Finite Group Databasebeta

← #30648974

Subgroups — #30648974

Order 16256.

418 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 121 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#22550481C211yesmaximal, characteristic, complemented
2#22550350C211yesmaximal, characteristic, complemented
3#22550201C212yesmaximal, complemented
4#22550177C212yesmaximal, complemented
5#22550072C212yesmaximal, complemented
6#22550012C212yesmaximal, complemented
7#22549989C212yesmaximal, complemented
8C4×D1016C212yesmaximal, complemented
9#433×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C2×D1016C2212yes—
12C2×D1016C2212yes—
13#16861206C2211yescharacteristic, complemented
14#16861201C2211yescharacteristic, complemented
15#16861194—22no—
16#16861183C2212yes—
17#16861182—22no—
18C4×D508C2212yes—
19C4×D508C2212yes—
20C4×D508C2212yescomplemented
21#16861171—22no—
22#16861168C2211yescharacteristic
23C22⋊C4×D127—22no—
24#16861143C2212yescomplemented
25#16861141C2212yescomplemented
26#16861131C2212yes—
27#16861131C2212yes—
28#16861131C2212yescomplemented
29#16861131C2212yescomplemented
30#117×C127C2211yescharacteristic
31C4⋊C8×C127C2212yescomplemented
32C22⋊C8×C127C2212yes—
33#97×C127C2211yescharacteristic, complemented
34C4×C8×C127C2211yescharacteristic, complemented
35C23×D127—21no—
36C2×D508—24no—
37C2×D508—21no—
38C2×D508—22no—
39C2×D508C2312yes—
40C2×D508C2312yes—
41C2×C22⋊D127—22no—
42C2×C4×D127—22no—
43C2×C4×D127—22no—
44C2×C4×D127—22no—
45D1016—24no—
46D1016—24no—
47C4⋊Dic127C2312yes—
48C4⋊Dic127C2312yes—
49Dic127⋊C4—22no—
50C22⋊Dic127—21no—
51(C2×D127)⋊C4—22no—
52(C2×D127)⋊C4—22no—
53(C2×D127)⋊C4—22no—
54(C2×D127)⋊C4—22no—
55C4×Dic127—21no—
56C22×C4×C127C2311yescharacteristic
57C2×C8×C127C2312yes—
58C2×C8×C127C2312yes—
59C4⋊C4×C127D412yes—
60C22⋊C4×C127D412yes—
61C42×C127C2311yescharacteristic
62C42×C127C2311yescharacteristic
63C22×D127—22no—
64C22×D127—24no—
65C22×D127—22no—
66C22×D127—82no—
67D508—24no—
68D508—22no—
69D508—22no—
70D508—22no—
71D508—82no—
72D508#4114yes—
73C22⋊D127—82no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C4×D127—82no—
77C4×D127—82no—
78C4×D127—82no—
79C23×C127C2×D411yescharacteristic
80C2×C4×C127—22no—
81C2×C4×C127C2×D412yes—
82C2×C4×C127C2×D412yes—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2411yescharacteristic
85C8×C127—22no—
86C8×C127—22no—
87C22×C127—21no—
88C22×C127—22no—
89C22×C127C22×D411yescharacteristic
90C2×D127—24no—
91C2×D127—42no—
92C2×D127—42no—
93C2×D127—42no—
94C2×D127—42no—
95C2×D127—42no—
96C2×D127—84no—
97C4×C127—21no—
98C4×C127—22no—
99C4×C127—22no—
100C4×C127C2×#4112yes—
101Dic127—42no—
102Dic127—42no—
103C2×C127—41no—
104C2×C127#52511yescharacteristic
105C2×C127#57612yes—
106D127—44no—
107D127—82no—
108#3122—1271nomaximal, Sylow
109C127#312211yescharacteristic, complemented, Sylow
110#540—1271no—
111#518—1271no—
112#482—1272no—
113#478—1272no—
