Finite Group Databasebeta

← #30648079

Subgroups — #30648079

Order 16256.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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#22550674C212yesmaximal, complemented
2#22550422C211yesmaximal, characteristic, complemented
3#22550059C212yesmaximal, complemented
4#22550054C212yesmaximal, complemented
5#22550049C212yesmaximal, complemented
6#22550044C212yesmaximal, complemented
7C2×#16861136C212yesmaximal, complemented
8#22549860C211yesmaximal, characteristic, complemented
9#585×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic, complemented
11#16861240—22no—
12#16861239C2214yescomplemented
13#16861235—22no—
14#16861234C2214yescomplemented
15#16861229—24no—
16C2×#14523232C2212yescomplemented
17#16861207C2211yescharacteristic, complemented
18#16861205C2211yescharacteristic, complemented
19#16861198C2211yescharacteristic, complemented
20#16861196C2211yescharacteristic, complemented
21#16861181—22no—
22C4×C22⋊D127—22no—
23C2×C4×Dic127C2211yescharacteristic, complemented
24#16861136C2214yescomplemented
25#16861136C2214yescomplemented
26#16861128C2212yescomplemented
27#16861126C2212yescomplemented
282−1+4×C127—22no—
292−1+4×C127C2212yes—
30H(5,𝔽2)×C127—22no—
31H(5,𝔽2)×C127—22no—
32H(5,𝔽2)×C127C2212yes—
33C2×#41×C127—24no—
34C2×#41×C127—22no—
35C2×#41×C127C2212yes—
36C2×#41×C127C2211yescharacteristic
37#14523248—22no—
38#14523248—22no—
39C2×Dic254C2311yescharacteristic
40C2×D508C2311yescharacteristic
41C2×C22⋊D127—22no—
42C2×C22⋊D127—21no—
43C22×Dic127—21no—
44C2×C4×D127—21no—
45#14523238—24no—
46#14523236—24no—
47#14523235—24no—
48#14523233—24no—
49#14523232—22no—
50#14523232C2314yes—
51C2×C127⋊C8—22no—
52Dic127⋊C4—22no—
53C22⋊Dic127—22no—
54(C2×D127)⋊C4—22no—
55C4×Dic127—21no—
56C4×Dic127C2311yescharacteristic
57C4×Dic127C2311yescharacteristic
58#41×C127—24no—
59#41×C127—28no—
60#41×C127—24no—
61#41×C127—24no—
62#41×C127—24no—
63#41×C127—22no—
64#41×C127—22no—
65#41×C127—24no—
66#41×C127—22no—
67#41×C127—22no—
68#41×C127D414yescomplemented
69#41×C127C2314yes—
70C2×Q8×C127—22no—
71C2×Q8×C127—24no—
72C2×Q8×C127D412yescomplemented
73C2×Q8×C127C2311yescharacteristic
74C2×D4×C127—28no—
75C2×D4×C127—22no—
76C2×D4×C127—24no—
77C2×D4×C127—22no—
78C2×D4×C127—24no—
79C2×D4×C127D412yes—
80C2×D4×C127D412yescomplemented
81C2×D4×C127C2311yescharacteristic
82C22×C4×C127—24no—
83C22×C4×C127—22no—
84C22×C4×C127D412yes—
85C22×C4×C127C2311yescharacteristic
86C22×D127—21no—
87Dic254—22no—
88D508—22no—
89C22⋊D127—44no—
90C22⋊D127—42no—
91C2×Dic127—21no—
92C2×Dic127—21no—
93C2×Dic127—21no—
94C2×Dic127—42no—
95C4×D127—41no—
96C4×D127—41no—
97C127⋊C8—24no—
98C23×C127—24no—
99C23×C127—22no—
100C23×C127C2×D412yes—
101C23×C127C2×D411yescharacteristic
102Q8×C127—24no—
103Q8×C127—22no—
104Q8×C127—22no—
105Q8×C127C2×D414yes—
106D4×C127—28no—
107D4×C127—24no—
108D4×C127—24no—
109D4×C127—24no—
110D4×C127—24no—
111D4×C127—22no—
112D4×C127—22no—
113D4×C127C2×D414yes—
114C2×C4×C127—24no—
115C2×C4×C127—28no—
116C2×C4×C127—22no—
117C2×C4×C127—24no—
118C2×C4×C127—22no—
119C2×C4×C127—22no—
120C2×C4×C127—24no—
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2×D411yescharacteristic
123C2×C4×C127C2×D412yes—
124C2×C4×C127C2×D412yes—
125C2×C4×C127C2411yescharacteristic
126C22×C127—22no—
127C22×C127—24no—
128C22×C127—41no—
129C22×C127—42no—
130C22×C127—82no—
131C22×C127—84no—
132C22×C127#12012yes—
133C22×C127C22×D411yescharacteristic
134C2×D127—42no—
135C2×D127—41no—
136C4×C127—24no—
137C4×C127—22no—
138C4×C127#12011yescharacteristic
139C4×C127#12011yescharacteristic
140C4×C127C22×D412yes—
141Dic127—22no—
142Dic127—41no—
143Dic127—41no—
144C2×C127—21no—
145C2×C127—22no—
146C2×C127—42no—
147C2×C127—44no—
