Finite Group Databasebeta

← #30648238

Subgroups — #30648238

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
1C2×#16861234C211yesmaximal, characteristic
2C22×#14523235C211yesmaximal, characteristic, complemented
3#22550350C211yesmaximal, characteristic, complemented
4#22550088C214yesmaximal, complemented
5#22550078C214yesmaximal, complemented
6#22549903C211yesmaximal, characteristic, complemented
7#22549885C211yesmaximal, characteristic, complemented
8#22549872C211yesmaximal, characteristic, complemented
9#536×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11#16861234—24no—
12C2×#14523235—24no—
13C2×#14523235C2212yes—
14C2×#14523235C2214yes—
15C2×#14523233C2212yes—
16C2×#14523232C2211yescharacteristic
17C22×C127⋊C8C2211yescharacteristic
18C4×D508C2214yes—
19#16861171—22no—
20#16861168C2211yescharacteristic
21C22⋊C4×D127—22no—
22C4⋊(C127⋊C8)C2214yescomplemented
23#16861133C2212yescomplemented
24#16861133C2214yescomplemented
25#16861130C2212yescomplemented
26C2×#41×C127C2211yescharacteristic
27C22×Q8×C127C2211yescharacteristic
28C4⋊Q8×C127C2212yescomplemented
29#124×C127C2212yescomplemented
30#123×C127—22no—
31C22⋊Q8×C127—22no—
32#117×C127C2211yescharacteristic
33C23×D127—21no—
34C2×D508—21no—
35C2×D508—22no—
36C2×D508D414yes—
37C2×D508C2312yes—
38C2×C4×D127—24no—
39#14523235—28no—
40#14523235—24no—
41#14523233—24no—
42#14523232—22no—
43C2×C127⋊C8—22no—
44C2×C127⋊C8C2312yes—
45C2×C127⋊C8C2312yes—
46C4⋊Dic127C2312yes—
47C22⋊Dic127—21no—
48(C2×D127)⋊C4—24no—
49C4×Dic127—21no—
50#41×C127—22no—
51#41×C127—22no—
52C2×Q8×C127—21no—
53C2×Q8×C127—22no—
54C2×Q8×C127—22no—
55C2×Q8×C127C2312yes—
56C2×Q8×C127C2311yescharacteristic
57C2×D4×C127—21no—
58C2×D4×C127C2311yescharacteristic
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62C4⋊C4×C127—24no—
63C4⋊C4×C127D412yes—
64C22⋊C4×C127—24no—
65C22⋊C4×C127D412yes—
66C42×C127C2312yes—
67C22×D127—22no—
68C22×D127—24no—
69C22×D127—82no—
70D508—24no—
71D508—22no—
72D508C2×D414yes—
73C2×Dic127—22no—
74C4×D127—84no—
75C127⋊C8—22no—
76C127⋊C8—22no—
77C23×C127—21no—
78C23×C127C2×D411yescharacteristic
79Q8×C127—22no—
80Q8×C127—21no—
81Q8×C127—22no—
82Q8×C127—21no—
83Q8×C127—42no—
84D4×C127—21no—
85D4×C127—21no—
86D4×C127—42no—
87C2×C4×C127—22no—
88C2×C4×C127—21no—
89C2×C4×C127—42no—
90C2×C4×C127—42no—
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2×D414yes—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127#4112yes—
95C2×C4×C127C2411yescharacteristic
96C22×C127—22no—
97C22×C127—41no—
98C22×C127—42no—
99C22×C127#12112yes—
100C22×C127C22×D411yescharacteristic
101C2×D127—24no—
102C2×D127—44no—
103C2×D127—84no—
104C4×C127—24no—
105C4×C127—41no—
106C4×C127—42no—
107C4×C127#12112yes—
108C4×C127C22×D411yescharacteristic
109C4×C127C2×#4111yescharacteristic
110Dic127—42no—
111C2×C127—22no—
112C2×C127—81no—
113C2×C127C2×#12111yescharacteristic
114C2×C127#57712yes—
115D127—44no—
116#3042—1271nomaximal, Sylow
117C127#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1271no—
