Finite Group Databasebeta

← #30640683

Subgroups — #30640683

Order 16256.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550676C211yesmaximal, characteristic
2#22550319C212yesmaximal, complemented
3#22550134C211yesmaximal, characteristic, complemented
4#22549687C212yesmaximal, complemented
5#349×C127C211yesmaximal, characteristic, complemented
6C2×#14523248—21no—
7#16861242—22no—
8#16861237—22no—
9#16861231—21no—
10#16861231C2212yes—
11C2×Dic508C2211yescharacteristic, complemented
12#16861188—21no—
13Dic1016—22no—
14#16861154—22no—
15#16861141C2211yescharacteristic, complemented
16#16861134—21no—
17C2×M16×C127C2211yescharacteristic
18M32×C127C2212yescomplemented
19#14523248—22no—
20#14523248—22no—
21#14523248—41no—
22#14523248—41no—
23C2×Dic254—21no—
24C2×Dic254—21no—
25C2×D508—21no—
26C2×C22⋊D127—41no—
27C2×C4×D127—41no—
28Dic508—21no—
29Dic508—22no—
30Dic508D412yes—
31#14523237—22no—
32#14523237—22no—
33D1016—21no—
34D1016D412yes—
35C4⋊Dic127—21no—
36Dic127⋊C4—41no—
37C22⋊Dic127—41no—
38C22×C4×C127D411yescharacteristic
39M16×C127—22no—
40C2×C8×C127D411yescharacteristic
41C2×C8×C127C2311yescharacteristic
42C16×C127—22no—
43C22×D127—41no—
44Dic254—22no—
45Dic254—41no—
46Dic254—41no—
47Dic254—41no—
48D508—22no—
49D508—41no—
50C22⋊D127—42no—
51C22⋊D127—42no—
52C2×Dic127—41no—
53C2×Dic127—41no—
54C2×Dic127—41no—
55C4×D127—42no—
56C4×D127—42no—
57C23×C127D811yescharacteristic
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127D811yescharacteristic
61C2×C4×C127C2×D411yescharacteristic
62C8×C127—21no—
63C8×C127C2×D412yes—
64C22×C127—21no—
65C22×C127—41no—
66C22×C127C2×D811yescharacteristic
67C2×D127—42no—
68C2×D127—82no—
69C4×C127—21no—
70C4×C127#12011yescharacteristic
71C4×C127AGL(1,ℤ/8)11yescharacteristic
72Dic127—42no—
73Dic127—81no—
74Dic127—81no—
75C2×C127—21no—
76C2×C127—41no—
77C2×C127#44711yescharacteristic
78D127—82no—
79#2191—1271nomaximal, Sylow
80C127#219111yescharacteristic, complemented, Sylow
81#575—1271no—
82#510—1272no—
83#470—1271no—
84#360—1272no—
85#349D12711yescharacteristic, complemented
86C2×#41—2541no—
87#137—2542no—
88AGL(1,ℤ/8)—2542no—
89#135—1272no—
90#135—2541no—
91C2×Q16—1271no—
92C2×M16C2×D12711yescharacteristic
93C22⋊Q8—2541no—
94Q32—2542no—
95SD32—2542no—
96M32C2×D12712yescomplemented
97#106—1271no—
98#103—2541no—
99#41—5081no—
100#41—5081no—
101#41—2542no—
102#41—2542no—
103C2×Q8—2541no—
104C2×Q8—2541no—
105C2×D4—5081no—
106C2×D4—2541no—
107C22×C4—5081no—
108C22×C4D50811yescharacteristic
109Q16—1272no—
110Q16—2541no—
111Q16—2542no—
112SD16—2542no—
113SD16—2542no—
114D8—1272no—
115D8—2541no—
116M16—22no—
117C2×C8D50811yescharacteristic
118C2×C8C22×D12711yescharacteristic
119C4⋊C4—5081no—
120C4⋊C4—2541no—
121C22⋊C4—5081no—
122C16—22no—
123C23—5081no—
124C23D101611yescharacteristic
125Q8—5081no—
126Q8—5081no—
127Q8—5081no—
128Q8—2542no—
129D4—5081no—
130D4—5082no—
131D4—5082no—
132D4—2542no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—5082no—
136C2×C4—5081no—
137C2×C4—5081no—
138C2×C4—5081no—
139C2×C4—5082no—
140C2×C4D101611yescharacteristic
141C2×C4C2×D50811yescharacteristic
142C8—21no—
143C8D4×D12712yes—
144C22—21no—
145C22—41no—
146C22—10162no—
147C22—5082no—
148C22C2×D101611yescharacteristic
149C4—21no—
150C4—10161no—
151C4—10161no—
152C4—5082no—
153C4#1686117911yescharacteristic
154C4#1686123711yescharacteristic
155C2—21no—
156C2—41no—
157C2—10162no—
158C2#2255001211yescharacteristic
subgroupG/H
centerC2#22550012
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#349×C127C2
socleC2×C127#447
prime radical#22550676C2
hypercenter#349D127
hypocenterC127#2191
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #22550319C2semidirect
3. #22550134C2semidirect
4. #22549687C2semidirect
5. #349×C127C2semidirect
11. C2×Dic508C22semidirect
15. #16861141C22semidirect
18. M32×C127C22semidirect
80. C127#2191semidirect
85. #349D127semidirect
96. M32C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
#225503192
#225501342
#225496872
#349×C1272
#2191127