Finite Group Databasebeta

← #30648825

Subgroups — #30648825

Order 16256.

444 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C2×C4×D508C211yesmaximal, characteristic
2#22550209C212yesmaximal, complemented
3#22550153C212yesmaximal, complemented
4#22550078C212yesmaximal, complemented
5#22550026C212yesmaximal
6#22549974C214yesmaximal, complemented
7#22549933C211yesmaximal, characteristic, complemented
8#533×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C4×D127—21no—
11C2×#14523235C2214yes—
12C4×D508—21no—
13C4×D508—21no—
14C4×D508C2212yes—
15C4×D508C2212yes—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×(C2×D127)⋊C4—21no—
18#16861142C2214yescomplemented
19C4⋊(C127⋊C8)C2212yescomplemented
20#16861133C2214yescomplemented
21#16861129C2214yes—
22C22⋊(C127⋊C8)C2212yes—
23#16861120C2212yescomplemented
24C4⋊Q8×C127C2211yescharacteristic, complemented
25C4⋊Q8×C127C2211yescharacteristic, complemented
26#125×C127—21no—
27C22⋊Q8×C127—21no—
28C22⋊Q8×C127C2212yes—
29C4×Q8×C127—21no—
30C4×Q8×C127C2212yescomplemented
31#117×C127—21no—
32C2×C42×C127C2211yescharacteristic
33C23×D127—21no—
34C2×D508—24no—
35C2×D508—21no—
36C2×D508C2312yes—
37C22×Dic127—21no—
38C2×C4×D127—22no—
39C2×C4×D127—22no—
40C2×C4×D127—44no—
41#14523235—28no—
42C2×C127⋊C8C2314yes—
43C4⋊Dic127—21no—
44C4⋊Dic127C2312yes—
45(C2×D127)⋊C4—22no—
46(C2×D127)⋊C4—22no—
47C2×Q8×C127—21no—
48C2×Q8×C127C2312yes—
49C22×C4×C127D411yescharacteristic
50C22×C4×C127D411yescharacteristic
51C22×C4×C127C2311yescharacteristic
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—22no—
56C4⋊C4×C127—22no—
57C4⋊C4×C127C2312yes—
58C22⋊C4×C127—22no—
59C42×C127—22no—
60C42×C127D411yescharacteristic
61C42×C127D411yescharacteristic
62C42×C127C2311yescharacteristic
63C42×C127C2311yescharacteristic
64C22×D127—24no—
65C22×D127—22no—
66C22×D127—42no—
67C22×D127—42no—
68D508—24no—
69D508—22no—
70D508#4114yes—
71C2×Dic127—22no—
72C2×Dic127—42no—
73C4×D127—44no—
74C4×D127—44no—
75C127⋊C8—24no—
76C23×C127C2×D411yescharacteristic
77Q8×C127—24no—
78Q8×C127—42no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
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#13712yes—
93C22×C127C22×D411yescharacteristic
94C2×D127—24no—
95C2×D127—48no—
96C2×D127—42no—
97C2×D127—42no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—42no—
103C4×C127—42no—
104C4×C127AGL(1,ℤ/8)12yes—
105C4×C127C2×#4112yes—
106Dic127—42no—
107C2×C127—22no—
108C2×C127#52511yescharacteristic
109C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C127C2×#13711yescharacteristic
111D127—44no—
112#3110—1271nomaximal, Sylow
113C127#311011yescharacteristic, complemented, Sylow
114#533D12711yescharacteristic, complemented
115C2×C4×D4—1271no—
116#484—1272no—
117#474—1272no—
118#461—1272no—
119#450—1272no—
120#440—1274no—
121#432—1271no—
122C22×D4—1271no—
123C23×C4—2541no—
124C2×SD16—1274no—
125C4⋊Q8C2×D12711yescharacteristic, complemented
126C4⋊Q8C2×D12711yescharacteristic, complemented
127#125—21no—
128C22⋊Q8—21no—
129C22⋊Q8C2×D12712yes—
130C4×Q8—21no—
131C4×Q8C2×D12712yescomplemented
132C4×D4—1272no—
133C4×D4—1272no—
134C4×D4—2541no—
135C4×D4—2541no—
136#117—21no—
137C2×C4⋊C4—1271no—
138C2×C22⋊C4—2541no—
139C2×C42C2×D12711yescharacteristic
140#107—1274no—
141C4⋊C8—1272no—
142#103—1274no—
143#102—1274no—
144C22⋊C8—1272no—
145#97—1272no—
146C24—2541no—
147C2×Q8—21no—
148C2×Q8C22×D12712yes—
149C2×D4—1272no—
150C2×D4—2544no—
151C2×D4—2541no—
152C22×C4—5084no—
153C22×C4—2541no—
154C22×C4—2542no—
155C22×C4—2542no—
156C22×C4C22⋊D12711yescharacteristic
157C22×C4C22⋊D12711yescharacteristic
158C22×C4C22×D12711yescharacteristic
159SD16—2548no—
160C2×C8—1274no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—1272no—
167C4⋊C4—2541no—
168C4⋊C4C22×D12712yes—
169C22⋊C4—22no—
170C22⋊C4—2542no—
171C22⋊C4—2542no—
172C42—22no—
173C42C22⋊D12711yescharacteristic
174C42C22⋊D12711yescharacteristic
175C42C22×D12711yescharacteristic
176C42C22×D12711yescharacteristic
177C23—5082no—
178C23—5082no—
179C23—2544no—
180C23—2542no—
181C23C2×C22⋊D12711yescharacteristic
182Q8—24no—
183Q8—42no—
184D4—1274no—
185D4—2544no—
186D4—2542no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—5084no—
193C2×C4—5082no—
194C2×C4—5084no—
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—5088no—
206C22—5082no—
207C22—5082no—
208C22—2544no—
209C22#1686124012yes—
210C22C22×C22⋊D12711yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—5082no—
218C4#1686123412yes—
219C4#41×D12712yes—
220C2—22no—
221C2—5084no—
222C2#2255038311yescharacteristic
223C2C2×#1686123411yescharacteristic
224C2C2×#1686124011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#533×C127C2
socleC22×C127C22×D4
prime radicalC22×D508C22
hypercenter#533D127
hypocenterC127#3110
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #22550209C2semidirect
3. #22550153C2semidirect
4. #22550078C2semidirect
6. #22549974C2semidirect
7. #22549933C2semidirect
8. #533×C127C2semidirect
18. #16861142C22semidirect
19. C4⋊(C127⋊C8)C22semidirect
20. #16861133C22semidirect
23. #16861120C22semidirect
24. C4⋊Q8×C127C22semidirect
25. C4⋊Q8×C127C22semidirect
30. C4×Q8×C127C22semidirect
113. C127#3110semidirect
114. #533D127semidirect
125. C4⋊Q8C2×D127semidirect
126. C4⋊Q8C2×D127semidirect
131. C4×Q8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D5082
#225502092
#225501532
#225500782
#225500262
#225499742
#225499332
#533×C1272
#3110127