Finite Group Databasebeta

← #30648361

Subgroups — #30648361

Order 16256.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D127C211yesmaximal, characteristic, complemented
3#22550388C211yesmaximal, characteristic
4C2×#16861189C211yesmaximal, characteristic, complemented
5#22550135C212yesmaximal, complemented
6#22550123C212yesmaximal, complemented
7#22550118C212yesmaximal, complemented
8#22550106C212yesmaximal, complemented
9#22549884C211yesmaximal, characteristic, complemented
10#22549875C211yesmaximal, characteristic, complemented
11#428×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×Dic254C2211yescharacteristic
14C22×C4×D127C2211yescharacteristic
15#16861242—24no—
16C2×Dic508C2212yescomplemented
17C2×#14523237C2212yescomplemented
18C2×#14523232C2211yescharacteristic
19C2×#14523231C2212yes—
20M16×D127—24no—
21C2×C8×D127C2212yes—
22#16861193—21no—
23#16861189—21no—
24#16861189—21no—
25#16861189C2212yes—
26#16861189C2211yescharacteristic
27#16861189C2212yes—
28#16861189C2211yescharacteristic
29#16861184C2211yescharacteristic
30#16861184C2211yescharacteristic
31#16861179—21no—
32C4×D508—21no—
33C4×D508—21no—
34C2×C4⋊Dic127—21no—
35C2×(C2×D127)⋊C4—21no—
36#16861132C2212yescomplemented
37#16861132C2212yescomplemented
38#16861132C2212yescomplemented
39#16861129C2212yescomplemented
40C2×M16×C127C2211yescharacteristic
41#117×C127C2211yescharacteristic
42C2×C4⋊C4×C127C2211yescharacteristic
43#107×C127C2212yescomplemented
44#106×C127C2212yescomplemented
45C23×D127D411yescharacteristic
46#14523248—22no—
47#14523248—22no—
48C2×Dic254—21no—
49C2×Dic254—22no—
50C2×Dic254—22no—
51C2×Dic254C2312yes—
52C2×Dic254C2311yescharacteristic
53C2×D508—21no—
54C2×D508—21no—
55C2×D508C2311yescharacteristic
56C2×C22⋊D127—21no—
57C22×Dic127—21no—
58C22×Dic127D411yescharacteristic
59C2×C4×D127—22no—
60C2×C4×D127—21no—
61C2×C4×D127—22no—
62C2×C4×D127D411yescharacteristic
63C2×C4×D127D411yescharacteristic
64C2×C4×D127C2311yescharacteristic
65C2×C4×D127C2311yescharacteristic
66Dic508—24no—
67#14523237—24no—
68#14523232—22no—
69#14523231—24no—
70C2×C127⋊C8C2312yes—
71C8×D127—24no—
72C4⋊Dic127—22no—
73C4⋊Dic127—22no—
74C4⋊Dic127—21no—
75C4⋊Dic127—21no—
76(C2×D127)⋊C4—22no—
77(C2×D127)⋊C4—22no—
78(C2×D127)⋊C4—21no—
79(C2×D127)⋊C4—21no—
80(C2×D127)⋊C4—21no—
81(C2×D127)⋊C4—21no—
82C22×C4×C127—21no—
83C22×C4×C127C2311yescharacteristic
84M16×C127D414yescomplemented
85C2×C8×C127C2312yes—
86C4⋊C4×C127—21no—
87C4⋊C4×C127C2311yescharacteristic
88C4⋊C4×C127C2311yescharacteristic
89C4⋊C4×C127C2312yes—
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—22no—
99Dic254—21no—
100Dic254—22no—
101Dic254—21no—
102Dic254—42no—
103D508—21no—
104D508—21no—
105D508—42no—
106D508—42no—
107C22⋊D127—42no—
108C2×Dic127—22no—
109C2×Dic127—21no—
110C2×Dic127—22no—
111C2×Dic127—42no—
112C2×Dic127C2×D411yescharacteristic
113C2×Dic127C2×D411yescharacteristic
114C4×D127—22no—
115C4×D127—22no—
116C4×D127—21no—
117C4×D127—21no—
118C4×D127—21no—
119C4×D127—21no—
120C4×D127—42no—
121C127⋊C8—22no—
122C23×C127C2×D411yescharacteristic
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×C127C2×D414yes—
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—41no—
147C4×C127—42no—
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#428D12711yescharacteristic, complemented
174#419—1271no—
175#417—1271no—
176C2×#41—1271no—
177C22×Q8—1271no—
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—2541no—
201C2×C4⋊C4C2×D12711yescharacteristic
202C2×C22⋊C4—2541no—
203#107C2×D12712yescomplemented
204#106C2×D12712yescomplemented
205#103—1272no—
206#103—1272no—
207#103—1272no—
208#102—1272no—
209C24—1271no—
210#41—2542no—
211#41—2542no—
212C2×Q8—1272no—
213C2×Q8—1271no—
214C2×Q8—2542no—
215C2×Q8—2542no—
216C2×Q8—2541no—
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—2542no—
235M16—2544no—
236M16D50814yescomplemented
237C2×C8—1272no—
238C2×C8—2544no—
239C2×C8C22×D12712yes—
240C4⋊C4—21no—
241C4⋊C4—2542no—
242C4⋊C4—2542no—
243C4⋊C4—2541no—
244C4⋊C4—2541no—
245C4⋊C4C22×D12712yes—
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—5082no—
264Q8—2542no—
265Q8—2541no—
266Q8—2542no—
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—2541no—
282C2×C4—2542no—
283C2×C4—2542no—
284C2×C4—2542no—
285C2×C4—2542no—
286C2×C4—2541no—
287C2×C4—2541no—
288C2×C4—2541no—
289C2×C4—2541no—
290C2×C4D4×D12711yescharacteristic
291C2×C4C2×D50812yes—
292C2×C4#1452324912yes—
293C2×C4C23×D12711yescharacteristic
294C8—2542no—
295C8C2×D50814yes—
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#1686118412yes—
309C22C2×D4×D12711yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—5081no—
314C4—5082no—
315C4—2541no—
316C4—2541no—
317C4#1686118412yes—
318C4C22×D50811yescharacteristic
319C4C2×#1452324911yescharacteristic
320C2—22no—
321C2—10161no—
322C2—5081no—
323C2—2542no—
324C2C2×#1686118411yescharacteristic
325C2#137×D12711yescharacteristic
326C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
prime radical#22550388C2
hypercenter#428D127
hypocenterC127#3056
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
2. C2×M16×D127C2semidirect
4. C2×#16861189C2semidirect
5. #22550135C2semidirect
6. #22550123C2semidirect
7. #22550118C2semidirect
8. #22550106C2semidirect
9. #22549884C2semidirect
10. #22549875C2semidirect
11. #428×C127C2semidirect
16. C2×Dic508C22semidirect
17. C2×#14523237C22semidirect
36. #16861132C22semidirect
37. #16861132C22semidirect
38. #16861132C22semidirect
39. #16861129C22semidirect
43. #107×C127C22semidirect
44. #106×C127C22semidirect
84. M16×C127D4semidirect
145. C2×D127#137semidirect
164. C127#3056semidirect
173. #428D127semidirect
203. #107C2×D127semidirect
204. #106C2×D127semidirect
236. M16D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C2×M16×D1272
#225503882
C2×#168611892
#225501352
#225501232
#225501182
#225501062
#225498842
#225498752
#428×C1272
#3056127