Finite Group Databasebeta

← #30648682

Subgroups — #30648682

Order 16256.

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

normal subgroups only

This group has 418 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#22550359C211yesmaximal, characteristic, complemented
2#22550197C212yesmaximal, complemented
3#22550197C212yesmaximal, complemented
4#22550192C212yesmaximal, complemented
5#22550187C212yesmaximal, complemented
6#22549919C211yesmaximal, characteristic, complemented
7#22549912C211yesmaximal, characteristic, complemented
8#22549881C211yesmaximal, characteristic, complemented
9C2×#16861130C211yesmaximal, characteristic, complemented
10#22549843C211yesmaximal, characteristic, complemented
11#535×C127C211yesmaximal, characteristic
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14Q8×Dic127—21no—
15D4×Dic127—22no—
16D4×Dic127—21no—
17#16861168C2211yescharacteristic
18#16861168C2212yescomplemented
19C2×C4⋊Dic127C2211yescharacteristic
20C2×C4⋊Dic127C2212yescomplemented
21#16861142C2211yescharacteristic, complemented
22#16861142C2211yescharacteristic, complemented
23#16861142C2212yescomplemented
24#16861140C2211yescharacteristic, complemented
25#16861140C2211yescharacteristic, complemented
26#16861140C2212yescomplemented
27#16861130C2212yes—
28#16861130C2212yes—
29#16861130C2212yes—
30#16861130C2212yes—
31C22⋊(C127⋊C8)C2212yescomplemented
32C22⋊(C127⋊C8)C2212yescomplemented
33C2×#41×C127C2211yescharacteristic
34C22×D4×C127C2211yescharacteristic
35#127×C127—21no—
36#124×C127—21no—
37#121×C127C2212yes—
38#121×C127C2212yes—
39#120×C127—22no—
40#117×C127C2211yescharacteristic
41C22×Dic127—21no—
42C22×Dic127—22no—
43#14523232—22no—
44C2×C127⋊C8—22no—
45C2×C127⋊C8C2311yescharacteristic
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2312yes—
48C4⋊Dic127—21no—
49C4⋊Dic127—22no—
50C4⋊Dic127C2312yes—
51C4⋊Dic127C2311yescharacteristic
52C4⋊Dic127C2311yescharacteristic
53C22⋊Dic127—21no—
54C22⋊Dic127—22no—
55C4×Dic127—21no—
56C4×Dic127—22no—
57C24×C127—21no—
58#41×C127—22no—
59#41×C127—22no—
60C2×Q8×C127D411yescharacteristic
61C2×D4×C127—22no—
62C2×D4×C127—22no—
63C2×D4×C127—21no—
64C2×D4×C127—22no—
65C2×D4×C127—22no—
66C2×D4×C127D412yes—
67C2×D4×C127D411yescharacteristic
68C2×D4×C127C2312yes—
69C22×C4×C127C2312yes—
70C22×C4×C127C2311yescharacteristic
71C4⋊C4×C127C2311yescharacteristic
72C4⋊C4×C127C2311yescharacteristic
73C22⋊C4×C127—21no—
74C22⋊C4×C127—22no—
75C22⋊C4×C127—22no—
76C42×C127—21no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C2×Dic127—81no—
81C2×Dic127—82no—
82C127⋊C8—21no—
83C127⋊C8—21no—
84C127⋊C8—22no—
85C23×C127—22no—
86C23×C127—42no—
87C23×C127—81no—
88C23×C127—81no—
89C23×C127C2×D412yes—
90C23×C127C2×D411yescharacteristic
91Q8×C127—22no—
92D4×C127—24no—
93D4×C127—22no—
94D4×C127—22no—
95D4×C127—22no—
96D4×C127—42no—
97D4×C127—82no—
98D4×C127—82no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—21no—
102C2×C4×C127—21no—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127#4112yes—
106C2×C4×C127#4112yes—
107C2×C4×C127C2411yescharacteristic
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—22no—
111C22×C127—22no—
112C22×C127—22no—
113C22×C127—42no—
114C22×C127—42no—
115C22×C127—42no—
116C22×C127—84no—
117C22×C127#12312yes—
118C22×C127C22×D411yescharacteristic
119C4×C127—22no—
120C4×C127—41no—
121C4×C127—41no—
122C4×C127#12312yes—
123C4×C127C2×#4111yescharacteristic
124C4×C127C2×#4111yescharacteristic
125Dic127—41no—
126Dic127—42no—
127Dic127—41no—
128C2×C127—22no—
