Finite Group Databasebeta

← #30650461

Subgroups — #30650461

Order 16256.

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

normal subgroups only

This group has 494 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×#16861230C212yesmaximal, complemented
2#22550548C211yesmaximal, characteristic, complemented
3#22550530C211yesmaximal, characteristic, complemented
4#22550249C211yesmaximal, characteristic, complemented
5#22550106C212yesmaximal, complemented
6#22550086C211yesmaximal, characteristic, complemented
7#22550071C211yesmaximal, characteristic, complemented
8#22550041C212yesmaximal, complemented
9#22550022C212yesmaximal, complemented
10C8×D508C211yesmaximal, characteristic, complemented
11C4×SD16×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yes—
13C2×#14523247C2212yes—
14#16861230—28no—
15C2×#14523238C2212yes—
16C2×#14523237C2212yes—
17C2×#14523237C2212yes—
18C2×#14523233C2212yes—
19C2×C8×D127C2212yes—
20#16861209C2211yescharacteristic
21#16861206—21no—
22#16861206C2211yescharacteristic
23#16861193—22no—
24#16861189C2212yes—
25#16861184—22no—
26#16861184C2212yes—
27#16861183—21no—
28C4×D508—21no—
29C4×D508C2211yescharacteristic
30C2×C4⋊Dic127—21no—
31C4⋊(C127⋊C8)C2211yescharacteristic
32#16861132C2212yes—
33#16861129C2212yes—
34(C2×D127)⋊C8C2212yes—
35C2×SD16×C127C2211yescharacteristic
36C4×Q8×C127C2211yescharacteristic
37C4×D4×C127C2211yescharacteristic
38#106×C127C2211yescharacteristic
39#103×C127C2211yescharacteristic
40#102×C127C2211yescharacteristic
41C4×C8×C127C2211yescharacteristic
42#14523249—24no—
43#14523249—42no—
44#14523247—24no—
45#14523247—42no—
46C2×Dic254C2312yes—
47C2×D508—22no—
48C2×D508—22no—
49C2×D508—21no—
50C2×D508—21no—
51C2×D508D411yescharacteristic
52C2×D508C2312yes—
53C2×C22⋊D127—22no—
54C22×Dic127—22no—
55C2×C4×D127—22no—
56C2×C4×D127C2312yes—
57#14523238—24no—
58#14523237—24no—
59#14523237—42no—
60#14523233—24no—
61C2×C127⋊C8C2312yes—
62C8×D127—24no—
63C4⋊Dic127—22no—
64C4⋊Dic127—21no—
65C4⋊Dic127—21no—
66C4⋊Dic127D411yescharacteristic
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4D412yes—
70C2×Q8×C127C2311yescharacteristic
71C2×D4×C127C2311yescharacteristic
72C22×C4×C127—21no—
73SD16×C127D414yescomplemented
74C2×C8×C127C2311yescharacteristic
75C2×C8×C127C2311yescharacteristic
76C4⋊C4×C127—21no—
77C4⋊C4×C127C2311yescharacteristic
78C4⋊C4×C127C2311yescharacteristic
79C22⋊C4×C127—21no—
80C42×C127—21no—
81C42×C127C2311yescharacteristic
82C22×D127—22no—
83C22×D127C2×D412yes—
84Dic254—22no—
85Dic254—22no—
86D508—22no—
87D508—22no—
88D508—22no—
89D508—44no—
90D508—42no—
91D508—81no—
92D508—82no—
93C22⋊D127—44no—
94C2×Dic127—22no—
95C2×Dic127—44no—
96C2×Dic127C2×D412yes—
97C4×D127—22no—
98C4×D127—22no—
99C4×D127—44no—
100C127⋊C8—22no—
101C23×C127—21no—
102Q8×C127—21no—
103Q8×C127C2×D412yes—
104D4×C127—21no—
105D4×C127C2×D412yes—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—42no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C8×C127—21no—
114C8×C127C2×D412yes—
115C22×C127—22no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127C22×D411yescharacteristic
119C2×D127—22no—
120C2×D127—22no—
121C2×D127—22no—
122C2×D127—42no—
123C2×D127—42no—
124C2×D127—42no—
125C4×C127—22no—
126C4×C127—21no—
127C4×C127—41no—
128C4×C127—41no—
129C4×C127#13512yescomplemented
130C4×C127C22×D411yescharacteristic
131C4×C127H(5,𝔽2)11yescharacteristic
132Dic127—22no—
133Dic127—42no—
134C2×C127—42no—
135C2×C127D4211yescharacteristic
136C2×C127C2×#13511yescharacteristic
137C2×C127#57711yescharacteristic
138D127—42no—
139D127—82no—
140#3284—1271nomaximal, Sylow
