Finite Group Databasebeta

← #30637801

Subgroups — #30637801

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#22550586C211yesmaximal, characteristic, complemented
2#22550248C211yesmaximal, characteristic, complemented
3#22550068C211yesmaximal, characteristic
4#22549624C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#460×C127C211yesmaximal, characteristic, complemented
8C2×#14523236—22no—
9C2×#14523233—21no—
10#16861209C2211yescharacteristic
11#16861201—21no—
12#16861187—22no—
13#16861185—21no—
14C4×D508—21no—
15C4×D508C2211yescharacteristic
16C4⋊C4×D127—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×Dic254D412yes—
26C2×D508D411yescharacteristic
27C2×C4×D127—22no—
28C2×C4×D127—21no—
29#14523236—44no—
30#14523233—42no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—22no—
37Dic127⋊C4—22no—
38(C2×D127)⋊C4—22no—
39(C2×D127)⋊C4—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—21no—
51Dic254—42no—
52D508—21no—
53D508SD1612yescomplemented
54C2×Dic127—21no—
55C2×Dic127—22no—
56C4×D127—41no—
57C4×D127—41no—
58C4×D127—82no—
59C127⋊C8—22no—
60C127⋊C8—41no—
61C23×C127—41no—
62Q8×C127—21no—
63Q8×C127—21no—
64D4×C127—42no—
65D4×C127—41no—
66D4×C127—41no—
67D4×C127—41no—
68D4×C127—41no—
69C2×C4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C8×C127—41no—
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—41no—
84C4×C127—41no—
85C4×C127C2×SD1611yescharacteristic
86C4×C127#13511yescharacteristic
87C4×C127AGL(1,ℤ/8)11yescharacteristic
88C4×C127#13711yescharacteristic
89Dic127—41no—
90Dic127—42no—
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#492—1271no—
100#460D12711yescharacteristic, complemented
101#459—1271no—
102#333—1271no—
103Q8⋊C8—1271no—
104D4⋊C8—1271no—
105C2×SD16—21no—
106C2×SD16—2542no—
107C2×D8—2541no—
108C4⋊Q8—1271no—
109#127C2×D12711yescharacteristic
110#125—2541no—
111C22⋊Q8—2542no—
112C22⋊Q8—2541no—
113C4×Q8C2×D12711yescharacteristic
114C4×D4—1271no—
115C4×D4—2541no—
116C2×C4⋊C4—2541no—
117C4⋊C8—1271no—
118C4⋊C8C2×D12711yescharacteristic
119#102—21no—
120#102—2541no—
121C4×C8—1271no—
122C2×Q8—1272no—
123C2×Q8C22⋊D12711yescharacteristic
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—1271no—
128C22×C4—2542no—
129C22×C4—2541no—
130SD16—42no—
131SD16—5084no—
132D8—5082no—
133C2×C8—21no—
134C2×C8—2541no—
135C2×C8—2541no—
136C4⋊C4—21no—
137C4⋊C4—1271no—
138C4⋊C4—2542no—
139C4⋊C4—2542no—
140C4⋊C4—2541no—
141C4⋊C4—2541no—
142C4⋊C4C22⋊D12711yescharacteristic
143C22⋊C4—2542no—
144C22⋊C4—2541no—
145C42—21no—
146C42C22×D12711yescharacteristic
147C23—41no—
148C23—2541no—
149Q8—21no—
150Q8—21no—
151Q8—5082no—
152D4—42no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—1272no—
158D4—2541no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—10162no—
162C2×C4—5081no—
163C2×C4—5081no—
164C2×C4—2541no—
165C2×C4—2542no—
166C2×C4D4×D12711yescharacteristic
167C2×C4D4×D12711yescharacteristic
168C2×C4C2×C22⋊D12711yescharacteristic
169C8—41no—
170C8—5081no—
171C8—2542no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—5081no—
176C22—5081no—
177C22—2542no—
178C22#1686117711yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—5081no—
183C4—5082no—
184C4SD16×D12711yescharacteristic
185C4#1686122911yescharacteristic
186C4#1686123511yescharacteristic
187C4#1686123811yescharacteristic
188C2—161no—
189C2—5082no—
190C2#2255001311yescharacteristic
191C2#2255001911yescharacteristic
192C2#2255004211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#460×C127C2
socleC22×C127#120
prime radical#22550068C2
hypercenter#460D127
hypocenterC127#1622
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. #22550586C2semidirect
2. #22550248C2semidirect
4. #22549624C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
6. #22549603C2semidirect
7. #460×C127C2semidirect
53. D508SD16semidirect
97. C127#1622semidirect
100. #460D127semidirect

Maximal subgroups

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

subgroupindex
#225505862
#225502482
#225500682
#225496242
Q8⋊(C127⋊C8)2
#225496032
#460×C1272
#1622127