Finite Group Databasebeta

← #30649192

Subgroups — #30649192

Order 16256.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861242C211yesmaximal, characteristic, complemented
2C2×#16861239C211yesmaximal, characteristic, complemented
3C2×#16861235C211yesmaximal, characteristic, complemented
4#22550422C211yesmaximal, characteristic, complemented
5C2×#16861196C211yesmaximal, characteristic, complemented
6#22550267C212yesmaximal, complemented
7#22550264C212yesmaximal, complemented
8#22550259C212yesmaximal, complemented
9#22550257C212yesmaximal, complemented
10M16×Dic127C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×Dic254C2211yescharacteristic
14#16861242—24no—
15#16861239—24no—
16#16861235—24no—
17C2×Dic508C2212yescomplemented
18C2×#14523238C2212yescomplemented
19C2×#14523237C2212yescomplemented
20C2×#14523236C2212yes—
21C2×#14523236C2212yescomplemented
22C2×#14523233C2212yes—
23C2×#14523232C2211yescharacteristic
24#16861207C2211yescharacteristic, complemented
25#16861205C2211yescharacteristic, complemented
26#16861198C2211yescharacteristic, complemented
27#16861196—22no—
28#16861196C2212yes—
29#16861196C2212yes—
30#16861196C2211yescharacteristic, complemented
31#16861181—22no—
32C4×C22⋊D127—22no—
33C2×C22⋊Dic127—22no—
34C2×C4×Dic127C2211yescharacteristic
35#16861119C2212yescomplemented
36C8×Dic127C2212yescomplemented
37C2×#41×C127C2211yescharacteristic
38C22×D4×C127C2211yescharacteristic
39AGL(1,ℤ/8)×C127—24no—
40C2×SD16×C127C2212yes—
41C2×D8×C127C2212yes—
42C2×M16×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45C2×Dic254—21no—
46C2×Dic254—22no—
47C2×Dic254—22no—
48C2×Dic254C2312yes—
49C2×Dic254C2311yescharacteristic
50C2×D508C2311yescharacteristic
51C2×C22⋊D127—22no—
52C2×C22⋊D127—21no—
53C22×Dic127—21no—
54C22×Dic127D412yes—
55C2×C4×D127—21no—
56Dic508—24no—
57#14523238—24no—
58#14523237—24no—
59#14523236—24no—
60#14523236—24no—
61#14523233—24no—
62#14523232D414yescomplemented
63C2×C127⋊C8C2312yes—
64Dic127⋊C4—22no—
65C22⋊Dic127—24no—
66C22⋊Dic127—24no—
67C22⋊Dic127—22no—
68(C2×D127)⋊C4—22no—
69C4×Dic127D412yes—
70C4×Dic127C2311yescharacteristic
71C4×Dic127C2311yescharacteristic
72C24×C127—21no—
73#41×C127—22no—
74#41×C127—22no—
75C2×Q8×C127C2311yescharacteristic
76C2×D4×C127—21no—
77C2×D4×C127—22no—
78C2×D4×C127—22no—
79C2×D4×C127—21no—
80C2×D4×C127C2312yes—
81C2×D4×C127C2311yescharacteristic
82C22×C4×C127—21no—
83C22×C4×C127C2311yescharacteristic
84SD16×C127—24no—
85D8×C127—24no—
86M16×C127D414yescomplemented
87C2×C8×C127C2312yes—
88C22×D127—21no—
89Dic254—21no—
90Dic254—22no—
91Dic254—22no—
92Dic254—21no—
93Dic254—42no—
94D508—21no—
95D508—21no—
96C22⋊D127—44no—
97C22⋊D127—42no—
98C2×Dic127—22no—
99C2×Dic127—21no—
100C2×Dic127—42no—
101C2×Dic127—42no—
102C2×Dic127C2×D412yes—
103C2×Dic127C2×D412yes—
104C4×D127—42no—
105C127⋊C8C2×D414yes—
106C23×C127—21no—
107C23×C127—22no—
108C23×C127—44no—
109C23×C127—42no—
110C23×C127C2×D411yescharacteristic
111Q8×C127—21no—
112Q8×C127—21no—
113D4×C127—21no—
114D4×C127—22no—
115D4×C127—22no—
116D4×C127—21no—
117D4×C127—42no—
118D4×C127—42no—
119C2×C4×C127—21no—
120C2×C4×C127—42no—
121C2×C4×C127C2×D411yescharacteristic
122C2×C4×C127C2×D412yes—
123C2×C4×C127C2×D412yes—
124C2×C4×C127C2411yescharacteristic
125C8×C127C2×D414yes—
126C22×C127—22no—
127C22×C127—41no—
128C22×C127—42no—
129C22×C127—44no—
130C22×C127—42no—
131C22×C127—84no—
132C22×C127#12712yes—
133C22×C127C22×D411yescharacteristic
134C2×D127—41no—
135C2×D127—42no—
136C4×C127—41no—
137C4×C127#12712yes—
138C4×C127C22×D411yescharacteristic
139C4×C127C22×D411yescharacteristic
140Dic127—22no—
141Dic127—22no—
142Dic127—41no—
143Dic127—42no—
144C2×C127—22no—
145C2×C127—81no—
146C2×C127—82no—
147C2×C127C2×#12711yescharacteristic
148C2×C127#57512yes—
