Finite Group Databasebeta

← #30648129

Subgroups — #30648129

Order 16256.

800 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16861254C211yesmaximal, characteristic, complemented
2#22550438C212yesmaximal, complemented
3#22550435C211yesmaximal, characteristic, complemented
4#22550067C214yesmaximal, complemented
5#22550064C214yesmaximal, complemented
6C2×#16861123C212yesmaximal, complemented
7C2×#99×C127C211yesmaximal, characteristic, complemented
8#16861254—24no—
9#16861254—22no—
10#16861254C2214yes—
11C2×#14523248—21no—
12C2×#14523247—22no—
13C22×C22⋊D127—21no—
14C22×C22⋊D127C2212yes—
15C22×C22⋊D127C2211yescharacteristic, complemented
16C2×D4×D127—22no—
17#16861193—22no—
18#16861193C2214yescomplemented
19#16861192—21no—
20#16861190C2212yescomplemented
21#16861183—24no—
22#16861181—22no—
23#16861179—22no—
24#16861179C2214yescomplemented
25#16861178—21no—
26#16861177C2212yescomplemented
27C2×C22⋊Dic127C2211yescharacteristic, complemented
28#16861123C2218yescomplemented
29C22×D4×C127C2211yescharacteristic
30C2×C22⋊C4×C127C2212yescomplemented
31#99×C127C2214yescomplemented
32C23×D127—21no—
33C23×D127D412yescomplemented
34C23×D127C2311yescharacteristic, complemented
35#14523248—24no—
36#14523248—24no—
37#14523247—28no—
38#14523247—28no—
39C2×Dic254—21no—
40C2×D508—21no—
41C2×D508—24no—
42C2×C22⋊D127—22no—
43C2×C22⋊D127—24no—
44C2×C22⋊D127—22no—
45C2×C22⋊D127—22no—
46C2×C22⋊D127—24no—
47C2×C22⋊D127—22no—
48C2×C22⋊D127—22no—
49C2×C22⋊D127—21no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127—21no—
52C2×C22⋊D127—22no—
53C2×C22⋊D127—22no—
54C2×C22⋊D127D414yescomplemented
55C2×C22⋊D127C2314yes—
56D4×D127—28no—
57D4×D127—28no—
58C22×Dic127—21no—
59C22×Dic127—21no—
60C22×Dic127D412yescomplemented
61C22×Dic127C2311yescharacteristic, complemented
62C2×C4×D127—22no—
63C4⋊Dic127—24no—
64Dic127⋊C4—22no—
65C22⋊Dic127—22no—
66C22⋊Dic127—21no—
67C22⋊Dic127C2312yescomplemented
68(C2×D127)⋊C4—24no—
69(C2×D127)⋊C4—24no—
70(C2×D127)⋊C4—22no—
71C24×C127D411yescharacteristic, complemented
72C24×C127C2311yescharacteristic
73C2×D4×C127—22no—
74C2×D4×C127—22no—
75C2×D4×C127D412yescomplemented
76C2×D4×C127C2312yes—
77C22×C4×C127—22no—
78C22×C4×C127D411yescharacteristic, complemented
79C22⋊C4×C127—22no—
80C22⋊C4×C127C2314yescomplemented
81C22×D127—22no—
82C22×D127—24no—
83C22×D127—22no—
84C22×D127—21no—
85C22×D127—22no—
86C22×D127—81no—
87C22×D127—82no—
88C22×D127—82no—
89C22×D127C2×D414yes—
90Dic254—24no—
91D508—24no—
92D508—84no—
93C22⋊D127—28no—
94C22⋊D127—24no—
95C22⋊D127—24no—
96C22⋊D127—24no—
97C22⋊D127—24no—
98C22⋊D127—41no—
99C22⋊D127—41no—
100C22⋊D127—42no—
101C22⋊D127—82no—
102C2×Dic127—24no—
103C2×Dic127—22no—
104C2×Dic127—22no—
105C2×Dic127—22no—
