Finite Group Databasebeta

← #30637800

Subgroups — #30637800

Order 16256.

236 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550583C211yesmaximal, characteristic, complemented
2#22550248C211yesmaximal, characteristic, complemented
3#22550073C211yesmaximal, characteristic, complemented
4#22549624C211yesmaximal, characteristic, complemented
5#22549606C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
7#459×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9C2×#14523236—22no—
10#16861209C2211yescharacteristic
11#16861200—21no—
12#16861188—21no—
13#16861186—22no—
14C4×Dic254C2211yescharacteristic
15D4×Dic127—21no—
16C2×C4⋊Dic127—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18#16861130—21no—
19C4×C127⋊C8C2211yescharacteristic
20C2×D8×C127—21no—
21#127×C127C2211yescharacteristic
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24#102×C127—21no—
25C2×Dic254D411yescharacteristic
26C2×Dic254D412yes—
27C22×Dic127—22no—
28#14523236—44no—
29#14523236—42no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C4⋊Dic127—22no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—22no—
37Dic127⋊C4—21no—
38C22⋊Dic127—22no—
39C4×Dic127—21no—
40C2×D4×C127—21no—
41C2×D4×C127—21no—
42C2×D4×C127—21no—
43C2×D4×C127D411yescharacteristic
44C22×C4×C127—21no—
45D8×C127—42no—
46C2×C8×C127—21no—
47C4⋊C4×C127D411yescharacteristic
48C22⋊C4×C127—21no—
49C42×C127C2311yescharacteristic
50Dic254—21no—
51Dic254—21no—
52Dic254—42no—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C2×Dic127—22no—
56C2×Dic127—82no—
57C127⋊C8—22no—
58C127⋊C8—41no—
59C23×C127—21no—
60C23×C127—41no—
61D4×C127—21no—
62D4×C127—42no—
63D4×C127—41no—
64D4×C127—41no—
65D4×C127—41no—
66D4×C127—41no—
67D4×C127SD1612yescomplemented
68C2×C4×C127—21no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C8×C127—41no—
75C22×C127—22no—
76C22×C127—41no—
77C22×C127—41no—
78C22×C127—81no—
79C22×C127—81no—
80C22×C127—81no—
81C22×C127#12011yescharacteristic
82C4×C127—21no—
83C4×C127—41no—
84C4×C127C2×SD1611yescharacteristic
85C4×C127#13511yescharacteristic
86C4×C127AGL(1,ℤ/8)11yescharacteristic
87C4×C127#13711yescharacteristic
88Dic127—41no—
89Dic127—41no—
90Dic127—42no—
91C2×C127—42no—
92C2×C127—161no—
93C2×C127#44811yescharacteristic
94C2×C127#44911yescharacteristic
95C2×C127#45311yescharacteristic
96#1622—1271nomaximal, Sylow
97C127#162211yescharacteristic, complemented, Sylow
98#554—1271no—
99#492—1271no—
100#460—1271no—
101#459D12711yescharacteristic
102#333—1271no—
103Q8⋊C8—1271no—
104D4⋊C8—1271no—
105C2×SD16—2541no—
106C2×SD16—2542no—
107C2×D8—21no—
108C4⋊Q8—1271no—
109#127C2×D12711yescharacteristic
110#125—2541no—
111C22⋊Q8—2542no—
112C22⋊Q8—2541no—
113C4×Q8—1271no—
114C4×D4—2541no—
115C4×D4C2×D12711yescharacteristic
116C2×C4⋊C4—2541no—
117C4⋊C8—1271no—
118C4⋊C8C2×D12711yescharacteristic
119#102—21no—
120#102—2541no—
121C4×C8—1271no—
122C2×Q8—1272no—
123C2×Q8—1271no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4C22⋊D12711yescharacteristic
128C22×C4—21no—
129C22×C4—2542no—
130SD16—5082no—
131SD16—5084no—
132D8—42no—
133C2×C8—21no—
134C2×C8—2541no—
135C2×C8—2541no—
136C4⋊C4—1271no—
137C4⋊C4—2542no—
138C4⋊C4—2542no—
139C4⋊C4—2541no—
140C4⋊C4—2541no—
141C4⋊C4—2541no—
142C4⋊C4C22⋊D12711yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—2542no—
145C42—2541no—
146C42C22×D12711yescharacteristic
147C23—21no—
148C23—41no—
149Q8—5082no—
150Q8—2541no—
151Q8—2541no—
152D4—21no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4#1452323612yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—10162no—
163C2×C4—2541no—
164C2×C4—2541no—
165C2×C4—2542no—
166C2×C4D4×D12711yescharacteristic
167C2×C4D4×D12711yescharacteristic
168C2×C4C2×C22⋊D12711yescharacteristic
169C8—41no—
170C8—5081no—
171C8—2542no—
172C22—22no—
173C22—41no—
174C22—41no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22#1686117711yescharacteristic
179C4—21no—
180C4—41no—
181C4—5081no—
182C4—5081no—
183C4—5082no—
184C4C2×#1452323611yescharacteristic
185C4#1686122811yescharacteristic
186C4#1686123311yescharacteristic
187C4#1686123911yescharacteristic
188C2—42no—
189C2—161no—
190C2#2255001411yescharacteristic
191C2#2255001811yescharacteristic
192C2#2255004211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting, prime radical#459×C127C2
socleC22×C127#120
hypercenter#459D127
hypocenterC127#1622
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #22550583C2semidirect
2. #22550248C2semidirect
3. #22550073C2semidirect
4. #22549624C2semidirect
5. #22549606C2semidirect
6. D4⋊(C127⋊C8)C2semidirect
67. D4×C127SD16semidirect
97. C127#1622semidirect
158. D4#14523236semidirect

Maximal subgroups

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

subgroupindex
#225505832
#225502482
#225500732
#225496242
#225496062
D4⋊(C127⋊C8)2
#459×C1272
#1622127