Finite Group Databasebeta

← #30648851

Subgroups — #30648851

Order 16256.

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

normal subgroups only

This group has 424 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#22550424C211yesmaximal, characteristic
2C4×Q8×D127C211yesmaximal, characteristic, complemented
3#22550188C212yesmaximal, complemented
4#22550172C211yesmaximal, characteristic, complemented
5#22550171C211yesmaximal, characteristic, complemented
6#22550073C211yesmaximal, characteristic, complemented
7#22550013C212yesmaximal, complemented
8#22549977C212yesmaximal, complemented
9#22549973C212yesmaximal, complemented
10#22549932C211yesmaximal, characteristic, complemented
11#460×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—21no—
13C2×Q8×D127C2211yescharacteristic
14C2×#14523236C2212yes—
15#16861209C2211yescharacteristic
16#16861196—21no—
17#16861180—21no—
18#16861180C2212yes—
19Q8×Dic127C2211yescharacteristic
20C4×Dic254—21no—
21C4×Dic254C2211yescharacteristic
22#16861174—21no—
23#16861174C2211yescharacteristic
24D4×Dic127—21no—
25D4×Dic127C2212yes—
26C4⋊C4×D127—21no—
27C4⋊C4×D127C2211yescharacteristic
28C42×D127—21no—
29C42×D127C2211yescharacteristic
30#16861143C2212yes—
31#16861142C2212yes—
32C4⋊(C127⋊C8)C2211yescharacteristic
33#16861134C2212yes—
34#16861133C2212yes—
35#16861130C2212yes—
36(C2×D127)⋊C8C2212yes—
37#16861119C2212yes—
38C2×SD16×C127C2212yes—
39#127×C127C2211yescharacteristic
40C4×Q8×C127C2211yescharacteristic
41C4⋊C8×C127C2211yescharacteristic
42#102×C127C2212yes—
43#14523247—44no—
44C2×Dic254—21no—
45C2×Dic254—21no—
46C2×Dic254C2311yescharacteristic
47Q8×D127—22no—
48Q8×D127—22no—
49C2×C22⋊D127—22no—
50C22×Dic127—22no—
51C2×C4×D127—21no—
52C2×C4×D127—21no—
53C2×C4×D127D411yescharacteristic
54C2×C4×D127D411yescharacteristic
55C2×C4×D127C2311yescharacteristic
56#14523236—24no—
57C2×C127⋊C8C2312yes—
58C4⋊Dic127—21no—
59C4⋊Dic127—21no—
60C4⋊Dic127C2312yes—
61C4⋊Dic127C2311yescharacteristic
62Dic127⋊C4—21no—
63Dic127⋊C4—21no—
64Dic127⋊C4—21no—
65C22⋊Dic127—22no—
66C22⋊Dic127—22no—
67C4×Dic127—21no—
68C4×Dic127—21no—
69C4×Dic127—21no—
70C4×Dic127D411yescharacteristic
71C4×Dic127D411yescharacteristic
72C4×Dic127C2311yescharacteristic
73C2×Q8×C127C2311yescharacteristic
74C2×D4×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127C2312yes—
77SD16×C127—24no—
78C2×C8×C127C2312yes—
79C4⋊C4×C127—21no—
80C4⋊C4×C127C2311yescharacteristic
81C42×C127—21no—
82C42×C127C2311yescharacteristic
83C22×D127C2×D411yescharacteristic
84Dic254—24no—
85Dic254—21no—
86Dic254—42no—
87Dic254#4112yescomplemented
88C22⋊D127—44no—
89C2×Dic127—21no—
90C2×Dic127—21no—
91C2×Dic127—22no—
92C2×Dic127—44no—
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C4×D127—24no—
97C4×D127—22no—
98C4×D127—22no—
99C4×D127—21no—
100C4×D127—21no—
101C4×D127—42no—
102C127⋊C8—22no—
103C23×C127—22no—
104Q8×C127—21no—
105Q8×C127#4112yescomplemented
106D4×C127—22no—
107D4×C127—22no—
108D4×C127—42no—
109D4×C127—42no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2411yescharacteristic
115C8×C127—22no—
116C22×C127—42no—
117C22×C127—42no—
118C22×C127—42no—
119C22×C127C22×D411yescharacteristic
120C2×D127—21no—
121C2×D127—21no—
122C2×D127AGL(1,ℤ/8)12yes—
123C4×C127—22no—
124C4×C127—21no—
125C4×C127—41no—
126C4×C127#13712yes—
127C4×C127C2×#4111yescharacteristic
128C4×C127C2×#4111yescharacteristic
129Dic127—22no—
130Dic127—21no—
131Dic127—21no—
132Dic127—21no—
133Dic127—41no—
134Dic127—42no—
135C2×C127—82no—
136C2×C127#52511yescharacteristic
