Finite Group Databasebeta

← #30650505

Subgroups — #30650505

Order 16256.

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

normal subgroups only

This group has 448 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#22550689C212yesmaximal, complemented
2#22550682C212yesmaximal, complemented
3#22550148C212yesmaximal, complemented
4#22550136C212yesmaximal, complemented
5#22550058C212yesmaximal, complemented
6#22550053C211yesmaximal, characteristic, complemented
7#22550048C211yesmaximal, characteristic, complemented
8#22549999C212yesmaximal, complemented
9#497×C127C211yesmaximal, characteristic, complemented
10#16861257C2212yes—
11#16861257C2211yescharacteristic, complemented
12#16861254C2211yescharacteristic
13#16861241—22no—
14#16861241—22no—
15#16861240C2212yes—
16#16861240C2212yes—
17#16861240C2212yescomplemented
18#16861238—22no—
19#16861236—22no—
20#16861235C2212yes—
21#16861232—22no—
22#16861231C2212yes—
23#16861230—22no—
24Q16×D127—22no—
25SD16×D127—22no—
26#16861215C2212yes—
27#16861215C2212yescomplemented
28#16861145C2212yescomplemented
29#16861137C2212yes—
30#16861137C2212yescomplemented
31#16861128C2211yescharacteristic, complemented
32#16861128C2212yescomplemented
33#16861126C2211yescharacteristic
34C2×Q16×C127C2212yescomplemented
35C2×SD16×C127C2212yes—
36C4⋊Q8×C127C2211yescharacteristic, complemented
37#124×C127C2211yescharacteristic
38#97×C127C2211yescharacteristic, complemented
39#14523249—22no—
40#14523249—22no—
41#14523249—22no—
42#14523248—42no—
43#14523248—41no—
44#14523248C2312yes—
45#14523248C2312yes—
46#14523247—22no—
47Q8×D127—22no—
48Q8×D127—22no—
49Q8×D127—22no—
50C2×C22⋊D127—41no—
51D4×D127—22no—
52Dic508D412yes—
53#14523238—22no—
54#14523238—22no—
55#14523238—22no—
56#14523237D412yes—
57#14523237D412yes—
58#14523236—22no—
59#14523235—22no—
60#14523235—22no—
61#14523235—22no—
62D1016D412yes—
63#14523233—22no—
64#14523232C2312yes—
65#14523232C2312yes—
66#14523231—22no—
67#14523231—22no—
68C8×D127—22no—
69C8×D127—22no—
70C2×Q8×C127C2311yescharacteristic
71C2×Q8×C127C2312yes—
72C2×D4×C127C2311yescharacteristic
73Q16×C127—22no—
74Q16×C127—22no—
75SD16×C127—22no—
76SD16×C127—22no—
77C2×C8×C127C2312yes—
78C4⋊C4×C127—41no—
79C22⋊C4×C127—41no—
80C42×C127C2311yescharacteristic
81C22×D127—41no—
82Dic254—41no—
83Dic254—42no—
84Dic254C2×D412yes—
85Dic254C2×D412yes—
86D508—41no—
87D508—42no—
88D508C2×D412yes—
89D508C2×D412yes—
90C22⋊D127—22no—
91C22⋊D127—22no—
92C22⋊D127—42no—
93C2×Dic127—41no—
94C4×D127—22no—
95C4×D127—22no—
96C4×D127—42no—
97C4×D127—42no—
98C4×D127—42no—
99C127⋊C8—22no—
100C127⋊C8—22no—
101C23×C127—21no—
102Q8×C127—22no—
103Q8×C127—22no—
104Q8×C127—22no—
105D4×C127—22no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127C2411yescharacteristic
110C8×C127C2×D412yes—
111C8×C127C2×D412yes—
112C22×C127—41no—
113C22×C127—81no—
114C22×C127H(5,𝔽2)11yescharacteristic
115C2×D127—22no—
116C2×D127—22no—
117C2×D127—161no—
118C4×C127—41no—
119C4×C127—41no—
120C4×C127—42no—
121C4×C127C22×D412yes—
122Dic127—22no—
123Dic127—22no—
124C2×C127—21no—
125C2×C127—81no—
126C2×C127D4211yescharacteristic
127D127—42no—
128D127—42no—
129#3289—1271nomaximal, Sylow
130C127#328911yescharacteristic, complemented, Sylow
