Finite Group Databasebeta

← #30648268

Subgroups — #30648268

Order 16256.

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

normal subgroups only

This group has 504 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
1C2×#16861237C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550440C211yesmaximal, characteristic, complemented
4#22550357C211yesmaximal, characteristic, complemented
5#22550101C212yesmaximal, complemented
6#22550094C212yesmaximal, complemented
7#22550075C212yesmaximal, complemented
8#22550070C212yesmaximal, complemented
9#22549904C211yesmaximal, characteristic, complemented
10C2×#16861129C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×D508C2211yescharacteristic
14#16861237—24no—
15#16861229—24no—
16C2×#14523238C2211yescharacteristic, complemented
17C2×#14523237C2212yes—
18C2×#14523236C2211yescharacteristic, complemented
19C2×#14523235C2211yescharacteristic, complemented
20C2×D1016C2212yes—
21C2×#14523233C2211yescharacteristic, complemented
22C22×C127⋊C8C2211yescharacteristic
23#16861205C2212yes—
24#16861198C2212yes—
25#16861183—21no—
26#16861183—21no—
27#16861177—22no—
28#16861174C2212yescomplemented
29#16861171C2212yescomplemented
30C4×C22⋊D127—22no—
31#16861168C2211yescharacteristic
32#16861167—21no—
33C4⋊C4×D127—21no—
34Dic127⋊C8C2212yescomplemented
35Dic127⋊C8C2212yescomplemented
36#16861129C2212yes—
37#16861129C2212yes—
38C2×#41×C127C2211yescharacteristic
39C2×M16×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#103×C127C2212yescomplemented
42#102×C127C2212yescomplemented
43C23×D127—21no—
44#14523248—22no—
45#14523248D414yescomplemented
46C2×Dic254C2311yescharacteristic
47C2×D508—22no—
48C2×D508—21no—
49C2×D508—22no—
50C2×D508C2312yes—
51C2×D508C2311yescharacteristic
52C2×C22⋊D127—22no—
53C2×C22⋊D127—22no—
54C2×C22⋊D127—21no—
55C2×C4×D127—22no—
56C2×C4×D127—21no—
57#14523238—22no—
58#14523237—24no—
59#14523236—22no—
60#14523235—22no—
61D1016—24no—
62#14523233—22no—
63C2×C127⋊C8—22no—
64C2×C127⋊C8C2311yescharacteristic
65C2×C127⋊C8C2311yescharacteristic
66C4⋊Dic127D411yescharacteristic
67C4⋊Dic127D411yescharacteristic
68Dic127⋊C4—22no—
69C22⋊Dic127D412yes—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—22no—
72(C2×D127)⋊C4—22no—
73C4×Dic127—22no—
74C4×Dic127C2312yes—
75#41×C127—22no—
76#41×C127—22no—
77C2×Q8×C127C2311yescharacteristic
78C2×D4×C127—21no—
79C2×D4×C127C2311yescharacteristic
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127C2311yescharacteristic
83M16×C127—22no—
84C2×C8×C127C2312yes—
85C4⋊C4×C127—21no—
86C4⋊C4×C127C2312yes—
87C22×D127—21no—
88C22×D127—22no—
89C22×D127—42no—
90C22×D127—81no—
91C22×D127—81no—
92Dic254—21no—
93Dic254C2×D412yes—
94D508—22no—
95D508—21no—
96D508—22no—
97D508—42no—
98D508C2×D412yes—
99C22⋊D127—24no—
100C22⋊D127—82no—
101C22⋊D127—82no—
102C2×Dic127—21no—
103C2×Dic127C2×D412yes—
104C2×Dic127C2×D412yes—
105C4×D127—24no—
106C4×D127—82no—
107C127⋊C8—21no—
108C127⋊C8—21no—
109C23×C127—21no—
110C23×C127C2×D411yescharacteristic
111Q8×C127—21no—
112Q8×C127—21no—
113D4×C127—21no—
114D4×C127—21no—
115D4×C127—42no—
116C2×C4×C127—22no—
117C2×C4×C127—21no—
118C2×C4×C127—42no—
119C2×C4×C127—81no—
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D412yes—
122C2×C4×C127#4112yes—
123C2×C4×C127C2411yescharacteristic
124C8×C127—22no—
125C22×C127—21no—
126C22×C127—21no—
127C22×C127—41no—
128C22×C127—41no—
129C22×C127—41no—
130C22×C127#12112yes—
131C22×C127C22×D411yescharacteristic
132C2×D127—22no—
133C2×D127—41no—
134C2×D127—41no—
135C2×D127—42no—
136C2×D127—42no—
137C2×D127—42no—
138C2×D127—84no—
139C4×C127—41no—
140C4×C127—42no—
141C4×C127#12112yes—
142C4×C127C22×D411yescharacteristic
143C4×C127C2×#4111yescharacteristic
144Dic127—22no—
145Dic127—22no—
146Dic127—22no—
147C2×C127—22no—
148C2×C127—81no—
149C2×C127C2×#12111yescharacteristic
150C2×C127#57611yescharacteristic
