Finite Group Databasebeta

← #30637689

Subgroups — #30637689

Order 16256.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550502C211yesmaximal, characteristic
2#22550034C212yesmaximal, complemented
3#22549788C211yesmaximal, characteristic
4#22549614C212yesmaximal, complemented
5#323×C127C211yesmaximal, characteristic
6C22×Dic254C2211yescharacteristic
7C2×Dic508—22no—
8#16861188—21no—
9#16861188C2212yes—
10#16861186—22no—
11#16861186—22no—
12C2×C22⋊Dic127—21no—
13C2×C22⋊Dic127—21no—
14#16861134—22no—
15C127⋊#95—21no—
16C127⋊#95C2211yescharacteristic
17C2×C22⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19C2×Dic254—21no—
20C2×Dic254—22no—
21C2×Dic254—42no—
22C2×Dic254D412yes—
23C22×Dic127—21no—
24C22×Dic127—21no—
25C22×Dic127—21no—
26C22×Dic127—21no—
27Dic508—44no—
28C4⋊Dic127—21no—
29C4⋊Dic127D412yes—
30Dic127⋊C4—22no—
31Dic127⋊C4—22no—
32C22⋊Dic127—22no—
33C22⋊Dic127—22no—
34C22⋊Dic127—42no—
35C24×C127D411yescharacteristic
36C22×C4×C127D411yescharacteristic
37C22×C4×C127C2311yescharacteristic
38C2×C8×C127—22no—
39C22⋊C4×C127—22no—
40Dic254—24no—
41Dic254—42no—
42Dic254—42no—
43Dic254—81no—
44C2×Dic127—22no—
45C2×Dic127—22no—
46C2×Dic127—22no—
47C2×Dic127—41no—
48C2×Dic127—41no—
49C2×Dic127—41no—
50C2×Dic127—41no—
51C2×Dic127—41no—
52C2×Dic127—81no—
53C2×Dic127—81no—
54C23×C127—21no—
55C23×C127—21no—
56C23×C127—41no—
57C23×C127—41no—
58C23×C127Q1612yescomplemented
59C23×C127C2×D411yescharacteristic
60C2×C4×C127—21no—
61C2×C4×C127—41no—
62C2×C4×C127—41no—
63C2×C4×C127C2×D412yes—
64C8×C127—42no—
65C22×C127—21no—
66C22×C127—21no—
67C22×C127—22no—
68C22×C127—42no—
69C22×C127—41no—
70C22×C127—41no—
71C22×C127—81no—
72C22×C127#12011yescharacteristic
73C22×C127C2×Q1611yescharacteristic
74C22×C127#13711yescharacteristic
75C4×C127—22no—
76C4×C127—41no—
77Dic127—42no—
78Dic127—42no—
79Dic127—42no—
80Dic127—81no—
81C2×C127—21no—
82C2×C127—21no—
83C2×C127—42no—
84C2×C127#45111yescharacteristic
85C2×C127#45511yescharacteristic
86C2×C127B(4,𝔽2)11yescharacteristic
87#1602—1271nomaximal, Sylow
88C127#160211yescharacteristic, complemented, Sylow
89#543—1271no—
90#451—1272no—
91#393—1271no—
92#328—1272no—
93#323D12711yescharacteristic
94C22×Q8—1271no—
95C2×Q16—2542no—
96C22⋊Q8—1272no—
97C22⋊Q8—2542no—
98C22⋊Q8—2542no—
99C22⋊Q8—2541no—
100C2×C22⋊C4—2541no—
101C2×C22⋊C4—2541no—
102C2×C22⋊C4C2×D12711yescharacteristic
103#103—2542no—
104C22⋊C8C2×D12712yes—
105#95—1271no—
106#95—2541no—
107C24D50811yescharacteristic
108C2×Q8—1272no—
109C2×Q8—5082no—
110C2×Q8—2541no—
111C2×Q8—2542no—
112C22×C4—2541no—
113C22×C4—2541no—
114C22×C4—2541no—
115C22×C4—2541no—
116C22×C4D50811yescharacteristic
117C22×C4C22×D12711yescharacteristic
118Q16—5084no—
119C2×C8—22no—
120C4⋊C4—1272no—
121C4⋊C4—2542no—
122C4⋊C4—2541no—
123C4⋊C4—2542no—
124C22⋊C4—22no—
125C22⋊C4—5082no—
126C22⋊C4—2542no—
127C22⋊C4—2542no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic50812yescomplemented
133C23C2×D50811yescharacteristic
134Q8—10161no—
135Q8—5082no—
136Q8—5082no—
137Q8—2544no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—10161no—
142C2×C4—10161no—
143C2×C4—5081no—
144C2×C4—5081no—
145C2×C4—5081no—
146C2×C4—5081no—
147C2×C4—5081no—
148C2×C4—2542no—
149C2×C4—2542no—
150C2×C4—2542no—
151C2×C4D4×D12712yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1686117911yescharacteristic
161C22C2×Dic50811yescharacteristic
162C22#1686124211yescharacteristic
163C4—22no—
164C4—41no—
165C4—10161no—
166C4—5082no—
167C4—5082no—
168C4—5082no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2255003411yescharacteristic
173C2#2255005611yescharacteristic
174C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#323×C127C2
socleC22×C127#120
prime radicalC24×C127D4
hypercenter#323D127
hypocenterC127#1602
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22550034C2semidirect
4. #22549614C2semidirect
58. C23×C127Q16semidirect
88. C127#1602semidirect
132. C23Dic508semidirect

Maximal subgroups

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

subgroupindex
#225505022
#225500342
#225497882
#225496142
#323×C1272
#1602127