Finite Group Databasebeta

← #30640677

Subgroups — #30640677

Order 16256.

226 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 47 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
1#22550673C211yesmaximal, characteristic, complemented
2#22550314C211yesmaximal, characteristic, complemented
3#22550311C211yesmaximal, characteristic, complemented
4#22550130C211yesmaximal, characteristic, complemented
5#22549685C211yesmaximal, characteristic, complemented
6#22549657C211yesmaximal, characteristic, complemented
7#360×C127C211yesmaximal, characteristic, complemented
8#41×D127—21no—
9#16861241—21no—
10#16861238—21no—
11#16861236—21no—
12#16861233—21no—
13#16861231C2211yescharacteristic
14#16861229—21no—
15C2×D1016C2211yescharacteristic, complemented
16C2×#14523231C2211yescharacteristic
17#16861184—21no—
18#16861154—21no—
19#16861152—21no—
20D2032—21no—
21#16861150—21no—
22#16861149C2211yescharacteristic, complemented
23#16861129—21no—
24#135×C127C2211yescharacteristic
25M32×C127C2211yescharacteristic, complemented
26#106×C127C2211yescharacteristic, complemented
27#14523249—21no—
28#14523248—21no—
29#14523248—41no—
30#14523247—21no—
31#14523247—41no—
32Q8×D127—21no—
33C2×D508—21no—
34C2×D508—41no—
35D4×D127—21no—
36D4×D127—41no—
37C2×C4×D127—41no—
38C2×C4×D127D411yescharacteristic
39Dic508D411yescharacteristic, complemented
40#14523238—21no—
41#14523237—21no—
42#14523236—21no—
43#14523235—21no—
44D1016—21no—
45D1016—21no—
46D1016D411yescharacteristic, complemented
47#14523233—21no—
48#14523231—21no—
49#14523231—21no—
50C2×C127⋊C8D411yescharacteristic
51(C2×D127)⋊C4—41no—
52C127⋊C16—21no—
53#41×C127—21no—
54Q16×C127D411yescharacteristic, complemented
55SD16×C127—21no—
56D8×C127D411yescharacteristic, complemented
57C2×C8×C127C2311yescharacteristic
58C4⋊C4×C127—21no—
59C16×C127—21no—
60C22×D127—41no—
61C22×D127—41no—
62C22×D127D811yescharacteristic
63Dic254—21no—
64Dic254—41no—
65D508—21no—
66D508—41no—
67D508—41no—
68D508—41no—
69D508—82no—
70C22⋊D127—41no—
71C22⋊D127—41no—
72C22⋊D127—41no—
73C2×Dic127—41no—
74C2×Dic127D811yescharacteristic
75C4×D127—21no—
76C4×D127—21no—
77C4×D127—41no—
78C4×D127—41no—
79C4×D127—41no—
80C127⋊C8—21no—
81Q8×C127—21no—
82D4×C127—21no—
83D4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127C2×D411yescharacteristic
87C8×C127C2×D411yescharacteristic
88C8×C127C2×D411yescharacteristic
89C22×C127—41no—
90C22×C127C2×D811yescharacteristic
91C2×D127—21no—
92C2×D127—41no—
93C2×D127—41no—
94C2×D127—82no—
95C2×D127—81no—
96C2×D127—81no—
97C2×D127—81no—
98C4×C127—41no—
99C4×C127—81no—
100C4×C127#12011yescharacteristic
101C4×C127AGL(1,ℤ/8)11yescharacteristic
102Dic127—21no—
103Dic127—41no—
104C2×C127—21no—
105C2×C127—81no—
106C2×C127#44711yescharacteristic
107D127—41no—
108D127—81no—
109D127—161no—
110#2190—1271nomaximal, Sylow
111C127#219011yescharacteristic, complemented, Sylow
112#575—1271no—
113#509—1271no—
114#509—1271no—
115#469—1271no—
116#360—1271no—
117#360D12711yescharacteristic, complemented
118#349—1271no—
119C2×#41—2541no—
120#137—2541no—
121#137—2541no—
122AGL(1,ℤ/8)—2541no—
123AGL(1,ℤ/8)—2541no—
124#135—1271no—
125#135—2541no—
126#135C2×D12711yescharacteristic
127C2×D8—1271no—
128C2×M16—1271no—
129#121—2541no—
130SD32—2541no—
131SD32—2541no—
132D16—2541no—
133D16—2541no—
134M32—1271no—
135M32C2×D12711yescharacteristic, complemented
136#106C2×D12711yescharacteristic, complemented
137#102—2541no—
138#41—21no—
139#41—5081no—
140#41—5081no—
141#41—2541no—
142#41—2541no—
143#41—2541no—
144C2×Q8—2541no—
145C2×D4—5081no—
146C2×D4—5081no—
147C2×D4—2541no—
148C2×D4—2541no—
149C22×C4—1271no—
150C22×C4—5081no—
151Q16—1271no—
152Q16—2541no—
153Q16D50811yescharacteristic, complemented
154SD16—21no—
155SD16—2541no—
156SD16—2541no—
157SD16—2541no—
158D8—1271no—
159D8—2541no—
160D8—2541no—
161D8—2541no—
162D8D50811yescharacteristic, complemented
163M16—2541no—
164M16—2541no—
165C2×C8—1271no—
166C2×C8C22×D12711yescharacteristic
167C4⋊C4—21no—
168C22⋊C4—5081no—
169C16—21no—
170C16—2541no—
171C23—1271no—
172C23—5081no—
173C23—5081no—
174Q8—21no—
175Q8—5081no—
176Q8—2541no—
177D4—21no—
178D4—41no—
179D4—10162no—
180D4—5081no—
181D4—5081no—
182D4—5081no—
183D4—5081no—
184D4—5081no—
185D4—5081no—
186D4—2541no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1271no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4D4×D12711yescharacteristic
197C8—2541no—
198C8D4×D12711yescharacteristic
199C8C2×D50811yescharacteristic
200C22—41no—
201C22—10161no—
202C22—10161no—
203C22—10162no—
204C22—10161no—
205C22—5081no—
206C22—5081no—
207C22—2541no—
208C22D8×D12711yescharacteristic
209C4—41no—
210C4—81no—
211C4—5081no—
212C4—2541no—
213C4#1686117911yescharacteristic
214C4#1686123611yescharacteristic
215C2—21no—
216C2—81no—
217C2—20321no—
218C2—10161no—
219C2—5081no—
220C2#2255001111yescharacteristic
subgroupG/H
centerC2#22550011
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#360×C127C2
socleC2×C127#447
hypercenter#360D127
hypocenterC127#2190
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
1. #22550673C2semidirect
2. #22550314C2semidirect
3. #22550311C2semidirect
4. #22550130C2semidirect
5. #22549685C2semidirect
6. #22549657C2semidirect
7. #360×C127C2semidirect
15. C2×D1016C22semidirect
22. #16861149C22semidirect
25. M32×C127C22semidirect
26. #106×C127C22semidirect
39. Dic508D4semidirect
46. D1016D4semidirect
54. Q16×C127D4semidirect
56. D8×C127D4semidirect
111. C127#2190semidirect
117. #360D127semidirect
135. M32C2×D127semidirect
136. #106C2×D127semidirect
153. Q16D508semidirect
162. D8D508semidirect

Maximal subgroups

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

subgroupindex
#225506732
#225503142
#225503112
#225501302
#225496852
#225496572
#360×C1272
#2190127