Finite Group Databasebeta

← #30648242

Subgroups — #30648242

Order 16256.

494 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861235C211yesmaximal, characteristic
2C22×#14523236C211yesmaximal, characteristic, complemented
3#22550452C211yesmaximal, characteristic, complemented
4#22550091C214yesmaximal, complemented
5#22550081C214yesmaximal, complemented
6#22549903C211yesmaximal, characteristic, complemented
7#22549884C211yesmaximal, characteristic, complemented
8#22549870C211yesmaximal, characteristic, complemented
9#518×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×Dic254C2211yescharacteristic
12#16861235—24no—
13C2×#14523236—24no—
14C2×#14523236C2212yes—
15C2×#14523236C2214yes—
16C2×#14523233C2212yes—
17C2×#14523232C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19#16861209C2212yescomplemented
20#16861197C2212yescomplemented
21#16861193—22no—
22#16861189—22no—
23C4⋊(C127⋊C8)C2214yescomplemented
24#16861132C2212yescomplemented
25#16861132C2214yescomplemented
26#16861129C2212yescomplemented
27C22×D4×C127C2211yescharacteristic
28C4×D4×C127—22no—
29C4×D4×C127C2214yes—
30#117×C127C2211yescharacteristic
31#117×C127C2211yescharacteristic
32C2×C22⋊C4×C127—22no—
33#14523248—22no—
34#14523248—22no—
35C2×Dic254—21no—
36C2×Dic254—22no—
37C2×Dic254—22no—
38C2×Dic254C2312yes—
39C2×Dic254C2311yescharacteristic
40C2×D508C2311yescharacteristic
41C2×C22⋊D127—21no—
42C22×Dic127—21no—
43C2×C4×D127—21no—
44#14523236—28no—
45#14523236—24no—
46#14523233—24no—
47#14523232—22no—
48C2×C127⋊C8—22no—
49C2×C127⋊C8C2312yes—
50C2×C127⋊C8C2312yes—
51C4⋊Dic127—24no—
52(C2×D127)⋊C4—24no—
53C24×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—22no—
56C2×D4×C127D414yes—
57C2×D4×C127C2312yes—
58C22×C4×C127—24no—
59C22×C4×C127C2311yescharacteristic
60C4⋊C4×C127D412yes—
61C4⋊C4×C127C2312yes—
62C22⋊C4×C127—21no—
63C22⋊C4×C127—24no—
64C22⋊C4×C127D412yes—
65C42×C127—21no—
66C42×C127C2312yes—
67C22×D127—21no—
68Dic254—22no—
69Dic254—21no—
70Dic254—22no—
71Dic254—21no—
72Dic254—42no—
73D508—21no—
74D508—21no—
75C22⋊D127—42no—
76C2×Dic127—22no—
77C2×Dic127—21no—
78C2×Dic127—42no—
79C4×D127—42no—
80C127⋊C8—22no—
81C127⋊C8—22no—
82C23×C127—22no—
83C23×C127—24no—
84C23×C127—82no—
85C23×C127C2×D411yescharacteristic
86D4×C127—24no—
87D4×C127—22no—
88D4×C127C2×D414yes—
89C2×C4×C127—22no—
90C2×C4×C127—84no—
91C2×C4×C127C2×D414yes—
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D412yes—
94C2×C4×C127#4112yes—
95C2×C4×C127C2411yescharacteristic
96C22×C127—22no—
97C22×C127—24no—
98C22×C127—44no—
99C22×C127—84no—
100C22×C127#12112yes—
101C22×C127C22×D411yescharacteristic
102C2×D127—41no—
103C2×D127—42no—
104C4×C127—24no—
105C4×C127—42no—
106C4×C127#12112yes—
107C4×C127C22×D411yescharacteristic
108C4×C127C2×#4111yescharacteristic
109Dic127—41no—
110Dic127—42no—
111C2×C127—22no—
112C2×C127—44no—
113C2×C127C2×#12111yescharacteristic
114C2×C127#57712yes—
115D127—81no—
116#3042—1271nomaximal, Sylow
