Finite Group Databasebeta

← #30649204

Subgroups — #30649204

Order 16256.

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

normal subgroups only

This group has 564 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×#16861229C211yesmaximal, characteristic, complemented
3C22×#14523236C211yesmaximal, characteristic, complemented
4#22550447C211yesmaximal, characteristic, complemented
5#22550443C211yesmaximal, characteristic, complemented
6#22550267C212yesmaximal, complemented
7#22550264C212yesmaximal, complemented
8#22550259C212yesmaximal, complemented
9#22550247C212yesmaximal, complemented
10#22549834C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×Dic254C2211yescharacteristic
14#16861242—24no—
15#16861229—24no—
16C2×Dic508C2212yescomplemented
17C2×#14523238C2211yescharacteristic, complemented
18C2×#14523237C2212yescomplemented
19C2×#14523236—24no—
20C2×#14523236C2212yes—
21C2×#14523236C2212yes—
22C2×#14523236C2211yescharacteristic, complemented
23C2×#14523235C2211yescharacteristic, complemented
24C2×#14523233C2211yescharacteristic, complemented
25C22×C127⋊C8C2211yescharacteristic
26#16861207C2212yescomplemented
27#16861205C2212yes—
28#16861196C2212yes—
29#16861196C2212yescomplemented
30#16861190—22no—
31#16861188—22no—
32#16861180—22no—
33#16861178—22no—
34#16861168C2211yescharacteristic
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×Dic254C2311yescharacteristic
49C2×Dic254C2312yes—
50C2×D508C2311yescharacteristic
51C2×C22⋊D127—24no—
52C2×C22⋊D127—21no—
53C22×Dic127—21no—
54C2×C4×D127—21no—
55Dic508—24no—
56#14523238—22no—
57#14523237—24no—
58#14523236—24no—
59#14523236—24no—
60#14523236—22no—
61#14523235—22no—
62#14523233—22no—
63C2×C127⋊C8D414yescomplemented
64C2×C127⋊C8C2311yescharacteristic
65C2×C127⋊C8C2311yescharacteristic
66C4⋊Dic127D412yes—
67Dic127⋊C4—24no—
68C22⋊Dic127—24no—
69C22⋊Dic127—24no—
70C22⋊Dic127D412yes—
71C4×Dic127C2312yes—
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—22no—
90Dic254—21no—
91Dic254—22no—
92Dic254—21no—
93Dic254—42no—
94D508—21no—
95D508—21no—
96C22⋊D127—42no—
97C22⋊D127—84no—
98C2×Dic127—21no—
99C2×Dic127—22no—
100C2×Dic127—42no—
101C2×Dic127C2×D414yes—
102C4×D127—42no—
103C127⋊C8C2×D412yes—
104C127⋊C8C2×D412yes—
105C23×C127—21no—
106C23×C127—22no—
107C23×C127—42no—
108C23×C127—82no—
109C23×C127C2×D411yescharacteristic
110Q8×C127—21no—
111Q8×C127—21no—
112D4×C127—21no—
113D4×C127—22no—
114D4×C127—22no—
115D4×C127—21no—
116D4×C127—42no—
117D4×C127—42no—
118C2×C4×C127—21no—
119C2×C4×C127—42no—
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2×D412yes—
123C2×C4×C127C2411yescharacteristic
124C8×C127C2×D414yes—
125C22×C127—22no—
126C22×C127—41no—
127C22×C127—42no—
128C22×C127—42no—
129C22×C127—44no—
130C22×C127—84no—
131C22×C127#12712yes—
132C22×C127C22×D411yescharacteristic
133C2×D127—41no—
134C2×D127—42no—
135C4×C127—41no—
136C4×C127#12712yes—
137C4×C127C22×D411yescharacteristic
138C4×C127C22×D411yescharacteristic
139Dic127—24no—
140Dic127—41no—
141Dic127—42no—
142C2×C127—22no—
143C2×C127—81no—
144C2×C127—82no—
145C2×C127C2×#12711yescharacteristic
146C2×C127#57712yes—
147D127—81no—
148#3154—1271nomaximal, Sylow
149C127#315411yescharacteristic, complemented, Sylow
150C2×#137—1271no—
151C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
152C2×#135—1271no—
153C22×SD16—1271no—
154#536—1271no—
155#535—1271no—
156#497—1272no—
157#496—1272no—
158#495—1272no—
