Finite Group Databasebeta

← #30648039

Subgroups — #30648039

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×#16861235C211yesmaximal, characteristic, complemented
3C2×#16861231C211yesmaximal, characteristic, complemented
4#22550457C211yesmaximal, characteristic, complemented
5#22550039C212yesmaximal, complemented
6#22550028C212yesmaximal, complemented
7#22550022C212yesmaximal, complemented
8#22550011C212yesmaximal, complemented
9#22549875C211yesmaximal, characteristic, complemented
10#22549842C211yesmaximal, characteristic, complemented
11C2×#102×C127C211yesmaximal, characteristic, complemented
12#16861254—24no—
13#16861254—24no—
14C2×#14523248C2211yescharacteristic
15C2×#14523248C2211yescharacteristic
16C2×#14523248C2211yescharacteristic, complemented
17C2×#14523247—21no—
18C2×#14523247C2212yes—
19C22×C22⋊D127—22no—
20C2×D4×D127—21no—
21C2×D4×D127C2212yes—
22#16861235—24no—
23#16861231—24no—
24C2×Dic508C2211yescharacteristic, complemented
25C2×#14523237C2211yescharacteristic, complemented
26C2×#14523237C2211yescharacteristic, complemented
27C2×#14523236C2212yescomplemented
28C2×D1016C2211yescharacteristic, complemented
29C2×#14523233C2212yescomplemented
30C2×#14523232C2211yescharacteristic, complemented
31#16861189—21no—
32#16861189C2212yescomplemented
33#16861184—21no—
34#16861184C2212yescomplemented
35#16861183—22no—
36#16861132C2212yescomplemented
37#16861129C2212yescomplemented
38(C2×D127)⋊C8C2212yescomplemented
39(C2×D127)⋊C8C2212yescomplemented
40C22×D4×C127C2211yescharacteristic
41C22×C8×C127C2211yescharacteristic, complemented
42C2×C4⋊C4×C127C2211yescharacteristic, complemented
43#102×C127C2212yescomplemented
44#102×C127C2212yescomplemented
45C23×D127—22no—
46#14523248—22no—
47#14523248—22no—
48#14523248—22no—
49#14523248—22no—
50#14523248—22no—
51#14523248D414yescomplemented
52#14523247—28no—
53#14523247—24no—
54#14523247—24no—
55C2×Dic254D411yescharacteristic
56C2×Dic254C2311yescharacteristic
57C2×Dic254C2311yescharacteristic
58C2×D508—22no—
59C2×D508D411yescharacteristic
60C2×D508C2311yescharacteristic
61C2×D508C2311yescharacteristic
62C2×C22⋊D127—216no—
63C2×C22⋊D127—22no—
64C2×C22⋊D127—22no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127D412yes—
68D4×D127—28no—
69D4×D127—24no—
70D4×D127—24no—
71C22×Dic127—22no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127C2312yes—
75Dic508—22no—
76#14523237—22no—
77#14523237—22no—
78#14523236—24no—
79D1016—22no—
80#14523233—24no—
81#14523232—22no—
82C2×C127⋊C8C2312yes—
83C4⋊Dic127—22no—
84(C2×D127)⋊C4—22no—
85(C2×D127)⋊C4—22no—
86C24×C127—21no—
87C2×D4×C127—21no—
88C2×D4×C127—22no—
89C2×D4×C127D414yescomplemented
90C2×D4×C127C2312yes—
91C22×C4×C127—21no—
92C22×C4×C127C2311yescharacteristic
93C2×C8×C127—22no—
94C2×C8×C127C2311yescharacteristic
95C2×C8×C127C2311yescharacteristic
96C4⋊C4×C127—21no—
97C4⋊C4×C127C2312yes—
98C22×D127—21no—
99C22×D127—28no—
100C22×D127—21no—
101C22×D127—82no—
102C22×D127—82no—
