Finite Group Databasebeta

← #30648265

Subgroups — #30648265

Order 16256.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861235C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550356C211yesmaximal, characteristic, complemented
4#22550095C212yesmaximal, complemented
5#22550088C212yesmaximal, complemented
6#22550073C212yesmaximal, complemented
7#22550068C212yesmaximal, complemented
8#22549903C211yesmaximal, characteristic, complemented
9#22549877C211yesmaximal, characteristic, complemented
10C2×#16861130C211yesmaximal, characteristic, complemented
11#535×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13#16861235—24no—
14#16861229—24no—
15C2×#14523238C2211yescharacteristic, complemented
16C2×#14523236C2212yes—
17C2×#14523236C2211yescharacteristic, complemented
18C2×#14523235C2211yescharacteristic, complemented
19C2×#14523233C2212yes—
20C2×#14523233C2211yescharacteristic, complemented
21C2×#14523232C2211yescharacteristic
22C22×C127⋊C8C2211yescharacteristic
23C4×Dic254C2212yescomplemented
24C4×D508C2212yescomplemented
25(C22⋊D127)⋊C4—22no—
26#16861167—21no—
27C2×C4⋊Dic127C2211yescharacteristic
28C4⋊C4×D127—21no—
29C4⋊(C127⋊C8)C2212yescomplemented
30C4⋊(C127⋊C8)C2212yescomplemented
31#16861133C2212yescomplemented
32#16861130C2212yes—
33#16861130C2212yes—
34#16861130C2212yescomplemented
35C2×#41×C127C2211yescharacteristic
36C22×D4×C127C2211yescharacteristic
37#127×C127C2212yes—
38#124×C127C2212yes—
39#121×C127—21no—
40#121×C127—21no—
41#120×C127—22no—
42#117×C127C2211yescharacteristic
43#14523248—22no—
44#14523248D414yescomplemented
45C2×Dic254C2311yescharacteristic
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—21no—
48C22×Dic127—21no—
49C2×C4×D127—21no—
50C2×C4×D127—22no—
51#14523238—22no—
52#14523236—24no—
53#14523236—22no—
54#14523235—22no—
55#14523233—24no—
56#14523233—22no—
57#14523232—22no—
58C2×C127⋊C8—22no—
59C2×C127⋊C8C2311yescharacteristic
60C2×C127⋊C8C2311yescharacteristic
61C2×C127⋊C8C2312yes—
62C4⋊Dic127—21no—
63C4⋊Dic127C2312yes—
64Dic127⋊C4—22no—
65(C2×D127)⋊C4—22no—
66C4×Dic127—22no—
67C24×C127—21no—
68#41×C127—22no—
69#41×C127—22no—
70C2×Q8×C127C2311yescharacteristic
71C2×D4×C127—21no—
72C2×D4×C127—22no—
73C2×D4×C127—21no—
74C2×D4×C127—22no—
75C2×D4×C127—22no—
76C2×D4×C127—22no—
77C2×D4×C127C2312yes—
78C2×D4×C127C2311yescharacteristic
79C22×C4×C127—21no—
80C22×C4×C127C2311yescharacteristic
81C4⋊C4×C127D411yescharacteristic
82C4⋊C4×C127D411yescharacteristic
83C22⋊C4×C127—22no—
84C22⋊C4×C127—22no—
85C22⋊C4×C127D412yes—
86C42×C127C2312yes—
87C22×D127—21no—
88Dic254—21no—
89Dic254C2×D412yes—
90D508—21no—
91D508C2×D412yes—
92C22⋊D127—24no—
93C2×Dic127—21no—
94C2×Dic127—22no—
95C2×Dic127—81no—
96C4×D127—24no—
97C4×D127—82no—
98C127⋊C8—21no—
99C127⋊C8—21no—
100C127⋊C8—22no—
101C23×C127—22no—
102C23×C127—21no—
103C23×C127—42no—
104C23×C127—81no—
105C23×C127—81no—
106C23×C127C2×D411yescharacteristic
107Q8×C127—21no—
108Q8×C127—21no—
109D4×C127—22no—
110D4×C127—21no—
111D4×C127—22no—
112D4×C127—21no—
113D4×C127—42no—
114D4×C127—42no—
115D4×C127—82no—
116D4×C127—82no—
117C2×C4×C127—21no—
118C2×C4×C127—42no—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2×D411yescharacteristic
123C2×C4×C127#4112yes—
124C2×C4×C127C2411yescharacteristic
125C22×C127—21no—
126C22×C127—21no—
127C22×C127—41no—
128C22×C127—42no—
129C22×C127—41no—
130C22×C127—42no—
131C22×C127—41no—
132C22×C127—42no—
133C22×C127—84no—
134C22×C127#12112yes—
135C22×C127C22×D411yescharacteristic
136C2×D127—22no—
137C2×D127—41no—
138C2×D127—41no—
139C4×C127—22no—
140C4×C127—22no—
141C4×C127—41no—
142C4×C127#12112yes—
143C4×C127C22×D411yescharacteristic
144C4×C127C2×#4111yescharacteristic
145Dic127—22no—
146Dic127—42no—
147C2×C127—22no—
148C2×C127—81no—
149C2×C127—82no—
150C2×C127C2×#12111yescharacteristic
151C2×C127#57611yescharacteristic
