Finite Group Databasebeta

← #30638001

Subgroups — #30638001

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#22550600C211yesmaximal, characteristic
2#22550272C211yesmaximal, characteristic, complemented
3#22550157C211yesmaximal, characteristic
4#22550087C211yesmaximal, characteristic, complemented
5#22549606C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic
7#332×C127C211yesmaximal, characteristic, complemented
8C2×Dic508—21no—
9#16861209C2211yescharacteristic
10#16861204—22no—
11#16861198—21no—
12#16861189—21no—
13Q8×Dic127—21no—
14C4×Dic254C2211yescharacteristic
15C4×D508C2211yescharacteristic
16#16861167—21no—
17#16861143—21no—
18#16861140—22no—
19#16861134—21no—
20#16861131—21no—
21C4×C127⋊C8C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25C2×Dic254—21no—
26C2×Dic254D411yescharacteristic
27C2×D508D411yescharacteristic
28C2×C4×D127—21no—
29Dic508—42no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D411yescharacteristic
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—21no—
37(C2×D127)⋊C4—22no—
38(C2×D127)⋊C4—22no—
39(C2×D127)⋊C4—21no—
40C4×Dic127—22no—
41C4×Dic127—21no—
42C2×Q8×C127—21no—
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127D412yes—
47C42×C127C2311yescharacteristic
48C22×D127—21no—
49Dic254—21no—
50Dic254—21no—
51Dic254—41no—
52Dic254—41no—
53D508—21no—
54D508—21no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—41no—
59C4×D127—41no—
60C4×D127—41no—
61C127⋊C8—22no—
62Q8×C127—41no—
63Q8×C127—41no—
64C2×C4×C127—22no—
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C8×C127—41no—
69C8×C127—41no—
70C22×C127#12011yescharacteristic
71C2×D127—41no—
72C2×D127—41no—
73C2×D127—41no—
74C4×C127—21no—
75C4×C127—42no—
76C4×C127#13511yescharacteristic
77C4×C127#13511yescharacteristic
78C4×C127AGL(1,ℤ/8)11yescharacteristic
79C4×C127#13711yescharacteristic
80Dic127—41no—
81Dic127—41no—
82Dic127—41no—
83Dic127—81no—
84C2×C127#44911yescharacteristic
85C2×C127#45211yescharacteristic
86C2×C127#45611yescharacteristic
87D127—81no—
88#1652—1271nomaximal, Sylow
89C127#165211yescharacteristic, complemented, Sylow
90#556—1271no—
91#498—1271no—
92#474—1271no—
93#462—1271no—
94#332D12711yescharacteristic, complemented
95Q8⋊C8—1271no—
96D4⋊C8—1271no—
97C2×Q16—2541no—
98C4⋊Q8—1271no—
99C4⋊Q8C2×D12711yescharacteristic
100#126—2542no—
101#124—2541no—
102C22⋊Q8—2541no—
103C4×Q8—1271no—
104C4×Q8—2541no—
105C4×D4—1271no—
106#117—2541no—
107#107—2541no—
108#106—2542no—
109C4⋊C8C2×D12711yescharacteristic
110C4⋊C8C2×D12711yescharacteristic
111#103—2541no—
112#102—2541no—
113C4×C8—1271no—
114C2×Q8—21no—
115C2×Q8—1271no—
116C2×Q8—2541no—
117C2×D4—1271no—
118C22×C4—2541no—
119Q16—5082no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—2541no—
123C4⋊C4—21no—
124C4⋊C4—1271no—
125C4⋊C4—1271no—
126C4⋊C4—2542no—
127C4⋊C4—2541no—
128C4⋊C4—2541no—
129C4⋊C4—2541no—
130C4⋊C4D50812yes—
131C22⋊C4—2542no—
132C22⋊C4—2542no—
133C22⋊C4—2541no—
134C42—2542no—
135C42—2541no—
136C42C22×D12711yescharacteristic
137C23—2541no—
138Q8—41no—
139Q8—41no—
140Q8—5081no—
141Q8—5081no—
142Q8—2541no—
143Q8—2541no—
144D4—2541no—
145D4—2541no—
146C2×C4—22no—
147C2×C4—5081no—
148C2×C4—5081no—
149C2×C4—5081no—
150C2×C4—2541no—
151C2×C4—2541no—
152C2×C4—2541no—
153C2×C4D4×D12711yescharacteristic
154C2×C4D4×D12711yescharacteristic
155C2×C4C2×D50811yescharacteristic
156C8—41no—
157C8—41no—
158C8—2542no—
159C22—5081no—
160C22—5081no—
161C22—5081no—
162C22#1686117911yescharacteristic
163C4—21no—
164C4—42no—
165C4—10161no—
166C4—5081no—
167C4—5081no—
168C4—5081no—
169C4#1686122811yescharacteristic
170C4#1686123211yescharacteristic
171C4#1686123611yescharacteristic
172C4#1686123811yescharacteristic
173C2—10161no—
174C2#2255002411yescharacteristic
175C2#2255003911yescharacteristic
176C2#2255006111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#332×C127C2
socleC22×C127#120
prime radicalC2×D508D4
hypercenter#332D127
hypocenterC127#1652
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22550272C2semidirect
4. #22550087C2semidirect
5. #22549606C2semidirect
7. #332×C127C2semidirect
89. C127#1652semidirect
94. #332D127semidirect

Maximal subgroups

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

subgroupindex
#225506002
#225502722
#225501572
#225500872
#225496062
#225496032
#332×C1272
#1652127