Finite Group Databasebeta

← #30648358

Subgroups — #30648358

Order 16256.

496 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16861240C211yesmaximal, characteristic, complemented
2C2×M16×D127C211yesmaximal, characteristic, complemented
3#22550388C211yesmaximal, characteristic
4C2×#16861185C211yesmaximal, characteristic, complemented
5#22550133C212yesmaximal, complemented
6#22550125C212yesmaximal, complemented
7#22550116C212yesmaximal, complemented
8#22550108C212yesmaximal, complemented
9#22549919C211yesmaximal, characteristic, complemented
10#22549878C211yesmaximal, characteristic, complemented
11#419×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×C4×D127C2211yescharacteristic
14#16861240—24no—
15C2×#14523238C2212yescomplemented
16C2×#14523235C2212yescomplemented
17C2×#14523232C2211yescharacteristic
18C2×#14523231C2212yes—
19M16×D127—24no—
20C2×C8×D127C2212yes—
21#16861193—21no—
22#16861189C2211yescharacteristic
23#16861189C2211yescharacteristic
24#16861185—21no—
25#16861185—21no—
26#16861185C2212yes—
27#16861185C2212yes—
28#16861184C2211yescharacteristic
29#16861184C2211yescharacteristic
30#16861179—21no—
31C4×D508—21no—
32C4×D508—21no—
33C2×C4⋊Dic127C2211yescharacteristic
34C2×Dic127⋊C4—21no—
35C2×(C2×D127)⋊C4—21no—
36#16861142C2212yescomplemented
37#16861140C2212yescomplemented
38#16861134C2212yescomplemented
39#16861131C2212yescomplemented
40C22×Q8×C127C2211yescharacteristic
41C2×M16×C127C2211yescharacteristic
42#117×C127C2211yescharacteristic
43#103×C127C2212yescomplemented
44#103×C127C2212yescomplemented
45C23×D127D411yescharacteristic
46#14523248—22no—
47#14523248—22no—
48C2×Dic254C2311yescharacteristic
49C2×D508—21no—
50C2×D508—21no—
51C2×D508C2311yescharacteristic
52C2×C22⋊D127—21no—
53C22×Dic127—21no—
54C22×Dic127D411yescharacteristic
55C2×C4×D127—22no—
56C2×C4×D127—21no—
57C2×C4×D127—22no—
58C2×C4×D127D411yescharacteristic
59C2×C4×D127D411yescharacteristic
60C2×C4×D127C2311yescharacteristic
61C2×C4×D127C2311yescharacteristic
62#14523238—24no—
63#14523235—24no—
64#14523232D414yescomplemented
65#14523231—24no—
66C2×C127⋊C8C2312yes—
67C8×D127—24no—
68C4⋊Dic127—21no—
69C4⋊Dic127—21no—
70C4⋊Dic127—21no—
71C4⋊Dic127C2312yes—
72Dic127⋊C4—22no—
73Dic127⋊C4—22no—
74(C2×D127)⋊C4—22no—
75(C2×D127)⋊C4—22no—
76(C2×D127)⋊C4—21no—
77(C2×D127)⋊C4—21no—
78(C2×D127)⋊C4—21no—
79(C2×D127)⋊C4—21no—
80C2×Q8×C127—21no—
81C2×Q8×C127—22no—
82C2×Q8×C127—22no—
83C2×Q8×C127C2312yes—
84C22×C4×C127—21no—
85C22×C4×C127C2311yescharacteristic
86M16×C127—22no—
87C2×C8×C127C2312yes—
88C4⋊C4×C127C2311yescharacteristic
89C4⋊C4×C127C2311yescharacteristic
90C22⋊C4×C127—21no—
91C42×C127—21no—
92C22×D127—22no—
93C22×D127—22no—
94C22×D127—21no—
95C22×D127—41no—