137C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C127C2×#13711yescharacteristic
139D127—22no—
140#3112—1271nomaximal, Sylow
141C127#311211yescharacteristic, complemented, Sylow
142#532—1271no—
143C2×C4×Q8—1271no—
144#480—1272no—
145#477—1271no—
146#477—1271no—
147#460—1271no—
148#460D12711yescharacteristic, complemented
149#448—1272no—
150#440—1272no—
151#440—1272no—
152#432—1271no—
153C2×#41—2541no—
154C22×Q8—1271no—
155C2×SD16—1272no—
156C2×SD16C2×D12712yes—
157C4⋊Q8—1271no—
158#127C2×D12711yescharacteristic
159#124—2541no—
160#121—1272no—
161#121—2541no—
162C4×Q8—1271no—
163C4×Q8—1271no—
164C4×Q8—1271no—
165C4×Q8—2541no—
166C4×Q8—2541no—
167C4×Q8C2×D12711yescharacteristic
168C4×D4—1272no—
169C4×D4—2541no—
170C2×C4⋊C4—1271no—
171C2×C4⋊C4—2541no—
172C2×C42—1271no—
173C2×C42—2541no—
174#107—1272no—
175#107—1272no—
176C4⋊C8—1271no—
177C4⋊C8C2×D12711yescharacteristic
178#103—1272no—
179#103—1272no—
180#102—1272no—
181#102C2×D12712yes—
182C22⋊C8—1272no—
183#97—1272no—
184#41—5084no—
185C2×Q8—1271no—
186C2×Q8—2541no—
187C2×Q8—2542no—
188C2×Q8—2542no—
189C2×Q8—2541no—
190C2×Q8C22×D12711yescharacteristic
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—2542no—
194C2×D4C22×D12712yes—
195C22×C4—1271no—
196C22×C4—1271no—
197C22×C4—1271no—
198C22×C4—2541no—
199C22×C4—2541no—
200C22×C4—2542no—
201SD16—24no—
202SD16—2544no—
203C2×C8—1272no—
204C2×C8C22×D12712yes—
205C4⋊C4—21no—
206C4⋊C4—1271no—
207C4⋊C4—1272no—
208C4⋊C4—2541no—
209C4⋊C4—2541no—
210C4⋊C4—2541no—
211C4⋊C4—2541no—
212C4⋊C4—2541no—
213C4⋊C4C22×D12711yescharacteristic
214C22⋊C4—2542no—
215C22⋊C4—2542no—
216C42—21no—
217C42—1271no—
218C42—1271no—
219C42—1271no—
220C42—2541no—
221C42—2541no—
222C42—2541no—
223C42C22×D12711yescharacteristic
224C23—22no—
225C23—1271no—
226Q8—21no—
227Q8—1272no—
228Q8—5082no—
229Q8—2544no—
230Q8—2541no—
231Q8#1452324712yescomplemented
232D4—22no—
233D4—22no—
234D4—42no—
235D4—42no—
236D4—5084no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—1271no—
240C2×C4—1271no—
241C2×C4—1271no—
242C2×C4—5082no—
243C2×C4—5084no—
244C2×C4—2542no—
245C2×C4—2544no—
246C2×C4—2541no—
247C2×C4—2542no—
248C2×C4—2541no—
249C2×C4—2542no—
250C2×C4—2541no—
251C2×C4—2541no—
252C2×C4D4×D12711yescharacteristic
253C2×C4D4×D12711yescharacteristic
254C2×C4C23×D12711yescharacteristic
255C8—22no—
256C8—2542no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—1272no—
261C22—2541no—
262C22—2541no—
263C22C2×D4×D12711yescharacteristic
264C4—22no—
265C4—21no—
266C4—41no—
267C4—5082no—
268C4—5081no—
269C4—2541no—
270C4—2542no—
271C4—2541no—
272C4—2541no—
273C4#1686123812yes—
274C4#41×D12711yescharacteristic
275C4C2×#1452324711yescharacteristic
276C2—82no—
277C2—2542no—
278C2#2255038611yescharacteristic
279C2AGL(1,ℤ/8)×D12711yescharacteristic
280C2C2×#1686123811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#460×C127C2
socleC22×C127C22×D4
prime radical#22550424C2
hypercenter#460D127
hypocenterC127#3112
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D127C2semidirect
3. #22550188C2semidirect
4. #22550172C2semidirect
5. #22550171C2semidirect
6. #22550073C2semidirect
7. #22550013C2semidirect
8. #22549977C2semidirect
9. #22549973C2semidirect
10. #22549932C2semidirect
11. #460×C127C2semidirect
87. Dic254#41semidirect
105. Q8×C127#41semidirect
141. C127#3112semidirect
148. #460D127semidirect
231. Q8#14523247semidirect

Maximal subgroups

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

subgroupindex
#225504242
C4×Q8×D1272
#225501882
#225501722
#225501712
#225500732
#225500132
#225499772
#225499732
#225499322
#460×C1272
#3112127