Finite Group Databasebeta

← C2×#22550171 #30648501

Subgroups — C2×#22550171 #30648501

Order 16256.

416 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 181 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×#16861209C211yesmaximal, characteristic
2#22550171C218yesmaximal, direct product factor
3C2×#16861142C212yesmaximal
4C2×C4⋊(C127⋊C8)C211yesmaximal, characteristic
5C2×#16861133C212yesmaximal
6C2×C4×Q8×C127C211yesmaximal, characteristic
7C22×Dic254—21no—
8C22×C127⋊C8C2212yes—
9#16861209—22no—
10#16861209C2214yes—
11C2×C4⋊Dic127—21no—
12C2×C4⋊Dic127C2212yes—
13#16861142C2218yes—
14C4⋊(C127⋊C8)C2214yes—
15#16861133C2218yes—
16C22×Q8×C127C2211yescharacteristic
17C4×Q8×C127—22no—
18C4×Q8×C127C2214yes—
19C2×C4⋊C4×C127—21no—
20C2×C4⋊C4×C127C2211yescharacteristic
21C2×C42×C127—21no—
22C2×C42×C127C2211yescharacteristic
23C2×Dic254—24no—
24C2×Dic254—44no—
25C22×Dic127—22no—
26C2×C127⋊C8—24no—
27C2×C127⋊C8C2314yes—
28C4⋊Dic127—22no—
29C4⋊Dic127—24no—
30C4⋊Dic127C2314yes—
31C2×Q8×C127—21no—
32C2×Q8×C127—22no—
33C2×Q8×C127Q814yescomplemented
34C2×Q8×C127C2312yes—
35C22×C4×C127—21no—
36C22×C4×C127—21no—
37C22×C4×C127D411yescharacteristic
38C22×C4×C127D411yescharacteristic
39C22×C4×C127C2311yescharacteristic
40C4⋊C4×C127—24no—
41C4⋊C4×C127—21no—
42C4⋊C4×C127C2312yes—
43C42×C127—24no—
44C42×C127D412yes—
45C42×C127C2312yes—
46Dic254—48no—
47C2×Dic127—24no—
48C2×Dic127—44no—
49C127⋊C8—24no—
50C23×C127C2×D411yescharacteristic
51Q8×C127—24no—
52Q8×C127—22no—
53Q8×C127C2×Q814yescomplemented
54C2×C4×C127—24no—
55C2×C4×C127—22no—
56C2×C4×C127—22no—
57C2×C4×C127—22no—
58C2×C4×C127—42no—
59C2×C4×C127Q1614yes—
60C2×C4×C127C2×D412yes—
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×D412yes—
63C2×C4×C127C2×Q812yes—
64C2×C4×C127#4112yes—
65C2×C4×C127C2411yescharacteristic
66C22×C127C22⋊Q812yes—
67C22×C127C2×Q1612yes—
68C22×C127AGL(1,ℤ/8)12yes—
69C22×C127C22×D411yescharacteristic
70C4×C127—24no—
71C4×C127—22no—
72C4×C127—42no—
73C4×C127C22⋊Q812yes—
74C4×C127C2×Q1614yes—
75C4×C127C22×Q811yescharacteristic
76C4×C127C2×#4111yescharacteristic
77Dic127—44no—
78C2×C127#47714yescomplemented
79C2×C127C2×C22⋊Q811yescharacteristic
80C2×C127C22×Q1611yescharacteristic
81C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
82C2×#477—1271nomaximal, Sylow
83C127C2×#47711yescharacteristic, complemented, Sylow
84C2×C4⋊Q8—1271no—
85C2×C4×Q8D12711yescharacteristic
86#477—1278no—
87C2×#107—1272no—
88C2×C4⋊C8—1271no—
89C2×#103—1272no—
90C22×Q8—2541no—
91C22×Q8C2×D12711yescharacteristic
92C22×C8—1272no—
93C4⋊Q8—1274no—
94C4⋊Q8—2542no—
95C4×Q8—22no—
96C4×Q8C2×D12714yes—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—1272no—
99C2×C4⋊C4—2541no—
100C2×C4⋊C4C2×D12711yescharacteristic
101C2×C42—21no—
102C2×C42C2×D12711yescharacteristic
103#107—1278no—
104C4⋊C8—1274no—
105#103—1278no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—5084no—
109C2×Q8—2544no—
110C2×Q8Dic25414yescomplemented
111C2×Q8C22×D12712yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—2542no—
115C22×C4C22⋊D12711yescharacteristic
116C22×C4C22⋊D12711yescharacteristic
117C22×C4C22×D12711yescharacteristic
118C2×C8—1274no—
119C2×C8—2544no—
120C4⋊C4—24no—
121C4⋊C4—21no—
122C4⋊C4—1274no—
123C4⋊C4—2542no—
124C4⋊C4—2544no—
125C4⋊C4C22×D12712yes—
126C42—24no—
127C42C22⋊D12712yes—
128C42C22×D12712yes—
129C23C2×C22⋊D12711yescharacteristic
130Q8—24no—
131Q8—22no—
132Q8—5088no—
133Q8C2×Dic25414yescomplemented
134C2×C4—24no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—5084no—
140C2×C4—2544no—
141C2×C4#1452323814yes—
142C2×C4C2×C22⋊D12712yes—
143C2×C4C2×C22⋊D12711yescharacteristic
144C2×C4C2×C22⋊D12712yes—
145C2×C4C2×Dic25412yes—
146C2×C4#1452324812yes—
147C2×C4C23×D12711yescharacteristic
148C8—2544no—
149C22#1686118812yes—
150C22C2×#1452323812yes—
151C22#1686123412yes—
152C22C22×C22⋊D12711yescharacteristic
153C4—24no—
154C4—22no—
155C4—42no—
156C4—5084no—
157C4#1686118812yes—
158C4C2×#1452323814yes—
159C4C22×Dic25411yescharacteristic
160C4C2×#1452324811yescharacteristic
161C2#2255017114yesdirect product factor
162C2C2×#1686118811yescharacteristic
163C2C22×#1452323811yescharacteristic
164C2C2×#1686123411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C4×Q8×C127C2
socle, prime radicalC23×C127C2×D4
hypercenterC2×C4×Q8D127
hypocenterC127C2×#477
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #22550171C2direct (N × H)
33. C2×Q8×C127Q8semidirect
53. Q8×C127C2×Q8semidirect
78. C2×C127#477semidirect
83. C127C2×#477semidirect
110. C2×Q8Dic254semidirect
133. Q8C2×Dic254semidirect
161. C2#22550171direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612092
#225501712
C2×#168611422
C2×C4⋊(C127⋊C8)2
C2×#168611332
C2×C4×Q8×C1272
C2×#477127