106C2×Dic127—22no—
107C2×Dic127—21no—
108C2×Dic127—81no—
109C2×Dic127C2×D414yes—
110C4×D127—28no—
111C23×C127—21no—
112C23×C127—21no—
113C23×C127—81no—
114C23×C127—81no—
115C23×C127C2×D412yes—
116C23×C127C2×D412yes—
117C23×C127C2×D412yes—
118C23×C127C2×D411yescharacteristic
119C23×C127C2×D411yescharacteristic
120C23×C127C2411yescharacteristic
121D4×C127—24no—
122D4×C127—24no—
123C2×C4×C127—21no—
124C2×C4×C127—21no—
125C2×C4×C127—24no—
126C2×C4×C127—82no—
127C2×C4×C127C2×D412yes—
128C22×C127—22no—
129C22×C127—42no—
130C22×C127—41no—
131C22×C127—41no—
132C22×C127—42no—
133C22×C127—41no—
134C22×C127—42no—
135C22×C127—42no—
136C22×C127—81no—
137C22×C127#12011yescharacteristic
138C22×C127#12011yescharacteristic
139C22×C127#12012yes—
140C22×C127C22×D412yes—
141C22×C127C22×D411yescharacteristic
142C2×D127—24no—
143C2×D127—44no—
144C2×D127—44no—
145C2×D127—42no—
146C2×D127—44no—
147C2×D127—41no—
148C2×D127—82no—
149C2×D127—84no—
150C4×C127—22no—
151C4×C127—44no—
152Dic127—24no—
153Dic127—42no—
154Dic127—41no—
155C2×C127—21no—
156C2×C127—22no—
157C2×C127—21no—
158C2×C127—22no—
159C2×C127—21no—
160C2×C127—42no—
161C2×C127C2×#12011yescharacteristic
162D127—44no—
163D127—81no—
164#3026—1271nomaximal, Sylow
165C127#302611yescharacteristic, complemented, Sylow
166C2×H(5,𝔽2)—1271no—
167#534—1271no—
168#534—1272no—
169#458—1274no—
170B(4,𝔽2)—1274no—
171C2×#99—1272no—
172C2×#99D12711yescharacteristic, complemented
173H(5,𝔽2)—1274no—
174H(5,𝔽2)—2544no—
175H(5,𝔽2)—2542no—
176C2×#41—2542no—
177C2×#41—2541no—
178C22×D4—1272no—
179C22×D4—1271no—
180C22×D4—2542no—
181C22×D4—2541no—
182C22×D4C2×D12711yescharacteristic
183#123—1272no—
184#123—1274no—
185#123—2542no—
186#123—2541no—
187#121—2544no—
188#121—2542no—
189#120—1272no—
190#120—1274no—
191#120—2542no—
192#120—2541no—
193C2×C22⋊C4—1271no—
194C2×C22⋊C4C2×D12712yescomplemented
195#99—1278no—
196#99C2×D12714yescomplemented
197C24—1272no—
198C24—1271no—
199C24—2541no—
200C24D50811yescharacteristic, complemented
201C24C22×D12711yescharacteristic
202#41—2544no—
203#41—2548no—
204#41—2548no—
205#41—2544no—
206C2×Q8—2541no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—1274no—
210C2×D4—1274no—
211C2×D4—2544no—
212C2×D4—2541no—
213C2×D4—2548no—
214C2×D4—2548no—
215C2×D4—2542no—
216C2×D4—2542no—
217C2×D4—2542no—
218C2×D4—2541no—
219C2×D4—2541no—
220C2×D4—2542no—
221C2×D4—2544no—
222C2×D4—2542no—
223C2×D4—2542no—
224C2×D4—2542no—
225C2×D4—2542no—
226C2×D4—2544no—
227C2×D4D50812yescomplemented
228C2×D4C22×D12712yes—
229C22×C4—22no—
230C22×C4—1272no—
231C22×C4—1271no—
232C22×C4—2542no—
233C22×C4—2541no—
234C22×C4—2541no—
235C22×C4D50811yescharacteristic, complemented
236C4⋊C4—2544no—
237C4⋊C4—2542no—
