Finite Group Databasebeta

← #30637799

Subgroups — #30637799

Order 16256.

236 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550587C211yesmaximal, characteristic, complemented
2#22550073C211yesmaximal, characteristic, complemented
3#22550068C211yesmaximal, characteristic
4#22549625C211yesmaximal, characteristic, complemented
5#22549608C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic, complemented
7#492×C127C211yesmaximal, characteristic, complemented
8C2×#14523236—21no—
9C2×#14523233—21no—
10#16861200—21no—
11#16861189—21no—
12#16861185—22no—
13C4×Dic254C2211yescharacteristic
14C4×D508C2211yescharacteristic
15#16861171—21no—
16C4⋊C4×D127—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861130—21no—
20#16861130—21no—
21C2×SD16×C127—22no—
22C4⋊Q8×C127C2211yescharacteristic
23#127×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254D411yescharacteristic
26C2×D508D411yescharacteristic
27C2×C4×D127—22no—
28C2×C4×D127—21no—
29#14523236—42no—
30#14523233—42no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C4⋊Dic127—22no—
34C4⋊Dic127D411yescharacteristic
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—22no—
37Dic127⋊C4—21no—
38(C2×D127)⋊C4—22no—
39(C2×D127)⋊C4—21no—
40C4×Dic127—21no—
41C2×Q8×C127D412yes—
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45SD16×C127—44no—
46C2×C8×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51Dic254—21no—
52Dic254—21no—
53D508—21no—
54D508SD1612yescomplemented
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C4×D127—41no—
59C4×D127—41no—
60C4×D127—82no—
61C127⋊C8—41no—
62C127⋊C8—41no—
63C23×C127—41no—
64Q8×C127—42no—
65D4×C127—42no—
66D4×C127—41no—
67D4×C127—41no—
68D4×C127—41no—
69D4×C127—41no—
70C2×C4×C127—22no—
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C8×C127—22no—
75C22×C127—81no—
76C22×C127—81no—
77C22×C127—81no—
78C22×C127#12011yescharacteristic
79C2×D127—22no—
80C2×D127—41no—
81C2×D127—41no—
82C4×C127—21no—
83C4×C127—42no—
84C4×C127C2×SD1611yescharacteristic
85C4×C127#13511yescharacteristic
86C4×C127AGL(1,ℤ/8)11yescharacteristic
87C4×C127#13711yescharacteristic
88Dic127—41no—
89Dic127—41no—
90Dic127—41no—
91C2×C127—161no—
92C2×C127#44811yescharacteristic
93C2×C127#44911yescharacteristic
94C2×C127#45311yescharacteristic
95D127—42no—
96#1622—1271nomaximal, Sylow
97C127#162211yescharacteristic, complemented, Sylow
98#554—1271no—
99#492D12711yescharacteristic, complemented
100#460—1271no—
101#459—1271no—
102#333—1271no—
103Q8⋊C8—1271no—
104D4⋊C8—1271no—
105C2×SD16—22no—
106C2×SD16—2541no—
107C2×D8—2541no—
108C4⋊Q8C2×D12711yescharacteristic
109#127C2×D12711yescharacteristic
110#125—2541no—
111C22⋊Q8—2541no—
112C22⋊Q8—2542no—
113C4×Q8—1271no—
114C4×D4—1271no—
115C4×D4—2541no—
116C2×C4⋊C4—2541no—
117C4⋊C8—1271no—
118C4⋊C8—1271no—
119#102—2541no—
120#102—2541no—
121C4×C8C2×D12711yescharacteristic
122C2×Q8—1271no—
123C2×Q8C22⋊D12712yes—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—1271no—
128C22×C4—2542no—
129C22×C4—2541no—
130SD16—44no—
131SD16—5082no—
132D8—5082no—
133C2×C8—21no—
134C2×C8—2541no—
135C2×C8—2541no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—1271no—
139C4⋊C4—1271no—
140C4⋊C4—2542no—
141C4⋊C4—2542no—
142C4⋊C4—2541no—
143C22⋊C4—2542no—
144C22⋊C4—2541no—
145C42—2541no—
146C42C22×D12711yescharacteristic
147C23—41no—
148C23—2541no—
149Q8—42no—
150Q8—2541no—
151Q8—2541no—
152D4—42no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—1272no—
158D4—2541no—
159C2×C4—22no—
160C2×C4—10162no—
161C2×C4—5081no—
162C2×C4—5081no—
163C2×C4—2541no—
164C2×C4—2541no—
165C2×C4—2541no—
166C2×C4D4×D12711yescharacteristic
167C2×C4D4×D12711yescharacteristic
168C2×C4C2×C22⋊D12711yescharacteristic
169C8—22no—
170C8—5081no—
171C8—5081no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—5081no—
176C22—5081no—
177C22—2542no—
178C22#1686117711yescharacteristic
179C4—21no—
180C4—42no—
181C4—5081no—
182C4—5081no—
183C4—5081no—
184C4SD16×D12711yescharacteristic
185C4#1686123011yescharacteristic
186C4#1686123611yescharacteristic
187C4#1686123811yescharacteristic
188C2—161no—
189C2—5082no—
190C2#2255001311yescharacteristic
191C2#2255002011yescharacteristic
192C2#2255004311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#492×C127C2
socleC22×C127#120
prime radical#22550068C2
hypercenter#492D127
hypocenterC127#1622
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #22550587C2semidirect
2. #22550073C2semidirect
4. #22549625C2semidirect
5. #22549608C2semidirect
6. #22549605C2semidirect
7. #492×C127C2semidirect
54. D508SD16semidirect
97. C127#1622semidirect
99. #492D127semidirect

Maximal subgroups

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

subgroupindex
#225505872
#225500732
#225500682
#225496252
#225496082
#225496052
#492×C1272
#1622127