96C22×D127C2×D411yescharacteristic
97C22×D127C2×D411yescharacteristic
98Dic254—21no—
99Dic254—21no—
100D508—21no—
101D508—21no—
102D508—42no—
103D508—42no—
104C22⋊D127—42no—
105C2×Dic127—22no—
106C2×Dic127—22no—
107C2×Dic127—21no—
108C2×Dic127—42no—
109C2×Dic127C2×D411yescharacteristic
110C2×Dic127C2×D411yescharacteristic
111C4×D127—22no—
112C4×D127—22no—
113C4×D127—21no—
114C4×D127—21no—
115C4×D127—21no—
116C4×D127—21no—
117C4×D127—42no—
118C127⋊C8C2×D414yes—
119C23×C127C2×D411yescharacteristic
120Q8×C127—22no—
121Q8×C127—22no—
122Q8×C127—42no—
123C2×C4×C127—21no—
124C2×C4×C127—21no—
125C2×C4×C127—22no—
126C2×C4×C127—42no—
127C2×C4×C127C2×D411yescharacteristic
128C2×C4×C127C2×D412yes—
129C2×C4×C127#4112yes—
130C2×C4×C127C2411yescharacteristic
131C8×C127—22no—
132C22×C127—21no—
133C22×C127—21no—
134C22×C127#12112yes—
135C22×C127C22×D411yescharacteristic
136C2×D127—21no—
137C2×D127—21no—
138C2×D127—21no—
139C2×D127—21no—
140C2×D127—21no—
141C2×D127—44no—
142C2×D127—41no—
143C2×D127—41no—
144C2×D127—41no—
145C2×D127#13712yescomplemented
146C4×C127—42no—
147C4×C127—41no—
148C4×C127—41no—
149C4×C127#12112yes—
150C4×C127C22×D411yescharacteristic
151C4×C127C2×#4111yescharacteristic
152Dic127—21no—
153Dic127—21no—
154Dic127—41no—
155Dic127—42no—
156C2×C127—22no—
157C2×C127C2×#12111yescharacteristic
158C2×C127C2×#13711yescharacteristic
159C2×C127#57511yescharacteristic
160D127—22no—
161D127—41no—
162D127—81no—
163#3056—1271nomaximal, Sylow
164C127#305611yescharacteristic, complemented, Sylow
165C2×#137—1271no—
166C22×M16—1271no—
167#525—1271no—
168C2×C22⋊Q8—1271no—
169#470—1272no—
170#468—1272no—
171#467—1272no—
172#465—1272no—
173#428—1271no—
174#419D12711yescharacteristic, complemented
175#417—1271no—
176C2×#41—1271no—
177C22×Q8C2×D12711yescharacteristic
178C23×C4—1271no—
179#137—2544no—
180C2×Q16—1272no—
181C2×SD16—1272no—
182C2×M16—1271no—
183C2×M16—1272no—
184C2×M16—2544no—
185C2×M16C2×D12711yescharacteristic
186C22×C8—1272no—
187#123—2541no—
188C22⋊Q8—1272no—
189C22⋊Q8—1272no—
190C22⋊Q8—1271no—
191C22⋊Q8—1271no—
192C22⋊Q8—2541no—
193C22⋊Q8—2541no—
194#121—1271no—
195#121—1271no—
196#120—2541no—
197C4×D4—2541no—
198C4×D4—2541no—
199#117C2×D12711yescharacteristic
200C2×C4⋊C4—1271no—
201C2×C4⋊C4—2541no—
202C2×C22⋊C4—2541no—
203#107—1272no—
204#106—1272no—
205#103—1272no—
206#103C2×D12712yescomplemented
207#103C2×D12712yescomplemented
208#102—1272no—
209C24—1271no—
210#41—2542no—
211#41—2542no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—1271no—
216C2×Q8C22×D12712yes—
217C2×D4—1271no—
218C2×D4—2541no—
219C2×D4—2541no—
220C2×D4—2541no—
221C22×C4—21no—
222C22×C4—1271no—
223C22×C4—1271no—
224C22×C4—1271no—
225C22×C4—1271no—
226C22×C4—1271no—
227C22×C4—2541no—
