Finite Group Databasebeta

← #30648829

Subgroups — #30648829

Order 16256.

444 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 133 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#22550433C211yesmaximal, characteristic, complemented
2#22550211C212yesmaximal, complemented
3#22550156C212yesmaximal, complemented
4#22550081C212yesmaximal, complemented
5#22550027C212yesmaximal
6#22549976C214yesmaximal, complemented
7#22549933C211yesmaximal, characteristic, complemented
8C2×C4×D4×C127C211yesmaximal, characteristic
9C2×#14523236C2214yes—
10#16861209C2211yescharacteristic, complemented
11#16861209C2211yescharacteristic, complemented
12#16861201—21no—
13#16861188—21no—
14#16861188C2212yes—
15C4×Dic254—21no—
16C4×Dic254C2212yescomplemented
17#16861168—21no—
18#16861142C2214yescomplemented
19C4⋊(C127⋊C8)C2212yescomplemented
20#16861132C2214yescomplemented
21#16861130C2214yes—
22C22⋊(C127⋊C8)C2212yes—
23#16861120C2212yescomplemented
24C22×D4×C127C2211yescharacteristic
25C23×C4×C127—21no—
26C4×D4×C127—21no—
27C4×D4×C127—21no—
28C4×D4×C127C2212yes—
29C4×D4×C127C2212yes—
30C2×C4⋊C4×C127C2211yescharacteristic
31C2×C22⋊C4×C127—21no—
32C2×C42×C127C2211yescharacteristic
33C2×Dic254—21no—
34C2×Dic254C2312yes—
35#14523236—28no—
36C2×C127⋊C8C2314yes—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127C2312yes—
41Dic127⋊C4—22no—
42Dic127⋊C4—22no—
43C22⋊Dic127—22no—
44C4×Dic127—22no—
45C24×C127—21no—
46C2×D4×C127—24no—
47C2×D4×C127—21no—
48C2×D4×C127C2312yes—
49C22×C4×C127—21no—
50C22×C4×C127—22no—
51C22×C4×C127—22no—
52C22×C4×C127—44no—
53C22×C4×C127D411yescharacteristic
54C22×C4×C127D411yescharacteristic
55C22×C4×C127C2311yescharacteristic
56C4⋊C4×C127—21no—
57C4⋊C4×C127C2312yes—
58C22⋊C4×C127—22no—
59C22⋊C4×C127—22no—
60C42×C127D411yescharacteristic
61C42×C127D411yescharacteristic
62C42×C127C2311yescharacteristic
63C42×C127C2311yescharacteristic
64Dic254—24no—
65Dic254—42no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C127⋊C8—24no—
69C23×C127—24no—
70C23×C127—22no—
71C23×C127—42no—
72C23×C127—42no—
73C23×C127C2×D411yescharacteristic
74D4×C127—24no—
75D4×C127—22no—
76D4×C127#4114yes—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—44no—
82C2×C4×C127—42no—
83C2×C4×C127—44no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2411yescharacteristic
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—24no—
93C22×C127—48no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127#13712yes—
97C22×C127C22×D411yescharacteristic
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—42no—
103C4×C127AGL(1,ℤ/8)12yes—
104C4×C127C2×#4112yes—
105Dic127—42no—
106Dic127—42no—
107C2×C127—22no—
108C2×C127—44no—
109C2×C127#52511yescharacteristic
110C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C127C2×#13711yescharacteristic
112#3110—1271nomaximal, Sylow
113C127#311011yescharacteristic, complemented, Sylow
114#533—1271no—
115C2×C4×D4D12711yescharacteristic
116#484—1272no—
117#474—1272no—
118#461—1272no—
119#450—1272no—
120#440—1274no—
121#432—1271no—
122C22×D4C2×D12711yescharacteristic
123C23×C4—21no—
124C2×SD16—1274no—
125C4⋊Q8—1271no—
126C4⋊Q8—1271no—
127#125—2541no—
128C22⋊Q8—1272no—
129C22⋊Q8—2541no—
130C4×Q8—1272no—
131C4×Q8—2541no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4C2×D12712yes—
135C4×D4C2×D12712yes—
136#117—2541no—
137C2×C4⋊C4C2×D12711yescharacteristic
138C2×C22⋊C4—21no—
139C2×C42C2×D12711yescharacteristic
140#107—1274no—
141C4⋊C8—1272no—
142#103—1274no—
143#102—1274no—
144C22⋊C8—1272no—
145#97—1272no—
146C24—21no—
147C2×Q8—1272no—
148C2×Q8—2541no—
149C2×D4—24no—
150C2×D4—21no—
151C2×D4C22×D12712yes—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—44no—
156C22×C4C22⋊D12711yescharacteristic
157C22×C4C22⋊D12711yescharacteristic
158C22×C4C22×D12711yescharacteristic
159SD16—2548no—
160C2×C8—1274no—
161C4⋊C4—21no—
162C4⋊C4—1272no—
163C4⋊C4—2542no—
164C4⋊C4—2542no—
165C4⋊C4—2541no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C4⋊C4C22×D12712yes—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—2542no—
172C42—2542no—
173C42C22⋊D12711yescharacteristic
174C42C22⋊D12711yescharacteristic
175C42C22×D12711yescharacteristic
176C42C22×D12711yescharacteristic
177C23—24no—
178C23—22no—
179C23—42no—
180C23—42no—
181C23C2×C22⋊D12711yescharacteristic
182Q8—5082no—
183Q8—2544no—
184D4—24no—
185D4—22no—
186D4#1452324814yes—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—2542no—
195C2×C4—2542no—
196C2×C4C2×C22⋊D12711yescharacteristic
197C2×C4C2×C22⋊D12711yescharacteristic
198C2×C4C2×C22⋊D12711yescharacteristic
199C2×C4C2×C22⋊D12711yescharacteristic
200C2×C4C2×C22⋊D12711yescharacteristic
201C2×C4C23×D12711yescharacteristic
202C8—2544no—
203C22—21no—
204C22—21no—
205C22—24no—
206C22—48no—
207C22—42no—
208C22—42no—
209C22#1686123912yes—
210C22C22×C22⋊D12711yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—42no—
216C4—5082no—
217C4—5082no—
218C4#1686123512yes—
219C4C2×#1452324812yes—
220C2—22no—
221C2—44no—
222C2#2255038711yescharacteristic
223C2C2×#1686123511yescharacteristic
224C2C2×#1686123911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C4×D4×C127C2
socleC22×C127C22×D4
prime radicalC22×D4×C127C22
hypercenterC2×C4×D4D127
hypocenterC127#3110
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #22550433C2semidirect
2. #22550211C2semidirect
3. #22550156C2semidirect
4. #22550081C2semidirect
6. #22549976C2semidirect
7. #22549933C2semidirect
10. #16861209C22semidirect
11. #16861209C22semidirect
16. C4×Dic254C22semidirect
18. #16861142C22semidirect
19. C4⋊(C127⋊C8)C22semidirect
20. #16861132C22semidirect
23. #16861120C22semidirect
113. C127#3110semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225502112
#225501562
#225500812
#225500272
#225499762
#225499332
C2×C4×D4×C1272
#3110127