103C22×D127C2×D412yes—
104Dic254—22no—
105Dic254—21no—
106Dic254—21no—
107Dic254—21no—
108Dic254C2×D412yes—
109D508—22no—
110D508—21no—
111D508—21no—
112D508—21no—
113D508—82no—
114D508C2×D412yes—
115C22⋊D127—28no—
116C22⋊D127—28no—
117C22⋊D127—24no—
118C22⋊D127—24no—
119C22⋊D127—42no—
120C2×Dic127—28no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127C2×D412yes—
124C4×D127—24no—
125C4×D127—22no—
126C4×D127—22no—
127C4×D127—42no—
128C127⋊C8—22no—
129C23×C127—22no—
130C23×C127—24no—
131C23×C127—81no—
132C23×C127—81no—
133C23×C127C2×D411yescharacteristic
134D4×C127—24no—
135D4×C127—22no—
136D4×C127C2×D414yes—
137C2×C4×C127—22no—
138C2×C4×C127—81no—
139C2×C4×C127C2×D411yescharacteristic
140C2×C4×C127C2×D412yes—
141C2×C4×C127C2×D412yes—
142C2×C4×C127C2411yescharacteristic
143C8×C127—21no—
144C8×C127—21no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—24no—
148C22×C127—42no—
149C22×C127—42no—
150C22×C127—84no—
151C22×C127#12012yes—
152C22×C127C22×D411yescharacteristic
153C2×D127—22no—
154C2×D127—22no—
155C2×D127—22no—
156C2×D127—22no—
157C2×D127—41no—
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—88no—
163C4×C127—42no—
164C4×C127#12012yes—
165C4×C127C22×D411yescharacteristic
166C4×C127C22×D411yescharacteristic
167Dic127—22no—
168Dic127—22no—
169Dic127—41no—
170C2×C127—22no—
171C2×C127—44no—
172C2×C127C2×#12011yescharacteristic
173C2×C127#57611yescharacteristic
174C2×C127#57711yescharacteristic
175D127—42no—
176D127—42no—
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#417—1271no—
189C2×#102D12711yescharacteristic, complemented
190#408—1271no—
191H(5,𝔽2)—2544no—
192H(5,𝔽2)—2544no—
193C2×#41—1271no—
194C2×#41—1272no—
195C2×#41—1271no—
196C2×#41—1271no—
197C2×#41—2541no—
198C22×D4—1272no—
199C22×D4—2541no—
200C22×D4—2542no—
201C22×D4C2×D12711yescharacteristic
202AGL(1,ℤ/8)—2544no—
203#135—2544no—
204C2×Q16—1271no—
205C2×SD16—1272no—
206C2×SD16—1271no—
207C2×SD16—1271no—
208C2×D8—1272no—
209C2×D8—1271no—
210C2×M16—1271no—
211C22×C8C2×D12711yescharacteristic, complemented
212C22⋊Q8—1272no—
213C22⋊Q8—2541no—
214#121—1272no—
215#121—2541no—
216#121—2542no—
217C2×C4⋊C4C2×D12711yescharacteristic, complemented
218#103—1272no—
219#102—1272no—
220#102C2×D12712yescomplemented
221#102C2×D12712yescomplemented
222C22⋊C8—1272no—
223C22⋊C8—1272no—
224C24—21no—
225C24—2542no—
226#41—1274no—
227#41—2542no—
228#41—2542no—
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×Q8—1271no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—1272no—
241C2×D4—1271no—
242C2×D4—1271no—
243C2×D4—1271no—
244C2×D4—2542no—
245C2×D4—2542no—
246C2×D4—2544no—
247C2×D4—2548no—
248C2×D4—2541no—
249C2×D4—2542no—
250C2×D4—25416no—
251C2×D4—2541no—
252C2×D4—2544no—
253C2×D4D50814yescomplemented
254C2×D4C22×D12712yes—
255C22×C4—21no—
256C22×C4—1272no—
257C22×C4—2542no—
258C22×C4—2541no—
259C22×C4—2541no—
260C22×C4C22×D12711yescharacteristic
261Q16—2542no—
262SD16—2542no—
263SD16—2542no—
264SD16—2544no—
