Finite Group Databasebeta

← #30648244

Subgroups — #30648244

Order 16256.

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

normal subgroups only

This group has 494 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
1C22×#14523236C211yesmaximal, characteristic, complemented
2#22550447C211yesmaximal, characteristic, complemented
3#22550351C211yesmaximal, characteristic, complemented
4#22550089C214yesmaximal, complemented
5#22550079C214yesmaximal, complemented
6#22549904C211yesmaximal, characteristic, complemented
7#22549886C211yesmaximal, characteristic, complemented
8#22549872C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11C2×#14523236—24no—
12C2×#14523236C2214yes—
13C22×C127⋊C8C2211yescharacteristic
14#16861207C2212yescomplemented
15#16861196C2212yescomplemented
16#16861190—22no—
17#16861188—22no—
18D4×Dic127—22no—
19D4×Dic127C2214yes—
20#16861168C2211yescharacteristic
21#16861168C2211yescharacteristic
22C2×C22⋊Dic127—22no—
23Dic127⋊C8C2214yescomplemented
24#16861134C2214yescomplemented
25#16861133C2212yescomplemented
26#16861130C2212yescomplemented
27C2×#41×C127C2211yescharacteristic
28C22×D4×C127C2211yescharacteristic
29AGL(1,ℤ/8)×C127—24no—
30C2×SD16×C127C2212yes—
31C2×D8×C127C2212yes—
32C2×M16×C127C2211yescharacteristic
33C2×Dic254—21no—
34C2×Dic254—22no—
35C2×Dic254—22no—
36C2×Dic254C2312yes—
37C22×Dic127—24no—
38C22×Dic127—21no—
39#14523236—28no—
40C2×C127⋊C8—22no—
41C2×C127⋊C8C2312yes—
42C4⋊Dic127D412yes—
43C4⋊Dic127C2312yes—
44Dic127⋊C4—24no—
45C22⋊Dic127—24no—
46C22⋊Dic127—24no—
47C22⋊Dic127—21no—
48C22⋊Dic127D412yes—
49C4×Dic127—21no—
50C4×Dic127C2312yes—
51C24×C127—21no—
52#41×C127—22no—
53#41×C127—22no—
54C2×Q8×C127C2311yescharacteristic
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—22no—
58C2×D4×C127D414yes—
59C2×D4×C127C2311yescharacteristic
60C2×D4×C127C2312yes—
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63SD16×C127—24no—
64D8×C127—24no—
65M16×C127—22no—
66C2×C8×C127C2312yes—
67Dic254—22no—
68Dic254—22no—
69Dic254—42no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—42no—
73C2×Dic127—84no—
74C2×Dic127C2×D414yes—
75C127⋊C8—22no—
76C23×C127—22no—
77C23×C127—24no—
78C23×C127—21no—
79C23×C127—82no—
80C23×C127C2×D411yescharacteristic
81Q8×C127—21no—
82Q8×C127—21no—
83D4×C127—24no—
84D4×C127—21no—
85D4×C127—22no—
86D4×C127—21no—
87D4×C127—42no—
88D4×C127C2×D414yes—
89C2×C4×C127—21no—
90C2×C4×C127—42no—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D412yes—
93C2×C4×C127#4112yes—
94C2×C4×C127C2411yescharacteristic
95C8×C127—22no—
96C22×C127—22no—
97C22×C127—24no—
98C22×C127—44no—
99C22×C127—41no—
100C22×C127—42no—
101C22×C127—84no—
102C22×C127#12112yes—
103C22×C127C22×D411yescharacteristic
104C4×C127—41no—
105C4×C127#12112yes—
106C4×C127C22×D411yescharacteristic
107C4×C127C2×#4111yescharacteristic
108Dic127—24no—
109Dic127—42no—
110Dic127—42no—
111C2×C127—22no—
112C2×C127—44no—
113C2×C127—81no—
