Finite Group Databasebeta

← #30648241

Subgroups — #30648241

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×#16861237C211yesmaximal, characteristic
2C22×#14523235C211yesmaximal, characteristic, complemented
3#22550450C211yesmaximal, characteristic, complemented
4#22550350C211yesmaximal, characteristic, complemented
5#22550094C214yesmaximal, complemented
6#22550080C214yesmaximal, complemented
7#22549904C211yesmaximal, characteristic, complemented
8#22549870C211yesmaximal, characteristic, complemented
9#419×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×D508C2211yescharacteristic
12#16861237—24no—
13C2×#14523237C2212yes—
14C2×#14523235—24no—
15C2×#14523235C2214yes—
16C2×D1016C2212yes—
17C22×C127⋊C8C2211yescharacteristic
18#16861207C2212yescomplemented
19#16861198C2212yescomplemented
20#16861192—22no—
21#16861185—22no—
22C4×D508—22no—
23#16861171C2214yes—
24#16861168C2211yescharacteristic
25C22⋊C4×D127—22no—
26Dic127⋊C8C2214yescomplemented
27#16861132C2212yescomplemented
28#16861129C2212yescomplemented
29C22×Q8×C127C2211yescharacteristic
30C2×M16×C127C2211yescharacteristic
31#117×C127C2211yescharacteristic
32#103×C127C2214yescomplemented
33C23×D127—21no—
34#14523248—22no—
35#14523248—22no—
36C2×Dic254C2311yescharacteristic
37C2×D508—21no—
38C2×D508—22no—
39C2×D508D414yes—
40C2×D508C2311yescharacteristic
41C2×D508C2312yes—
42C2×C22⋊D127—21no—
43C2×C4×D127—24no—
44C2×C4×D127—21no—
45#14523237—24no—
46#14523235—28no—
47D1016—24no—
48C2×C127⋊C8—22no—
49C2×C127⋊C8C2312yes—
50C4⋊Dic127D412yes—
51Dic127⋊C4—24no—
52C22⋊Dic127D412yes—
53(C2×D127)⋊C4—24no—
54(C2×D127)⋊C4—24no—
55C4×Dic127C2312yes—
56C2×Q8×C127—21no—
57C2×Q8×C127—22no—
58C2×Q8×C127—22no—
59C2×Q8×C127C2312yes—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62M16×C127—22no—
63C2×C8×C127C2312yes—
64C4⋊C4×C127C2312yes—
65C22⋊C4×C127—21no—
66C42×C127—21no—
67C22×D127—22no—
68C22×D127—24no—
69C22×D127—21no—
70C22×D127—82no—
71Dic254—21no—
72Dic254—21no—
73D508—24no—
74D508—21no—
75D508—22no—
76D508—21no—
77D508C2×D414yes—
78C22⋊D127—42no—
79C2×Dic127—21no—
80C2×Dic127C2×D414yes—
81C4×D127—42no—
82C4×D127—84no—
83C127⋊C8—22no—
84C23×C127C2×D411yescharacteristic
85Q8×C127—22no—
86Q8×C127—22no—
87Q8×C127—42no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
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#12112yes—
98C22×C127C22×D411yescharacteristic
99C2×D127—24no—
100C2×D127—44no—
101C2×D127—41no—
102C2×D127—42no—
103C2×D127—84no—
104C4×C127—42no—
105C4×C127—42no—
106C4×C127#12112yes—
107C4×C127C22×D411yescharacteristic
108C4×C127C2×#4111yescharacteristic
109Dic127—24no—
110Dic127—41no—
111C2×C127—22no—
112C2×C127C2×#12111yescharacteristic
113C2×C127#57712yes—
114D127—44no—
115D127—81no—
116#3042—1271nomaximal, Sylow
117C127#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1271no—
119C22×SD16—1271no—
120#536—1271no—
121#518—1271no—
122#463—1274no—
123#461—1274no—
124#424—1271no—
125#419D12711yescharacteristic, complemented
126#416—1271no—
127C2×#41—1271no—
128C22×Q8C2×D12711yescharacteristic
129C22×D4—1271no—
130AGL(1,ℤ/8)—2544no—
131C2×SD16—1274no—
132C2×SD16—1272no—
133C2×SD16—2544no—
134C2×D8—1272no—
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#117C2×D12711yescharacteristic
145C2×C22⋊C4—2542no—
146C4⋊C8—1274no—
147#103—1272no—
148#103C2×D12714yescomplemented
149#102—1272no—
150C24—2541no—
151#41—2542no—
152#41—2542no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1271no—
157C2×Q8C22×D12712yes—
158C2×D4—1274no—
159C2×D4—1272no—
160C2×D4—1271no—
161C2×D4—2541no—
162C2×D4—2541no—
163C2×D4—2542no—
164C22×C4—21no—
165C22×C4—2544no—
166C22×C4—2541no—
167C22×C4C22×D12711yescharacteristic
168SD16—2544no—
169SD16—2548no—
170D8—2544no—
171M16—22no—
172C2×C8—1272no—
173C2×C8—2542no—
174C2×C8C22×D12712yes—
175C4⋊C4—1272no—
176C4⋊C4—2544no—
177C4⋊C4C22×D12712yes—
178C22⋊C4—21no—
179C22⋊C4—1272no—
180C22⋊C4—2544no—
181C22⋊C4—2544no—
182C42—21no—
183C42—1272no—
184C23—10162no—
185C23—2544no—
186C23—2541no—
187C23—2542no—
188C23D4×D12711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—2541no—
193Q8—2541no—
194D4—1274no—
195D4—5082no—
196D4—2541no—
197D4—2541no—
198D4—2542no—
199D4—2544no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—1274no—
204C2×C4—10164no—
205C2×C4—5082no—
206C2×C4—2541no—
207C2×C4D4×D12712yes—
208C2×C4D4×D12711yescharacteristic
209C2×C4#1452324812yes—
210C2×C4C23×D12711yescharacteristic
211C8—22no—
212C8—2542no—
213C22—22no—
214C22—10164no—
215C22—5081no—
216C22—5082no—
217C22—5084no—
218C22—2544no—
219C22#1686118212yes—
220C22C2×D4×D12711yescharacteristic
221C4—42no—
222C4—42no—
223C4—5081no—
224C4—2544no—
225C4#1686118212yes—
226C4C2×D4×D12711yescharacteristic
227C4C2×#1452324811yescharacteristic
228C2—22no—
229C2—10161no—
230C2—5084no—
231C2C2×#1686118211yescharacteristic
232C2#2255068512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861237C2
hypercenter#419D127
hypocenterC127#3042
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#14523235C2semidirect
3. #22550450C2semidirect
4. #22550350C2semidirect
5. #22550094C2semidirect
6. #22550080C2semidirect
7. #22549904C2semidirect
8. #22549870C2semidirect
9. #419×C127C2semidirect
18. #16861207C22semidirect
19. #16861198C22semidirect
26. Dic127⋊C8C22semidirect
27. #16861132C22semidirect
28. #16861129C22semidirect
32. #103×C127C22semidirect
117. C127#3042semidirect
125. #419D127semidirect
148. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C22×#145232352
#225504502
#225503502
#225500942
#225500802
#225499042
#225498702
#419×C1272
#3042127