Finite Group Databasebeta

← #30639887

Subgroups — #30639887

Order 16256.

282 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 67 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#22550507C211yesmaximal, characteristic, complemented
2#22549882C212yesmaximal
3#22549777C211yesmaximal, characteristic
4#22549637C212yesmaximal, complemented
5#343×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7#16861208—22no—
8#16861200—22no—
9#16861188—22no—
10#16861168C2212yes—
11C2×(C2×D127)⋊C4—22no—
12C2×C4×Dic127C2211yescharacteristic
13#16861131—22no—
14#16861131—22no—
15C2×M16×C127C2212yes—
16C2×C4⋊C4×C127C2211yescharacteristic
17C23×D127—21no—
18C2×Dic254—22no—
19C2×D508—22no—
20C2×D508—22no—
21C2×D508—21no—
22C2×D508Q812yescomplemented
23C22×Dic127—21no—
24C4⋊Dic127—22no—
25C4⋊Dic127D412yes—
26C4⋊Dic127D412yes—
27Dic127⋊C4—24no—
28C22⋊Dic127—22no—
29(C2×D127)⋊C4—44no—
30C4×Dic127—22no—
31C4×Dic127—22no—
32C22×C4×C127D412yes—
33C22×C4×C127C2311yescharacteristic
34M16×C127—22no—
35M16×C127—22no—
36C2×C8×C127—22no—
37C4⋊C4×C127—22no—
38C22×D127—22no—
39C22×D127—42no—
40C22×D127—42no—
41C22×D127—41no—
42C22×D127—41no—
43Dic254—82no—
44D508—24no—
45D508—42no—
46C2×Dic127—22no—
47C2×Dic127—22no—
48C2×Dic127—22no—
49C2×Dic127—41no—
50C2×Dic127—41no—
51C23×C127C2×D411yescharacteristic
52C2×C4×C127—22no—
53C2×C4×C127—42no—
54C2×C4×C127C2×D412yes—
55C2×C4×C127C2×Q811yescharacteristic
56C2×C4×C127#4111yescharacteristic
57C2×C4×C127#4112yes—
58C8×C127—42no—
59C22×C127—21no—
60C22×C127—21no—
61C22×C127#12011yescharacteristic
62C22×C127C22⋊Q811yescharacteristic
63C22×C127#12411yescharacteristic
64C2×D127—44no—
65C2×D127—42no—
66C2×D127—82no—
67C2×D127—81no—
68C2×D127—81no—
69C4×C127—42no—
70C4×C127C22⋊Q812yes—
71C4×C127#12412yes—
72Dic127—42no—
73Dic127—42no—
74Dic127—42no—
75C2×C127—21no—
76C2×C127—21no—
77C2×C127#39311yescharacteristic
78C2×C127#45311yescharacteristic
79C2×C127#45411yescharacteristic
80D127—82no—
81#2027—1271nomaximal, Sylow
82C127#202711yescharacteristic, complemented, Sylow
83#544—1271no—
84#418—1272no—
85#390—1271no—
86#343D12711yescharacteristic, complemented
87#339—1272no—
88C22×D4—1271no—
89C2×M16C2×D12712yes—
90C4⋊Q8—2542no—
91#125—2542no—
92C22⋊Q8—2542no—
93#117—1272no—
94C2×C4⋊C4C2×D12711yescharacteristic
95C2×C22⋊C4—2542no—
96C2×C42—1271no—
97#102—2542no—
98#102—2542no—
99C24—2541no—
100C2×Q8—2542no—
101C2×D4—1272no—
102C2×D4—2542no—
103C2×D4—2541no—
104C2×D4—2542no—
105C22×C4—2541no—
106C22×C4D50812yes—
107C22×C4C22×D12711yescharacteristic
108M16—22no—
109M16—22no—
110C2×C8—22no—
111C4⋊C4—22no—
112C4⋊C4—1272no—
113C4⋊C4—1272no—
114C4⋊C4—2544no—
115C4⋊C4—2542no—
116C22⋊C4—5084no—
117C22⋊C4—2542no—
118C42—2542no—
119C42—2542no—
120C23—5082no—
121C23—5082no—
122C23—5081no—
123C23—5081no—
124C23—2542no—
125C23C2×D50811yescharacteristic
126Q8—10162no—
127D4—5082no—
128D4—2544no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—5081no—
132C2×C4—5081no—
133C2×C4—2542no—
134C2×C4—2542no—
135C2×C4—2542no—
136C2×C4D4×D12712yes—
137C2×C4Q8×D12711yescharacteristic
138C2×C4#1452324711yescharacteristic
139C2×C4#1452324812yes—
140C8—42no—
141C22—21no—
142C22—21no—
143C22—10162no—
144C22—10161no—
145C22—10161no—
146C22—5084no—
147C22—5082no—
148C22#1686117911yescharacteristic
149C22#1686118911yescharacteristic
150C22#1686119711yescharacteristic
151C4—42no—
152C4—5082no—
153C4—5082no—
154C4—5082no—
155C4#1686118712yes—
156C4#1686119512yes—
157C2—21no—
158C2—21no—
159C2—10162no—
160C2#2254978911yescharacteristic
161C2#2255004611yescharacteristic
162C2#2255005111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#343×C127C2
socleC22×C127#120
prime radicalC22×D508C22
hypercenter#343D127
hypocenterC127#2027
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. #22550507C2semidirect
4. #22549637C2semidirect
5. #343×C127C2semidirect
22. C2×D508Q8semidirect
82. C127#2027semidirect
86. #343D127semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225498822
#225497772
#225496372
#343×C1272
#2027127