Finite Group Databasebeta

← #30639770

Subgroups — #30639770

Order 16256.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16861235C211yesmaximal, characteristic, complemented
2#22550454C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic, complemented
5#22549779C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861235—22no—
10#16861235—22no—
11C2×#14523236—21no—
12C2×#14523233—21no—
13C2×#14523232C2211yescharacteristic
14#16861188—21no—
15#16861185—21no—
16#16861183—21no—
17#16861181—22no—
18#16861180—21no—
19C2×C4⋊Dic127C2211yescharacteristic
20C2×C22⋊Dic127—22no—
21C2×C4×Dic127C2211yescharacteristic
22#16861136—22no—
23#16861135—22no—
24C2×#41×C127C2211yescharacteristic
25C22×D4×C127C2211yescharacteristic
26AGL(1,ℤ/8)×C127—22no—
27AGL(1,ℤ/8)×C127—22no—
28C2×SD16×C127—21no—
29C2×D8×C127—21no—
30C2×M16×C127C2211yescharacteristic
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35C2×Dic254D411yescharacteristic
36C2×D508D411yescharacteristic
37C2×C22⋊D127—22no—
38C2×C22⋊D127—21no—
39C22×Dic127—21no—
40C22×Dic127D412yes—
41C2×C4×D127—21no—
42#14523236—24no—
43#14523233—24no—
44#14523232—21no—
45#14523232D412yescomplemented
46C2×C127⋊C8—21no—
47C4⋊Dic127—21no—
48C4⋊Dic127—21no—
49Dic127⋊C4—22no—
50C22⋊Dic127—22no—
51C22⋊Dic127—44no—
52(C2×D127)⋊C4—22no—
53C4×Dic127—21no—
54C4×Dic127—21no—
55C24×C127—21no—
56#41×C127—21no—
57#41×C127—21no—
58#41×C127—21no—
59#41×C127—21no—
60C2×Q8×C127D411yescharacteristic
61C2×D4×C127—21no—
62C2×D4×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127—21no—
67C2×D4×C127D411yescharacteristic
68C2×D4×C127D412yescomplemented
69C22×C4×C127—21no—
70C22×C4×C127C2311yescharacteristic
71SD16×C127—24no—
72D8×C127—24no—
73M16×C127—21no—
74M16×C127D412yescomplemented
75C2×C8×C127—21no—
76C22×D127—21no—
77Dic254—21no—
78Dic254—21no—
79D508—21no—
80D508—21no—
81C22⋊D127—41no—
82C22⋊D127—41no—
83C22⋊D127—82no—
84C2×Dic127—22no—
85C2×Dic127—22no—
86C2×Dic127—21no—
87C2×Dic127—41no—
88C2×Dic127—41no—
89C2×Dic127—42no—
90C4×D127—41no—
91C4×D127—41no—
92C127⋊C8—22no—
93C23×C127—21no—
94C23×C127—22no—
95C23×C127—42no—
96C23×C127—42no—
97C23×C127—41no—
98C23×C127—41no—
99C23×C127C2×D411yescharacteristic
100Q8×C127—21no—
101Q8×C127—21no—
102D4×C127—21no—
103D4×C127—22no—
104D4×C127—22no—
105D4×C127—21no—
106D4×C127—41no—
107D4×C127—41no—
108D4×C127—41no—
109D4×C127—41no—
110C2×C4×C127—21no—
111C2×C4×C127—41no—
112C2×C4×C127—41no—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127#4111yescharacteristic
119C8×C127—22no—
120C22×C127—21no—
121C22×C127—21no—
122C22×C127—44no—
123C22×C127—41no—
124C22×C127—42no—
125C22×C127—41no—
126C22×C127—41no—
127C22×C127—82no—
128C22×C127—81no—
129C22×C127—81no—
130C22×C127#12011yescharacteristic
131C22×C127#12111yescharacteristic
132C22×C127#12711yescharacteristic
133C2×D127—41no—
134C2×D127—41no—
135C2×D127—41no—
136C4×C127—41no—
137C4×C127#12011yescharacteristic
138C4×C127#12011yescharacteristic
139C4×C127#12111yescharacteristic
140C4×C127#12111yescharacteristic
141Dic127—42no—
142Dic127—41no—
143Dic127—42no—
144C2×C127—21no—
145C2×C127—21no—
146C2×C127—81no—
147C2×C127—82no—
148C2×C127#39211yescharacteristic
149C2×C127#45311yescharacteristic
150C2×C127#45411yescharacteristic
151D127—81no—
152#2006—1271nomaximal, Sylow
153C127#200611yescharacteristic, complemented, Sylow
154C2×AGL(1,ℤ/8)—1271no—