141C127#328411yescharacteristic, complemented, Sylow
142C2×#135—1272no—
143#550—1271no—
144#547—1271no—
145#492—1271no—
146#465—1272no—
147#462—1271no—
148#459—1271no—
149#452—1272no—
150#449—1272no—
151C4×SD16D12711yescharacteristic, complemented
152C8×D4—1271no—
153C2×#41—1272no—
154C2×#41—1272no—
155#135—2548no—
156C2×Q16—1272no—
157C2×SD16—1272no—
158C2×SD16—1272no—
159C2×SD16C2×D12711yescharacteristic
160C2×D8—1272no—
161C22×C8—1272no—
162C4⋊Q8—1271no—
163#127—1271no—
164#127—2541no—
165#123—2542no—
166C22⋊Q8—1272no—
167#121—1272no—
168#121—2542no—
169#121—2541no—
170C4×Q8C2×D12711yescharacteristic
171C4×D4—1271no—
172C4×D4—2541no—
173C4×D4C2×D12711yescharacteristic
174C2×C4⋊C4—2541no—
175#106C2×D12711yescharacteristic
176C4⋊C8—1271no—
177#103—1272no—
178#103C2×D12711yescharacteristic
179#102—1272no—
180#102C2×D12711yescharacteristic
181C22⋊C8—1272no—
182C4×C8C2×D12711yescharacteristic
183#41—5082no—
184#41—5082no—
185#41—2544no—
186#41—2544no—
187C2×Q8—1272no—
188C2×Q8C22×D12711yescharacteristic
189C2×D4—1271no—
190C2×D4—1272no—
191C2×D4—2542no—
192C2×D4—2542no—
193C2×D4—2542no—
194C2×D4—2541no—
195C2×D4—2541no—
196C2×D4C22×D12711yescharacteristic
197C22×C4—21no—
198C22×C4—1272no—
199C22×C4—2542no—
200C22×C4—2542no—
201Q16—2544no—
202SD16—5082no—
203SD16—2544no—
204SD16D50814yescomplemented
205D8—2544no—
206C2×C8—1272no—
207C2×C8—2544no—
208C2×C8C22×D12711yescharacteristic
209C2×C8C22×D12711yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—1271no—
212C4⋊C4—2542no—
213C4⋊C4—2541no—
214C4⋊C4—2541no—
215C4⋊C4C22×D12711yescharacteristic
216C4⋊C4C22×D12711yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—1272no—
219C22⋊C4—2542no—
220C22⋊C4—2542no—
221C42—21no—
222C42C22×D12711yescharacteristic
223C23—21no—
224C23—1272no—
225C23—2542no—
226Q8—21no—
227Q8—2542no—
228Q8—2542no—
229Q8C2×D50812yes—
230D4—21no—
231D4—10161no—
232D4—10162no—
233D4—5084no—
234D4—5084no—
235D4—5082no—
236D4—2542no—
237D4—2542no—
238D4—2542no—
239D4C2×D50812yes—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—1272no—
245C2×C4—5084no—
246C2×C4—5084no—
247C2×C4—2542no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4D4×D12711yescharacteristic
251C2×C4D4×D12711yescharacteristic
252C2×C4C23×D12711yescharacteristic
253C8—21no—
254C8—2542no—
255C8C2×D50812yes—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—5082no—
260C22—5082no—
261C22—5082no—
262C22—2542no—
263C22—2542no—
264C22—2542no—
265C22C2×D4×D12711yescharacteristic
266C4—21no—
267C4—22no—
268C4—41no—
269C4—41no—
270C4—5082no—
271C4—2542no—
272C4#1686123012yescomplemented
273C4C22×D50811yescharacteristic
274C4#1686125511yescharacteristic
275C2—42no—
276C2—10162no—
277C2—5082no—
278C2D4×D50811yescharacteristic
279C2C2×#1686123011yescharacteristic
280C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×SD16×C127C2
socleC22×C127C22×D4
hypercenterC4×SD16D127
hypocenterC127#3284
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16861230C2semidirect
2. #22550548C2semidirect
3. #22550530C2semidirect
4. #22550249C2semidirect
5. #22550106C2semidirect
6. #22550086C2semidirect
7. #22550071C2semidirect
8. #22550041C2semidirect
9. #22550022C2semidirect
10. C8×D508C2semidirect
11. C4×SD16×C127C2semidirect
73. SD16×C127D4semidirect
129. C4×C127#135semidirect
141. C127#3284semidirect
151. C4×SD16D127semidirect
204. SD16D508semidirect
272. C4#16861230semidirect

Maximal subgroups

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

subgroupindex
C2×#168612302
#225505482
#225505302
#225502492
#225501062
#225500862
#225500712
#225500412
#225500222
C8×D5082
C4×SD16×C1272
#3284127