152C2×C127#57711yescharacteristic
153D127—42no—
154#3045—1271nomaximal, Sylow
155C127#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)—1271no—
157C2×#135—1271no—
158#535D12711yescharacteristic, complemented
159#520—1271no—
160#464—1272no—
161#463—1272no—
162#460—1272no—
163#459—1272no—
164#424—1271no—
165#417—1271no—
166C2×#102—1271no—
167C2×#41—1271no—
168C2×#41C2×D12711yescharacteristic
169C22×D4C2×D12711yescharacteristic
170AGL(1,ℤ/8)—2544no—
171#135—2544no—
172C2×Q16—1271no—
173C2×SD16—1271no—
174C2×SD16—1272no—
175C2×SD16—1271no—
176C2×D8—1272no—
177C2×D8—1271no—
178C2×M16—1271no—
179C22×C8—1271no—
180#127C2×D12712yes—
181#124C2×D12712yes—
182#121—21no—
183#121—21no—
184#120—22no—
185C4×Q8—1272no—
186C4×D4—1272no—
187C4×D4—2542no—
188#117—2541no—
189#117C2×D12711yescharacteristic
190C2×C4⋊C4—1271no—
191C2×C4⋊C4—2541no—
192C4⋊C8—1272no—
193C4⋊C8—1272no—
194#103—1272no—
195#102—1272no—
196#102—1272no—
197#102—1272no—
198C24—21no—
199#41—22no—
200#41—22no—
201#41—1274no—
202#41—2542no—
203C2×Q8—1271no—
204C2×Q8C22×D12711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—1271no—
212C2×D4—2541no—
213C2×D4C22×D12712yes—
214C2×D4C22×D12711yescharacteristic
215C22×C4—21no—
216C22×C4—2541no—
217C22×C4—2542no—
218C22×C4—2541no—
219C22×C4C22×D12711yescharacteristic
220Q16—2542no—
221SD16—2542no—
222SD16—2544no—
223SD16—2542no—
224D8—2542no—
225D8—2544no—
226M16—2542no—
227C2×C8—1271no—
228C2×C8—1271no—
229C2×C8—1272no—
230C2×C8—2542no—
231C4⋊C4—1272no—
232C4⋊C4—2542no—
233C4⋊C4—2541no—
234C4⋊C4C22⋊D12711yescharacteristic
235C4⋊C4C22⋊D12711yescharacteristic
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—2542no—
239C22⋊C4C22⋊D12712yes—
240C42—2542no—
241C42C22×D12712yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—2541no—
248C23C2×C22⋊D12711yescharacteristic
249Q8—21no—
250Q8—21no—
251Q8—1272no—
252Q8—2541no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—21no—
257D4—42no—
258D4—42no—
259D4—82no—
260D4—82no—
261D4—1272no—
262D4—2544no—
263D4—2541no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—10162no—
267C2×C4—10161no—
268C2×C4—2542no—
269C2×C4—2541no—
270C2×C4—2544no—
271C2×C4D4×D12712yes—
272C2×C4C2×C22⋊D12712yes—
273C2×C4C2×C22⋊D12712yes—
274C2×C4C2×C22⋊D12711yescharacteristic
275C2×C4#1452324712yes—
276C2×C4C23×D12711yescharacteristic
277C8—2541no—
278C8—2541no—
279C8—2542no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—42no—
288C22—84no—
289C22—5081no—
290C22—5081no—
291C22—2542no—
292C22#1686118012yes—
293C22C22×C22⋊D12711yescharacteristic
294C4—22no—
295C4—22no—
296C4—41no—
297C4—5082no—
298C4—2542no—
299C4#1686118012yes—
300C4C2×D4×D12711yescharacteristic
301C4C2×#1452324711yescharacteristic
302C2—22no—
303C2—82no—
304C2—81no—
305C2—5082no—
306C2C2×#1686118011yescharacteristic
307C2#2255067911yescharacteristic
308C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#535×C127C2
socleC22×C127C22×D4
hypercenter#535D127
hypocenterC127#3045
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. C2×#16861229C2semidirect
3. #22550356C2semidirect
4. #22550095C2semidirect
5. #22550088C2semidirect
6. #22550073C2semidirect
7. #22550068C2semidirect
8. #22549903C2semidirect
9. #22549877C2semidirect
10. C2×#16861130C2semidirect
11. #535×C127C2semidirect
15. C2×#14523238C22semidirect
17. C2×#14523236C22semidirect
18. C2×#14523235C22semidirect
20. C2×#14523233C22semidirect
23. C4×Dic254C22semidirect
24. C4×D508C22semidirect
29. C4⋊(C127⋊C8)C22semidirect
30. C4⋊(C127⋊C8)C22semidirect
31. #16861133C22semidirect
34. #16861130C22semidirect
44. #14523248D4semidirect
155. C127#3045semidirect
158. #535D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
C2×#168612292
#225503562
#225500952
#225500882
#225500732
#225500682
#225499032
#225498772
C2×#168611302
#535×C1272
#3045127