228C22×C4—2541no—
229C22×C4—2542no—
230C22×C4—2542no—
231C22×C4C22×D12711yescharacteristic
232Q16—2544no—
233SD16—2544no—
234M16—22no—
235M16—1274no—
236M16—2544no—
237C2×C8—1272no—
238C2×C8—2544no—
239C2×C8C22×D12712yes—
240C4⋊C4—1272no—
241C4⋊C4—2542no—
242C4⋊C4—2542no—
243C4⋊C4—2541no—
244C4⋊C4—2541no—
245C4⋊C4—2541no—
246C4⋊C4C22×D12711yescharacteristic
247C4⋊C4C22×D12711yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—2541no—
250C22⋊C4—2541no—
251C22⋊C4—2542no—
252C22⋊C4—2542no—
253C22⋊C4—2541no—
254C22⋊C4—2541no—
255C42—21no—
256C23—1271no—
257C23—1271no—
258C23—5081no—
259C23—2542no—
260C23—2541no—
261C23—2542no—
262C23D4×D12711yescharacteristic
263Q8—22no—
264Q8—22no—
265Q8—42no—
266Q8—2541no—
267Q8—2541no—
268D4—5082no—
269D4—5082no—
270D4—5082no—
271D4—2541no—
272D4—2541no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—1271no—
278C2×C4—1271no—
279C2×C4—5082no—
280C2×C4—5082no—
281C2×C4—2542no—
282C2×C4—2541no—
283C2×C4—2542no—
284C2×C4—2541no—
285C2×C4—2541no—
286C2×C4—2541no—
287C2×C4—2541no—
288C2×C4—2542no—
289C2×C4—2542no—
290C2×C4D4×D12712yes—
291C2×C4D4×D12711yescharacteristic
292C2×C4#1452324812yes—
293C2×C4C23×D12711yescharacteristic
294C8—22no—
295C8—1274no—
296C22—21no—
297C22—21no—
298C22—1272no—
299C22—5084no—
300C22—5081no—
301C22—5081no—
302C22—5081no—
303C22—2541no—
304C22—2541no—
305C22—2541no—
306C22—2541no—
307C22—2541no—
308C22#1686118212yes—
309C22C2×D4×D12711yescharacteristic
310C4—42no—
311C4—41no—
312C4—41no—
313C4—5081no—
314C4—5082no—
315C4—2541no—
316C4—2541no—
317C4#1686118212yes—
318C4C2×D4×D12711yescharacteristic
319C4C2×#1452324811yescharacteristic
320C2—22no—
321C2—10161no—
322C2—5081no—
323C2—2542no—
324C2C2×#1686118211yescharacteristic
325C2#137×D12711yescharacteristic
326C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
prime radical#22550388C2
hypercenter#419D127
hypocenterC127#3056
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
2. C2×M16×D127C2semidirect
4. C2×#16861185C2semidirect
5. #22550133C2semidirect
6. #22550125C2semidirect
7. #22550116C2semidirect
8. #22550108C2semidirect
9. #22549919C2semidirect
10. #22549878C2semidirect
11. #419×C127C2semidirect
15. C2×#14523238C22semidirect
16. C2×#14523235C22semidirect
36. #16861142C22semidirect
37. #16861140C22semidirect
38. #16861134C22semidirect
39. #16861131C22semidirect
43. #103×C127C22semidirect
44. #103×C127C22semidirect
64. #14523232D4semidirect
145. C2×D127#137semidirect
164. C127#3056semidirect
174. #419D127semidirect
206. #103C2×D127semidirect
207. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×M16×D1272
#225503882
C2×#168611852
#225501332
#225501252
#225501162
#225501082
#225499192
#225498782
#419×C1272
#3056127