117C127#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1271no—
119C22×SD16—1271no—
120#536—1271no—
121#518D12711yescharacteristic, complemented
122#463—1274no—
123#461—1274no—
124#424—1271no—
125#419—1271no—
126#416—1271no—
127C2×#41—1271no—
128C22×Q8—1271no—
129C22×D4C2×D12711yescharacteristic
130AGL(1,ℤ/8)—2544no—
131C2×SD16—1274no—
132C2×SD16—1272no—
133C2×SD16—2544no—
134C2×D8—1272no—
135C2×M16—1271no—
136C22×C8—1271no—
137C4⋊Q8—1272no—
138#124—1272no—
139#123—2542no—
140C22⋊Q8—2542no—
141C4×D4—22no—
142C4×D4C2×D12714yes—
143#117C2×D12711yescharacteristic
144#117C2×D12711yescharacteristic
145C2×C22⋊C4—22no—
146C4⋊C8—1274no—
147#103—1272no—
148#103—1274no—
149#102—1272no—
150C24—21no—
151#41—2542no—
152#41—2542no—
153C2×Q8—1272no—
154C2×Q8—1271no—
155C2×Q8—2542no—
156C2×Q8—2542no—
157C2×Q8—2541no—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—1271no—
161C2×D4—2541no—
162C2×D4D50814yes—
163C2×D4C22×D12712yes—
164C22×C4—24no—
165C22×C4—2541no—
166C22×C4—2541no—
167C22×C4C22×D12711yescharacteristic
168SD16—2548no—
169SD16—2544no—
170D8—2544no—
171M16—2542no—
172C2×C8—1272no—
173C2×C8—1272no—
174C2×C8—2542no—
175C4⋊C4—2544no—
176C4⋊C4C22⋊D12712yes—
177C4⋊C4C22×D12712yes—
178C22⋊C4—21no—
179C22⋊C4—24no—
180C22⋊C4—2544no—
181C22⋊C4C22⋊D12712yes—
182C42—21no—
183C42C22×D12712yes—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—2541no—
188C23C2×C22⋊D12711yescharacteristic
189Q8—5082no—
190Q8—2542no—
191Q8—2541no—
192Q8—2542no—
193Q8—2541no—
194D4—22no—
195D4—24no—
196D4—5082no—
197D4—2541no—
198D4—2541no—
199D4C2×D50814yes—
200C2×C4—22no—
201C2×C4—84no—
202C2×C4—5082no—
203C2×C4—5082no—
204C2×C4—2542no—
205C2×C4—2541no—
206C2×C4C2×C22⋊D12714yes—
207C2×C4C2×C22⋊D12711yescharacteristic
208C2×C4C2×D50812yes—
209C2×C4#1452324812yes—
210C2×C4C23×D12711yescharacteristic
211C8—2542no—
212C8—2542no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—5081no—
218C22—5082no—
219C22#1686118312yes—
220C22C22×C22⋊D12711yescharacteristic
221C4—24no—
222C4—42no—
223C4—5081no—
224C4—5082no—
225C4#1686118312yes—
226C4C22×D50811yescharacteristic
227C4C2×#1452324811yescharacteristic
228C2—22no—
229C2—44no—
230C2—10161no—
231C2C2×#1686118311yescharacteristic
232C2#2255068712yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#518×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861235C2
hypercenter#518D127
hypocenterC127#3042
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C22×#14523236C2semidirect
3. #22550452C2semidirect
4. #22550091C2semidirect
5. #22550081C2semidirect
6. #22549903C2semidirect
7. #22549884C2semidirect
8. #22549870C2semidirect
9. #518×C127C2semidirect
19. #16861209C22semidirect
20. #16861197C22semidirect
23. C4⋊(C127⋊C8)C22semidirect
24. #16861132C22semidirect
25. #16861132C22semidirect
26. #16861129C22semidirect
117. C127#3042semidirect
121. #518D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
C22×#145232362
#225504522
#225500912
#225500812
#225499032
#225498842
#225498702
#518×C1272
#3042127