Finite Group Databasebeta

← #30637384

Subgroups — #30637384

Order 16256.

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

normal subgroups only

This group has 212 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#22549928C211yesmaximal, characteristic
3C4⋊C8×D127C211yesmaximal, characteristic
4#22549625C211yesmaximal, characteristic, complemented
5#22549623C211yesmaximal, characteristic, complemented
6#22549621C211yesmaximal, characteristic, complemented
7#333×C127C211yesmaximal, characteristic, complemented
8C2×C8×D127—21no—
9#16861209C2211yescharacteristic
10#16861200—21no—
11#16861189—21no—
12#16861185C412yes—
13Q8×Dic127C412yes—
14#16861174—21no—
15#16861167—21no—
16C42×D127C2211yescharacteristic
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19Dic127⋊C8—21no—
20(C2×D127)⋊C8—21no—
21C8×Dic127—21no—
22C4⋊Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25C2×Dic254—21no—
26C2×C4×D127C2×C411yescharacteristic
27C2×C4×D127D411yescharacteristic
28C2×C4×D127D411yescharacteristic
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C8×D127—42no—
32C4⋊Dic127—22no—
33C4⋊Dic127—21no—
34C4⋊Dic127C2×C412yes—
35Dic127⋊C4—22no—
36(C2×D127)⋊C4—22no—
37C4×Dic127—22no—
38C4×Dic127C2×C411yescharacteristic
39C4×Dic127D411yescharacteristic
40C4×Dic127D411yescharacteristic
41C2×Q8×C127C2×C412yes—
42C2×C8×C127—21no—
43C2×C8×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C42×C127C2311yescharacteristic
47C22×D127C22⋊C411yescharacteristic
48Dic254—41no—
49Dic254—41no—
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—42no—
62C2×C4×C127—22no—
63C2×C4×C127C22×C411yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C8×C127—41no—
67C8×C127—41no—
68C22×C127C2×C22⋊C411yescharacteristic
69C2×D127—21no—
70C2×D127—21no—
71C2×D127#10312yes—
72C4×C127—21no—
73C4×C127—42no—
74C4×C127C2×SD1611yescharacteristic
75C4×C127C2×Q1611yescharacteristic
76C4×C127#13511yescharacteristic
77C4×C127#13511yescharacteristic
78Dic127—21no—
79Dic127—21no—
80Dic127—42no—
81Dic127#10312yes—
82C2×C127#41311yescharacteristic
83C2×C127C2×#10311yescharacteristic
84C2×C127#41611yescharacteristic
85D127—22no—
86#1542—1271nomaximal, Sylow
87C127#154211yescharacteristic, complemented, Sylow
88#533—1271no—
89#431—1271no—
90C2×C4⋊C8—1271no—
91#333—1271no—
92#333D12711yescharacteristic, complemented
93#332—1271no—
94#332—1271no—
95C22×C8—2541no—
96C4⋊Q8—1271no—
97C4⋊Q8C2×D12711yescharacteristic
98#125—2541no—
99C22⋊Q8—1272no—
100C22⋊Q8—2541no—
101C4×Q8—1272no—
102C4×Q8—2541no—
103#117—2541no—
104C2×C42—1271no—
105C4⋊C8—1271no—
106C4⋊C8—1271no—
107C4⋊C8—2541no—
108C4⋊C8C2×D12711yescharacteristic
109C4⋊C8C2×D12711yescharacteristic
110C22⋊C8—2541no—
111C4×C8—2541no—
112C2×Q8—2541no—
113C2×Q8C4×D12712yes—
114C22×C4—1271no—
115C22×C4—1271no—
116C22×C4—1271no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—5082no—
120C2×C8—2541no—
121C2×C8—2541no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—1272no—
125C4⋊C4—2542no—
126C4⋊C4—2542no—
127C4⋊C4—2541no—
128C22⋊C4—2542no—
129C42—1271no—
130C42—1271no—
131C42—1271no—
132C42—2542no—
133C42C22×D12711yescharacteristic
134C23—1271no—
135Q8—42no—
136Q8—5081no—
137Q8—5081no—
138C2×C4—22no—
139C2×C4—1271no—
140C2×C4—1272no—
141C2×C4—1272no—
142C2×C4—1271no—
143C2×C4—1271no—
144C2×C4—5081no—
145C2×C4—2542no—
146C2×C4—2541no—
147C2×C4—2541no—
148C2×C4C2×C4×D12711yescharacteristic
149C2×C4D4×D12711yescharacteristic
150C2×C4D4×D12711yescharacteristic
151C8—41no—
152C8—41no—
153C8—5081no—
154C8—5081no—
155C22—1272no—
156C22—2541no—
157C22—2541no—
158C22C22⋊C4×D12711yescharacteristic
159C4—21no—
160C4—42no—
161C4—1272no—
162C4—5082no—
163C4—2541no—
164C4—2541no—
165C4SD16×D12711yescharacteristic
166C4Q16×D12711yescharacteristic
167C4#1686123011yescharacteristic
168C4#1686123211yescharacteristic
169C2—2542no—
170C2#2254985711yescharacteristic
171C2#103×D12711yescharacteristic
172C2#2254987111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#333×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D127C22⋊C4
cosocleC42×C127C23
hypercenter#333D127
hypocenterC127#1542
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
4. #22549625C2semidirect
5. #22549623C2semidirect
6. #22549621C2semidirect
7. #333×C127C2semidirect
87. C127#1542semidirect
92. #333D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
#225499282
C4⋊C8×D1272
#225496252
#225496232
#225496212
#333×C1272
#1542127