Finite Group Databasebeta

← #30637984

Subgroups — #30637984

Order 16256.

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

normal subgroups only

This group has 200 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#22550596C211yesmaximal, characteristic
2#22550279C211yesmaximal, characteristic, complemented
3#22550158C211yesmaximal, characteristic, complemented
4#22549623C211yesmaximal, characteristic, complemented
5#22549606C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic
7#461×C127C211yesmaximal, characteristic
8#16861209C2211yescharacteristic
9#16861202—22no—
10#16861196—21no—
11#16861188—21no—
12C4×Dic254C2211yescharacteristic
13#16861174—21no—
14#16861168—21no—
15#16861142—22no—
16#16861140—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18#16861132—21no—
19C4×C127⋊C8C2211yescharacteristic
20C2×SD16×C127—21no—
21C4⋊Q8×C127C2211yescharacteristic
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24#103×C127—21no—
25C2×Dic254—21no—
26C2×Dic254D411yescharacteristic
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127D412yes—
31C4⋊Dic127D411yescharacteristic
32Dic127⋊C4—22no—
33Dic127⋊C4—21no—
34C22⋊Dic127—22no—
35C22⋊Dic127—22no—
36C4×Dic127—22no—
37C4×Dic127—21no—
38C2×Q8×C127—21no—
39C2×D4×C127D411yescharacteristic
40C22×C4×C127—21no—
41SD16×C127—42no—
42C2×C8×C127—21no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C22⋊C4×C127—21no—
47C42×C127C2311yescharacteristic
48Dic254—21no—
49Dic254—21no—
50Dic254—41no—
51Dic254—41no—
52C2×Dic127—21no—
53C2×Dic127—21no—
54C2×Dic127—22no—
55C127⋊C8—22no—
56C127⋊C8—41no—
57C23×C127—21no—
58Q8×C127—41no—
59Q8×C127—41no—
60D4×C127—21no—
61D4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C8×C127—41no—
70C22×C127—41no—
71C22×C127—41no—
72C22×C127—41no—
73C22×C127#12011yescharacteristic
74C4×C127—21no—
75C4×C127—41no—
76C4×C127—81no—
77C4×C127#13511yescharacteristic
78C4×C127#13511yescharacteristic
79C4×C127AGL(1,ℤ/8)11yescharacteristic
80C4×C127#13711yescharacteristic
81Dic127—42no—
82Dic127—41no—
83Dic127—41no—
84C2×C127—81no—
85C2×C127#44911yescharacteristic
86C2×C127#45211yescharacteristic
87C2×C127#45611yescharacteristic
88#1650—1271nomaximal, Sylow
89C127#165011yescharacteristic, complemented, Sylow
90#556—1271no—
91#500—1271no—
92#475—1271no—
93#461D12711yescharacteristic
94#332—1271no—
95Q8⋊C8—1271no—
96D4⋊C8—1271no—
97C2×SD16—21no—
98C4⋊Q8—1271no—
99C4⋊Q8C2×D12711yescharacteristic
100#126—2542no—
101#124—2541no—
102C22⋊Q8—2541no—
103C4×Q8—1271no—
104C4×Q8—2541no—
105C4×D4C2×D12711yescharacteristic
106#117—2541no—
107#107—2542no—
108#106—2541no—
109C4⋊C8—1271no—
110C4⋊C8C2×D12711yescharacteristic
111#103—21no—
112#103—2541no—
113C4×C8—1271no—
114C2×Q8—21no—
115C2×Q8—1271no—
116C2×Q8—2541no—
117C2×D4C22⋊D12711yescharacteristic
118C22×C4—21no—
119SD16—42no—
120C2×C8—21no—
121C2×C8—2541no—
122C2×C8—2541no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—1271no—
126C4⋊C4—1272no—
127C4⋊C4—2542no—
128C4⋊C4—2541no—
129C4⋊C4—2541no—
130C4⋊C4C22⋊D12711yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—2542no—
133C22⋊C4—2542no—
134C42—2542no—
135C42—2541no—
136C42C22×D12711yescharacteristic
137C23—21no—
138Q8—41no—
139Q8—41no—
140Q8—5081no—
141Q8—5081no—
142Q8—2541no—
143Q8—2541no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—2541no—
151C2×C4—2542no—
152C2×C4—2541no—
153C2×C4D4×D12711yescharacteristic
154C2×C4D4×D12711yescharacteristic
155C2×C4C2×C22⋊D12711yescharacteristic
156C8—41no—
157C8—5081no—
158C8—2542no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#1686117711yescharacteristic
163C4—21no—
164C4—41no—
165C4—81no—
166C4—5082no—
167C4—5081no—
168C4—5081no—
169C4#1686122911yescharacteristic
170C4#1686123011yescharacteristic
171C4#1686123511yescharacteristic
172C4#1686123811yescharacteristic
173C2—81no—
174C2#2255001911yescharacteristic
175C2#2255003511yescharacteristic
176C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#461×C127C2
socleC22×C127#120
prime radicalC2×D4×C127D4
hypercenter#461D127
hypocenterC127#1650
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #22550279C2semidirect
3. #22550158C2semidirect
4. #22549623C2semidirect
5. #22549606C2semidirect
89. C127#1650semidirect

Maximal subgroups

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

subgroupindex
#225505962
#225502792
#225501582
#225496232
#225496062
D4⋊(C127⋊C8)2
#461×C1272
#1650127