119C22×SD16—1271no—
120#536D12711yescharacteristic, complemented
121#518—1271no—
122#463—1274no—
123#461—1274no—
124#424—1271no—
125#419—1271no—
126#416—1271no—
127C2×#41C2×D12711yescharacteristic
128C22×Q8C2×D12711yescharacteristic
129C22×D4—1271no—
130AGL(1,ℤ/8)—2544no—
131C2×SD16—1274no—
132C2×SD16—1272no—
133C2×SD16—2544no—
134C2×D8—1272no—
135C2×M16—1271no—
136C22×C8—1271no—
137C4⋊Q8C2×D12712yescomplemented
138#124C2×D12712yescomplemented
139#123—22no—
140C22⋊Q8—22no—
141C4×D4—1274no—
142C4×D4—2542no—
143#117—1271no—
144#117C2×D12711yescharacteristic
145C2×C22⋊C4—2542no—
146C4⋊C8—1274no—
147#103—1272no—
148#103—1274no—
149#102—1272no—
150C24—2541no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8C22×D12712yes—
157C2×Q8C22×D12711yescharacteristic
158C2×D4—21no—
159C2×D4—1274no—
160C2×D4—1272no—
161C2×D4—2541no—
162C2×D4—2542no—
163C2×D4C22×D12711yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—2544no—
167C22×C4C22×D12711yescharacteristic
168SD16—2548no—
169SD16—2544no—
170D8—2544no—
171M16—2542no—
172C2×C8—1272no—
173C2×C8—1272no—
174C2×C8—2542no—
175C4⋊C4—24no—
176C4⋊C4—1272no—
177C4⋊C4C22⋊D12712yes—
178C22⋊C4—24no—
179C22⋊C4—2541no—
180C22⋊C4—2544no—
181C22⋊C4C22⋊D12712yes—
182C42—2541no—
183C42C22×D12712yes—
184C23—21no—
185C23—10162no—
186C23—2544no—
187C23—2542no—
188C23C2×C22⋊D12711yescharacteristic
189Q8—22no—
190Q8—21no—
191Q8—22no—
192Q8—21no—
193Q8—42no—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—1274no—
198D4—2542no—
199D4—2544no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—10164no—
205C2×C4—2542no—
206C2×C4D4×D12712yes—
207C2×C4C2×C22⋊D12714yes—
208C2×C4C2×C22⋊D12711yescharacteristic
209C2×C4#1452324712yes—
210C2×C4C23×D12711yescharacteristic
211C8—2542no—
212C8—2542no—
213C22—22no—
214C22—41no—
215C22—42no—
216C22—10164no—
217C22—5084no—
218C22—2544no—
219C22#1686118012yes—
220C22C22×C22⋊D12711yescharacteristic
221C4—24no—
222C4—41no—
223C4—42no—
224C4—5082no—
225C4#1686118012yes—
226C4C2×D4×D12711yescharacteristic
227C4C2×#1452324711yescharacteristic
228C2—22no—
229C2—81no—
230C2—5084no—
231C2C2×#1686118011yescharacteristic
232C2#2255068412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#536×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861234C2
hypercenter#536D127
hypocenterC127#3042
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C22×#14523235C2semidirect
3. #22550350C2semidirect
4. #22550088C2semidirect
5. #22550078C2semidirect
6. #22549903C2semidirect
7. #22549885C2semidirect
8. #22549872C2semidirect
9. #536×C127C2semidirect
22. C4⋊(C127⋊C8)C22semidirect
23. #16861133C22semidirect
24. #16861133C22semidirect
25. #16861130C22semidirect
28. C4⋊Q8×C127C22semidirect
29. #124×C127C22semidirect
117. C127#3042semidirect
120. #536D127semidirect
137. C4⋊Q8C2×D127semidirect
138. #124C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C22×#145232352
#225503502
#225500882
#225500782
#225499032
#225498852
#225498722
#536×C1272
#3042127