129C2×C127—42no—
130C2×C127—82no—
131C2×C127C2×#12311yescharacteristic
132C2×C127#57611yescharacteristic
133C2×C127#57711yescharacteristic
134#3096—1271nomaximal, Sylow
135C127#309611yescharacteristic, complemented, Sylow
136#535D12711yescharacteristic
137#520—1271no—
138#482—1272no—
139#482—1272no—
140#481—1272no—
141#480—1272no—
142#428—1271no—
143#427—1271no—
144#418—1271no—
145C2×#102—1271no—
146#408—1271no—
147C2×#41C2×D12711yescharacteristic
148C22×D4C2×D12711yescharacteristic
149C2×M16—1271no—
150C22×C8—1271no—
151#127—21no—
152#124—21no—
153#121C2×D12712yes—
154#121C2×D12712yes—
155#120—22no—
156C4×Q8—2541no—
157C4×D4—2542no—
158C4×D4—2541no—
159#117—1271no—
160#117—1272no—
161#117C2×D12711yescharacteristic
162C2×C4⋊C4—1271no—
163C2×C4⋊C4—1272no—
164#107—1271no—
165#107—1271no—
166#107—1272no—
167#106—1271no—
168#106—1271no—
169#106—1272no—
170#102—1272no—
171#102—1272no—
172#102—1272no—
173#102—1272no—
174C22⋊C8—1272no—
175C22⋊C8—1272no—
176C24—21no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D12711yescharacteristic
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4C22⋊D12712yes—
186C2×D4C22⋊D12711yescharacteristic
187C2×D4C22×D12712yes—
188C22×C4—2542no—
189C22×C4—2541no—
190C22×C4C22×D12712yes—
191C22×C4C22×D12711yescharacteristic
192M16—2542no—
193C2×C8—1271no—
194C2×C8—1272no—
195C2×C8—1271no—
196C2×C8—2542no—
197C4⋊C4—1272no—
198C4⋊C4—1271no—
199C4⋊C4—1271no—
200C4⋊C4—2542no—
201C4⋊C4—2541no—
202C4⋊C4C22×D12711yescharacteristic
203C4⋊C4C22×D12711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—2542no—
208C22⋊C4—2541no—
209C42—21no—
210C42—2542no—
211C42—2541no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23C2×C22⋊D12712yes—
217C23C2×C22⋊D12711yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4—22no—
223D4—42no—
224D4—82no—
225D4—82no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—10161no—
231C2×C4—10162no—
232C2×C4—2541no—
233C2×C4—2542no—
234C2×C4—2541no—
235C2×C4C2×C22⋊D12712yes—
236C2×C4C2×C22⋊D12711yescharacteristic
237C2×C4#1452324712yes—
238C2×C4#1452324712yes—
239C2×C4C23×D12711yescharacteristic
240C8—2541no—
241C8—2541no—
242C8—2542no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—84no—
252C22#1686119012yes—
253C22C22×C22⋊D12711yescharacteristic
254C4—22no—
255C4—41no—
256C4—41no—
257C4—5082no—
258C4—5081no—
259C4—5081no—
260C4#1686119012yes—
261C4C2×#1452324711yescharacteristic
262C4C2×#1452324711yescharacteristic
263C2—22no—
264C2—42no—
265C2—82no—
266C2C2×#1686119011yescharacteristic
267C2#2255067911yescharacteristic
268C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#535×C127C2
socleC22×C127C22×D4
hypercenter#535D127
hypocenterC127#3096
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #22550359C2semidirect
2. #22550197C2semidirect
3. #22550197C2semidirect
4. #22550192C2semidirect
5. #22550187C2semidirect
6. #22549919C2semidirect
7. #22549912C2semidirect
8. #22549881C2semidirect
9. C2×#16861130C2semidirect
10. #22549843C2semidirect
18. #16861168C22semidirect
20. C2×C4⋊Dic127C22semidirect
21. #16861142C22semidirect
22. #16861142C22semidirect
23. #16861142C22semidirect
24. #16861140C22semidirect
25. #16861140C22semidirect
26. #16861140C22semidirect
31. C22⋊(C127⋊C8)C22semidirect
32. C22⋊(C127⋊C8)C22semidirect
135. C127#3096semidirect

Maximal subgroups

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

subgroupindex
#225503592
#225501972
#225501922
#225501872
#225499192
#225499122
#225498812
C2×#168611302
#225498432
#535×C1272
#3096127