159#492—1272no—
160#405—1271no—
161C2×#41—1271no—
162C2×#41C2×D12711yescharacteristic
163C22×Q8—1271no—
164C22×D4C2×D12711yescharacteristic
165#137—2544no—
166AGL(1,ℤ/8)—24no—
167#135—2544no—
168C2×Q16—1271no—
169C2×Q16—1272no—
170C2×SD16—1271no—
171C2×SD16—1272no—
172C2×SD16—1272no—
173C2×SD16—1271no—
174C2×SD16—1272no—
175C2×SD16—2544no—
176C2×SD16C2×D12712yes—
177C2×D8—1271no—
178C2×D8C2×D12712yes—
179C2×M16C2×D12711yescharacteristic
180C22×C8—1271no—
181C4⋊Q8—1272no—
182#127—1272no—
183#124—1272no—
184#124—1272no—
185#123—2542no—
186C22⋊Q8—2542no—
187#121—2542no—
188#120—2542no—
189#117—1271no—
190#97—1272no—
191C4×C8—1272no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—2542no—
196#41—2542no—
197C2×Q8—1271no—
198C2×Q8—1272no—
199C2×Q8—2542no—
200C2×Q8—2542no—
201C2×Q8—2541no—
202C2×Q8C22×D12711yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—1271no—
208C2×D4—2544no—
209C2×D4—2541no—
210C2×D4C22×D12711yescharacteristic
211C2×D4C22×D12712yes—
212C22×C4—21no—
213C22×C4—2541no—
214C22×C4—2541no—
215C22×C4C22×D12711yescharacteristic
216Q16—2542no—
217Q16—2544no—
218SD16—24no—
219SD16—2544no—
220SD16—2544no—
221SD16—2542no—
222SD16—2542no—
223SD16—2544no—
224D8—24no—
225D8—2542no—
226M16C22⋊D12714yescomplemented
227C2×C8—1274no—
228C2×C8—1271no—
229C2×C8—1271no—
230C2×C8C22×D12712yes—
231C4⋊C4—1272no—
232C4⋊C4—2544no—
233C22⋊C4—1272no—
234C22⋊C4—2544no—
235C22⋊C4—2544no—
236C42—1272no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—82no—
241C23—2541no—
242C23D4×D12711yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—5082no—
246Q8—2542no—
247Q8—2541no—
248Q8—2542no—
249Q8—2541no—
250D4—21no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—10164no—
257D4—5082no—
258D4—2541no—
259D4—2541no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—1274no—
263C2×C4—5082no—
264C2×C4—5082no—
265C2×C4—2541no—
266C2×C4—2542no—
267C2×C4D4×D12711yescharacteristic
268C2×C4D4×D12712yes—
269C2×C4C2×C22⋊D12712yes—
270C2×C4C23×D12711yescharacteristic
271C8—1272no—
272C8—1272no—
273C8C2×C22⋊D12714yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—84no—
280C22—5081no—
281C22—5082no—
282C22#1686120512yes—
283C22C2×D4×D12711yescharacteristic
284C4—41no—
285C4—5081no—
286C4—5082no—
287C4—2544no—
288C4#1686120512yes—
289C4C2×D4×D12711yescharacteristic
290C4C22×C22⋊D12711yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—10161no—
295C2C2×#1686120511yescharacteristic
296C2#2255068612yes—
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#3154
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×#16861229C2semidirect
3. C22×#14523236C2semidirect
4. #22550447C2semidirect
5. #22550443C2semidirect
6. #22550267C2semidirect
7. #22550264C2semidirect
8. #22550259C2semidirect
9. #22550247C2semidirect
10. #22549834C2semidirect
11. C2×AGL(1,ℤ/8)×C127C2semidirect
16. C2×Dic508C22semidirect
17. C2×#14523238C22semidirect
18. C2×#14523237C22semidirect
22. C2×#14523236C22semidirect
23. C2×#14523235C22semidirect
24. C2×#14523233C22semidirect
26. #16861207C22semidirect
29. #16861196C22semidirect
35. #16861119C22semidirect
36. C8×Dic127C22semidirect
63. C2×C127⋊C8D4semidirect
86. M16×C127D4semidirect
149. C127#3154semidirect
151. C2×AGL(1,ℤ/8)D127semidirect
226. 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×#168612292
C22×#145232362
#225504472
#225504432
#225502672
#225502642
#225502592
#225502472
#225498342
C2×AGL(1,ℤ/8)×C1272
#3154127