Finite Group Databasebeta

← #30650525

Subgroups — #30650525

Order 16256.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550687C212yesmaximal, complemented
2#22550686C212yesmaximal, complemented
3#22550248C211yesmaximal, characteristic, complemented
4#22550138C212yesmaximal, complemented
5#22550137C212yesmaximal, complemented
6#22550060C212yesmaximal, complemented
7#22550043C211yesmaximal, characteristic, complemented
8#22549990C212yesmaximal, complemented
9#453×C127C211yesmaximal, characteristic, complemented
10#16861256C2212yes—
11#16861254C2211yescharacteristic, complemented
12#16861242C2212yescomplemented
13#16861239—22no—
14#16861239C2212yescomplemented
15#16861238—22no—
16#16861235—22no—
17#16861235C2212yes—
18#16861233—22no—
19#16861229—22no—
20#16861229C2212yes—
21#16861228—22no—
22C2×#14523236—22no—
23C2×#14523236C2212yescomplemented
24C2×#14523236C2212yescomplemented
25SD16×D127—22no—
26#16861217C2212yescomplemented
27#16861216C2212yescomplemented
28#16861209C2211yescharacteristic, complemented
29#16861144C2212yescomplemented
30#16861137C2212yescomplemented
31#16861127C2212yescomplemented
32#16861126C2211yescharacteristic, complemented
33C4×C127⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C127C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C127C2212yes—
36#127×C127C2211yescharacteristic
37#104×C127C2212yescomplemented
38#100×C127C2211yescharacteristic, complemented
39#14523248—22no—
40#14523248C2312yes—
41#14523247—22no—
42#14523247—22no—
43#14523247—42no—
44C2×Dic254—22no—
45C2×Dic254C2312yes—
46Q8×D127—22no—
47C2×C22⋊D127—41no—
48D4×D127—22no—
49Dic508—22no—
50#14523238—22no—
51#14523238D412yescomplemented
52#14523237—22no—
53#14523236—22no—
54#14523236—22no—
55#14523236—22no—
56#14523236—22no—
57#14523236—22no—
58#14523236—22no—
59#14523236D412yescomplemented
60#14523235D412yescomplemented
61#14523233—22no—
62#14523233D412yescomplemented
63#14523232—22no—
64#14523232C2312yes—
65#14523231—22no—
66C2×C127⋊C8—22no—
67C2×C127⋊C8C2312yes—
68C8×D127—22no—
69C4⋊Dic127—41no—
70#41×C127—22no—
71#41×C127C2312yes—
72C2×D4×C127—22no—
73C2×D4×C127—41no—
74C2×D4×C127—41no—
75C2×D4×C127C2311yescharacteristic
76C2×D4×C127C2311yescharacteristic
77SD16×C127—22no—
78D8×C127—22no—
79M16×C127C2312yes—
80C42×C127C2311yescharacteristic
81C22×D127—41no—
82Dic254—22no—
83Dic254—22no—
84Dic254—42no—
85Dic254C2×D412yes—
86D508C2×D412yes—
87C22⋊D127—22no—
88C22⋊D127—42no—
89C22⋊D127—42no—
90C2×Dic127—22no—
91C2×Dic127—41no—
92C2×Dic127—42no—
93C4×D127—22no—
94C4×D127—42no—
95C127⋊C8—22no—
96C127⋊C8C2×D412yes—
97C127⋊C8C2×D412yes—
98C23×C127—21no—
99C23×C127—21no—
100C23×C127—41no—
101Q8×C127C2×D412yes—
102D4×C127—22no—
103D4×C127—22no—
104D4×C127—22no—
105D4×C127—42no—
106D4×C127—82no—
107D4×C127C2×D412yes—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127—41no—
111C2×C4×C127C2411yescharacteristic
112C8×C127—22no—
113C22×C127—22no—
114C22×C127—41no—
115C22×C127—41no—
116C22×C127—81no—
117C22×C127—81no—
118C22×C127—161no—
119C22×C127H(5,𝔽2)11yescharacteristic
120C2×D127—22no—
121C2×D127—161no—
122C4×C127—22no—
123C4×C127—22no—
124C4×C127C22×D412yes—
125Dic127—22no—
126Dic127—42no—
127C2×C127—21no—
128C2×C127—42no—
129C2×C127—81no—
130C2×C127—81no—
131C2×C127D4211yescharacteristic
132D127—42no—
133#3291—1271nomaximal, Sylow
134C127#329111yescharacteristic, complemented, Sylow
135#577—1272no—
136#577—1272no—
137#492—1271no—
138#471—1272no—