155C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
156#537—1271no—
157C2×#104—1271no—
158C2×#104—1271no—
159#390—1271no—
160#343—1271no—
161C2×#41—1271no—
162C2×#41C2×D12711yescharacteristic
163C22×D4C2×D12711yescharacteristic
164AGL(1,ℤ/8)—22no—
165AGL(1,ℤ/8)—22no—
166AGL(1,ℤ/8)—2542no—
167AGL(1,ℤ/8)—2542no—
168C2×SD16—21no—
169C2×SD16—2541no—
170C2×D8—21no—
171C2×D8—2541no—
172C2×M16—1271no—
173C2×M16C2×D12711yescharacteristic
174C22⋊Q8—2541no—
175C22⋊Q8—2541no—
176#121—2541no—
177#121—2542no—
178#121—2541no—
179C2×C4⋊C4—1271no—
180C2×C22⋊C4—2542no—
181C2×C42—1271no—
182#104—2542no—
183#104—2542no—
184C24—21no—
185#41—21no—
186#41—21no—
187#41—21no—
188#41—21no—
189#41—2541no—
190#41—2541no—
191#41—2541no—
192#41—2541no—
193C2×Q8—1271no—
194C2×Q8C22⋊D12711yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—1271no—
202C2×D4—2542no—
203C2×D4—2541no—
204C2×D4C22⋊D12711yescharacteristic
205C2×D4C22⋊D12712yescomplemented
206C22×C4—21no—
207C22×C4—1272no—
208C22×C4—2541no—
209C22×C4—2541no—
210C22×C4C22×D12711yescharacteristic
211SD16—24no—
212SD16—2544no—
213D8—24no—
214D8—2544no—
215M16—21no—
216M16—1272no—
217M16—2541no—
218M16C22⋊D12712yescomplemented
219C2×C8—21no—
220C2×C8—2541no—
221C4⋊C4—2542no—
222C4⋊C4—2541no—
223C4⋊C4—2541no—
224C22⋊C4—5084no—
225C22⋊C4—2542no—
226C22⋊C4—2542no—
227C42—2541no—
228C42—2541no—
229C23—21no—
230C23—22no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—2541no—
236C23D4×D12711yescharacteristic
237Q8—21no—
238Q8—21no—
239Q8—2541no—
240Q8—2541no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—21no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—10162no—
250D4—5081no—
251D4—5081no—
252D4—2541no—
253D4—2541no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—5081no—
258C2×C4—5081no—
259C2×C4—5081no—
260C2×C4—5081no—
261C2×C4—5082no—
262C2×C4—2542no—
263C2×C4—2542no—
264C2×C4—2541no—
265C2×C4D4×D12711yescharacteristic
266C2×C4D4×D12711yescharacteristic
267C2×C4C2×C22⋊D12711yescharacteristic
268C2×C4C2×C22⋊D12711yescharacteristic
269C2×C4C2×C22⋊D12711yescharacteristic
270C2×C4#1452324711yescharacteristic
271C8—22no—
272C8—2542no—
273C22—21no—
274C22—21no—
275C22—44no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22—5081no—
284C22—5081no—
285C22—5081no—
286C22#1686117711yescharacteristic
287C22#1686118111yescharacteristic
288C22#1686120511yescharacteristic
289C4—41no—
290C4—5082no—
291C4—5081no—
292C4—5082no—
293C4#1686117711yescharacteristic
294C4#1686117811yescharacteristic
295C4#1686118011yescharacteristic
296C4#1686118111yescharacteristic
297C2—21no—
298C2—21no—
299C2—82no—
300C2—81no—
301C2—10161no—
302C2#2254978411yescharacteristic
303C2#2255004211yescharacteristic
304C2#2255004711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127#120
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#2006
LCSG ▷ C22×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×#16861235C2semidirect
2. #22550454C2semidirect
3. C2×#16861136C2semidirect
4. C2×#16861135C2semidirect
5. #22549779C2semidirect
6. #22549644C2semidirect
7. C2×AGL(1,ℤ/8)×C127C2semidirect
45. #14523232D4semidirect
68. C2×D4×C127D4semidirect
74. M16×C127D4semidirect
153. C127#2006semidirect
155. C2×AGL(1,ℤ/8)D127semidirect
205. C2×D4C22⋊D127semidirect
218. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225504542
C2×#168611362
C2×#168611352
#225497792
#225496442
C2×AGL(1,ℤ/8)×C1272
#2006127