238C22⋊C4—22no—
239C22⋊C4—1272no—
240C22⋊C4—2544no—
241C22⋊C4—2544no—
242C22⋊C4—2541no—
243C22⋊C4—2542no—
244C22⋊C4—2542no—
245C22⋊C4C22×D12714yescomplemented
246C23—21no—
247C23—21no—
248C23—81no—
249C23—81no—
250C23—1274no—
251C23—10162no—
252C23—10162no—
253C23—10161no—
254C23—2544no—
255C23—2542no—
256C23—2542no—
257C23—2541no—
258C23—2542no—
259C23D4×D12712yes—
260C23D4×D12712yes—
261C23C2×D50812yes—
262C23C2×D50811yescharacteristic
263C23C2×D50811yescharacteristic
264C23C23×D12711yescharacteristic
265Q8—2544no—
266D4—24no—
267D4—24no—
268D4—10164no—
269D4—10162no—
270D4—5081no—
271D4—5081no—
272D4—5082no—
273D4—2544no—
274D4—2548no—
275D4—2544no—
276D4—2544no—
277D4—2544no—
278D4—2544no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—82no—
283C2×C4—1274no—
284C2×C4—10161no—
285C2×C4—2548no—
286C2×C4—2542no—
287C2×C4—2542no—
288C2×C4—2541no—
289C2×C4—2544no—
290C2×C4—2542no—
291C2×C4—2542no—
292C2×C4C2×D50812yes—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—10162no—
303C22—10164no—
304C22—5084no—
305C22—5082no—
306C22—5084no—
307C22—5084no—
308C22—5081no—
309C22—2544no—
310C22#1686117911yescharacteristic
311C22#1686117911yescharacteristic
312C22#1686117912yes—
313C22C2×D4×D12712yes—
314C22C22×D50811yescharacteristic
315C4—22no—
316C4—44no—
317C4—5082no—
318C4—5081no—
319C4—2544no—
320C2—22no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—10161no—
327C2—5084no—
328C2C2×#1686117911yescharacteristic
subgroupG/H
centerC2C2×#16861179
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC2×#99×C127C2
socleC2×C127C2×#120
hypercenterC2×#99D127
hypocenterC127#3026
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#16861254C2semidirect
2. #22550438C2semidirect
3. #22550435C2semidirect
4. #22550067C2semidirect
5. #22550064C2semidirect
6. C2×#16861123C2semidirect
7. C2×#99×C127C2semidirect
15. C22×C22⋊D127C22semidirect
18. #16861193C22semidirect
20. #16861190C22semidirect
24. #16861179C22semidirect
26. #16861177C22semidirect
27. C2×C22⋊Dic127C22semidirect
28. #16861123C22semidirect
30. C2×C22⋊C4×C127C22semidirect
31. #99×C127C22semidirect
33. C23×D127D4semidirect
34. C23×D127C23semidirect
54. C2×C22⋊D127D4semidirect
60. C22×Dic127D4semidirect
61. C22×Dic127C23semidirect
67. C22⋊Dic127C23semidirect
71. C24×C127D4semidirect
75. C2×D4×C127D4semidirect
78. C22×C4×C127D4semidirect
80. C22⋊C4×C127C23semidirect
165. C127#3026semidirect
172. C2×#99D127semidirect
194. C2×C22⋊C4C2×D127semidirect
196. #99C2×D127semidirect
200. C24D508semidirect
227. C2×D4D508semidirect
235. C22×C4D508semidirect
245. C22⋊C4C22×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
#225504382
#225504352
#225500672
#225500642
C2×#168611232
C2×#99×C1272
#3026127