Finite Group Databasebeta

← #30639883

Subgroups — #30639883

Order 16256.

282 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 71 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#22549880C212yesmaximal
2#22549777C211yesmaximal, characteristic
3#22549645C211yesmaximal, characteristic, complemented
4#22549635C212yesmaximal, complemented
5#544×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C2×#14523232C2212yes—
8C2×(C2×D127)⋊C4—22no—
9C2×C4×Dic127C2211yescharacteristic
10#16861129—22no—
11#16861129—22no—
12C4⋊Q8×C127—22no—
13#125×C127—22no—
14C22⋊Q8×C127—22no—
15#117×C127C2212yes—
16C2×C4⋊C4×C127C2211yescharacteristic
17C23×D127—21no—
18C2×D508—22no—
19C2×D508—22no—
20C2×D508—21no—
21C2×D508Q812yescomplemented
22C22×Dic127—21no—
23#14523232—22no—
24#14523232—22no—
25C2×C127⋊C8—22no—
26(C2×D127)⋊C4—44no—
27C4×Dic127—22no—
28C2×Q8×C127—22no—
29C22×C4×C127D412yes—
30C22×C4×C127C2311yescharacteristic
31C4⋊C4×C127—22no—
32C4⋊C4×C127—24no—
33C4⋊C4×C127—22no—
34C4⋊C4×C127D412yes—
35C4⋊C4×C127D412yes—
36C22⋊C4×C127—22no—
37C42×C127—22no—
38C22×D127—22no—
39C22×D127—42no—
40C22×D127—42no—
41C22×D127—41no—
42C22×D127—41no—
43D508—24no—
44D508—42no—
45C2×Dic127—22no—
46C2×Dic127—41no—
47C2×Dic127—41no—
48C127⋊C8—42no—
49C23×C127C2×D411yescharacteristic
50Q8×C127—82no—
51C2×C4×C127—22no—
52C2×C4×C127—22no—
53C2×C4×C127—22no—
54C2×C4×C127—42no—
55C2×C4×C127C2×D412yes—
56C2×C4×C127C2×Q811yescharacteristic
57C2×C4×C127#4112yes—
58C2×C4×C127#4111yescharacteristic
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×C127—42no—
71C4×C127—42no—
72C4×C127C22⋊Q812yes—
73C4×C127#12412yes—
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#544D12711yescharacteristic, complemented
84#418—1272no—
85#390—1271no—
86#343—1271no—
87#339—1272no—
88C22×D4—1271no—
89C2×M16—1272no—
90C4⋊Q8—22no—
91#125—22no—
92C22⋊Q8—22no—
93#117C2×D12712yes—
94C2×C4⋊C4C2×D12711yescharacteristic
95C2×C22⋊C4—2542no—
96C2×C42—1271no—
97#102—2542no—
98#102—2542no—
99C24—2541no—
100C2×Q8—22no—
101C2×D4—1272no—
102C2×D4—2542no—
103C2×D4—2541no—
104C2×D4—2542no—
105C22×C4—2541no—
106C22×C4C22⋊D12712yes—
107C22×C4C22×D12711yescharacteristic
108M16—2542no—
109M16—2542no—
110C2×C8—2542no—
111C4⋊C4—22no—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4C22⋊D12712yes—
115C4⋊C4C22⋊D12712yes—
116C22⋊C4—22no—
117C22⋊C4—5084no—
118C42—22no—
119C42—2542no—
120C23—5082no—
121C23—5082no—
122C23—5081no—
123C23—5081no—
124C23—2542no—
125C23C2×C22⋊D12711yescharacteristic
126Q8—82no—
127D4—5082no—
128D4—2544no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—42no—
133C2×C4—5081no—
134C2×C4—5081no—
135C2×C4—2542no—
136C2×C4C2×C22⋊D12712yes—
137C2×C4Q8×D12711yescharacteristic
138C2×C4#1452324912yes—
139C2×C4#1452324911yescharacteristic
140C8—5082no—
141C22—21no—
142C22—21no—
143C22—10162no—
144C22—10161no—
145C22—10161no—
146C22—5084no—
147C22—5082no—
148C22#1686117811yescharacteristic
149C22#1686118511yescharacteristic
150C22#1686119811yescharacteristic
151C4—42no—
152C4—42no—
153C4—42no—
154C4—5082no—
155C4#1686118512yes—
156C4#1686119812yes—
157C2—21no—
158C2—21no—
159C2—10162no—
160C2#2254978711yescharacteristic
161C2#2255004411yescharacteristic
162C2#2255004911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#544×C127C2
socleC22×C127#120
prime radicalC22×D508C22
hypercenter#544D127
hypocenterC127#2027
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
3. #22549645C2semidirect
4. #22549635C2semidirect
5. #544×C127C2semidirect
21. C2×D508Q8semidirect
82. C127#2027semidirect
83. #544D127semidirect

Maximal subgroups

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

subgroupindex
#225498802
#225497772
#225496452
#225496352
#544×C1272
#2027127