Finite Group Databasebeta

← #30648454

Subgroups — #30648454

Order 16256.

466 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#22550674C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3#22550642C211yesmaximal, characteristic, complemented
4#22550143C214yesmaximal, complemented
5#22550136C214yesmaximal, complemented
6#22549925C211yesmaximal, characteristic, complemented
7#22549860C211yesmaximal, characteristic, complemented
8C2×#16861126C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11#16861240C2212yescomplemented
12#16861239—21no—
13#16861235—22no—
14#16861235C2214yes—
15#16861235C2212yescomplemented
16#16861234—21no—
17#16861229—22no—
18C2×#14523236—21no—
19C2×#14523233—21no—
20#16861217—22no—
21#16861215C2214yescomplemented
22C2×#14523232C2211yescharacteristic
23C2×#14523232C2211yescharacteristic
24C2×#14523232C2211yescharacteristic
25M16×D127—22no—
26#16861145C2214yescomplemented
27#16861128C2212yescomplemented
28#16861126C2214yescomplemented
29#16861126C2212yescomplemented
30C2×#41×C127C2211yescharacteristic
31C22×D4×C127C2211yescharacteristic
32AGL(1,ℤ/8)×C127—22no—
33AGL(1,ℤ/8)×C127—22no—
34C2×SD16×C127C2212yescomplemented
35C2×D8×C127C2212yescomplemented
36C2×M16×C127C2211yescharacteristic
37#14523248—21no—
38#14523248—21no—
39#14523248D412yes—
40#14523248C2312yes—
41C2×Dic254D411yescharacteristic
42C2×D508D411yescharacteristic
43C2×C22⋊D127—21no—
44C2×C4×D127—21no—
45#14523238—22no—
46#14523236—24no—
47#14523236—22no—
48#14523235—22no—
49#14523233—24no—
50#14523233—22no—
51#14523232—21no—
52#14523232—21no—
53#14523232—21no—
54#14523232C2312yes—
55#14523232C2312yes—
56#14523232C2312yes—
57#14523231—24no—
58C2×C127⋊C8—21no—
59C2×C127⋊C8—21no—
60C2×C127⋊C8—21no—
61C8×D127—24no—
62C24×C127—21no—
63#41×C127—21no—
64#41×C127—21no—
65#41×C127—21no—
66#41×C127—21no—
67C2×Q8×C127C2311yescharacteristic
68C2×D4×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127—21no—
72C2×D4×C127—21no—
73C2×D4×C127—21no—
74C2×D4×C127C2312yes—
75C2×D4×C127C2311yescharacteristic
76C22×C4×C127—21no—
77C22×C4×C127C2311yescharacteristic
78SD16×C127—24no—
79D8×C127—24no—
80M16×C127D412yes—
81M16×C127D412yes—
82C2×C8×C127C2312yes—
83C22×D127—21no—
84Dic254—21no—
85Dic254C2×D412yes—
86D508—21no—
87D508C2×D412yes—
88C22⋊D127—22no—
89C22⋊D127—22no—
90C2×Dic127—21no—
91C4×D127—22no—
92C4×D127—22no—
93C127⋊C8—22no—
94C127⋊C8—22no—
95C127⋊C8—22no—
96C23×C127—21no—
97C23×C127—22no—
98C23×C127—41no—
99C23×C127—41no—
100C23×C127—82no—
101C23×C127#4111yescharacteristic
102Q8×C127—21no—
103Q8×C127—21no—
104D4×C127—21no—
105D4×C127—22no—
106D4×C127—22no—
107D4×C127—21no—
108D4×C127—41no—
109D4×C127—41no—
110D4×C127—41no—
111D4×C127—41no—
112C2×C4×C127—21no—
113C2×C4×C127—41no—
114C2×C4×C127—41no—
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127#4111yescharacteristic
120C2×C4×C127C2411yescharacteristic
121C8×C127C2×D414yes—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127—42no—
125C22×C127—81no—
126C22×C127—82no—
127C22×C127—82no—
128C22×C127—82no—
129C22×C127#12111yescharacteristic
130C22×C127#12111yescharacteristic
131C22×C127C2×#4111yescharacteristic
132C2×D127—22no—
133C2×D127—81no—
134C4×C127—41no—
135C4×C127#12111yescharacteristic
136C4×C127#12111yescharacteristic
137C4×C127C22×D411yescharacteristic
138C4×C127C22×D411yescharacteristic
139Dic127—22no—
140C2×C127—21no—
141C2×C127—21no—
142C2×C127—21no—
143C2×C127—81no—
144C2×C127—82no—
145C2×C127C2×#12111yescharacteristic
146D127—42no—
147#3068—1271nomaximal, Sylow
148C127#306811yescharacteristic, complemented, Sylow
149#575—1271no—
150C2×AGL(1,ℤ/8)—1271no—
151C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