148C2×C127C2×#12011yescharacteristic
149D127—81no—
150#3018—1271nomaximal, Sylow
151C127#301811yescharacteristic, complemented, Sylow
152#585D12711yescharacteristic, complemented
153#575—1272no—
154#532—1271no—
155#456—1272no—
156#455—1272no—
157#454—1272no—
158#453—1272no—
159C2×#104—1272no—
160#413—1271no—
1612−1+4—22no—
1622−1+4C2×D12712yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D12712yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—1271no—
169C2×#41C2×D12712yes—
170C2×#41C2×D12711yescharacteristic
171#137—1274no—
172#137—2542no—
173AGL(1,ℤ/8)—1274no—
174AGL(1,ℤ/8)—2542no—
175#135—2544no—
176C2×M16—1272no—
177C4⋊Q8—1271no—
178#127—1271no—
179#124—1271no—
180#124—1271no—
181#121—2542no—
182C4×D4—2542no—
183C2×C42—1271no—
184#104—1274no—
185#104—1274no—
186#101—1272no—
187#100—1272no—
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—2542no—
199#41—2542no—
200#41C22⋊D12714yescomplemented
201#41C22×D12714yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—1271no—
205C2×Q8C22⋊D12712yescomplemented
206C2×Q8C22×D12711yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—1271no—
213C2×D4—2542no—
214C2×D4—2541no—
215C2×D4C22⋊D12712yes—
216C2×D4C22⋊D12712yescomplemented
217C2×D4C22×D12711yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—2541no—
221C22×C4—2541no—
222C22×C4C22⋊D12712yes—
223C22×C4C22×D12711yescharacteristic
224Q16—2544no—
225SD16—2544no—
226SD16—2544no—
227D8—2544no—
228M16—1274no—
229M16—2542no—
230C2×C8—2542no—
231C4⋊C4—2542no—
232C22⋊C4—2542no—
233C22⋊C4—2542no—
234C42—1271no—
235C42—1271no—
236C42—2541no—
237C23—24no—
238C23—22no—
239C23—2541no—
240C23C2×C22⋊D12712yes—
241C23C2×C22⋊D12711yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—2542no—
246Q8C2×C22⋊D12714yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—5082no—
255D4—5084no—
256D4—2542no—
257D4C2×C22⋊D12714yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—5082no—
266C2×C4—5081no—
267C2×C4—5081no—
268C2×C4—2541no—
269C2×C4—2541no—
270C2×C4—2541no—
271C2×C4C2×C22⋊D12712yes—
272C2×C4C2×C22⋊D12712yes—
273C2×C4C2×C22⋊D12712yes—
274C2×C4C2×C22⋊D12711yescharacteristic
275C2×C4C23×D12711yescharacteristic
276C8—2544no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—5082no—
284C22—5081no—
285C22#1686117812yes—
286C22C22×C22⋊D12711yescharacteristic
287C4—24no—
288C4—22no—
289C4—5081no—
290C4—5081no—
291C4—2542no—
292C4#1686117811yescharacteristic
293C4#1686117811yescharacteristic
294C4C22×C22⋊D12712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—10161no—
300C2C2×#1686117811yescharacteristic
subgroupG/H
centerC4#16861178
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#585×C127C2
socleC2×C127C2×#120
hypercenter#585D127
hypocenterC127#3018
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #22550674C2semidirect
2. #22550422C2semidirect
3. #22550059C2semidirect
4. #22550054C2semidirect
5. #22550049C2semidirect
6. #22550044C2semidirect
7. C2×#16861136C2semidirect
8. #22549860C2semidirect
9. #585×C127C2semidirect
10. C2×#14523248C22semidirect
12. #16861239C22semidirect
14. #16861234C22semidirect
16. C2×#14523232C22semidirect
17. #16861207C22semidirect
18. #16861205C22semidirect
19. #16861198C22semidirect
20. #16861196C22semidirect
23. C2×C4×Dic127C22semidirect
24. #16861136C22semidirect
25. #16861136C22semidirect
26. #16861128C22semidirect
27. #16861126C22semidirect
68. #41×C127D4semidirect
72. C2×Q8×C127D4semidirect
80. C2×D4×C127D4semidirect
151. C127#3018semidirect
152. #585D127semidirect
200. #41C22⋊D127semidirect
205. C2×Q8C22⋊D127semidirect
216. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225504222
#225500592
#225500542
#225500492
#225500442
C2×#168611362
#225498602
#585×C1272
#3018127