Finite Group Databasebeta

← #30648849

Subgroups — #30648849

Order 16256.

424 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 119 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#22550421C211yesmaximal, characteristic
2C2×#16861174C211yesmaximal, characteristic, complemented
3#22550189C211yesmaximal, characteristic, complemented
4#22550170C212yesmaximal, complemented
5#22550075C212yesmaximal, complemented
6#22550017C211yesmaximal, characteristic, complemented
7#22550014C211yesmaximal, characteristic, complemented
8#22549975C212yesmaximal, complemented
9#22549973C212yesmaximal, complemented
10#22549930C211yesmaximal, characteristic, complemented
11#480×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—21no—
13C22×Dic254C2211yescharacteristic
14C2×#14523237C2212yes—
15C2×#14523236C2212yes—
16#16861208C2211yescharacteristic
17#16861205C2211yescharacteristic
18#16861195—21no—
19#16861183C2211yescharacteristic
20#16861181—21no—
21#16861174—21no—
22#16861174—21no—
23#16861174C2212yes—
24#16861174C2212yes—
25D4×Dic127C2211yescharacteristic
26#16861171C2211yescharacteristic
27(C22⋊D127)⋊C4—21no—
28C2×Dic127⋊C4—21no—
29C2×C4×Dic127—21no—
30C2×C4×Dic127C2211yescharacteristic
31#16861142C2212yes—
32Dic127⋊C8C2212yes—
33#16861134C2212yes—
34#16861132C2212yes—
35#16861129C2212yes—
36C22⋊(C127⋊C8)C2211yescharacteristic
37#16861119C2212yes—
38#121×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic
40#107×C127C2212yes—
41#102×C127C2212yes—
42C22⋊C8×C127C2211yescharacteristic
43#14523247—44no—
44C2×Dic254—24no—
45C2×Dic254—21no—
46C2×Dic254—21no—
47C2×Dic254C2312yes—
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—21no—
50C2×C22⋊D127—21no—
51C2×C22⋊D127—21no—
52C22×Dic127—21no—
53C22×Dic127—21no—
54C22×Dic127D411yescharacteristic
55C22×Dic127D411yescharacteristic
56C2×C4×D127—21no—
57#14523237—24no—
58#14523236—24no—
59C2×C127⋊C8C2312yes—
60C4⋊Dic127C2311yescharacteristic
61Dic127⋊C4—21no—
62Dic127⋊C4—22no—
63Dic127⋊C4—22no—
64C22⋊Dic127—21no—
65(C2×D127)⋊C4—21no—
66(C2×D127)⋊C4—21no—
67C4×Dic127—22no—
68C4×Dic127—22no—
69C4×Dic127D411yescharacteristic
70C4×Dic127D411yescharacteristic
71C4×Dic127C2311yescharacteristic
72C4×Dic127C2311yescharacteristic
73C2×D4×C127—21no—
74C2×D4×C127C2311yescharacteristic
75C22×C4×C127—21no—
76C22×C4×C127C2311yescharacteristic
77C2×C8×C127C2312yes—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—21no—
82C22×D127—21no—
83Dic254—24no—
84Dic254—22no—
85Dic254—42no—
86Dic254#4114yescomplemented
87D508—22no—
88C22⋊D127—42no—
89C22⋊D127—42no—
90C22⋊D127—42no—
91C2×Dic127—24no—
92C2×Dic127—22no—
93C2×Dic127—22no—
94C2×Dic127—22no—
95C2×Dic127—21no—
96C2×Dic127—42no—
97C2×Dic127C2×D411yescharacteristic
98C2×Dic127C2×D411yescharacteristic
99C2×Dic127C2×D411yescharacteristic
100C2×Dic127C2×D411yescharacteristic
101C4×D127—42no—
102C127⋊C8—22no—
103C23×C127—21no—
104C23×C127C2×D411yescharacteristic
105D4×C127—22no—
106D4×C127—42no—
107C2×C4×C127—22no—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127—42no—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C8×C127—22no—
114C22×C127—21no—
115C22×C127—21no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127—41no—
119C22×C127AGL(1,ℤ/8)12yes—
120C22×C127C22×D411yescharacteristic
121C2×D127—41no—
122C2×D127—41no—
123C2×D127—41no—
124C4×C127—21no—
125C4×C127—42no—
126C4×C127—41no—
127C4×C127C2×#4112yes—
128Dic127—24no—
129Dic127—21no—
130Dic127—21no—
131Dic127—21no—
132Dic127—41no—
133Dic127#13712yes—
134C2×C127—22no—
