Finite Group Databasebeta

← #30637750

Subgroups — #30637750

Order 16256.

288 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550192C212yesmaximal, complemented
2#22549812C211yesmaximal, characteristic, complemented
3#22549610C212yesmaximal, complemented
4#22549600C211yesmaximal, characteristic, complemented
5#534×C127C211yesmaximal, characteristic
6C2×C4⋊Dic127C2211yescharacteristic, complemented
7C2×Dic127⋊C4—21no—
8C2×C22⋊Dic127—21no—
9#16861140—22no—
10#16861130—22no—
11C22⋊(C127⋊C8)C2212yescomplemented
12C127⋊#95C2211yescharacteristic, complemented
13C22×D4×C127—21no—
14#123×C127—22no—
15#121×C127—21no—
16#121×C127C2212yes—
17#120×C127—22no—
18C2×C22⋊C4×C127C2211yescharacteristic
19C22×Dic127—21no—
20C22×Dic127—21no—
21C2×C127⋊C8—22no—
22C4⋊Dic127—22no—
23Dic127⋊C4—42no—
24C22⋊Dic127—42no—
25C24×C127—21no—
26C24×C127D411yescharacteristic
27C2×D4×C127—22no—
28C2×D4×C127—22no—
29C2×D4×C127—21no—
30C2×D4×C127—42no—
31C2×D4×C127—42no—
32C2×D4×C127D412yes—
33C22×C4×C127—21no—
34C22×C4×C127D411yescharacteristic
35C22×C4×C127C2311yescharacteristic
36C4⋊C4×C127—21no—
37C4⋊C4×C127D412yes—
38C22⋊C4×C127—22no—
39C22⋊C4×C127—22no—
40C22⋊C4×C127—22no—
41C2×Dic127—22no—
42C2×Dic127—41no—
43C2×Dic127—41no—
44C2×Dic127—41no—
45C2×Dic127—41no—
46C2×Dic127—41no—
47C127⋊C8—42no—
48C23×C127—21no—
49C23×C127—21no—
50C23×C127—22no—
51C23×C127—41no—
52C23×C127—41no—
53C23×C127—42no—
54C23×C127—81no—
55C23×C127—81no—
56C23×C127SD1612yes—
57C23×C127C2×D411yescharacteristic
58D4×C127—42no—
59D4×C127—44no—
60D4×C127—42no—
61D4×C127—42no—
62D4×C127—81no—
63C2×C4×C127—21no—
64C2×C4×C127—22no—
65C2×C4×C127—41no—
66C2×C4×C127—42no—
67C2×C4×C127—41no—
68C2×C4×C127C2×D412yes—
69C22×C127—21no—
70C22×C127—21no—
71C22×C127—22no—
72C22×C127—42no—
73C22×C127—41no—
74C22×C127—41no—
75C22×C127—42no—
76C22×C127—42no—
77C22×C127—42no—
78C22×C127—81no—
79C22×C127—84no—
80C22×C127#12011yescharacteristic
81C22×C127C2×SD1611yescharacteristic
82C22×C127AGL(1,ℤ/8)11yescharacteristic
83C4×C127—22no—
84C4×C127—42no—
85C4×C127—41no—
86Dic127—42no—
87Dic127—81no—
88C2×C127—21no—
89C2×C127—21no—
90C2×C127—42no—
91C2×C127—82no—
92C2×C127#45011yescharacteristic
93C2×C127#45311yescharacteristic
94C2×C127#45811yescharacteristic
95#1613—1271nomaximal, Sylow
96C127#161311yescharacteristic, complemented, Sylow
97#534D12711yescharacteristic
98#481—1272no—
99#399—1271no—
100#327—1272no—
101#323—1271no—
102C22×D4—21no—
103#123—22no—
104#121—21no—
105#121C2×D12712yes—
106#120—22no—
107C2×C4⋊C4—1271no—
108C2×C4⋊C4—2541no—
109C2×C22⋊C4—2541no—
110C2×C22⋊C4C2×D12711yescharacteristic
111#106—2542no—
112#102—2542no—
113C22⋊C8—1272no—
114#95—1271no—
115C24—21no—
116C24C22⋊D12711yescharacteristic
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4C22⋊D12712yes—
123C22×C4—21no—
124C22×C4—2541no—
125C22×C4—2541no—
126C22×C4C22⋊D12711yescharacteristic
127C22×C4C22×D12711yescharacteristic
128C2×C8—2542no—
129C4⋊C4—21no—
130C4⋊C4—5082no—
131C4⋊C4—2542no—
132C4⋊C4C22⋊D12712yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—5082no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—81no—
144C23—81no—
145C23#1452323612yes—
146C23C2×C22⋊D12711yescharacteristic
147D4—42no—
148D4—44no—
149D4—42no—
150D4—42no—
151D4—81no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—5081no—
158C2×C4—5081no—
159C2×C4—5081no—
160C2×C4—5081no—
161C2×C4—5081no—
162C2×C4—2542no—
163C2×C4C2×C22⋊D12712yes—
164C8—5082no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—81no—
175C22—84no—
176C22#1686117811yescharacteristic
177C22C2×#1452323611yescharacteristic
178C22#1686123511yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—10161no—
183C4—5082no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—82no—
188C2#2255002711yescharacteristic
189C2#2255004411yescharacteristic
190C2#2255006611yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#534×C127C2
socleC22×C127#120
hypercenter#534D127
hypocenterC127#1613
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #22550192C2semidirect
2. #22549812C2semidirect
3. #22549610C2semidirect
4. #22549600C2semidirect
6. C2×C4⋊Dic127C22semidirect
11. C22⋊(C127⋊C8)C22semidirect
12. C127⋊#95C22semidirect
96. C127#1613semidirect

Maximal subgroups

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

subgroupindex
#225501922
#225498122
#225496102
#225496002
#534×C1272
#1613127