Finite Group Databasebeta

← #30650447

Subgroups — #30650447

Order 16256.

484 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550530C211yesmaximal, characteristic, complemented
3#22550529C211yesmaximal, characteristic, complemented
4#22550255C211yesmaximal, characteristic, complemented
5#22550116C212yesmaximal, complemented
6#22550082C211yesmaximal, characteristic, complemented
7#22550041C212yesmaximal, complemented
8#22550035C212yesmaximal, complemented
9C4×#14523233C211yesmaximal, characteristic, complemented
10#22549941C211yesmaximal, characteristic, complemented
11#461×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yes—
13C2×#14523247C2212yes—
14#16861230—28no—
15C2×#14523238C2212yes—
16C2×#14523238C2212yes—
17C2×#14523237C2212yes—
18C2×#14523233C2211yescharacteristic
19C2×C8×D127C2212yes—
20#16861209C2211yescharacteristic
21#16861194—22no—
22#16861193—22no—
23#16861189C2212yes—
24#16861185C2212yes—
25#16861184—21no—
26#16861183—21no—
27C4×D508C2211yescharacteristic
28C4×D508C2211yescharacteristic
29C2×C4⋊Dic127—21no—
30C4⋊C4×D127—21no—
31#16861142C2211yescharacteristic
32#16861134C2212yes—
33#16861130C2211yescharacteristic
34#16861129C2211yescharacteristic
35(C2×D127)⋊C8C2212yes—
36C4×C127⋊C8C2211yescharacteristic
37C2×SD16×C127C2212yes—
38C4⋊Q8×C127C2211yescharacteristic
39C4×D4×C127C2211yescharacteristic
40C4⋊C8×C127C2211yescharacteristic
41#103×C127C2212yes—
42#14523249—24no—
43#14523249—42no—
44#14523247—24no—
45#14523247—42no—
46C2×Dic254C2312yes—
47C2×D508—22no—
48C2×D508D411yescharacteristic
49C2×D508C2311yescharacteristic
50C2×C22⋊D127—22no—
51C22×Dic127—22no—
52C2×C4×D127—22no—
53C2×C4×D127—21no—
54C2×C4×D127C2312yes—
55#14523238—24no—
56#14523238—42no—
57#14523237—24no—
58#14523233D414yescomplemented
59C2×C127⋊C8C2311yescharacteristic
60C2×C127⋊C8C2311yescharacteristic
61C8×D127—24no—
62C4⋊Dic127—22no—
63C4⋊Dic127—21no—
64C4⋊Dic127—21no—
65C4⋊Dic127D411yescharacteristic
66C4⋊Dic127C2311yescharacteristic
67Dic127⋊C4—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—21no—
71(C2×D127)⋊C4D412yes—
72C2×Q8×C127C2312yes—
73C2×D4×C127C2311yescharacteristic
74C22×C4×C127—21no—
75SD16×C127—24no—
76C2×C8×C127C2312yes—
77C4⋊C4×C127—21no—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C22⋊C4×C127—21no—
81C42×C127C2311yescharacteristic
82C22×D127—21no—
83C22×D127C2×D412yes—
84Dic254—22no—
85Dic254—22no—
86D508—22no—
87D508—21no—
88D508—44no—
89D508C2×D412yes—
90C22⋊D127—44no—
91C2×Dic127—21no—
92C2×Dic127—22no—
93C2×Dic127—44no—
94C2×Dic127C2×D412yes—
95C4×D127—22no—
96C4×D127—22no—
97C4×D127—44no—
98C4×D127—42no—
99C127⋊C8—21no—
100C127⋊C8C2×D412yes—
101C23×C127—21no—
102Q8×C127—22no—
103Q8×C127—22no—
104D4×C127—21no—
105D4×C127C2×D412yes—
106C2×C4×C127—21no—
107C2×C4×C127—22no—
108C2×C4×C127—42no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—22no—
113C22×C127—22no—
114C22×C127—41no—
115C22×C127—41no—
116C22×C127C22×D411yescharacteristic
117C2×D127—22no—
118C2×D127—22no—
119C2×D127—22no—
120C2×D127—22no—
121C2×D127—41no—
122C2×D127—41no—
123C4×C127—21no—
124C4×C127—42no—
125C4×C127—41no—
126C4×C127#13512yescomplemented
127C4×C127C22×D411yescharacteristic
128C4×C127H(5,𝔽2)11yescharacteristic
129Dic127—22no—
130Dic127—42no—
131Dic127—41no—
132C2×C127—42no—
133C2×C127D4211yescharacteristic
134C2×C127C2×#13511yescharacteristic
135C2×C127#57811yescharacteristic
136D127—42no—
137D127—42no—
138#3283—1271nomaximal, Sylow
