Finite Group Databasebeta

← #30648666

Subgroups — #30648666

Order 16256.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16861168C211yesmaximal, characteristic, complemented
2#22550197C214yesmaximal, complemented
3#22550192C212yesmaximal, complemented
4#22550187C212yesmaximal, complemented
5#22549916C212yesmaximal, complemented
6#22549881C212yesmaximal, complemented
7#22549840C211yesmaximal, characteristic, complemented
8C2×#121×C127C211yesmaximal, characteristic
9C2×#14523232C2212yes—
10#16861168—21no—
11#16861168—21no—
12#16861168C2212yes—
13#16861168C2212yescomplemented
14C2×C4⋊Dic127C2211yescharacteristic, complemented
15C2×C4⋊Dic127C2211yescharacteristic, complemented
16C2×C22⋊Dic127—21no—
17C2×C4×Dic127—21no—
18#16861142C2214yescomplemented
19#16861140C2214yescomplemented
20#16861130C2214yescomplemented
21#16861130C2214yescomplemented
22C22⋊(C127⋊C8)C2212yescomplemented
23C22⋊(C127⋊C8)C2212yescomplemented
24C22×D4×C127—21no—
25C22×D4×C127C2211yescharacteristic
26C23×C4×C127C2211yescharacteristic
27#121×C127—21no—
28#121×C127—21no—
29#121×C127C2212yes—
30#121×C127C2212yes—
31C2×C4⋊C4×C127C2211yescharacteristic
32C2×C22⋊C4×C127—21no—
33C22×Dic127—21no—
34C22×Dic127—21no—
35#14523232—24no—
36C2×C127⋊C8C2314yes—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127C2312yes—
40C4⋊Dic127C2312yes—
41C22⋊Dic127—22no—
42C22⋊Dic127—22no—
43C4×Dic127—22no—
44C4×Dic127—22no—
45C24×C127—21no—
46C24×C127D411yescharacteristic
47C2×D4×C127—22no—
48C2×D4×C127—22no—
49C2×D4×C127—21no—
50C2×D4×C127—24no—
51C2×D4×C127—44no—
52C2×D4×C127C2312yes—
53C22×C4×C127—24no—
54C22×C4×C127—21no—
55C22×C4×C127D411yescharacteristic
56C22×C4×C127D411yescharacteristic
57C22×C4×C127D411yescharacteristic
58C22×C4×C127C2311yescharacteristic
59C22×C4×C127C2311yescharacteristic
60C22×C4×C127C2311yescharacteristic
61C4⋊C4×C127—21no—
62C4⋊C4×C127C2312yes—
63C22⋊C4×C127—22no—
64C22⋊C4×C127—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—42no—
68C2×Dic127—42no—
69C127⋊C8—24no—
70C23×C127—22no—
71C23×C127—22no—
72C23×C127—22no—
73C23×C127—41no—
74C23×C127—42no—
75C23×C127—42no—
76C23×C127—42no—
77C23×C127C2×D411yescharacteristic
78C23×C127C2×D411yescharacteristic
79C23×C127C2×D411yescharacteristic
80D4×C127—24no—
81D4×C127—44no—
82D4×C127—44no—
83D4×C127—42no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—42no—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127#4114yes—
95C2×C4×C127C2411yescharacteristic
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—21no—
99C22×C127—21no—
100C22×C127—21no—
101C22×C127—21no—
102C22×C127—21no—
103C22×C127—44no—
104C22×C127—42no—
105C22×C127—42no—
106C22×C127—42no—
107C22×C127—84no—
108C22×C127#12312yes—
109C22×C127AGL(1,ℤ/8)12yes—
110C22×C127C22×D411yescharacteristic
111C4×C127—21no—
112C4×C127—21no—
113C4×C127—42no—
114C4×C127#12312yes—
115C4×C127C2×#4112yes—
116Dic127—42no—
117Dic127—42no—
118C2×C127—22no—
119C2×C127—22no—
120C2×C127—41no—
121C2×C127—82no—
122C2×C127C2×#12311yescharacteristic
123C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
124C2×C127#57511yescharacteristic
125#3094—1271nomaximal, Sylow