114C2×C127C2×#12111yescharacteristic
115C2×C127#57712yes—
116#3042—1271nomaximal, Sylow
117C127#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)D12711yescharacteristic
119C22×SD16—1271no—
120#536—1271no—
121#518—1271no—
122#463—1274no—
123#461—1274no—
124#424—1271no—
125#419—1271no—
126#416—1271no—
127C2×#41C2×D12711yescharacteristic
128C22×Q8—1271no—
129C22×D4C2×D12711yescharacteristic
130AGL(1,ℤ/8)—24no—
131C2×SD16—1274no—
132C2×SD16—2544no—
133C2×SD16C2×D12712yes—
134C2×D8C2×D12712yes—
135C2×M16C2×D12711yescharacteristic
136C22×C8—1271no—
137C4⋊Q8—1272no—
138#124—1272no—
139#123—2542no—
140C22⋊Q8—2542no—
141C4×D4—1274no—
142C4×D4—2542no—
143#117—1271no—
144#117—1271no—
145C2×C22⋊C4—2542no—
146C4⋊C8—1274no—
147#103—1272no—
148#103—1274no—
149#102—1272no—
150C24—21no—
151#41—22no—
152#41—22no—
153C2×Q8—1272no—
154C2×Q8—2542no—
155C2×Q8—2542no—
156C2×Q8—2541no—
157C2×Q8C22×D12711yescharacteristic
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4C22⋊D12714yes—
162C2×D4C22×D12711yescharacteristic
163C2×D4C22×D12712yes—
164C22×C4—21no—
165C22×C4—2544no—
166C22×C4—2541no—
167C22×C4C22×D12711yescharacteristic
168SD16—24no—
169SD16—2548no—
170D8—24no—
171M16—22no—
172C2×C8—1272no—
173C2×C8—2542no—
174C2×C8C22×D12712yes—
175C4⋊C4—1272no—
176C4⋊C4—1272no—
177C4⋊C4—2544no—
178C22⋊C4—1272no—
179C22⋊C4—2544no—
180C22⋊C4—2544no—
181C22⋊C4—2541no—
182C42—1272no—
183C42—2541no—
184C23—22no—
185C23—24no—
186C23—21no—
187C23—82no—
188C23D4×D12711yescharacteristic
189Q8—21no—
190Q8—21no—
191Q8—5082no—
192Q8—2542no—
193Q8—2542no—
194D4—21no—
195D4—22no—
196D4—21no—
197D4—24no—
198D4—42no—
199D4C2×C22⋊D12714yes—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—1274no—
203C2×C4—10164no—
204C2×C4—5082no—
205C2×C4—2542no—
206C2×C4—2542no—
207C2×C4D4×D12711yescharacteristic
208C2×C4C2×C22⋊D12712yes—
209C2×C4#1452324712yes—
210C2×C4C23×D12711yescharacteristic
211C8—22no—
212C8—2542no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—41no—
217C22—42no—
218C22—84no—
219C22#1686118112yes—
220C22C2×D4×D12711yescharacteristic
221C4—41no—
222C4—5082no—
223C4—5082no—
224C4—2544no—
225C4#1686118112yes—
226C4C22×C22⋊D12711yescharacteristic
227C4C2×#1452324711yescharacteristic
228C2—22no—
229C2—44no—
230C2—81no—
231C2C2×#1686118111yescharacteristic
232C2#2255068612yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3042
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×#14523236C2semidirect
2. #22550447C2semidirect
3. #22550351C2semidirect
4. #22550089C2semidirect
5. #22550079C2semidirect
6. #22549904C2semidirect
7. #22549886C2semidirect
8. #22549872C2semidirect
14. #16861207C22semidirect
15. #16861196C22semidirect
23. Dic127⋊C8C22semidirect
24. #16861134C22semidirect
25. #16861133C22semidirect
26. #16861130C22semidirect
117. C127#3042semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
#225504472
#225503512
#225500892
#225500792
#225499042
#225498862
#225498722
C2×AGL(1,ℤ/8)×C1272
#3042127