Finite Group Databasebeta

← #30648310

Subgroups — #30648310

Order 16256.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16861229C211yesmaximal, characteristic, complemented
2C22×C8×D127C211yesmaximal, characteristic, complemented
3#22550388C211yesmaximal, characteristic, complemented
4#22550383C211yesmaximal, characteristic, complemented
5#22550116C212yesmaximal, complemented
6#22550111C212yesmaximal, complemented
7#22550108C212yesmaximal, complemented
8#22550104C212yesmaximal, complemented
9#22549912C211yesmaximal, characteristic, complemented
10#22549873C211yesmaximal, characteristic, complemented
11#416×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×C4×D127C2211yescharacteristic
14#16861229—24no—
15C2×#14523238C2211yescharacteristic, complemented
16C2×#14523236C2211yescharacteristic, complemented
17C2×#14523235C2211yescharacteristic, complemented
18C2×#14523233C2211yescharacteristic, complemented
19C22×C127⋊C8C2211yescharacteristic
20C2×C8×D127—24no—
21C2×C8×D127C2212yes—
22C2×C8×D127C2212yes—
23#16861193—21no—
24#16861190—21no—
25#16861189C2212yes—
26#16861185C2212yes—
27#16861184C2212yes—
28#16861180C2212yes—
29#16861179—21no—
30#16861177—21no—
31C4×D508—22no—
32C4×C22⋊D127—22no—
33#16861168C2211yescharacteristic
34#16861142C2212yescomplemented
35#16861140C2212yescomplemented
36#16861134C2212yescomplemented
37#16861131C2212yescomplemented
38C2×#41×C127C2211yescharacteristic
39C22×C8×C127C2211yescharacteristic
40#117×C127C2211yescharacteristic
41#103×C127C2212yescomplemented
42#102×C127C2212yescomplemented
43C23×D127D411yescharacteristic
44#14523248—22no—
45#14523248—22no—
46C2×Dic254C2311yescharacteristic
47C2×D508—22no—
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—22no—
50C2×C22⋊D127—21no—
51C22×Dic127D411yescharacteristic
52C2×C4×D127—22no—
53C2×C4×D127—21no—
54C2×C4×D127—22no—
55C2×C4×D127D412yes—
56C2×C4×D127C2312yes—
57#14523238—22no—
58#14523236—22no—
59#14523235—22no—
60#14523233—22no—
61C2×C127⋊C8D414yescomplemented
62C2×C127⋊C8C2311yescharacteristic
63C2×C127⋊C8C2311yescharacteristic
64C8×D127—24no—
65C8×D127—24no—
66C4⋊Dic127—22no—
67C4⋊Dic127C2312yes—
68Dic127⋊C4—22no—
69C22⋊Dic127—22no—
70C22⋊Dic127—21no—
71(C2×D127)⋊C4—22no—
72(C2×D127)⋊C4—22no—
73(C2×D127)⋊C4—22no—
74C4×Dic127—21no—
75#41×C127—22no—
76#41×C127—22no—
77C2×Q8×C127C2311yescharacteristic
78C2×D4×C127—21no—
79C2×D4×C127C2311yescharacteristic
80C22×C4×C127—21no—
81C22×C4×C127C2311yescharacteristic
82C2×C8×C127—22no—
83C2×C8×C127C2312yes—
84C4⋊C4×C127C2312yes—
85C22⋊C4×C127—21no—
86C42×C127—21no—
87C22×D127—22no—
88C22×D127—24no—
89C22×D127—21no—
90C22×D127C2×D412yes—
91Dic254—21no—
92Dic254—21no—
93D508—21no—
94D508—21no—
95D508—44no—
96C22⋊D127—42no—
97C22⋊D127—44no—
98C2×Dic127—22no—
99C2×Dic127—22no—
100C2×Dic127—21no—
101C2×Dic127C2×D412yes—
102C4×D127—24no—
103C4×D127—22no—
104C4×D127—22no—
105C4×D127—42no—
106C127⋊C8C2×D412yes—
107C127⋊C8C2×D412yes—
108C23×C127—21no—
109C23×C127C2×D411yescharacteristic
110Q8×C127—21no—
111Q8×C127—21no—
112D4×C127—21no—
113D4×C127—21no—
114D4×C127—42no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—42no—
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127#4112yes—
121C2×C4×C127C2411yescharacteristic
122C8×C127—22no—
123C22×C127—22no—
124C22×C127—41no—
125C22×C127—42no—
126C22×C127#12112yes—
127C22×C127C22×D411yescharacteristic
128C2×D127—28no—
129C2×D127—22no—
130C2×D127—22no—
131C2×D127—41no—
132C2×D127—42no—
133C2×D127#13514yescomplemented
134C4×C127—41no—
135C4×C127—42no—
136C4×C127#12112yes—
137C4×C127C22×D411yescharacteristic
138C4×C127C2×#4111yescharacteristic
139Dic127—22no—
140Dic127—41no—
141Dic127—42no—
142C2×C127—22no—
143C2×C127—81no—
144C2×C127C2×#12111yescharacteristic
145C2×C127C2×#13512yes—
146D127—24no—
147D127—81no—
148#3050—1271nomaximal, Sylow