149D127—81no—
150#3153—1271nomaximal, Sylow
151C127#315311yescharacteristic, complemented, Sylow
152C2×#137—1271no—
153C2×#137—1271no—
154C2×AGL(1,ℤ/8)—1271no—
155C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
156#532—1271no—
157C2×#124—1271no—
158#497—1272no—
159#496—1272no—
160#495—1272no—
161#495—1272no—
162C4×M16—1271no—
163C2×#41—1271no—
164C2×#41C2×D12711yescharacteristic
165C22×Q8—1271no—
166C22×D4C2×D12711yescharacteristic
167#137—2544no—
168#137—2544no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—2544no—
171C2×Q16—1272no—
172C2×Q16—1272no—
173C2×SD16—1272no—
174C2×SD16—1272no—
175C2×SD16—1272no—
176C2×SD16C2×D12712yes—
177C2×D8—1272no—
178C2×D8C2×D12712yes—
179C2×M16—1271no—
180C2×M16C2×D12711yescharacteristic
181C4⋊Q8—1271no—
182#127—1271no—
183#124—1271no—
184#124—1272no—
185#124—1272no—
186#124—1271no—
187#124—2542no—
188#121—2542no—
189C4×D4—2542no—
190C2×C22⋊C4—2542no—
191C2×C42—1271no—
192#97—1272no—
193C4×C8—1272no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—2542no—
198#41—2542no—
199C2×Q8—1272no—
200C2×Q8—1271no—
201C2×Q8—2542no—
202C2×Q8—2542no—
203C2×Q8—2541no—
204C2×Q8C22×D12711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—1271no—
210C2×D4—2542no—
211C2×D4—2541no—
212C2×D4C22×D12712yes—
213C2×D4C22×D12711yescharacteristic
214C22×C4—21no—
215C22×C4—1272no—
216C22×C4—2541no—
217C22×C4—2541no—
218C22×C4C22×D12711yescharacteristic
219Q16—2544no—
220Q16—2544no—
221SD16—24no—
222SD16—2544no—
223SD16—2544no—
224SD16—2544no—
225D8—24no—
226D8—2544no—
227M16—1274no—
228M16C22⋊D12714yescomplemented
229C2×C8—1272no—
230C2×C8C22×D12712yes—
231C4⋊C4—2542no—
232C22⋊C4—2542no—
233C22⋊C4—2544no—
234C22⋊C4—2544no—
235C22⋊C4—2542no—
236C42—1272no—
237C42—1271no—
238C42—1271no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—2541no—
244C23D4×D12711yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—5082no—
248Q8—2541no—
249Q8—2542no—
250Q8—2542no—
251Q8—2541no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—5082no—
259D4—5084no—
260D4—2541no—
261D4—2541no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—1272no—
265C2×C4—1272no—
266C2×C4—5082no—
267C2×C4—5082no—
268C2×C4—5082no—
269C2×C4—2542no—
270C2×C4—2541no—
271C2×C4D4×D12711yescharacteristic
272C2×C4D4×D12712yes—
273C2×C4C2×C22⋊D12712yes—
274C2×C4C23×D12711yescharacteristic
275C8—1274no—
276C8C2×C22⋊D12714yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—5081no—
284C22—5082no—
285C22#1686120512yes—
286C22C2×D4×D12711yescharacteristic
287C4—41no—
288C4—5081no—
289C4—5082no—
290C4—2542no—
291C4—2542no—
292C4#1686120512yes—
293C4C2×D4×D12711yescharacteristic
294C4C22×C22⋊D12711yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—10161no—
299C2C2×#1686120511yescharacteristic
300C2#2255067212yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3153
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
2. C2×#16861239C2semidirect
3. C2×#16861235C2semidirect
4. #22550422C2semidirect
5. C2×#16861196C2semidirect
6. #22550267C2semidirect
7. #22550264C2semidirect
8. #22550259C2semidirect
9. #22550257C2semidirect
10. M16×Dic127C2semidirect
11. C2×AGL(1,ℤ/8)×C127C2semidirect
17. C2×Dic508C22semidirect
18. C2×#14523238C22semidirect
19. C2×#14523237C22semidirect
21. C2×#14523236C22semidirect
24. #16861207C22semidirect
25. #16861205C22semidirect
26. #16861198C22semidirect
30. #16861196C22semidirect
35. #16861119C22semidirect
36. C8×Dic127C22semidirect
62. #14523232D4semidirect
86. M16×C127D4semidirect
151. C127#3153semidirect
155. C2×AGL(1,ℤ/8)D127semidirect
228. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C2×#168612392
C2×#168612352
#225504222
C2×#168611962
#225502672
#225502642
#225502592
#225502572
M16×Dic1272
C2×AGL(1,ℤ/8)×C1272
#3153127