Finite Group Databasebeta

← #30649277

Subgroups — #30649277

Order 16256.

380 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 157 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#22550286C214yesmaximal, complemented
3#22550279C212yesmaximal, complemented
4#22550272C212yesmaximal, complemented
5#22549919C214yesmaximal
6C4×#14523232C211yesmaximal, characteristic
7#533×C127C211yesmaximal, characteristic
8C22×Dic254—21no—
9C2×#14523232C2212yes—
10#16861209—21no—
11#16861209—21no—
12#16861209C2212yes—
13#16861209C2212yes—
14C2×C4⋊Dic127—21no—
15C2×C4⋊Dic127C2212yes—
16#16861142C2218yes—
17#16861140C2218yes—
18#16861120C2212yes—
19C4×C127⋊C8C2212yes—
20C4⋊Q8×C127C2211yescharacteristic
21C4⋊Q8×C127C2211yescharacteristic
22#125×C127C2212yes—
23C22⋊Q8×C127—21no—
24C22⋊Q8×C127—21no—
25C4×Q8×C127—21no—
26C4×Q8×C127—21no—
27#117×C127C2212yes—
28C2×C42×C127C2211yescharacteristic
29C2×Dic254—22no—
30C2×Dic254—22no—
31C2×Dic254—44no—
32C22×Dic127—22no—
33#14523232Q818yescomplemented
34C2×C127⋊C8C2314yes—
35C4⋊Dic127—22no—
36C4⋊Dic127—22no—
37C4⋊Dic127—22no—
38C4⋊Dic127C2314yes—
39C2×Q8×C127—21no—
40C2×Q8×C127—21no—
41C22×C4×C127D411yescharacteristic
42C22×C4×C127D411yescharacteristic
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—22no—
45C4⋊C4×C127—22no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127C2312yes—
49C4⋊C4×C127C2312yes—
50C22⋊C4×C127—22no—
51C42×C127—22no—
52C42×C127D411yescharacteristic
53C42×C127D411yescharacteristic
54C42×C127C2311yescharacteristic
55C42×C127C2311yescharacteristic
56Dic254—44no—
57Dic254—44no—
58C2×Dic127—24no—
59C2×Dic127—44no—
60C127⋊C8C2×Q818yescomplemented
61C23×C127C2×D411yescharacteristic
62Q8×C127—42no—
63Q8×C127—42no—
64C2×C4×C127—22no—
65C2×C4×C127—22no—
66C2×C4×C127—22no—
67C2×C4×C127—22no—
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×Q814yes—
74C2×C4×C127C2411yescharacteristic
75C22×C127—21no—
76C22×C127—21no—
77C22×C127C4⋊Q812yes—
78C22×C127C22×D411yescharacteristic
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—42no—
82C4×C127—42no—
83C4×C127C4⋊Q812yes—
84C4×C127AGL(1,ℤ/8)12yes—
85C4×C127#13712yes—
86C4×C127C22×Q812yes—
87Dic127—44no—
88C2×C127—22no—
89C2×C127C2×C4⋊Q811yescharacteristic
90C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
91C2×C127C2×#13711yescharacteristic
92#3165—1271nomaximal, Sylow
93C127#316511yescharacteristic, complemented, Sylow
94#533D12711yescharacteristic
95C2×C4⋊Q8—1271no—
96#501—1274no—
97#500—1272no—
98#498—1272no—
99#428—1274no—
100C4×M16—1271no—
101C22×Q8—2541no—
102C2×M16—1272no—
103C4⋊Q8—1272no—
104C4⋊Q8—1272no—
105C4⋊Q8—2541no—
106C4⋊Q8—2541no—
107C4⋊Q8C2×D12711yescharacteristic
108C4⋊Q8C2×D12711yescharacteristic
109#125C2×D12712yes—
110C22⋊Q8—21no—
111C22⋊Q8—21no—
112C4×Q8—21no—
113C4×Q8—21no—
114#117C2×D12712yes—
115C2×C4⋊C4—1272no—
116C2×C4⋊C4—2541no—
117C2×C42C2×D12711yescharacteristic
118#107—1278no—
119#106—1278no—
120#97—1272no—
121C4×C8—1272no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—5084no—
125C2×Q8—2542no—
126C2×Q8—2542no—
127C22×C4—2542no—
128C22×C4C22⋊D12711yescharacteristic
129C22×C4C22⋊D12711yescharacteristic
130C22×C4C22×D12711yescharacteristic
131M16—1278no—
132C2×C8—1274no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—1274no—
138C4⋊C4—2542no—
139C4⋊C4—2542no—
140C4⋊C4—2542no—
141C4⋊C4C22×D12712yes—
142C4⋊C4C22×D12712yes—
143C22⋊C4—22no—
144C42—22no—
145C42C22⋊D12711yescharacteristic
146C42C22⋊D12711yescharacteristic
147C42C22×D12711yescharacteristic
148C42C22×D12711yescharacteristic
149C23C2×C22⋊D12711yescharacteristic
150Q8—42no—
151Q8—42no—
152Q8—5084no—
153Q8—5084no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—5084no—
159C2×C4—2544no—
160C2×C4C2×C22⋊D12711yescharacteristic
161C2×C4C2×C22⋊D12711yescharacteristic
162C2×C4C2×C22⋊D12711yescharacteristic
163C2×C4C2×C22⋊D12711yescharacteristic
164C2×C4C2×C22⋊D12711yescharacteristic
165C2×C4Q8×D12714yes—
166C2×C4C23×D12711yescharacteristic
167C8—1278no—
168C22—21no—
169C22—21no—
170C22#1686120712yes—
171C22C22×C22⋊D12711yescharacteristic
172C4—21no—
173C4—21no—
174C4—42no—
175C4—42no—
176C4—5084no—
177C4#1686120712yes—
178C4#1686123412yes—
179C4#1686123912yes—
180C4C2×Q8×D12712yes—
181C2—22no—
182C2C2×#1686120711yescharacteristic
183C2C2×#1686123411yescharacteristic
184C2C2×#1686123911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#533×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenter#533D127
hypocenterC127#3165
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #22550286C2semidirect
3. #22550279C2semidirect
4. #22550272C2semidirect
33. #14523232Q8semidirect
60. C127⋊C8C2×Q8semidirect
93. C127#3165semidirect

Maximal subgroups

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

subgroupindex
C2×#168612092
#225502862
#225502792
#225502722
#225499192
C4×#145232322
#533×C1272
#3165127