149C127#305011yescharacteristic, complemented, Sylow
150C2×#135—1271no—
151C23×C8—1271no—
152#525—1271no—
153#525—1271no—
154#467—1272no—
155#466—1272no—
156#465—1272no—
157#465—1272no—
158#427—1271no—
159#416—1271no—
160#416D12711yescharacteristic, complemented
161C2×#41—1271no—
162C2×#41C2×D12711yescharacteristic
163C23×C4—1271no—
164#135—2544no—
165C2×Q16—1271no—
166C2×SD16—1271no—
167C2×SD16—1271no—
168C2×D8—1271no—
169C22×C8—1271no—
170C22×C8—1272no—
171C22×C8—1272no—
172C22×C8—2544no—
173C22×C8C2×D12711yescharacteristic
174#123—2541no—
175#123—2541no—
176C22⋊Q8—1272no—
177C22⋊Q8—1272no—
178#121—1272no—
179#121—1272no—
180#120—2541no—
181#120—2541no—
182C4×D4—2542no—
183C4×D4—2542no—
184#117—1271no—
185#117C2×D12711yescharacteristic
186#107—1272no—
187#106—1272no—
188#103—1272no—
189#103C2×D12712yescomplemented
190#102—1272no—
191#102C2×D12712yescomplemented
192C24—1271no—
193#41—22no—
194#41—22no—
195#41—2542no—
196#41—2542no—
197C2×Q8—1271no—
198C2×Q8C22×D12711yescharacteristic
199C2×D4—21no—
200C2×D4—1271no—
201C2×D4—2542no—
202C2×D4—2542no—
203C2×D4—2541no—
204C2×D4C22×D12711yescharacteristic
205C22×C4—21no—
206C22×C4—1271no—
207C22×C4—1272no—
208C22×C4—1272no—
209C22×C4—2541no—
210C22×C4—2542no—
211C22×C4—2542no—
212C22×C4C22×D12711yescharacteristic
213Q16—2542no—
214SD16—2542no—
215SD16—2542no—
216D8—2542no—
217C2×C8—22no—
218C2×C8—1274no—
219C2×C8—1271no—
220C2×C8—1271no—
221C2×C8—2544no—
222C2×C8—2544no—
223C2×C8C22×D12712yes—
224C4⋊C4—1272no—
225C4⋊C4—2542no—
226C4⋊C4—2542no—
227C4⋊C4C22×D12712yes—
228C22⋊C4—21no—
229C22⋊C4—2542no—
230C22⋊C4—2542no—
231C22⋊C4—2542no—
232C22⋊C4—2542no—
233C22⋊C4—2541no—
234C42—21no—
235C42—2541no—
236C23—21no—
237C23—1272no—
238C23—2544no—
239C23—2541no—
240C23—2542no—
241C23D4×D12711yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—2541no—
245Q8—2541no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—5084no—
250D4—5084no—
251D4—5082no—
252D4—2541no—
253D4—2541no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—1272no—
258C2×C4—5082no—
259C2×C4—2542no—
260C2×C4—2541no—
261C2×C4—2542no—
262C2×C4—2542no—
263C2×C4—2542no—
264C2×C4—2544no—
265C2×C4D4×D12712yes—
266C2×C4D4×D12711yescharacteristic
267C2×C4#1452324812yes—
268C2×C4C23×D12711yescharacteristic
269C8—22no—
270C8—1272no—
271C8—1272no—
272C22—22no—
273C22—41no—
274C22—42no—
275C22—1274no—
276C22—5081no—
277C22—5082no—
278C22—2548no—
279C22—2542no—
280C22—2542no—
281C22#1686118212yes—
282C22C2×D4×D12711yescharacteristic
283C4—41no—
284C4—42no—
285C4—5081no—
286C4—5082no—
287C4—2542no—
288C4#1686118212yes—
289C4C2×D4×D12711yescharacteristic
290C4C2×#1452324811yescharacteristic
291C2—22no—
292C2—81no—
293C2—10161no—
294C2—2544no—
295C2C2×#1686118211yescharacteristic
296C2#135×D12712yes—
subgroupG/H
centerC2×C4D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#416×C127C2
socleC22×C127C22×D4
hypercenter#416D127
hypocenterC127#3050
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#16861229C2semidirect
2. C22×C8×D127C2semidirect
3. #22550388C2semidirect
4. #22550383C2semidirect
5. #22550116C2semidirect
6. #22550111C2semidirect
7. #22550108C2semidirect
8. #22550104C2semidirect
9. #22549912C2semidirect
10. #22549873C2semidirect
11. #416×C127C2semidirect
15. C2×#14523238C22semidirect
16. C2×#14523236C22semidirect
17. C2×#14523235C22semidirect
18. C2×#14523233C22semidirect
34. #16861142C22semidirect
35. #16861140C22semidirect
36. #16861134C22semidirect
37. #16861131C22semidirect
41. #103×C127C22semidirect
42. #102×C127C22semidirect
61. C2×C127⋊C8D4semidirect
133. C2×D127#135semidirect
149. C127#3050semidirect
160. #416D127semidirect
189. #103C2×D127semidirect
191. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
C22×C8×D1272
#225503882
#225503832
#225501162
#225501112
#225501082
#225501042
#225499122
#225498732
#416×C1272
#3050127