Finite Group Databasebeta

← #30639248

Subgroups — #30639248

Order 16256.

312 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861240C211yesmaximal, characteristic
2C2×#16861215C211yesmaximal, characteristic
3#22549875C211yesmaximal, characteristic
4C2×#16861128C211yesmaximal, characteristic, complemented
5#22549645C211yesmaximal, characteristic, complemented
6#22549641C211yesmaximal, characteristic, complemented
7C2×#103×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—22no—
10#16861240C414yescomplemented
11C2×#14523238—21no—
12C2×#14523235—21no—
13#16861215—22no—
14#16861215—22no—
15C2×#14523232C2211yescharacteristic
16C2×#14523232C2211yescharacteristic
17C2×#14523232C2211yescharacteristic
18C2×#14523231—21no—
19C2×C8×D127—21no—
20#16861132—21no—
21#16861129—21no—
22#16861128—22no—
23C22×Q8×C127C2211yescharacteristic
24C22×C8×C127C2211yescharacteristic
25C2×C4⋊C4×C127C2211yescharacteristic
26#103×C127—21no—
27#103×C127—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248C2×C412yes—
31#14523248D412yes—
32C2×Dic254D411yescharacteristic
33C2×D508D411yescharacteristic
34C2×C22⋊D127—21no—
35C2×C4×D127—21no—
36#14523238—24no—
37#14523235—24no—
38#14523232—21no—
39#14523232—21no—
40#14523232—21no—
41#14523232—21no—
42#14523232C2×C412yes—
43#14523232D412yes—
44#14523231—24no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8—21no—
47C2×C127⋊C8—21no—
48C8×D127—24no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127C2×C412yes—
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58C2×C8×C127—21no—
59C2×C8×C127—21no—
60C2×C8×C127D411yescharacteristic
61C2×C8×C127D411yescharacteristic
62C4⋊C4×C127—21no—
63C4⋊C4×C127—21no—
64C22×D127—21no—
65Dic254—21no—
66Dic254C22⋊C412yes—
67D508—21no—
68D508C22⋊C412yes—
69C22⋊D127—22no—
70C22⋊D127—22no—
71C2×Dic127—21no—
72C4×D127—22no—
73C4×D127—22no—
74C127⋊C8—22no—
75C127⋊C8—22no—
76C127⋊C8—41no—
77C23×C127#4111yescharacteristic
78Q8×C127—22no—
79Q8×C127—22no—
80Q8×C127—41no—
81Q8×C127—41no—
82C2×C4×C127—22no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127—81no—
87C2×C4×C127C22×C411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127#4111yescharacteristic
93C8×C127—21no—
94C8×C127—21no—
95C22×C127—21no—
96C22×C127—21no—
97C22×C127C4×D411yescharacteristic
98C22×C127#12111yescharacteristic
99C22×C127#12311yescharacteristic
100C2×D127—22no—
101C2×D127—41no—
102C2×D127—41no—
103C4×C127—42no—
104C4×C127—81no—
105C4×C127C2×C22⋊C411yescharacteristic
106C4×C127C4×D411yescharacteristic
107C4×C127#12011yescharacteristic
108C4×C127#12111yescharacteristic
109Dic127—22no—
110C2×C127—21no—
111C2×C127—21no—
112C2×C127#38611yescharacteristic
113C2×C127#47111yescharacteristic
114C2×C127#47311yescharacteristic
115D127—42no—
116#1908—1271nomaximal, Sylow
117C127#190811yescharacteristic, complemented, Sylow
118C2×#137—1271no—
119C2×#131—1271no—
120#417—1271no—
121C2×#103D12711yescharacteristic, complemented
122C2×#101—1271no—
123#343—1271no—
124#341—1271no—
125C2×#41—1271no—
126C22×Q8C2×D12711yescharacteristic
127#137—1274no—
128#137—2542no—
129C2×Q16—2541no—
130C2×SD16—2541no—
131#131—2542no—
132#131—2542no—
133C2×M16—1271no—
134C2×M16—1271no—
135C2×M16—1271no—
136C2×M16—2541no—
137C22×C8—2541no—
138C22×C8C2×D12711yescharacteristic
139C2×C4⋊C4C2×D12711yescharacteristic
140#103—21no—
141#103—21no—
142#103—2541no—
143#102—2541no—
144#101—2542no—
145#41—1272no—
146#41—1272no—
147#41—2541no—
148#41—2541no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—1271no—
155C2×Q8C4×D12712yes—
156C2×D4—1271no—
157C2×D4—2541no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2541no—
161C22×C4C22×D12711yescharacteristic
162Q16—2544no—
163SD16—2544no—
164M16—1272no—
165M16—1272no—
166M16—2544no—
167M16—2541no—
168M16—2541no—
169M16—2541no—
170M16—2541no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—2544no—
174C2×C8—2541no—
175C2×C8—2541no—
176C2×C8—2541no—
177C2×C8C22⋊D12711yescharacteristic
178C2×C8C22⋊D12711yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—21no—
181C23—2541no—
182C23#1452324711yescharacteristic
183Q8—22no—
184Q8—22no—
185Q8—41no—
186Q8—41no—
187Q8—1272no—
188Q8—2541no—
189D4—1272no—
190D4—2542no—
191D4—2542no—
192D4—2541no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—81no—
198C2×C4—2541no—
199C2×C4—2542no—
200C2×C4—2542no—
201C2×C4C2×C4×D12711yescharacteristic
202C2×C4D4×D12711yescharacteristic
203C2×C4D4×D12711yescharacteristic
204C2×C4D4×D12711yescharacteristic
205C2×C4C2×C22⋊D12711yescharacteristic
206C2×C4#1452324811yescharacteristic
207C8—21no—
208C8—21no—
209C8—5081no—
210C8—2542no—
211C8—2542no—
212C22—21no—
213C22—21no—
214C22—5081no—
215C22—5081no—
216C22—2542no—
217C22(C22⋊D127)⋊C411yescharacteristic
218C22#1686118011yescharacteristic
219C22#1686119111yescharacteristic
220C4—42no—
221C4—81no—
222C4—2542no—
223C4C22⋊C4×D12711yescharacteristic
224C4C4×C22⋊D12711yescharacteristic
225C4#1686117711yescharacteristic
226C4#1686118211yescharacteristic
227C2—21no—
228C2—21no—
229C2—5082no—
230C2#2254975211yescharacteristic
231C2#2255013611yescharacteristic
232C2#2255014811yescharacteristic
subgroupG/H
centerC22#16861180
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socleC22×C127#121
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenterC2×#103D127
hypocenterC127#1908
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
4. C2×#16861128C2semidirect
5. #22549645C2semidirect
6. #22549641C2semidirect
7. C2×#103×C127C2semidirect
10. #16861240C4semidirect
117. C127#1908semidirect
121. C2×#103D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612152
#225498752
C2×#168611282
#225496452
#225496412
C2×#103×C1272
#1908127