131#578—1272no—
132#577—1272no—
133#497D12711yescharacteristic, complemented
134#473—1272no—
135#471—1272no—
136#456—1272no—
137#455—1271no—
138#454—1271no—
139#444—1272no—
1402−1+4—1272no—
1412−1+4—1271no—
142H(5,𝔽2)—1271no—
143#137—1272no—
144#137—1272no—
145#137—1272no—
146#137—2542no—
147#137—2542no—
148#137—2542no—
149AGL(1,ℤ/8)—1272no—
150AGL(1,ℤ/8)—2542no—
151#135—1272no—
152#135—2542no—
153#135—2542no—
154C2×Q16—2542no—
155C2×Q16C2×D12712yescomplemented
156C2×SD16—2542no—
157C2×SD16C2×D12712yes—
158#131—1272no—
159#131—1272no—
160C4⋊Q8C2×D12711yescharacteristic, complemented
161#124C2×D12711yescharacteristic
162#108—1272no—
163#104—1272no—
164#104—1272no—
165#101—1271no—
166#101—1272no—
167#100—1271no—
168#97C2×D12711yescharacteristic, complemented
169#41—1272no—
170#41—1272no—
171#41—5081no—
172#41—5082no—
173#41—2542no—
174#41—2542no—
175#41—2542no—
176#41—2542no—
177C2×Q8—2542no—
178C2×Q8—2542no—
179C2×Q8—2542no—
180C2×Q8C22×D12711yescharacteristic
181C2×Q8C22×D12712yes—
182C2×D4—5081no—
183C2×D4—2542no—
184C2×D4C22×D12711yescharacteristic
185Q16—22no—
186Q16—22no—
187Q16—1272no—
188Q16—2542no—
189Q16—2542no—
190Q16—2542no—
191SD16—22no—
192SD16—22no—
193SD16—1272no—
194SD16—1272no—
195SD16—2542no—
196SD16—2542no—
197SD16—2542no—
198SD16—2542no—
199D8—1272no—
200D8—2542no—
201M16—1272no—
202M16—1272no—
203M16—2542no—
204M16—2542no—
205C2×C8—2542no—
206C2×C8—2542no—
207C2×C8C22×D12712yes—
208C4⋊C4—41no—
209C22⋊C4—41no—
210C42C22×D12711yescharacteristic
211C23—21no—
212C23—5081no—
213Q8—22no—
214Q8—22no—
215Q8—22no—
216Q8—1272no—
217Q8—1272no—
218Q8—5082no—
219Q8—5081no—
220D4—22no—
221D4—1272no—
222D4—1272no—
223D4—5081no—
224D4—5082no—
225D4—5082no—
226D4—2542no—
227D4—2542no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—5082no—
232C2×C4—5082no—
233C2×C4—5082no—
234C2×C4—5081no—
235C2×C4—2542no—
236C2×C4—2542no—
237C2×C4C23×D12711yescharacteristic
238C8—2542no—
239C8—2542no—
240C8D4×D12712yes—
241C8D4×D12712yes—
242C22—41no—
243C22—81no—
244C22—20321no—
245C22—2542no—
246C22—2542no—
247C22#1686125411yescharacteristic
248C4—41no—
249C4—41no—
250C4—42no—
251C4—2542no—
252C4—2542no—
253C4C2×D4×D12712yes—
254C2—21no—
255C2—81no—
256C2—5082no—
257C2—5082no—
258C2#2255051111yescharacteristic
subgroupG/H
centerC2#22550511
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#497×C127C2
socleC2×C127D42
hypercenter#497D127
hypocenterC127#3289
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #22550689C2semidirect
2. #22550682C2semidirect
3. #22550148C2semidirect
4. #22550136C2semidirect
5. #22550058C2semidirect
6. #22550053C2semidirect
7. #22550048C2semidirect
8. #22549999C2semidirect
9. #497×C127C2semidirect
11. #16861257C22semidirect
17. #16861240C22semidirect
27. #16861215C22semidirect
28. #16861145C22semidirect
30. #16861137C22semidirect
31. #16861128C22semidirect
32. #16861128C22semidirect
34. C2×Q16×C127C22semidirect
36. C4⋊Q8×C127C22semidirect
38. #97×C127C22semidirect
130. C127#3289semidirect
133. #497D127semidirect
155. C2×Q16C2×D127semidirect
160. C4⋊Q8C2×D127semidirect
168. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506892
#225506822
#225501482
#225501362
#225500582
#225500532
#225500482
#225499992
#497×C1272
#3289127