Finite Group Databasebeta

← #30637788

Subgroups — #30637788

Order 16256.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550611C211yesmaximal, characteristic, complemented
2#22550252C211yesmaximal, characteristic
3#22550073C211yesmaximal, characteristic, complemented
4#22549619C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549606C211yesmaximal, characteristic, complemented
7#460×C127C211yesmaximal, characteristic, complemented
8C2×#14523236—21no—
9C2×#14523233—22no—
10#16861206C2211yescharacteristic
11#16861198—21no—
12#16861197—21no—
13#16861194—22no—
14C4×Dic254C2211yescharacteristic
15C4×D508—21no—
16#16861171—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18#16861130—21no—
19C4×C127⋊C8C2211yescharacteristic
20C2×SD16×C127—21no—
21#127×C127C2211yescharacteristic
22C4×Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24#102×C127—21no—
25C2×Dic254D411yescharacteristic
26C2×D508—21no—
27C2×D508—21no—
28C2×D508D412yes—
29C2×C4×D127—22no—
30#14523236—42no—
31#14523233—44no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127D411yescharacteristic
35Dic127⋊C4—21no—
36(C2×D127)⋊C4—22no—
37(C2×D127)⋊C4—22no—
38(C2×D127)⋊C4—22no—
39C4×Dic127—21no—
40C2×Q8×C127D411yescharacteristic
41C2×D4×C127—21no—
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44SD16×C127—42no—
45C2×C8×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127D411yescharacteristic
48C42×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—22no—
51Dic254—21no—
52Dic254—21no—
53D508—42no—
54D508—41no—
55D508—41no—
56D508—41no—
57D508—41no—
58C2×Dic127—21no—
59C2×Dic127—21no—
60C4×D127—82no—
61C127⋊C8—22no—
62C127⋊C8—41no—
63C23×C127—41no—
64Q8×C127—21no—
65Q8×C127—21no—
66D4×C127—42no—
67D4×C127—41no—
68D4×C127—41no—
69D4×C127—41no—
70D4×C127—41no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C8×C127—41no—
77C22×C127—81no—
78C22×C127—81no—
79C22×C127—81no—
80C22×C127#12011yescharacteristic
81C2×D127—42no—
82C2×D127—42no—
83C2×D127—42no—
84C4×C127—21no—
85C4×C127—41no—
86C4×C127—41no—
87C4×C127#13511yescharacteristic
88C4×C127#13511yescharacteristic
89C4×C127AGL(1,ℤ/8)11yescharacteristic
90C4×C127AGL(1,ℤ/8)11yescharacteristic
91Dic127—41no—
92Dic127—41no—
93C2×C127—161no—
94C2×C127#44911yescharacteristic
95C2×C127#44911yescharacteristic
96C2×C127#45311yescharacteristic
97D127—82no—
98#1620—1271nomaximal, Sylow
99C127#162011yescharacteristic, complemented, Sylow
100#559—1271no—
101#493—1271no—
102#460—1271no—
103#460D12711yescharacteristic, complemented
104#331—1271no—
105Q8⋊C8—1271no—
106Q8⋊C8—1271no—
107C2×SD16—21no—
108C2×SD16—2541no—
109C2×D8—2542no—
110#127—1271no—
111#127C2×D12711yescharacteristic
112#124—2541no—
113#124—2541no—
114#123—2542no—
115C4×Q8—1271no—
116C4×Q8C2×D12711yescharacteristic
117C4×D4—2541no—
118C4×D4—2541no—
119C4⋊C8—1271no—
120C4⋊C8C2×D12711yescharacteristic
121#102—21no—
122#102—2541no—
123C4×C8—1271no—
124C2×Q8—1271no—
125C2×Q8C22⋊D12711yescharacteristic
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—1272no—
130C2×D4—2541no—
131C2×D4—2541no—
132C22×C4—2542no—
133SD16—42no—
134SD16—5082no—
135D8—5084no—
136C2×C8—21no—
137C2×C8—2541no—
138C2×C8—2541no—
139C4⋊C4—21no—
140C4⋊C4—1271no—
141C4⋊C4—2541no—
142C4⋊C4C22⋊D12711yescharacteristic
143C22⋊C4—2542no—
144C22⋊C4—2542no—
145C22⋊C4—2542no—
146C42—21no—
147C42—2541no—
148C42C22×D12711yescharacteristic
149C23—41no—
150C23—2542no—
151Q8—21no—
152Q8—21no—
153Q8—2541no—
154Q8—2541no—
155D4—42no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—5082no—
161D4—5081no—
162D4—5081no—
163D4—5081no—
164D4—5081no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—10162no—
168C2×C4—2541no—
169C2×C4—2541no—
170C2×C4D4×D12711yescharacteristic
171C2×C4D4×D12711yescharacteristic
172C2×C4C2×C22⋊D12711yescharacteristic
173C8—41no—
174C8—5081no—
175C8—2542no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—5082no—
180C22—5082no—
181C22—5082no—
182C22#1686117711yescharacteristic
183C4—21no—
184C4—41no—
185C4—41no—
186C4—5081no—
187C4—5081no—
188C4#1686122911yescharacteristic
189C4#1686123011yescharacteristic
190C4#1686123511yescharacteristic
191C4#1686123611yescharacteristic
192C2—161no—
193C2—10162no—
194C2#2255001911yescharacteristic
195C2#2255002011yescharacteristic
196C2#2255004211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#460×C127C2
socleC22×C127#120
prime radical#22550252C2
hypercenter#460D127
hypocenterC127#1620
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #22550611C2semidirect
3. #22550073C2semidirect
4. #22549619C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
6. #22549606C2semidirect
7. #460×C127C2semidirect
99. C127#1620semidirect
103. #460D127semidirect

Maximal subgroups

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

subgroupindex
#225506112
#225502522
#225500732
#225496192
Q8⋊(C127⋊C8)2
#225496062
#460×C1272
#1620127