Finite Group Databasebeta

← #30637706

Subgroups — #30637706

Order 16256.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 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#22550507C211yesmaximal, characteristic
2#22550034C212yesmaximal
3#22549796C211yesmaximal, characteristic
4#22549614C212yesmaximal
5#324×C127C211yesmaximal, characteristic
6C22×Dic254C2211yescharacteristic
7C2×Dic508—22no—
8#16861208—22no—
9#16861200—22no—
10#16861188—21no—
11#16861188C2212yes—
12#16861168—21no—
13C2×Dic127⋊C4—21no—
14#16861134—22no—
15C127⋊#95—21no—
16C127⋊#95C2211yescharacteristic
17C2×C4⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19C2×Dic254—21no—
20C2×Dic254—22no—
21C2×Dic254—42no—
22C2×Dic254D412yes—
23C22×Dic127—21no—
24C22×Dic127—21no—
25Dic508—44no—
26C4⋊Dic127—21no—
27C4⋊Dic127—21no—
28C4⋊Dic127—21no—
29C4⋊Dic127D412yes—
30Dic127⋊C4—22no—
31Dic127⋊C4—22no—
32Dic127⋊C4—42no—
33C22⋊Dic127—22no—
34C4×Dic127—22no—
35C22×C4×C127D411yescharacteristic
36C22×C4×C127D411yescharacteristic
37C22×C4×C127C2311yescharacteristic
38C2×C8×C127—22no—
39C4⋊C4×C127—22no—
40Dic254—24no—
41Dic254—42no—
42Dic254—42no—
43Dic254—81no—
44C2×Dic127—22no—
45C2×Dic127—22no—
46C2×Dic127—22no—
47C2×Dic127—22no—
48C2×Dic127—41no—
49C2×Dic127—41no—
50C2×Dic127—41no—
51C2×Dic127—41no—
52C23×C127C2×D411yescharacteristic
53C2×C4×C127—21no—
54C2×C4×C127—21no—
55C2×C4×C127—21no—
56C2×C4×C127—41no—
57C2×C4×C127—41no—
58C2×C4×C127Q1612yescomplemented
59C2×C4×C127C2×D412yes—
60C8×C127—42no—
61C22×C127—21no—
62C22×C127—21no—
63C22×C127#12011yescharacteristic
64C22×C127C2×Q1611yescharacteristic
65C22×C127#13711yescharacteristic
66C4×C127—22no—
67C4×C127—22no—
68C4×C127—41no—
69Dic127—42no—
70Dic127—42no—
71Dic127—42no—
72Dic127—42no—
73C2×C127—21no—
74C2×C127—21no—
75C2×C127#45111yescharacteristic
76C2×C127#45611yescharacteristic
77C2×C127B(4,𝔽2)11yescharacteristic
78#1605—1271nomaximal, Sylow
79C127#160511yescharacteristic, complemented, Sylow
80#544—1271no—
81#451—1272no—
82#395—1271no—
83#328—1272no—
84#324D12711yescharacteristic
85C22×Q8—1271no—
86C2×Q16—2542no—
87C4⋊Q8—2542no—
88#125—2542no—
89C22⋊Q8—1272no—
90C22⋊Q8—2541no—
91#117—2541no—
92C2×C4⋊C4—2541no—
93C2×C4⋊C4C2×D12711yescharacteristic
94#103—2542no—
95C22⋊C8C2×D12712yes—
96#95—1271no—
97#95—2541no—
98C2×Q8—1272no—
99C2×Q8—5082no—
100C2×Q8—2541no—
101C2×Q8—2542no—
102C22×C4—2541no—
103C22×C4—2541no—
104C22×C4D50811yescharacteristic
105C22×C4D50811yescharacteristic
106C22×C4C22×D12711yescharacteristic
107Q16—5084no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—1272no—
111C4⋊C4—5082no—
112C4⋊C4—2542no—
113C4⋊C4—2541no—
114C4⋊C4—2541no—
115C4⋊C4—2541no—
116C4⋊C4—2542no—
117C22⋊C4—2542no—
118C42—2542no—
119C23C2×D50811yescharacteristic
120Q8—10161no—
121Q8—5082no—
122Q8—5082no—
123Q8—2544no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—5081no—
130C2×C4—5081no—
131C2×C4—5081no—
132C2×C4—5081no—
133C2×C4—2542no—
134C2×C4—2542no—
135C2×C4—2542no—
136C2×C4—2542no—
137C2×C4Dic50812yescomplemented
138C2×C4D4×D12712yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#1686117911yescharacteristic
143C22C2×Dic50811yescharacteristic
144C22#1686124211yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—5082no—
149C4—5082no—
150C4—5082no—
151C4—5082no—
152C2—21no—
153C2—21no—
154C2#2255003411yescharacteristic
155C2#2255006111yescharacteristic
156C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
hypercenter#324D127
hypocenterC127#1605
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
58. C2×C4×C127Q16semidirect
79. C127#1605semidirect
137. C2×C4Dic508semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225500342
#225497962
#225496142
#324×C1272
#1605127