114#459—1272no—
115#447—1272no—
116#442—1272no—
117C4×D8—1272no—
118#433D12711yescharacteristic, complemented
119C22×D4—1271no—
120C2×D8—1272no—
121C2×D8—1272no—
122#127—1271no—
123#125—1271no—
124#123—2542no—
125#121—1272no—
126#121—2542no—
127C4×D4—1272no—
128C4×D4—1272no—
129C4×D4—1272no—
130C4×D4—2542no—
131#117—1271no—
132#117C2×D12711yescharacteristic
133C2×C22⋊C4—2542no—
134#107—1272no—
135#106—1272no—
136C4⋊C8C2×D12712yescomplemented
137#102—1272no—
138#102—1272no—
139#102—1272no—
140#102—1272no—
141C22⋊C8C2×D12712yes—
142#97C2×D12711yescharacteristic, complemented
143C4×C8C2×D12711yescharacteristic, complemented
144C24—2541no—
145C2×D4—1272no—
146C2×D4—1272no—
147C2×D4—2544no—
148C2×D4—2542no—
149C2×D4—2542no—
150C2×D4—2541no—
151C22×C4—2542no—
152C22×C4—2542no—
153C22×C4—2542no—
154C22×C4C22×D12711yescharacteristic
155D8—2544no—
156D8—2544no—
157C2×C8C22×D12712yes—
158C2×C8C22×D12712yes—
159C4⋊C4—1272no—
160C4⋊C4—1272no—
161C4⋊C4—2542no—
162C4⋊C4D50812yes—
163C22⋊C4—2541no—
164C22⋊C4—2542no—
165C22⋊C4—2542no—
166C22⋊C4—2542no—
167C22⋊C4—2542no—
168C22⋊C4D50812yes—
169C42—2541no—
170C42C22×D12711yescharacteristic
171C42C22×D12711yescharacteristic
172C23—10162no—
173C23—2544no—
174C23—2542no—
175C23—2542no—
176C23C2×D50811yescharacteristic
177D4—1274no—
178D4—10162no—
179D4—10162no—
180D4—2544no—
181D4—2542no—
182D4—2542no—
183D4—2542no—
184C2×C4—22no—
185C2×C4—10162no—
186C2×C4—10162no—
187C2×C4—10162no—
188C2×C4—2542no—
189C2×C4—2542no—
190C2×C4C2×D50811yescharacteristic
191C2×C4C2×D50812yes—
192C2×C4C2×D50812yes—
193C2×C4C23×D12711yescharacteristic
194C8—22no—
195C8—22no—
196C22—21no—
197C22—22no—
198C22—10164no—
199C22—5082no—
200C22—5082no—
201C22—5082no—
202C22—5082no—
203C22—5082no—
204C22—2544no—
205C22C22×D50811yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—5082no—
210C4—5082no—
211C4#41×D12712yes—
212C2—41no—
213C2—10162no—
214C2—5084no—
215C2#2255038811yescharacteristic
216C2#2255068112yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#433×C127C2
socleC22×C127C22×D4
hypercenter#433D127
hypocenterC127#3122
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #22550481C2semidirect
2. #22550350C2semidirect
3. #22550201C2semidirect
4. #22550177C2semidirect
5. #22550072C2semidirect
6. #22550012C2semidirect
7. #22549989C2semidirect
8. C4×D1016C2semidirect
9. #433×C127C2semidirect
13. #16861206C22semidirect
14. #16861201C22semidirect
20. C4×D508C22semidirect
24. #16861143C22semidirect
25. #16861141C22semidirect
28. #16861131C22semidirect
29. #16861131C22semidirect
31. C4⋊C8×C127C22semidirect
33. #97×C127C22semidirect
34. C4×C8×C127C22semidirect
109. C127#3122semidirect
118. #433D127semidirect
136. C4⋊C8C2×D127semidirect
142. #97C2×D127semidirect
143. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504812
#225503502
#225502012
#225501772
#225500722
#225500122
#225499892
C4×D10162
#433×C1272
#3122127