Finite Group Databasebeta

← #30649102

Subgroups — #30649102

Order 16256.

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

normal subgroups only

This group has 484 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
1#22550429C211yesmaximal, characteristic, complemented
2C2×#16861205C211yesmaximal, characteristic
3#22550239C214yesmaximal, complemented
4#22550221C214yesmaximal, complemented
5#22549882C211yesmaximal, characteristic, complemented
6#22549881C211yesmaximal, characteristic, complemented
7#22549880C211yesmaximal, characteristic, complemented
8M16×Dic127C211yesmaximal, characteristic, complemented
9#418×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C22×C22⋊D127—22no—
12C2×#14523232C2211yescharacteristic
13#16861208C2212yescomplemented
14#16861205—22no—
15#16861205C2212yes—
16#16861205C2212yes—
17#16861200C2212yescomplemented
18#16861186—22no—
19C4×Dic254—22no—
20#16861168C2211yescharacteristic
21C2×C4×Dic127C2211yescharacteristic
22#16861131C2214yescomplemented
23#16861130C2214yescomplemented
24#16861129C2214yescomplemented
25#16861119C2212yescomplemented
26C8×Dic127C2212yescomplemented
27C22×D4×C127C2211yescharacteristic
28C2×M16×C127C2211yescharacteristic
29#117×C127C2211yescharacteristic
30#102×C127C2214yescomplemented
31C23×D127—21no—
32C2×Dic254—22no—
33C2×D508—21no—
34C2×D508—22no—
35C2×D508—22no—
36C2×D508C2312yes—
37C2×C22⋊D127—24no—
38C2×C22⋊D127—24no—
39C2×C22⋊D127—48no—
40C22×Dic127D412yes—
41#14523232—22no—
42C2×C127⋊C8C2312yes—
43C4⋊Dic127—22no—
44C4⋊Dic127C2312yes—
45Dic127⋊C4—22no—
46Dic127⋊C4—22no—
47C22⋊Dic127—21no—
48C4×Dic127—21no—
49C4×Dic127D412yes—
50C4×Dic127C2311yescharacteristic
51C4×Dic127C2311yescharacteristic
52C24×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—22no—
55C2×D4×C127—22no—
56C2×D4×C127C2312yes—
57C22×C4×C127C2311yescharacteristic
58M16×C127—22no—
59C2×C8×C127C2312yes—
60C4⋊C4×C127C2312yes—
61C22⋊C4×C127—21no—
62C42×C127—21no—
63C22×D127—22no—
64C22×D127—44no—
65C22×D127—42no—
66Dic254—44no—
67D508—22no—
68D508—22no—
69D508—42no—
70C22⋊D127—48no—
71C22⋊D127—48no—
72C2×Dic127—24no—
73C2×Dic127—22no—
74C2×Dic127C2×D412yes—
75C2×Dic127C2×D412yes—
76C127⋊C8—22no—
77C23×C127—22no—
78C23×C127—44no—
79C23×C127—42no—
80C23×C127C2×D411yescharacteristic
81D4×C127—22no—
82D4×C127—22no—
83D4×C127—42no—
84C2×C4×C127—22no—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4112yes—
87C2×C4×C127#4112yes—
88C2×C4×C127C2411yescharacteristic
89C8×C127—22no—
90C22×C127—22no—
91C22×C127—42no—
92C22×C127—44no—
93C22×C127—84no—
94C22×C127#12412yes—
95C22×C127C22×D411yescharacteristic
96C2×D127—42no—
97C2×D127—44no—
98C2×D127—84no—
99C4×C127—42no—
100C4×C127#12412yes—
101C4×C127C2×#4111yescharacteristic
102C4×C127C2×#4111yescharacteristic
103Dic127—22no—
104Dic127—42no—
105Dic127AGL(1,ℤ/8)14yes—
106C2×C127—22no—
107C2×C127—82no—
108C2×C127C2×#12411yescharacteristic
109C2×C127C2×AGL(1,ℤ/8)12yes—
110D127—82no—
111#3138—1271nomaximal, Sylow