135C2×C127—81no—
136C2×C127#52511yescharacteristic
137C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C127C2×#13711yescharacteristic
139D127—81no—
140#3112—1271nomaximal, Sylow
141C127#311211yescharacteristic, complemented, Sylow
142#532—1271no—
143C2×C4×Q8—1271no—
144#480—1271no—
145#480D12711yescharacteristic, complemented
146#477—1272no—
147#460—1272no—
148#448—1271no—
149#448—1271no—
150#440—1272no—
151#440—1272no—
152#432—1271no—
153C2×#41—2541no—
154C22×Q8—1271no—
155C2×SD16—1272no—
156C2×SD16—1272no—
157C4⋊Q8—1271no—
158#127—1271no—
159#124—2541no—
160#121—1271no—
161#121—2541no—
162#121C2×D12711yescharacteristic
163C4×Q8—1272no—
164C4×Q8—1272no—
165C4×Q8—2541no—
166C4×Q8—2541no—
167C4×D4—1271no—
168C4×D4—1271no—
169C4×D4—2541no—
170C2×C4⋊C4—2541no—
171C2×C4⋊C4C2×D12711yescharacteristic
172C2×C42—1271no—
173C2×C42—2541no—
174#107—1272no—
175#107C2×D12712yes—
176C4⋊C8—1272no—
177#103—1272no—
178#103—1272no—
179#102—1272no—
180#102C2×D12712yes—
181C22⋊C8—1271no—
182C22⋊C8C2×D12711yescharacteristic
183#97—1272no—
184#41—5084no—
185C2×Q8—1272no—
186C2×Q8—2541no—
187C2×Q8—2544no—
188C2×Q8—2541no—
189C2×D4—21no—
190C2×D4—1271no—
191C2×D4—2541no—
192C2×D4—2541no—
193C2×D4—2541no—
194C2×D4C22×D12711yescharacteristic
195C22×C4—21no—
196C22×C4—1271no—
197C22×C4—1271no—
198C22×C4—2541no—
199C22×C4—2541no—
200C22×C4—2541no—
201C22×C4C22×D12711yescharacteristic
202SD16—2544no—
203SD16—2544no—
204C2×C8—1272no—
205C2×C8C22×D12712yes—
206C4⋊C4—21no—
207C4⋊C4—1271no—
208C4⋊C4—2542no—
209C4⋊C4—2542no—
210C4⋊C4—2541no—
211C4⋊C4C22×D12712yes—
212C4⋊C4C22×D12711yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—2541no—
215C22⋊C4—2541no—
216C22⋊C4—2541no—
217C42—1271no—
218C42—1271no—
219C42—1271no—
220C42—1271no—
221C42—2542no—
222C42—2542no—
223C23—21no—
224C23—2541no—
225C23D4×D12711yescharacteristic
226Q8—1274no—
227Q8—5082no—
228Q8—2544no—
229Q8—2542no—
230D4—22no—
231D4—42no—
232D4—5082no—
233D4—5082no—
234D4—5082no—
235D4—2542no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—1271no—
241C2×C4—1271no—
242C2×C4—1271no—
243C2×C4—1271no—
244C2×C4—5082no—
245C2×C4—5082no—
246C2×C4—2542no—
247C2×C4—2541no—
248C2×C4—2544no—
249C2×C4—2542no—
250C2×C4—2542no—
251C2×C4D4×D12711yescharacteristic
252C2×C4C23×D12711yescharacteristic
253C8—22no—
254C8—2542no—
255C22—21no—
256C22—21no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—5081no—
261C22—5081no—
262C22—5081no—
263C22#1686123312yes—
264C22C2×D4×D12711yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—1272no—
269C4—5081no—
270C4—2544no—
271C4—2541no—
272C4—2541no—
273C4—2541no—
274C4#41×D12712yes—
275C2—22no—
276C2—81no—
277C2—10161no—
278C2#2255038411yescharacteristic
279C2C2×#1686123311yescharacteristic
280C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#480×C127C2
socleC22×C127C22×D4
prime radical#22550421C2
hypercenter#480D127
hypocenterC127#3112
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#16861174C2semidirect
3. #22550189C2semidirect
4. #22550170C2semidirect
5. #22550075C2semidirect
6. #22550017C2semidirect
7. #22550014C2semidirect
8. #22549975C2semidirect
9. #22549973C2semidirect
10. #22549930C2semidirect
11. #480×C127C2semidirect
86. Dic254#41semidirect
141. C127#3112semidirect
145. #480D127semidirect

Maximal subgroups

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

subgroupindex
#225504212
C2×#168611742
#225501892
#225501702
#225500752
#225500172
#225500142
#225499752
#225499732
#225499302
#480×C1272
#3112127