Finite Group Databasebeta

← #30648419

Subgroups — #30648419

Order 16256.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16861215C211yesmaximal, characteristic, complemented
3#22550457C211yesmaximal, characteristic
4#22550450C211yesmaximal, characteristic, complemented
5#22550133C212yesmaximal, complemented
6#22550125C212yesmaximal, complemented
7#22550116C212yesmaximal, complemented
8#22550108C212yesmaximal, complemented
9#22549916C211yesmaximal, characteristic, complemented
10#22549873C211yesmaximal, characteristic, complemented
11C2×#103×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C2×#14523248C2211yescharacteristic
14#16861240—24no—
15C2×#14523238C2212yes—
16C2×#14523235C2212yes—
17#16861215—24no—
18C2×#14523232C2211yescharacteristic
19C2×#14523231C2212yes—
20C2×C8×D127C2212yes—
21#16861207—21no—
22#16861198—21no—
23#16861192—22no—
24#16861189—21no—
25#16861189C2212yes—
26#16861185C2212yes—
27#16861185C2212yes—
28#16861184—21no—
29#16861184C2212yes—
30#16861183—22no—
31#16861168C2211yescharacteristic
32#16861142C2212yes—
33#16861140C2212yes—
34#16861134C2211yescharacteristic
35#16861134C2211yescharacteristic
36#16861131C2211yescharacteristic
37#16861131C2211yescharacteristic
38C22×Q8×C127C2211yescharacteristic
39C22×C8×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#103×C127C2212yes—
42#103×C127C2212yes—
43#14523248—22no—
44#14523248—22no—
45#14523248—22no—
46#14523248—22no—
47C2×Dic254D411yescharacteristic
48C2×Dic254C2311yescharacteristic
49C2×D508—22no—
50C2×D508D411yescharacteristic
51C2×D508C2311yescharacteristic
52C2×C22⋊D127—21no—
53C2×C22⋊D127D412yes—
54C2×C4×D127—21no—
55C2×C4×D127C2312yes—
56#14523238—24no—
57#14523235—24no—
58#14523232D414yescomplemented
59#14523231—24no—
60C2×C127⋊C8C2312yes—
61C8×D127—24no—
62C4⋊Dic127—22no—
63C4⋊Dic127C2311yescharacteristic
64C4⋊Dic127C2311yescharacteristic
65Dic127⋊C4—22no—
66Dic127⋊C4—22no—
67C22⋊Dic127—21no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—22no—
72C4×Dic127—21no—
73C2×Q8×C127—21no—
74C2×Q8×C127—22no—
75C2×Q8×C127—22no—
76C2×Q8×C127C2312yes—
77C22×C4×C127—21no—
78C22×C4×C127—21no—
79C22×C4×C127C2311yescharacteristic
80C2×C8×C127—22no—
81C2×C8×C127C2311yescharacteristic
82C2×C8×C127C2311yescharacteristic
83C4⋊C4×C127—21no—
84C4⋊C4×C127C2312yes—
85C22×D127—21no—
86C22×D127C2×D412yes—
87Dic254—22no—
88Dic254—21no—
89Dic254—21no—
90D508—22no—
91D508—21no—
92D508—21no—
93D508—82no—
94C22⋊D127—24no—
95C22⋊D127—42no—
96C2×Dic127—21no—
97C2×Dic127—21no—
98C2×Dic127—21no—
99C2×Dic127C2×D412yes—
100C4×D127—22no—
101C4×D127—22no—
102C4×D127—42no—
103C127⋊C8C2×D414yes—
104C23×C127C2×D411yescharacteristic
105Q8×C127—22no—
106Q8×C127—22no—
107Q8×C127—42no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—42no—
111C2×C4×C127—81no—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D412yes—
114C2×C4×C127#4112yes—
115C2×C4×C127C2411yescharacteristic
116C8×C127—21no—
117C8×C127—21no—
118C22×C127—21no—
119C22×C127—21no—
120C22×C127#12112yes—
121C22×C127C22×D411yescharacteristic
122C2×D127—22no—
123C2×D127—22no—
124C2×D127—22no—
125C2×D127—41no—
126C2×D127—41no—
127C2×D127—41no—
128C4×C127—42no—
129C4×C127—42no—
130C4×C127#12112yes—
131C4×C127C22×D411yescharacteristic
132C4×C127C2×#4111yescharacteristic
133Dic127—22no—
134Dic127—41no—
135Dic127—41no—
136Dic127—41no—
137C2×C127—22no—
138C2×C127C2×#12111yescharacteristic
139C2×C127#57711yescharacteristic
140C2×C127#57811yescharacteristic