112C127#313811yescharacteristic, complemented, Sylow
113#533—1271no—
114C2×#127—1271no—
115#490—1274no—
116#486—1274no—
117#418—1271no—
118#418—1271no—
119#418—1271no—
120#418D12711yescharacteristic, complemented
121C4×M16—1271no—
122C22×D4—1271no—
123C22×D4—2542no—
124C22×D4C2×D12711yescharacteristic
125C2×M16—1271no—
126C2×M16C2×D12711yescharacteristic
127C4⋊Q8—1272no—
128#127—1272no—
129#127—1272no—
130#127—2542no—
131#125—1272no—
132C22⋊Q8—2542no—
133C4×Q8—2542no—
134#117—1271no—
135#117C2×D12711yescharacteristic
136C2×C42—1271no—
137#102—1274no—
138#102—1274no—
139#102—1274no—
140#102C2×D12714yescomplemented
141#97—1272no—
142C4×C8—1272no—
143C24—21no—
144C24—2541no—
145C2×Q8—2542no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—1272no—
150C2×D4—5088no—
151C2×D4—2544no—
152C2×D4—2544no—
153C2×D4—2541no—
154C2×D4—2542no—
155C2×D4—2542no—
156C2×D4C22×D12712yes—
157C22×C4—1272no—
158C22×C4C22×D12711yescharacteristic
159M16—22no—
160M16—2542no—
161C2×C8—1272no—
162C2×C8C22×D12712yes—
163C4⋊C4—1272no—
164C4⋊C4—2542no—
165C4⋊C4—2542no—
166C4⋊C4—2542no—
167C4⋊C4C22×D12712yes—
168C22⋊C4—21no—
169C22⋊C4—2541no—
170C42—21no—
171C42—1272no—
172C42—1271no—
173C42—1271no—
174C42—2541no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—5084no—
179C23—5082no—
180C23—2542no—
181C23D4×D12711yescharacteristic
182Q8—5084no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—5088no—
187D4—5088no—
188D4—5082no—
189D4—2542no—
190D4—2542no—
191C2×C4—22no—
192C2×C4—1272no—
193C2×C4—1272no—
194C2×C4—2544no—
195C2×C4—2542no—
196C2×C4D4×D12711yescharacteristic
197C2×C4#1452324712yes—
198C2×C4#1452324812yes—
199C2×C4C23×D12711yescharacteristic
200C8—22no—
201C8—2542no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—10164no—
207C22—5082no—
208C22—5084no—
209C22#1686119512yes—
210C22C2×D4×D12711yescharacteristic
211C4—42no—
212C4—1274no—
213C4—5082no—
214C4—2542no—
215C4#1686119512yes—
216C4C2×#1452324711yescharacteristic
217C4C2×#1452324811yescharacteristic
218C2—22no—
219C2—82no—
220C2—10162no—
221C2C2×#1686119511yescharacteristic
222C2AGL(1,ℤ/8)×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#418×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861205C2
hypercenter#418D127
hypocenterC127#3138
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #22550429C2semidirect
3. #22550239C2semidirect
4. #22550221C2semidirect
5. #22549882C2semidirect
6. #22549881C2semidirect
7. #22549880C2semidirect
8. M16×Dic127C2semidirect
9. #418×C127C2semidirect
13. #16861208C22semidirect
17. #16861200C22semidirect
22. #16861131C22semidirect
23. #16861130C22semidirect
24. #16861129C22semidirect
25. #16861119C22semidirect
26. C8×Dic127C22semidirect
30. #102×C127C22semidirect
112. C127#3138semidirect
120. #418D127semidirect
140. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504292
C2×#168612052
#225502392
#225502212
#225498822
#225498812
#225498802
M16×Dic1272
#418×C1272
#3138127