151C2×C127#57711yescharacteristic
152D127—42no—
153D127—82no—
154#3045—1271nomaximal, Sylow
155C127#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)—1271no—
157C2×#135—1271no—
158#535—1271no—
159#520—1271no—
160#464—1272no—
161#463—1272no—
162#460—1272no—
163#459—1272no—
164#424—1271no—
165#417D12711yescharacteristic, complemented
166C2×#102—1271no—
167C2×#41—1271no—
168C2×#41C2×D12711yescharacteristic
169C22×D4—1271no—
170AGL(1,ℤ/8)—2544no—
171#135—2544no—
172C2×Q16—1271no—
173C2×SD16—1271no—
174C2×SD16—1271no—
175C2×SD16—1272no—
176C2×D8—1271no—
177C2×D8—1272no—
178C2×M16C2×D12711yescharacteristic
179C22×C8—1271no—
180#127—1272no—
181#124—1272no—
182#121—2541no—
183#121—2541no—
184#120—2542no—
185C4×Q8—1272no—
186C4×D4—1272no—
187C4×D4—2542no—
188#117—1271no—
189#117—2541no—
190C2×C4⋊C4—2541no—
191C2×C4⋊C4C2×D12711yescharacteristic
192C4⋊C8—1272no—
193C4⋊C8—1272no—
194#103C2×D12712yescomplemented
195#102—1272no—
196#102—1272no—
197#102C2×D12712yescomplemented
198C24—2541no—
199#41—22no—
200#41—22no—
201#41—1274no—
202#41—2542no—
203C2×Q8—1271no—
204C2×Q8C22×D12711yescharacteristic
205C2×D4—21no—
206C2×D4—1271no—
207C2×D4—1272no—
208C2×D4—2542no—
209C2×D4—2542no—
210C2×D4—2541no—
211C2×D4—2541no—
212C2×D4—2542no—
213C2×D4—2542no—
214C2×D4C22×D12711yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—2542no—
218C22×C4—2541no—
219C22×C4C22×D12711yescharacteristic
220Q16—2542no—
221SD16—2544no—
222SD16—2542no—
223SD16—2542no—
224D8—2544no—
225D8—2542no—
226M16—22no—
227C2×C8—1271no—
228C2×C8—1271no—
229C2×C8—2542no—
230C2×C8C22×D12712yes—
231C4⋊C4—21no—
232C4⋊C4—1271no—
233C4⋊C4—1271no—
234C4⋊C4—2542no—
235C4⋊C4C22×D12712yes—
236C22⋊C4—1272no—
237C22⋊C4—2542no—
238C22⋊C4—2542no—
239C22⋊C4—2542no—
240C42—1272no—
241C42—2542no—
242C23—21no—
243C23—10161no—
244C23—10161no—
245C23—5082no—
246C23—2542no—
247C23—2541no—
248C23D4×D12711yescharacteristic
249Q8—21no—
250Q8—21no—
251Q8—1272no—
252Q8—2541no—
253D4—21no—
254D4—21no—
255D4—42no—
256D4—1272no—
257D4—10162no—
258D4—10162no—
259D4—5082no—
260D4—2544no—
261D4—2542no—
262D4—2542no—
263D4—2541no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—1272no—
269C2×C4—1272no—
270C2×C4—10162no—
271C2×C4—2541no—
272C2×C4—2544no—
273C2×C4D4×D12712yes—
274C2×C4D4×D12711yescharacteristic
275C2×C4#1452324812yes—
276C2×C4C23×D12711yescharacteristic
277C8—22no—
278C8—2541no—
279C8—2541no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—10164no—
286C22—5082no—
287C22—5082no—
288C22—5082no—
289C22—5081no—
290C22—5081no—
291C22—2542no—
292C22#1686118212yes—
293C22C2×D4×D12711yescharacteristic
294C4—41no—
295C4—42no—
296C4—2542no—
297C4—2542no—
298C4—2542no—
299C4#1686118212yes—
300C4C2×D4×D12711yescharacteristic
301C4C2×#1452324811yescharacteristic
302C2—22no—
303C2—81no—
304C2—10162no—
305C2—5082no—
306C2C2×#1686118211yescharacteristic
307C2#2255068011yescharacteristic
308C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
hypercenter#417D127
hypocenterC127#3045
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. C2×#16861229C2semidirect
3. #22550440C2semidirect
4. #22550357C2semidirect
5. #22550101C2semidirect
6. #22550094C2semidirect
7. #22550075C2semidirect
8. #22550070C2semidirect
9. #22549904C2semidirect
10. C2×#16861129C2semidirect
11. #417×C127C2semidirect
16. C2×#14523238C22semidirect
18. C2×#14523236C22semidirect
19. C2×#14523235C22semidirect
21. C2×#14523233C22semidirect
28. #16861174C22semidirect
29. #16861171C22semidirect
34. Dic127⋊C8C22semidirect
35. Dic127⋊C8C22semidirect
41. #103×C127C22semidirect
42. #102×C127C22semidirect
45. #14523248D4semidirect
155. C127#3045semidirect
165. #417D127semidirect
194. #103C2×D127semidirect
197. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C2×#168612292
#225504402
#225503572
#225501012
#225500942
#225500752
#225500702
#225499042
C2×#168611292
#417×C1272
#3045127