Finite Group Databasebeta

← #30637095

Subgroups — #30637095

Order 16256.

270 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 77 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×#16861188C211yesmaximal, characteristic
2C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
3#22549601C214yesmaximal, complemented
4#407×C127C211yesmaximal, characteristic
5C22×Dic254C411yescharacteristic
6C22×C127⋊C8—21no—
7#16861188—21no—
8#16861188—21no—
9#16861188—21no—
10#16861188—21no—
11C2×C4⋊Dic127C411yescharacteristic
12C2×Dic127⋊C4C2212yes—
13C2×C22⋊Dic127C412yes—
14C22⋊(C127⋊C8)—21no—
15C22⋊(C127⋊C8)C2212yes—
16C23×C4×C127C2211yescharacteristic
17C2×M16×C127—21no—
18C22⋊C8×C127—21no—
19C22⋊C8×C127C2212yes—
20C2×Dic254—21no—
21C2×Dic254—21no—
22C2×Dic254—42no—
23C22×Dic127C2×C412yes—
24C22×Dic127C2×C412yes—
25C2×C127⋊C8—22no—
26C2×C127⋊C8—42no—
27C4⋊Dic127—21no—
28C4⋊Dic127—21no—
29Dic127⋊C4—22no—
30Dic127⋊C4—22no—
31C22⋊Dic127—22no—
32C22⋊Dic127—22no—
33C24×C127C2×C411yescharacteristic
34C22×C4×C127—42no—
35C22×C4×C127C2×C411yescharacteristic
36C22×C4×C127D411yescharacteristic
37C22×C4×C127D411yescharacteristic
38C22×C4×C127D411yescharacteristic
39C22×C4×C127D411yescharacteristic
40C22×C4×C127C2311yescharacteristic
41M16×C127—42no—
42C2×C8×C127—22no—
43Dic254—42no—
44Dic254—42no—
45C2×Dic127—22no—
46C2×Dic127—22no—
47C2×Dic127—24no—
48C2×Dic127—42no—
49C127⋊C8—42no—
50C23×C127—22no—
51C23×C127—22no—
52C23×C127—41no—
53C23×C127C22⋊C411yescharacteristic
54C23×C127C22⋊C411yescharacteristic
55C23×C127C22×C411yescharacteristic
56C2×C4×C127—22no—
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—41no—
60C2×C4×C127—41no—
61C2×C4×C127—41no—
62C2×C4×C127C22⋊C411yescharacteristic
63C2×C4×C127C22⋊C411yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C8×C127—42no—
67C22×C127—21no—
68C22×C127—21no—
69C22×C127—21no—
70C22×C127—21no—
71C22×C127—21no—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127—44no—
75C22×C127#10112yes—
76C22×C127C2×C22⋊C411yescharacteristic
77C22×C127#13112yes—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127#9912yes—
82Dic127—42no—
83Dic127—42no—
84C2×C127—22no—
85C2×C127—22no—
86C2×C127—41no—
87C2×C127#40811yescharacteristic
88C2×C127C2×#9911yescharacteristic
89C2×C127C2×#10111yescharacteristic
90#1472—1271nomaximal, Sylow
91C127#147211yescharacteristic, complemented, Sylow
92C2×C22⋊Q8—1271no—
93#407D12711yescharacteristic
94C2×C22⋊C8—1271no—
95#324—1274no—
96C22×Q8—1271no—
97C23×C4C2×D12711yescharacteristic
98C2×M16—21no—
99C22×C8—2541no—
100C22⋊Q8—2541no—
101C22⋊Q8—2541no—
102C22⋊Q8—2541no—
103C22⋊Q8—2541no—
104C2×C4⋊C4—1271no—
105C2×C4⋊C4—1272no—
106C2×C22⋊C4—1272no—
107C22⋊C8—21no—
108C22⋊C8—1272no—
109C22⋊C8—2541no—
110C22⋊C8C2×D12712yes—
111C24C4×D12711yescharacteristic
112C2×Q8—5082no—
113C2×Q8—2541no—
114C2×Q8—2541no—
115C22×C4—42no—
116C22×C4—1272no—
117C22×C4—1272no—
118C22×C4C4×D12711yescharacteristic
119C22×C4C22⋊D12711yescharacteristic
120C22×C4C22⋊D12711yescharacteristic
121C22×C4D50811yescharacteristic
122C22×C4D50811yescharacteristic
123C22×C4C22×D12711yescharacteristic
124M16—42no—
125C2×C8—22no—
126C2×C8—5082no—
127C2×C8—2542no—
128C4⋊C4—2541no—
129C4⋊C4—2541no—
130C4⋊C4—2542no—
131C4⋊C4—2542no—
132C22⋊C4—2542no—
133C22⋊C4—2542no—
134C23—22no—
135C23—22no—
136C23—41no—
137C23(C2×D127)⋊C411yescharacteristic
138C23(C2×D127)⋊C411yescharacteristic
139C23C2×C4×D12711yescharacteristic
140Q8—5082no—
141Q8—5082no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—5082no—
149C2×C4—2544no—
150C2×C4—2542no—
151C2×C4—2542no—
152C2×C4(C2×D127)⋊C411yescharacteristic
153C2×C4(C2×D127)⋊C411yescharacteristic
154C2×C4C2×C22⋊D12711yescharacteristic
155C2×C4C2×D50811yescharacteristic
156C8—42no—
157C8—5082no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—44no—
166C22#1686112712yes—
167C22C2×(C2×D127)⋊C411yescharacteristic
168C22#1686121712yes—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—5082no—
173C4—5082no—
174C4#1686112312yes—
175C2—22no—
176C2—22no—
177C2—41no—
178C2#2254984411yescharacteristic
179C2C2×#1686112311yescharacteristic
180C2C2×#1686112711yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC24×C127C2×C4
cosocleC22×C4×C127C23
hypercenter#407D127
hypocenterC127#1472
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. #22549601C2semidirect
91. C127#1472semidirect

Maximal subgroups

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

subgroupindex
C2×#168611882
C2×C22⋊(C127⋊C8)2
#225496012
#407×C1272
#1472127