yescharacteristic, complemented
218#103C2×D12712yescomplemented
219#102—1272no—
220#102—1272no—
221#102C2×D12712yescomplemented
222C22⋊C8—1272no—
223C22⋊C8—1272no—
224C24—2542no—
225C24—2541no—
226#41—22no—
227#41—2548no—
228#41—2542no—
229#41—2542no—
230#41—2544no—
231#41—2544no—
232#41—2542no—
233#41—2542no—
234#41D50814yescomplemented
235C2×Q8—1271no—
236C2×Q8—1271no—
237C2×Q8C22×D12711yescharacteristic
238C2×D4—21no—
239C2×D4—1272no—
240C2×D4—1274no—
241C2×D4—1271no—
242C2×D4—1272no—
243C2×D4—1271no—
244C2×D4—2542no—
245C2×D4—2542no—
246C2×D4—2548no—
247C2×D4—2542no—
248C2×D4—2544no—
249C2×D4—25416no—
250C2×D4—2541no—
251C2×D4—2544no—
252C2×D4—2541no—
253C2×D4—2542no—
254C2×D4C22×D12711yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—1272no—
258C22×C4—2542no—
259C22×C4—2541no—
260C22×C4C22×D12711yescharacteristic
261Q16—2542no—
262SD16—2544no—
263SD16—2542no—
264SD16—2542no—
265D8—2544no—
266D8—2542no—
267M16—22no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—2542no—
271C2×C8C22×D12712yes—
272C4⋊C4—21no—
273C4⋊C4—2542no—
274C4⋊C4C22×D12712yes—
275C22⋊C4—2542no—
276C22⋊C4—2542no—
277C23—21no—
278C23—1272no—
279C23—10161no—
280C23—10161no—
281C23—10162no—
282C23—10162no—
283C23—2548no—
284C23—2544no—
285C23—2541no—
286C23—2542no—
287C23D4×D12711yescharacteristic
288Q8—21no—
289Q8—2541no—
290Q8—2541no—
291Q8—2542no—
292Q8C2×D50812yes—
293D4—24no—
294D4—21no—
295D4—1274no—
296D4—10162no—
297D4—5082no—
298D4—2548no—
299D4—2548no—
300D4—2544no—
301D4—2541no—
302D4—2541no—
303D4—2542no—
304D4—2544no—
305D4—2542no—
306D4C2×D50812yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—1272no—
312C2×C4—5082no—
313C2×C4—2548no—
314C2×C4—2541no—
315C2×C4—2542no—
316C2×C4—2542no—
317C2×C4D4×D12712yes—
318C2×C4D4×D12711yescharacteristic
319C2×C4C2×D50812yes—
320C2×C4C23×D12711yescharacteristic
321C8—22no—
322C8—2541no—
323C8—2541no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—10164no—
330C22—10168no—
331C22—5081no—
332C22—5081no—
333C22—5081no—
334C22—5082no—
335C22—5082no—
336C22—2544no—
337C22—2542no—
338C22—2542no—
339C22—2542no—
340C22#1686117912yes—
341C22C2×D4×D12711yescharacteristic
342C4—22no—
343C4—42no—
344C4—5081no—
345C4—2542no—
346C4#1686117912yes—
347C4C2×D4×D12711yescharacteristic
348C4C22×D50811yescharacteristic
349C2—22no—
350C2—42no—
351C2—10161no—
352C2—5084no—
353C2—5082no—
354C2C2×#1686117911yescharacteristic
355C2#2255068011yescharacteristic
356C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
hypercenter#417D127
hypocenterC127#3012
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16861255C2semidirect
2. C2×#16861237C2semidirect
3. C2×#16861229C2semidirect
4. #22550457C2semidirect
5. #22550041C2semidirect
6. #22550028C2semidirect
7. #22550024C2semidirect
8. #22550011C2semidirect
9. C2×#16861129C2semidirect
10. #22549844C2semidirect
11. #417×C127C2semidirect
17. C2×#14523248C22semidirect
24. C2×#14523238C22semidirect
25. C2×#14523237C22semidirect
26. C2×#14523236C22semidirect
27. C2×#14523235C22semidirect
28. C2×D1016C22semidirect
29. C2×#14523233C22semidirect
30. C22×C127⋊C8C22semidirect
32. #16861189C22semidirect
34. #16861184C22semidirect
36. #16861129C22semidirect
37. #16861129C22semidirect
38. (C2×D127)⋊C8C22semidirect
39. (C2×D127)⋊C8C22semidirect
41. C2×M16×C127C22semidirect
42. C2×C4⋊C4×C127C22semidirect
43. #103×C127C22semidirect
44. #102×C127C22semidirect
62. C2×D508D4semidirect
87. #41×C127D4semidirect
179. C127#3012semidirect
188. #417D127semidirect
210. C2×M16C2×D127semidirect
217. C2×C4⋊C4C2×D127semidirect
218. #103C2×D127semidirect
221. #102C2×D127semidirect
234. #41D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
C2×#168612372
C2×#168612292
#225504572
#225500412
#225500282
#225500242
#225500112
C2×#168611292
#225498442
#417×C1272
#3012127