141D127—42no—
142D127—81no—
143#3063—1271nomaximal, Sylow
144C127#306311yescharacteristic, complemented, Sylow
145C2×#137—1271no—
146C2×#131—1271no—
147#537—1271no—
148#536—1271no—
149#470—1272no—
150#468—1272no—
151#467—1272no—
152#465—1272no—
153#428—1271no—
154#416—1271no—
155C2×#103D12711yescharacteristic, complemented
156C2×#41—1271no—
157C2×#41—1271no—
158C22×Q8C2×D12711yescharacteristic
159#137—2544no—
160C2×Q16—1272no—
161C2×SD16—1272no—
162#131—2544no—
163C2×M16—1271no—
164C2×M16—1272no—
165C22×C8—1272no—
166C22×C8C2×D12711yescharacteristic
167C4⋊Q8—2541no—
168#124—2541no—
169#123—2542no—
170C22⋊Q8—1272no—
171C22⋊Q8—1272no—
172C22⋊Q8—1272no—
173C22⋊Q8—2541no—
174#121—1272no—
175#121—2541no—
176#121—2542no—
177#117—1271no—
178C2×C4⋊C4C2×D12711yescharacteristic
179#107—1272no—
180#106—1272no—
181#103—1271no—
182#103—1271no—
183#103C2×D12712yes—
184#103C2×D12712yes—
185#102—1271no—
186#102—1271no—
187#41—2542no—
188#41—2542no—
189#41—2542no—
190#41—2542no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—1271no—
195C2×Q8—1271no—
196C2×Q8C22×D12712yes—
197C2×D4—1272no—
198C2×D4—1271no—
199C2×D4—1271no—
200C2×D4—2542no—
201C2×D4—2541no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—1272no—
205C22×C4—2541no—
206C22×C4C22×D12711yescharacteristic
207Q16—2544no—
208SD16—2544no—
209M16—1274no—
210M16—2544no—
211C2×C8—22no—
212C2×C8—1272no—
213C2×C8—2544no—
214C2×C8C22×D12711yescharacteristic
215C2×C8C22×D12711yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—1271no—
218C4⋊C4—1271no—
219C4⋊C4—2542no—
220C4⋊C4—2542no—
221C4⋊C4—2542no—
222C4⋊C4C22×D12712yes—
223C22⋊C4—2542no—
224C22⋊C4—2542no—
225C22⋊C4—2542no—
226C22⋊C4—2542no—
227C22⋊C4—2541no—
228C42—2541no—
229C23—1272no—
230C23—2541no—
231C23D4×D12711yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—42no—
235Q8—2541no—
236Q8—2541no—
237Q8—2542no—
238D4—10162no—
239D4—5082no—
240D4—2544no—
241D4—2541no—
242D4—2541no—
243D4—2542no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—1272no—
249C2×C4—5082no—
250C2×C4—2541no—
251C2×C4—2541no—
252C2×C4—2541no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4D4×D12712yes—
256C2×C4D4×D12711yescharacteristic
257C2×C4#1452324812yes—
258C2×C4C23×D12711yescharacteristic
259C8—21no—
260C8—21no—
261C8—1274no—
262C22—21no—
263C22—21no—
264C22—5081no—
265C22—5081no—
266C22—5081no—
267C22—2542no—
268C22—2542no—
269C22—2542no—
270C22#1686118212yes—
271C22C2×D4×D12711yescharacteristic
272C4—42no—
273C4—42no—
274C4—5081no—
275C4—5081no—
276C4—5081no—
277C4—2542no—
278C4#1686118212yes—
279C4C2×D4×D12711yescharacteristic
280C4C2×#1452324811yescharacteristic
281C2—22no—
282C2—10161no—
283C2—5082no—
284C2C2×#1686118211yescharacteristic
285C2#2255068211yescharacteristic
286C2#2255068911yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#103×C127C2
socleC22×C127C22×D4
prime radical#22550457C2
hypercenterC2×#103D127
hypocenterC127#3063
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
2. C2×#16861215C2semidirect
4. #22550450C2semidirect
5. #22550133C2semidirect
6. #22550125C2semidirect
7. #22550116C2semidirect
8. #22550108C2semidirect
9. #22549916C2semidirect
10. #22549873C2semidirect
11. C2×#103×C127C2semidirect
58. #14523232D4semidirect
144. C127#3063semidirect
155. C2×#103D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612152
#225504572
#225504502
#225501332
#225501252
#225501162
#225501082
#225499162
#225498732
C2×#103×C1272
#3063127