139#471—1272no—
140#456—1272no—
141#453—1271no—
142#453D12711yescharacteristic, complemented
143#443—1272no—
1442−1+4—1272no—
145H(5,𝔽2)—1271no—
146H(5,𝔽2)C2×D12711yescharacteristic, complemented
147#137—1272no—
148#137—1272no—
149#137—2542no—
150#137—2542no—
151AGL(1,ℤ/8)—1272no—
152AGL(1,ℤ/8)—2542no—
153AGL(1,ℤ/8)—2542no—
154AGL(1,ℤ/8)C2×D12712yes—
155#135—1272no—
156#135—2542no—
157#135—2542no—
158C2×SD16—1272no—
159C2×SD16—1272no—
160C2×SD16—2542no—
161C2×SD16—2542no—
162#131—1272no—
163#131—1272no—
164C4⋊Q8—1271no—
165#127C2×D12711yescharacteristic
166#108—1272no—
167#104—1272no—
168#104C2×D12712yescomplemented
169#101—1272no—
170#100—1271no—
171#100C2×D12711yescharacteristic, complemented
172C4×C8—1271no—
173#41—22no—
174#41—1272no—
175#41—5082no—
176#41—2542no—
177#41—2542no—
178#41—2542no—
179#41C22×D12712yes—
180C2×Q8—1272no—
181C2×Q8—2542no—
182C2×Q8—2542no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—5081no—
187C2×D4—2542no—
188C2×D4C22×D12711yescharacteristic
189C2×D4C22×D12711yescharacteristic
190Q16—1272no—
191Q16—2542no—
192Q16—2542no—
193SD16—22no—
194SD16—1272no—
195SD16—1272no—
196SD16—2542no—
197SD16—2542no—
198SD16—2542no—
199SD16—2542no—
200SD16—2542no—
201SD16—2542no—
202SD16—2542no—
203D8—22no—
204D8—1272no—
205D8—2542no—
206M16—1272no—
207M16—2542no—
208M16—2542no—
209M16C22×D12712yes—
210C2×C8—1272no—
211C2×C8—2542no—
212C2×C8—2542no—
213C4⋊C4—5081no—
214C42C22×D12711yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—5081no—
219Q8—1272no—
220Q8—5082no—
221Q8—2542no—
222Q8—2542no—
223Q8D4×D12712yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—1272no—
230D4—5082no—
231D4—5082no—
232D4—2542no—
233D4D4×D12712yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—5082no—
238C2×C4—5082no—
239C2×C4—5081no—
240C2×C4—2542no—
241C2×C4—2542no—
242C2×C4C23×D12711yescharacteristic
243C8—22no—
244C8—1272no—
245C8—1272no—
246C8—2542no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—20321no—
254C22—2542no—
255C22#1686125411yescharacteristic
256C4—22no—
257C4—22no—
258C4—5082no—
259C4—2542no—
260C4C2×D4×D12712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—5082no—
266C2#2255050811yescharacteristic
subgroupG/H
centerC2#22550508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#453×C127C2
socleC2×C127D42
hypercenter#453D127
hypocenterC127#3291
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22550687C2semidirect
2. #22550686C2semidirect
3. #22550248C2semidirect
4. #22550138C2semidirect
5. #22550137C2semidirect
6. #22550060C2semidirect
7. #22550043C2semidirect
8. #22549990C2semidirect
9. #453×C127C2semidirect
11. #16861254C22semidirect
12. #16861242C22semidirect
14. #16861239C22semidirect
23. C2×#14523236C22semidirect
24. C2×#14523236C22semidirect
26. #16861217C22semidirect
27. #16861216C22semidirect
28. #16861209C22semidirect
29. #16861144C22semidirect
30. #16861137C22semidirect
31. #16861127C22semidirect
32. #16861126C22semidirect
33. C4×C127⋊C8C22semidirect
34. H(5,𝔽2)×C127C22semidirect
37. #104×C127C22semidirect
38. #100×C127C22semidirect
51. #14523238D4semidirect
59. #14523236D4semidirect
60. #14523235D4semidirect
62. #14523233D4semidirect
134. C127#3291semidirect
142. #453D127semidirect
146. H(5,𝔽2)C2×D127semidirect
168. #104C2×D127semidirect
171. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506872
#225506862
#225502482
#225501382
#225501372
#225500602
#225500432
#225499902
#453×C1272
#3291127