126C127#309411yescharacteristic, complemented, Sylow
127C2×#121D12711yescharacteristic
128C2×#117—1271no—
129#482—1274no—
130#481—1272no—
131#480—1272no—
132#428—1272no—
133#418—1272no—
134#407—1271no—
135C22×D4—21no—
136C22×D4C2×D12711yescharacteristic
137C23×C4C2×D12711yescharacteristic
138C2×M16—1272no—
139#121—21no—
140#121—21no—
141#121C2×D12712yes—
142#121C2×D12712yes—
143#117—1272no—
144#117—1272no—
145#117—2541no—
146#117—2541no—
147C2×C4⋊C4—1271no—
148C2×C4⋊C4—1271no—
149C2×C4⋊C4C2×D12711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—2541no—
152C2×C42—2541no—
153#107—1274no—
154#106—1274no—
155#102—1274no—
156#102—1274no—
157C22⋊C8—1272no—
158C22⋊C8—1272no—
159C24—21no—
160C24C22⋊D12711yescharacteristic
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—24no—
165C2×D4—44no—
166C2×D4C22×D12712yes—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—2541no—
170C22×C4—2541no—
171C22×C4C22⋊D12711yescharacteristic
172C22×C4C22⋊D12711yescharacteristic
173C22×C4C22⋊D12711yescharacteristic
174C22×C4C22×D12711yescharacteristic
175C22×C4C22×D12711yescharacteristic
176C22×C4C22×D12711yescharacteristic
177M16—2544no—
178C2×C8—1274no—
179C4⋊C4—21no—
180C4⋊C4—1272no—
181C4⋊C4—1272no—
182C4⋊C4—2541no—
183C4⋊C4—2541no—
184C4⋊C4C22×D12712yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—2542no—
188C22⋊C4—2542no—
189C42—2542no—
190C42—2542no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23C2×C22⋊D12711yescharacteristic
199C23C2×C22⋊D12711yescharacteristic
200C23C2×C22⋊D12711yescharacteristic
201D4—24no—
202D4—44no—
203D4—44no—
204D4—42no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—5082no—
213C2×C4—5082no—
214C2×C4—2542no—
215C2×C4—2542no—
216C2×C4C2×C22⋊D12711yescharacteristic
217C2×C4C2×C22⋊D12711yescharacteristic
218C2×C4C2×C22⋊D12711yescharacteristic
219C2×C4#1452324714yes—
220C2×C4C23×D12711yescharacteristic
221C8—2544no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—84no—
234C22#1686119012yes—
235C22#1686123512yes—
236C22C22×C22⋊D12711yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—5082no—
241C4—5082no—
242C4#1686119012yes—
243C4C2×#1452324712yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—82no—
248C2C2×#1686119011yescharacteristic
249C2C2×#1686123511yescharacteristic
250C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC2×#121×C127C2
socleC22×C127C22×D4
hypercenterC2×#121D127
hypocenterC127#3094
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16861168C2semidirect
2. #22550197C2semidirect
3. #22550192C2semidirect
4. #22550187C2semidirect
5. #22549916C2semidirect
6. #22549881C2semidirect
7. #22549840C2semidirect
13. #16861168C22semidirect
14. C2×C4⋊Dic127C22semidirect
15. C2×C4⋊Dic127C22semidirect
18. #16861142C22semidirect
19. #16861140C22semidirect
20. #16861130C22semidirect
21. #16861130C22semidirect
22. C22⋊(C127⋊C8)C22semidirect
23. C22⋊(C127⋊C8)C22semidirect
126. C127#3094semidirect

Maximal subgroups

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

subgroupindex
C2×#168611682
#225501972
#225501922
#225501872
#225499162
#225498812
#225498402
C2×#121×C1272
#3094127