Finite Group Databasebeta

← #30637063

Subgroups — #30637063

Order 16256.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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×#16861167C211yesmaximal, characteristic
2#22549930C212yesmaximal, complemented
3#22549602C211yesmaximal, characteristic, complemented
4#22549599C212yesmaximal
5#324×C127C211yesmaximal, characteristic, complemented
6C22×C4×D127C411yescharacteristic
7#16861167—22no—
8#16861167—22no—
9C2×C4⋊Dic127C411yescharacteristic
10C2×(C2×D127)⋊C4C411yescharacteristic
11C2×(C2×D127)⋊C4C2211yescharacteristic
12C2×C4×Dic127C411yescharacteristic
13C2×C4×Dic127C2211yescharacteristic
14Dic127⋊C8—22no—
15C22⋊(C127⋊C8)C2212yes—
16#16861119—22no—
17C2×C4⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19C23×D127C2×C411yescharacteristic
20C22×Dic127C2×C411yescharacteristic
21C22×Dic127C2×C411yescharacteristic
22C22×Dic127C2×C411yescharacteristic
23C2×C4×D127—22no—
24C2×C4×D127—21no—
25C2×C4×D127—21no—
26C2×C4×D127—22no—
27C2×C127⋊C8—22no—
28C4⋊Dic127—22no—
29(C2×D127)⋊C4—22no—
30(C2×D127)⋊C4—22no—
31C4×Dic127—21no—
32C4×Dic127—21no—
33C4×Dic127D412yes—
34C4×Dic127D412yes—
35C22×C4×C127C2×C411yescharacteristic
36C22×C4×C127C2×C411yescharacteristic
37C22×C4×C127C2311yescharacteristic
38C2×C8×C127—22no—
39C4⋊C4×C127—22no—
40C22×D127—21no—
41C22×D127—21no—
42C22×D127—41no—
43C22×D127—41no—
44C22×D127M1612yes—
45C2×Dic127—21no—
46C2×Dic127—21no—
47C2×Dic127—21no—
48C2×Dic127—41no—
49C2×Dic127—41no—
50C2×Dic127C22⋊C411yescharacteristic
51C2×Dic127C22⋊C411yescharacteristic
52C2×Dic127C22⋊C412yes—
53C4×D127—24no—
54C4×D127—24no—
55C127⋊C8—42no—
56C23×C127C22×C411yescharacteristic
57C2×C4×C127—21no—
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127—41no—
61C2×C4×C127—41no—
62C2×C4×C127M1612yes—
63C2×C4×C127C2×D412yes—
64C8×C127—42no—
65C22×C127—21no—
66C22×C127—21no—
67C22×C127C2×C22⋊C411yescharacteristic
68C22×C127C2×M1611yescharacteristic
69C22×C127C2×M1611yescharacteristic
70C2×D127—22no—
71C2×D127—42no—
72C2×D127—41no—
73C2×D127—41no—
74C2×D127—42no—
75C4×C127—22no—
76C4×C127—22no—
77C4×C127—41no—
78Dic127—21no—
79Dic127—21no—
80Dic127—22no—
81Dic127—41no—
82C2×C127—21no—
83C2×C127—21no—
84C2×C127#40711yescharacteristic
85C2×C127#41011yescharacteristic
86C2×C127#41311yescharacteristic
87D127—42no—
88#1464—1271nomaximal, Sylow
89C127#146411yescharacteristic, complemented, Sylow
90C2×#117—1271no—
91#432—1272no—
92#324—1271no—
93#324D12711yescharacteristic, complemented
94#323—1272no—
95C23×C4—1271no—
96#117—2542no—
97#117—2542no—
98C2×C4⋊C4—1271no—
99C2×C4⋊C4C2×D12711yescharacteristic
100C2×C22⋊C4—1271no—
101C2×C22⋊C4—1271no—
102C2×C42—1271no—
103C2×C42—1271no—
104C4⋊C8—2542no—
105C22⋊C8—1272no—
106C22⋊C8C2×D12712yes—
107#97—2542no—
108C24—1271no—
109C22×C4—1271no—
110C22×C4—1271no—
111C22×C4—1271no—
112C22×C4—2541no—
113C22×C4—2541no—
114C22×C4—2542no—
115C22×C4—2542no—
116C22×C4C4×D12711yescharacteristic
117C22×C4C4×D12711yescharacteristic
118C22×C4C22×D12711yescharacteristic
119C2×C8—22no—
120C2×C8—2542no—
121C4⋊C4—22no—
122C4⋊C4—2542no—
123C22⋊C4—2542no—
124C22⋊C4—2542no—
125C42—1272no—
126C42—1272no—
127C42—2541no—
128C42—2541no—
129C23—1272no—
130C23—5081no—
131C23—5081no—
132C23—2541no—
133C23—2541no—
134C23C2×C4×D12711yescharacteristic
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—1271no—
141C2×C4—1271no—
142C2×C4—1272no—
143C2×C4—5081no—
144C2×C4—5081no—
145C2×C4—2544no—
146C2×C4—2544no—
147C2×C4—2541no—
148C2×C4—2541no—
149C2×C4—2541no—
150C2×C4#1452323112yes—
151C2×C4D4×D12712yes—
152C8—42no—
153C8—5082no—
154C22—21no—
155C22—21no—
156C22—5082no—
157C22—5082no—
158C22—5081no—
159C22—5081no—
160C22—2542no—
161C22C22⋊C4×D12711yescharacteristic
162C22M16×D12711yescharacteristic
163C22C2×#1452323111yescharacteristic
164C4—22no—
165C4—22no—
166C4—41no—
167C4—5081no—
168C4—2541no—
169C4—2541no—
170C4—2542no—
171C2—21no—
172C2—21no—
173C2—5082no—
174C2#2254983811yescharacteristic
175C2#2254984811yescharacteristic
176C2#2254985911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×D127C2×C4
cosocleC22×C4×C127C23
hypercenter#324D127
hypocenterC127#1464
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #22549930C2semidirect
3. #22549602C2semidirect
5. #324×C127C2semidirect
89. C127#1464semidirect
93. #324D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611672
#225499302
#225496022
#225495992
#324×C1272
#1464127