Finite Group Databasebeta

← #30648415

Subgroups — #30648415

Order 16256.

472 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861217C211yesmaximal, characteristic, complemented
2#22550454C211yesmaximal, characteristic
3#22550450C211yesmaximal, characteristic, complemented
4#22550131C212yesmaximal, complemented
5#22550119C212yesmaximal, complemented
6#22550114C212yesmaximal, complemented
7#22550102C212yesmaximal, complemented
8#22549918C211yesmaximal, characteristic, complemented
9#22549872C211yesmaximal, characteristic, complemented
10C2×#16861133C211yesmaximal, characteristic, complemented
11C2×#137×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13#16861217—24no—
14C2×#14523231C2212yes—
15C22×C127⋊C8C2211yescharacteristic
16C2×C8×D127C2212yes—
17#16861207—21no—
18#16861198—21no—
19#16861192—22no—
20#16861188—21no—
21#16861185C2212yes—
22#16861185C2212yes—
23#16861185C2212yes—
24#16861183—21no—
25#16861181—22no—
26#16861180C2212yes—
27#16861168C2211yescharacteristic
28C2×C4⋊Dic127C2211yescharacteristic
29#16861143C2212yes—
30#16861141C2212yes—
31#16861133C2212yes—
32#16861133C2211yescharacteristic
33#16861133C2212yes—
34#16861133C2211yescharacteristic
35#16861130C2211yescharacteristic
36#16861130C2211yescharacteristic
37C2×#41×C127C2211yescharacteristic
38C22×Q8×C127C2211yescharacteristic
39#137×C127—24no—
40C2×Q16×C127C2212yes—
41C2×SD16×C127C2212yes—
42C2×M16×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45C2×Dic254D411yescharacteristic
46C2×D508D411yescharacteristic
47C2×C22⋊D127—22no—
48C2×C22⋊D127D412yes—
49C22×Dic127—21no—
50C2×C4×D127C2312yes—
51#14523231—24no—
52C2×C127⋊C8—22no—
53C2×C127⋊C8C2311yescharacteristic
54C2×C127⋊C8C2311yescharacteristic
55C8×D127—24no—
56C4⋊Dic127—21no—
57C4⋊Dic127C2311yescharacteristic
58C4⋊Dic127C2311yescharacteristic
59C4⋊Dic127C2312yes—
60Dic127⋊C4—22no—
61Dic127⋊C4—22no—
62Dic127⋊C4—22no—
63C22⋊Dic127—22no—
64C22⋊Dic127—21no—
65(C2×D127)⋊C4—22no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—22no—
68C4×Dic127—21no—
69#41×C127—22no—
70#41×C127—22no—
71C2×Q8×C127—21no—
72C2×Q8×C127—22no—
73C2×Q8×C127—22no—
74C2×Q8×C127C2312yes—
75C2×Q8×C127C2311yescharacteristic
76C2×D4×C127—21no—
77C2×D4×C127C2311yescharacteristic
78C22×C4×C127—21no—
79C22×C4×C127—21no—
80C22×C4×C127C2311yescharacteristic
81Q16×C127—24no—
82SD16×C127—24no—
83M16×C127D414yescomplemented
84C2×C8×C127C2312yes—
85C22×D127C2×D412yes—
86Dic254—22no—
87D508—22no—
88C22⋊D127—24no—
89C22⋊D127—82no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C2×Dic127—22no—
93C2×Dic127—81no—
94C2×Dic127C2×D412yes—
95C4×D127—22no—
96C4×D127—22no—
97C127⋊C8—21no—
98C127⋊C8—21no—
99C23×C127—21no—
100C23×C127C2×D411yescharacteristic
101Q8×C127—22no—
102Q8×C127—21no—
103Q8×C127—22no—
104Q8×C127—21no—
105Q8×C127—42no—
106D4×C127—21no—
107D4×C127—21no—
108D4×C127—42no—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127—42no—
112C2×C4×C127—42no—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D412yes—
115C2×C4×C127#4112yes—
116C2×C4×C127C2411yescharacteristic
117C8×C127C2×D414yes—
118C22×C127—21no—
119C22×C127—21no—
120C22×C127—41no—
121C22×C127—41no—
122C22×C127—41no—
123C22×C127#12112yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—22no—
126C2×D127—22no—
127C2×D127—22no—
128C4×C127—41no—
129C4×C127—42no—
130C4×C127#12112yes—
131C4×C127C22×D411yescharacteristic
