Finite Group Databasebeta

← #30637997

Subgroups — #30637997

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#22550153C211yesmaximal, characteristic
3#22550083C211yesmaximal, characteristic, complemented
4#22549622C211yesmaximal, characteristic, complemented
5#22549608C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic
7#498×C127C211yesmaximal, characteristic, complemented
8C2×#14523238—21no—
9#16861204—22no—
10#16861198—21no—
11#16861189—21no—
12Q8×Dic127—21no—
13C4×Dic254C2211yescharacteristic
14C4×D508C2211yescharacteristic
15#16861167—21no—
16#16861142—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861133—21no—
20#16861129—21no—
21C4⋊Q8×C127C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23#106×C127—22no—
24C4×C8×C127C2211yescharacteristic
25C2×Dic254D411yescharacteristic
26C2×D508D411yescharacteristic
27C2×C4×D127—21no—
28#14523238—42no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127D411yescharacteristic
33C4⋊Dic127D411yescharacteristic
34Dic127⋊C4—21no—
35(C2×D127)⋊C4—22no—
36(C2×D127)⋊C4—22no—
37(C2×D127)⋊C4—21no—
38C4×Dic127—22no—
39C4×Dic127—21no—
40C2×Q8×C127—21no—
41C2×Q8×C127—21no—
42C2×C8×C127—21no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127D412yes—
47C42×C127C2311yescharacteristic
48C22×D127—21no—
49Dic254—21no—
50Dic254—21no—
51D508—21no—
52D508—21no—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C2×Dic127—21no—
56C4×D127—41no—
57C4×D127—41no—
58C127⋊C8—41no—
59C127⋊C8—41no—
60Q8×C127—41no—
61Q8×C127—41no—
62Q8×C127—41no—
63Q8×C127—41no—
64C2×C4×C127—22no—
65C2×C4×C127—41no—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C8×C127—22no—
70C22×C127#12011yescharacteristic
71C2×D127—41no—
72C2×D127—41no—
73C2×D127—41no—
74C4×C127—21no—
75C4×C127—42no—
76C4×C127—81no—
77C4×C127#13511yescharacteristic
78C4×C127#13511yescharacteristic
79C4×C127AGL(1,ℤ/8)11yescharacteristic
80C4×C127#13711yescharacteristic
81Dic127—41no—
82Dic127—41no—
83Dic127—41no—
84C2×C127#44911yescharacteristic
85C2×C127#45211yescharacteristic
86C2×C127#45611yescharacteristic
87D127—81no—
88#1652—1271nomaximal, Sylow
89C127#165211yescharacteristic, complemented, Sylow
90#556—1271no—
91#498D12711yescharacteristic, complemented
92#474—1271no—
93#462—1271no—
94#332—1271no—
95Q8⋊C8—1271no—
96D4⋊C8—1271no—
97C2×Q16—2541no—
98C4⋊Q8C2×D12711yescharacteristic
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—22no—
109C4⋊C8—1271no—
110C4⋊C8—1271no—
111#103—2541no—
112#102—2541no—
113C4×C8C2×D12711yescharacteristic
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—1271no—
117C2×D4—1271no—
118C22×C4—2541no—
119Q16—5082no—
120C2×C8—21no—
121C2×C8—2541no—
122C2×C8—2541no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—1271no—
127C4⋊C4—1271no—
128C4⋊C4—2542no—
129C4⋊C4—2541no—
130C4⋊C4C22⋊D12712yes—
131C22⋊C4—2542no—
132C22⋊C4—2542no—
133C22⋊C4—2541no—
134C42—2542no—
135C42—2541no—
136C42C22×D12711yescharacteristic
137C23—2541no—
138Q8—41no—
139Q8—41no—
140Q8—41no—
141Q8—41no—
142Q8—2541no—
143Q8—2541no—
144D4—2541no—
145D4—2541no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—5081no—
149C2×C4—5081no—
150C2×C4—2541no—
151C2×C4—2541no—
152C2×C4—2541no—
153C2×C4D4×D12711yescharacteristic
154C2×C4D4×D12711yescharacteristic
155C2×C4C2×C22⋊D12711yescharacteristic
156C8—22no—
157C8—5081no—
158C8—5081no—
159C22—5081no—
160C22—5081no—
161C22—5081no—
162C22#1686117711yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—5081no—
167C4—5081no—
168C4—5081no—
169C4#1686123011yescharacteristic
170C4#1686123011yescharacteristic
171C4#1686123611yescharacteristic
172C4#1686123811yescharacteristic
173C2—10161no—
174C2#2255002011yescharacteristic
175C2#2255003511yescharacteristic
176C2#2255005811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#498×C127C2
socleC22×C127#120
prime radicalC2×D508D4
hypercenter#498D127
hypocenterC127#1652
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
3. #22550083C2semidirect
4. #22549622C2semidirect
5. #22549608C2semidirect
7. #498×C127C2semidirect
89. C127#1652semidirect
91. #498D127semidirect

Maximal subgroups

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

subgroupindex
#225506002
#225501532
#225500832
#225496222
#225496082
#225496052
#498×C1272
#1652127