265D8—2542no—
266D8—2544no—
267M16—2542no—
268C2×C8—22no—
269C2×C8—1272no—
270C2×C8C22×D12711yescharacteristic
271C2×C8C22×D12711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2542no—
274C4⋊C4C22×D12712yes—
275C22⋊C4—2542no—
276C22⋊C4—2542no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—1272no—
282C23—10162no—
283C23—10162no—
284C23—2548no—
285C23—2541no—
286C23—2541no—
287C23D4×D12711yescharacteristic
288Q8—1272no—
289Q8—2541no—
290Q8—2541no—
291Q8—2541no—
292Q8—2542no—
293D4—22no—
294D4—24no—
295D4—1272no—
296D4—10162no—
297D4—5082no—
298D4—2548no—
299D4—2548no—
300D4—2544no—
301D4—2544no—
302D4—2541no—
303D4—2541no—
304D4—2541no—
305D4—2542no—
306D4C2×D50814yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—1272no—
310C2×C4—5082no—
311C2×C4—2541no—
312C2×C4—2548no—
313C2×C4—2541no—
314C2×C4—2542no—
315C2×C4—2542no—
316C2×C4—2544no—
317C2×C4D4×D12712yes—
318C2×C4D4×D12711yescharacteristic
319C2×C4C2×D50812yes—
320C2×C4C23×D12711yescharacteristic
321C8—21no—
322C8—21no—
323C8—2542no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—10168no—
331C22—5081no—
332C22—5081no—
333C22—5081no—
334C22—5081no—
335C22—5081no—
336C22—2542no—
337C22—2542no—
338C22—2542no—
339C22—2542no—
340C22#1686117912yes—
341C22C2×D4×D12711yescharacteristic
342C4—42no—
343C4—5081no—
344C4—2542no—
345C4—2542no—
346C4#1686117912yes—
347C4C2×D4×D12711yescharacteristic
348C4C22×D50811yescharacteristic
349C2—22no—
350C2—44no—
351C2—10161no—
352C2—5082no—
353C2—5082no—
354C2C2×#1686117911yescharacteristic
355C2#2255067711yescharacteristic
356C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC22×C127C22×D4
hypercenterC2×#102D127
hypocenterC127#3012
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#16861254C2semidirect
2. C2×#16861235C2semidirect
3. C2×#16861231C2semidirect
4. #22550457C2semidirect
5. #22550039C2semidirect
6. #22550028C2semidirect
7. #22550022C2semidirect
8. #22550011C2semidirect
9. #22549875C2semidirect
10. #22549842C2semidirect
11. C2×#102×C127C2semidirect
16. C2×#14523248C22semidirect
24. C2×Dic508C22semidirect
25. C2×#14523237C22semidirect
26. C2×#14523237C22semidirect
27. C2×#14523236C22semidirect
28. C2×D1016C22semidirect
29. C2×#14523233C22semidirect
30. C2×#14523232C22semidirect
32. #16861189C22semidirect
34. #16861184C22semidirect
36. #16861132C22semidirect
37. #16861129C22semidirect
38. (C2×D127)⋊C8C22semidirect
39. (C2×D127)⋊C8C22semidirect
41. C22×C8×C127C22semidirect
42. C2×C4⋊C4×C127C22semidirect
43. #102×C127C22semidirect
44. #102×C127C22semidirect
51. #14523248D4semidirect
89. C2×D4×C127D4semidirect
179. C127#3012semidirect
189. C2×#102D127semidirect
211. C22×C8C2×D127semidirect
217. C2×C4⋊C4C2×D127semidirect
220. #102C2×D127semidirect
221. #102C2×D127semidirect
253. C2×D4D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
C2×#168612352
C2×#168612312
#225504572
#225500392
#225500282
#225500222
#225500112
#225498752
#225498422
C2×#102×C1272
#3012127