Finite Group Databasebeta

← #30637419

Subgroups — #30637419

Order 16256.

216 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 73 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#22550431C211yesmaximal, characteristic
2#22549898C211yesmaximal, characteristic
3C4⋊C8×D127C211yesmaximal, characteristic
4#22549624C211yesmaximal, characteristic, complemented
5#22549624C211yesmaximal, characteristic, complemented
6#22549622C211yesmaximal, characteristic, complemented
7#332×C127C211yesmaximal, characteristic, complemented
8C2×#14523231—21no—
9C2×C8×D127—21no—
10#16861209C2211yescharacteristic
11#16861200—21no—
12#16861189C412yes—
13#16861185—21no—
14Q8×Dic127—21no—
15#16861174C412yes—
16#16861167—21no—
17C42×D127C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19C4⋊(C127⋊C8)C2211yescharacteristic
20Dic127⋊C8—21no—
21Dic127⋊C8—21no—
22C4⋊Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25C2×Dic254C2×C412yes—
26C2×C4×D127C2×C411yescharacteristic
27C2×C4×D127D411yescharacteristic
28C2×C4×D127D411yescharacteristic
29#14523231—42no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C8×D127—42no—
33C4⋊Dic127—22no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36Dic127⋊C4—22no—
37(C2×D127)⋊C4—22no—
38C4×Dic127—22no—
39C4×Dic127C2×C411yescharacteristic
40C4×Dic127D411yescharacteristic
41C4×Dic127D411yescharacteristic
42C2×Q8×C127—21no—
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127C2×C412yes—
47C42×C127C2311yescharacteristic
48C22×D127C22⋊C411yescharacteristic
49Dic254—42no—
50C2×Dic127—22no—
51C2×Dic127C22⋊C411yescharacteristic
52C2×Dic127C22⋊C411yescharacteristic
53C2×Dic127C22⋊C411yescharacteristic
54C4×D127—21no—
55C4×D127—21no—
56C4×D127—41no—
57C4×D127SD1612yes—
58C4×D127Q1612yes—
59C127⋊C8—41no—
60C127⋊C8—41no—
61Q8×C127—41no—
62Q8×C127—41no—
63C2×C4×C127—22no—
64C2×C4×C127C22×C411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C8×C127—41no—
68C8×C127—41no—
69C22×C127C2×C22⋊C411yescharacteristic
70C2×D127—21no—
71C2×D127—21no—
72C2×D127#10312yes—
73C4×C127—21no—
74C4×C127—42no—
75C4×C127C2×SD1611yescharacteristic
76C4×C127C2×Q1611yescharacteristic
77C4×C127AGL(1,ℤ/8)11yescharacteristic
78C4×C127#13711yescharacteristic
79Dic127—21no—
80Dic127—21no—
81Dic127—42no—
82Dic127#10312yes—
83C2×C127#41311yescharacteristic
84C2×C127C2×#10311yescharacteristic
85C2×C127#41711yescharacteristic
86D127—22no—
87#1550—1271nomaximal, Sylow
88C127#155011yescharacteristic, complemented, Sylow
89#533—1271no—
90#423—1271no—
91C2×C4⋊C8—1271no—
92#333—1271no—
93#333—1271no—
94#332—1271no—
95#332D12711yescharacteristic, complemented
96C2×M16—2541no—
97C22×C8—2541no—
98C4⋊Q8—1271no—
99C4⋊Q8C2×D12711yescharacteristic
100#125—2541no—
101C22⋊Q8—1272no—
102C22⋊Q8—2541no—
103C4×Q8—1272no—
104C4×Q8—2541no—
105#117—2541no—
106C2×C42—1271no—
107C4⋊C8—1271no—
108C4⋊C8—1271no—
109C4⋊C8—2541no—
110C4⋊C8—2541no—
111C4⋊C8C2×D12711yescharacteristic
112C4⋊C8C2×D12711yescharacteristic
113C2×Q8—21no—
114C2×Q8—1272no—
115C22×C4—1271no—
116C22×C4—1271no—
117C22×C4—1271no—
118M16—5082no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—5082no—
122C2×C8—2541no—
123C2×C8—2541no—
124C4⋊C4—21no—
125C4⋊C4—2542no—
126C4⋊C4—2542no—
127C4⋊C4—2541no—
128C4⋊C4—2541no—
129C4⋊C4C4×D12712yes—
130C22⋊C4—2542no—
131C42—1271no—
132C42—1271no—
133C42—1271no—
134C42—2542no—
135C42C22×D12711yescharacteristic
136C23—1271no—
137Q8—41no—
138Q8—41no—
139Q8—5082no—
140C2×C4—22no—
141C2×C4—1271no—
142C2×C4—1272no—
143C2×C4—1272no—
144C2×C4—1271no—
145C2×C4—1271no—
146C2×C4—5081no—
147C2×C4—2542no—
148C2×C4—2541no—
149C2×C4—2541no—
150C2×C4C2×C4×D12711yescharacteristic
151C2×C4D4×D12711yescharacteristic
152C2×C4D4×D12711yescharacteristic
153C8—41no—
154C8—41no—
155C8—5081no—
156C8—5081no—
157C22—1272no—
158C22—2541no—
159C22—2541no—
160C22C22⋊C4×D12711yescharacteristic
161C4—21no—
162C4—42no—
163C4—1272no—
164C4—5082no—
165C4—2541no—
166C4—2541no—
167C4SD16×D12711yescharacteristic
168C4Q16×D12711yescharacteristic
169C4#1686123311yescharacteristic
170C4#1686123811yescharacteristic
171C2—2542no—
172C2#2254985911yescharacteristic
173C2#103×D12711yescharacteristic
174C2#2254987411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#332×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D127C22⋊C4
cosocleC42×C127C23
hypercenter#332D127
hypocenterC127#1550
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
4. #22549624C2semidirect
5. #22549624C2semidirect
6. #22549622C2semidirect
7. #332×C127C2semidirect
88. C127#1550semidirect
95. #332D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
#225498982
C4⋊C8×D1272
#225496242
#225496222
#332×C1272
#1550127