152#568—1271no—
153#472—1274no—
154#471—1274no—
155#430—1271no—
156#413—1271no—
157C2×#100—1271no—
158C2×#41—1271no—
159C2×#41C2×D12711yescharacteristic
160C22×D4C2×D12711yescharacteristic
161#137—1272no—
162#137—2541no—
163AGL(1,ℤ/8)—22no—
164AGL(1,ℤ/8)—22no—
165AGL(1,ℤ/8)—1274no—
166AGL(1,ℤ/8)—1272no—
167AGL(1,ℤ/8)—2542no—
168AGL(1,ℤ/8)—2541no—
169#135—2542no—
170C2×SD16—2541no—
171C2×SD16C2×D12712yescomplemented
172C2×D8—2541no—
173C2×D8C2×D12712yescomplemented
174#131—1274no—
175#131—2542no—
176C2×M16—1271no—
177C2×M16—1271no—
178C2×M16—1271no—
179C2×M16—2542no—
180C2×M16C2×D12711yescharacteristic
181#108—1274no—
182#101—1272no—
183#100—1274no—
184#100—1272no—
185C24—21no—
186#41—21no—
187#41—21no—
188#41—21no—
189#41—21no—
190#41—1272no—
191#41—1272no—
192#41—2541no—
193#41—2541no—
194C2×Q8—1271no—
195C2×Q8C22×D12711yescharacteristic
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—1271no—
203C2×D4—2541no—
204C2×D4C22×D12711yescharacteristic
205C2×D4C22×D12712yes—
206C22×C4—21no—
207C22×C4—2541no—
208C22×C4C22×D12711yescharacteristic
209Q16—2542no—
210SD16—24no—
211SD16—2544no—
212SD16—2542no—
213SD16—2542no—
214D8—24no—
215D8—2544no—
216D8—2542no—
217M16—1272no—
218M16—1272no—
219M16—1272no—
220M16—2541no—
221M16—2541no—
222M16—2541no—
223M16—2544no—
224M16C22⋊D12712yes—
225M16C22⋊D12712yes—
226C2×C8—2541no—
227C2×C8—2541no—
228C2×C8—2541no—
229C2×C8—2544no—
230C2×C8C22×D12712yes—
231C23—21no—
232C23—22no—
233C23—41no—
234C23—41no—
235C23—82no—
236C23—2541no—
237C23#1452324711yescharacteristic
238Q8—21no—
239Q8—21no—
240Q8—1272no—
241Q8—2541no—
242D4—22no—
243D4—21no—
244D4—21no—
245D4—22no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—1272no—
251D4—2542no—
252D4—2542no—
253D4—2541no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—2541no—
258C2×C4—2542no—
259C2×C4—2542no—
260C2×C4D4×D12711yescharacteristic
261C2×C4D4×D12711yescharacteristic
262C2×C4C2×C22⋊D12711yescharacteristic
263C2×C4C2×C22⋊D12711yescharacteristic
264C2×C4#1452324711yescharacteristic
265C2×C4C23×D12711yescharacteristic
266C8—2542no—
267C8—2542no—
268C8—2542no—
269C8C2×C22⋊D12714yes—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—82no—
274C22—81no—
275C22—82no—
276C22—82no—
277C22—10161no—
278C22—2542no—
279C22#1686118011yescharacteristic
280C22#1686118011yescharacteristic
281C22C2×#1452324711yescharacteristic
282C4—41no—
283C4—2542no—
284C4#1686118011yescharacteristic
285C4#1686118011yescharacteristic
286C4C2×D4×D12711yescharacteristic
287C4C22×C22⋊D12711yescharacteristic
288C2—21no—
289C2—21no—
290C2—21no—
291C2—82no—
292C2—81no—
293C2—5082no—
294C2C2×#1686118011yescharacteristic
subgroupG/H
centerC2C2×#16861180
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC2×C127C2×#121
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3068
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22550674C2semidirect
2. C2×#16861235C2semidirect
3. #22550642C2semidirect
4. #22550143C2semidirect
5. #22550136C2semidirect
6. #22549925C2semidirect
7. #22549860C2semidirect
8. C2×#16861126C2semidirect
9. C2×AGL(1,ℤ/8)×C127C2semidirect
11. #16861240C22semidirect
15. #16861235C22semidirect
21. #16861215C22semidirect
26. #16861145C22semidirect
27. #16861128C22semidirect
28. #16861126C22semidirect
29. #16861126C22semidirect
34. C2×SD16×C127C22semidirect
35. C2×D8×C127C22semidirect
148. C127#3068semidirect
151. C2×AGL(1,ℤ/8)D127semidirect
171. C2×SD16C2×D127semidirect
173. C2×D8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
C2×#168612352
#225506422
#225501432
#225501362
#225499252
#225498602
C2×#168611262
C2×AGL(1,ℤ/8)×C1272
#3068127