Finite Group Databasebeta

← #30649049

Subgroups — #30649049

Order 16256.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550431C211yesmaximal, characteristic, complemented
2#22550424C211yesmaximal, characteristic
3#22550229C214yesmaximal, complemented
4#22550228C211yesmaximal, characteristic, complemented
5#22550227C211yesmaximal, characteristic, complemented
6#22550222C211yesmaximal, characteristic, complemented
7#22549869C214yesmaximal, complemented
8C4×C8×D127C211yesmaximal, characteristic, complemented
9#486×C127C211yesmaximal, characteristic, complemented
10C2×#14523247C2212yes—
11C2×C8×D127C2212yes—
12#16861209C2211yescharacteristic
13#16861209C2211yescharacteristic
14#16861200C2212yes—
15#16861196C2212yes—
16#16861189—21no—
17#16861185—21no—
18#16861180—21no—
19#16861180—21no—
20Q8×Dic127—21no—
21#16861174—21no—
22D4×Dic127—21no—
23D4×Dic127—21no—
24#16861167C2212yes—
25C42×D127C2211yescharacteristic
26#16861134C2214yes—
27#16861132C2214yes—
28#16861130C2214yes—
29C4×C127⋊C8C2211yescharacteristic
30C8×Dic127C2212yes—
31C4⋊Q8×C127C2211yescharacteristic
32#127×C127C2211yescharacteristic
33#102×C127C2214yes—
34C4×C8×C127C2211yescharacteristic
35#14523247—28no—
36C2×Dic254—21no—
37C2×Dic254C2312yes—
38C2×C22⋊D127—22no—
39C22×Dic127—22no—
40C2×C4×D127D411yescharacteristic
41C2×C4×D127D411yescharacteristic
42C2×C4×D127C2311yescharacteristic
43C2×C127⋊C8C2312yes—
44C8×D127—24no—
45C4⋊Dic127—22no—
46C4⋊Dic127—21no—
47C4⋊Dic127—21no—
48C4⋊Dic127—21no—
49C4⋊Dic127C2312yes—
50Dic127⋊C4—22no—
51C22⋊Dic127—22no—
52C22⋊Dic127—22no—
53(C2×D127)⋊C4—22no—
54C4×Dic127—22no—
55C4×Dic127D411yescharacteristic
56C4×Dic127D411yescharacteristic
57C4×Dic127C2311yescharacteristic
58C2×Q8×C127—21no—
59C2×D4×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127C2312yes—
62C2×C8×C127C2312yes—
63C4⋊C4×C127—21no—
64C4⋊C4×C127C2312yes—
65C42×C127C2311yescharacteristic
66C22×D127C2×D411yescharacteristic
67Dic254—24no—
68Dic254—42no—
69C22⋊D127—44no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—44no—
73C2×Dic127C2×D411yescharacteristic
74C2×Dic127C2×D411yescharacteristic
75C2×Dic127C2×D411yescharacteristic
76C4×D127—22no—
77C4×D127—22no—
78C4×D127#4114yes—
79C127⋊C8—22no—
80C23×C127—22no—
81Q8×C127—42no—
82D4×C127—24no—
83D4×C127—42no—
84D4×C127—42no—
85C2×C4×C127—22no—
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2411yescharacteristic
89C8×C127—22no—
90C22×C127—42no—
91C22×C127—42no—
92C22×C127—42no—
93C22×C127C22×D411yescharacteristic
94C2×D127—21no—
95C2×D127—21no—
96C2×D127#12412yes—
97C4×C127—42no—
98C4×C127#13512yes—
99C4×C127#13512yes—
100C4×C127C2×#4112yes—
101Dic127—21no—
102Dic127—21no—
103Dic127—42no—
104Dic127—42no—
105Dic127#12412yes—
106C2×C127—82no—
107C2×C127C2×#12411yescharacteristic
108C2×C127C2×#13511yescharacteristic
109C2×C127C2×#13511yescharacteristic
110D127—22no—
111#3131—1271nomaximal, Sylow
112C127#313111yescharacteristic, complemented, Sylow
113#533—1271no—
114#532—1271no—
115#488—1274no—
116#487—1271no—
117#487—1271no—
118#486—1271no—
119#486D12711yescharacteristic, complemented
120#416—1274no—
121C2×C4×C8—1271no—
122C2×#41—1272no—
123C22×C8—1272no—
124C4⋊Q8—1271no—
125C4⋊Q8—1271no—
126C4⋊Q8C2×D12711yescharacteristic
127#127C2×D12711yescharacteristic
128#125—1272no—
129#124—1272no—
130C22⋊Q8—2541no—
131C22⋊Q8—2541no—
132#121—2541no—
133#121—2541no—
134C4×Q8—2541no—
135C4×Q8—2541no—
136C4×D4—2541no—
137C4×D4—2541no—
138#117—1272no—
139C2×C42—1271no—
140#103—1274no—
141#103—1274no—
142#102—1274no—
143#102C2×D12714yes—
144C4×C8—1272no—
145C4×C8—1271no—
146C4×C8C2×D12711yescharacteristic
147#41—2548no—
148C2×Q8—21no—
149C2×Q8—1272no—
150C2×Q8—2541no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—2542no—
154C2×D4C22×D12712yes—
155C22×C4—1271no—
156C22×C4—1271no—
157C22×C4—1271no—
158C22×C4—2542no—
159C2×C8—1272no—
160C2×C8—2544no—
161C2×C8C22×D12712yes—
162C4⋊C4—21no—
163C4⋊C4—1272no—
164C4⋊C4—2542no—
165C4⋊C4—2542no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C4⋊C4—2541no—
169C4⋊C4C22×D12712yes—
170C22⋊C4—2542no—
171C22⋊C4—2542no—
172C22⋊C4—2542no—
173C42—1271no—
174C42—1271no—
175C42—1271no—
176C42—2542no—
177C42C22×D12711yescharacteristic
178C23—22no—
179C23—1271no—
180Q8—42no—
181Q8—5082no—
182Q8—2544no—
183D4—24no—
184D4—42no—
185D4—42no—
186D4—5084no—
187C2×C4—22no—
188C2×C4—1271no—
189C2×C4—1274no—
190C2×C4—1271no—
191C2×C4—1271no—
192C2×C4—5084no—
193C2×C4—2542no—
194C2×C4—2542no—
195C2×C4—2542no—
196C2×C4—2542no—
197C2×C4D4×D12711yescharacteristic
198C2×C4D4×D12711yescharacteristic
199C2×C4C23×D12711yescharacteristic
200C8—22no—
201C8—2542no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—1272no—
206C22—2541no—
207C22—2541no—
208C22C2×D4×D12711yescharacteristic
209C4—42no—
210C4—1272no—
211C4—5082no—
212C4—5082no—
213C4—2541no—
214C4—2541no—
215C4#1686122812yes—
216C4#1686123012yes—
217C4#41×D12712yes—
218C2—82no—
219C2—2542no—
220C2#124×D12711yescharacteristic
221C2C2×#1686122811yescharacteristic
222C2C2×#1686123011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#486×C127C2
socleC22×C127C22×D4
prime radical#22550424C2
hypercenter#486D127
hypocenterC127#3131
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #22550431C2semidirect
3. #22550229C2semidirect
4. #22550228C2semidirect
5. #22550227C2semidirect
6. #22550222C2semidirect
7. #22549869C2semidirect
8. C4×C8×D127C2semidirect
9. #486×C127C2semidirect
112. C127#3131semidirect
119. #486D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
#225504242
#225502292
#225502282
#225502272
#225502222
#225498692
C4×C8×D1272
#486×C1272
#3131127