132C4×C127C2×#4111yescharacteristic
133Dic127—22no—
134Dic127—41no—
135Dic127—42no—
136Dic127—41no—
137C2×C127—22no—
138C2×C127—81no—
139C2×C127C2×#12111yescharacteristic
140C2×C127#57711yescharacteristic
141C2×C127#57811yescharacteristic
142D127—42no—
143#3063—1271nomaximal, Sylow
144C127#306311yescharacteristic, complemented, Sylow
145C2×#137D12711yescharacteristic, complemented
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×#103—1271no—
156C2×#41—1271no—
157C2×#41C2×D12711yescharacteristic
158C22×Q8C2×D12711yescharacteristic
159#137—24no—
160C2×Q16C2×D12712yes—
161C2×SD16C2×D12712yes—
162#131—2544no—
163C2×M16—1272no—
164C2×M16C2×D12711yescharacteristic
165C22×C8—1271no—
166C22×C8—1272no—
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⋊C4—1271no—
179#107—1272no—
180#106—1272no—
181#103—1272no—
182#103—1272no—
183#103—1271no—
184#103—1271no—
185#102—1271no—
186#102—1271no—
187#41—22no—
188#41—22no—
189#41—2542no—
190#41—2542no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—1271no—
195C2×Q8C22×D12712yes—
196C2×Q8C22×D12711yescharacteristic
197C2×D4—21no—
198C2×D4—1272no—
199C2×D4—1271no—
200C2×D4—2542no—
201C2×D4C22×D12711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—1272no—
205C22×C4—2541no—
206C22×C4C22×D12711yescharacteristic
207Q16—24no—
208SD16—24no—
209M16—2544no—
210M16C22⋊D12714yescomplemented
211C2×C8—1271no—
212C2×C8—1271no—
213C2×C8—2542no—
214C2×C8—2544no—
215C2×C8C22×D12712yes—
216C4⋊C4—1271no—
217C4⋊C4—1271no—
218C4⋊C4—1272no—
219C4⋊C4—2542no—
220C4⋊C4—2542no—
221C4⋊C4—2542no—
222C4⋊C4—2541no—
223C22⋊C4—2542no—
224C22⋊C4—2542no—
225C22⋊C4—2542no—
226C22⋊C4—2542no—
227C22⋊C4—2541no—
228C42—2541no—
229C23—21no—
230C23—1272no—
231C23D4×D12711yescharacteristic
232Q8—22no—
233Q8—21no—
234Q8—22no—
235Q8—21no—
236Q8—42no—
237Q8—2542no—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—10162no—
242D4—2544no—
243D4—2542no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—1272no—
249C2×C4—10161no—
250C2×C4—2541no—
251C2×C4—2541no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4D4×D12711yescharacteristic
256C2×C4C2×C22⋊D12712yes—
257C2×C4#1452324712yes—
258C2×C4C23×D12711yescharacteristic
259C8—2541no—
260C8—2541no—
261C8C2×C22⋊D12714yes—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—2542no—
268C22—2542no—
269C22—2542no—
270C22#1686118012yes—
271C22C2×D4×D12711yescharacteristic
272C4—41no—
273C4—42no—
274C4—5082no—
275C4—5081no—
276C4—5081no—
277C4—2542no—
278C4#1686118012yes—
279C4C22×C22⋊D12711yescharacteristic
280C4C2×#1452324711yescharacteristic
281C2—22no—
282C2—81no—
283C2—5082no—
284C2C2×#1686118011yescharacteristic
285C2#2255068511yescharacteristic
286C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
prime radical#22550454C2
hypercenterC2×#137D127
hypocenterC127#3063
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16861217C2semidirect
3. #22550450C2semidirect
4. #22550131C2semidirect
5. #22550119C2semidirect
6. #22550114C2semidirect
7. #22550102C2semidirect
8. #22549918C2semidirect
9. #22549872C2semidirect
10. C2×#16861133C2semidirect
11. C2×#137×C127C2semidirect
83. M16×C127D4semidirect
144. C127#3063semidirect
145. C2×#137D127semidirect
210. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612172
#225504542
#225504502
#225501312
#225501192
#225501142
#225501022
#225499182
#225498722
C2×#168611332
C2×#137×C1272
#3063127