139C127#328311yescharacteristic, complemented, Sylow
140C2×#135—1272no—
141#547—1271no—
142#547—1271no—
143#494—1271no—
144#467—1272no—
145#461—1271no—
146#461D12711yescharacteristic, complemented
147#452—1272no—
148#452—1272no—
149C4×D8—1271no—
150C8×D4—1271no—
151C2×#41—1272no—
152C2×#41—1272no—
153#135—2548no—
154C2×Q16—1272no—
155C2×Q16—1272no—
156C2×SD16—1272no—
157C2×SD16C2×D12712yes—
158C2×D8—1271no—
159C22×C8—1272no—
160C4⋊Q8—1271no—
161C4⋊Q8C2×D12711yescharacteristic
162#123—2542no—
163#123—2542no—
164C22⋊Q8—1272no—
165C22⋊Q8—1272no—
166#121—2541no—
167#121—2541no—
168C4×D4—1271no—
169C4×D4—1271no—
170C4×D4C2×D12711yescharacteristic
171C2×C4⋊C4—2541no—
172C2×C4⋊C4—2541no—
173#107—1271no—
174C4⋊C8C2×D12711yescharacteristic
175#103—1272no—
176#103C2×D12712yes—
177#102—1271no—
178#102—1271no—
179C22⋊C8—1272no—
180C4×C8—1271no—
181#41—5082no—
182#41—5082no—
183#41—2544no—
184#41—2544no—
185C2×Q8—1272no—
186C2×Q8C22×D12712yes—
187C2×D4—1271no—
188C2×D4—1271no—
189C2×D4—2542no—
190C2×D4—2542no—
191C2×D4C22×D12711yescharacteristic
192C22×C4—21no—
193C22×C4—1272no—
194C22×C4—2542no—
195C22×C4—2542no—
196C22×C4—2541no—
197Q16—5082no—
198Q16—2544no—
199SD16—24no—
200SD16—2544no—
201D8—1274no—
202C2×C8—1271no—
203C2×C8—1271no—
204C2×C8—2544no—
205C2×C8C22×D12712yes—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—1271no—
209C4⋊C4—1271no—
210C4⋊C4—2542no—
211C4⋊C4—2542no—
212C4⋊C4—2541no—
213C4⋊C4—2541no—
214C4⋊C4C22×D12711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—1272no—
217C22⋊C4—2542no—
218C22⋊C4—2542no—
219C22⋊C4—2541no—
220C42C22×D12711yescharacteristic
221C23—21no—
222C23—1272no—
223C23—2541no—
224Q8—22no—
225Q8—22no—
226Q8—2542no—
227Q8—2542no—
228D4—21no—
229D4—1272no—
230D4—5084no—
231D4—5084no—
232D4—2542no—
233D4—2541no—
234D4D4×D12712yes—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—1272no—
239C2×C4—5084no—
240C2×C4—5084no—
241C2×C4—5082no—
242C2×C4—2541no—
243C2×C4—2542no—
244C2×C4—2542no—
245C2×C4—2542no—
246C2×C4D4×D12711yescharacteristic
247C2×C4D4×D12711yescharacteristic
248C2×C4C23×D12711yescharacteristic
249C8—22no—
250C8—1272no—
251C8—2541no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—5081no—
256C22—5081no—
257C22—2542no—
258C22—2542no—
259C22—2542no—
260C22—2542no—
261C22C2×D4×D12711yescharacteristic
262C4—21no—
263C4—41no—
264C4—42no—
265C4—5081no—
266C4—5082no—
267C4—2542no—
268C4#1686123012yescomplemented
269C4C2×D4×D12711yescharacteristic
270C4#1686125511yescharacteristic
271C2—42no—
272C2—5082no—
273C2—5082no—
274C2#2255051011yescharacteristic
275C2C2×#1686123011yescharacteristic
276C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#461×C127C2
socleC22×C127C22×D4
hypercenter#461D127
hypocenterC127#3283
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#16861230C2semidirect
2. #22550530C2semidirect
3. #22550529C2semidirect
4. #22550255C2semidirect
5. #22550116C2semidirect
6. #22550082C2semidirect
7. #22550041C2semidirect
8. #22550035C2semidirect
9. C4×#14523233C2semidirect
10. #22549941C2semidirect
11. #461×C127C2semidirect
58. #14523233D4semidirect
126. C4×C127#135semidirect
139. C127#3283semidirect
146. #461D127semidirect
268. C4#16861230semidirect

Maximal subgroups

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

subgroupindex
C2×#168612302
#225505302
#225505292
#225502552
#225501162
#225500822
#225500412
#225500352
C4×#145232332
#225499412
#461×C1272
#3283127