Finite Group Databasebeta

← #30640759

Subgroups — #30640759

Order 16256.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550606C211yesmaximal, characteristic, complemented
2#22550059C212yesmaximal
3#22549675C211yesmaximal, characteristic, complemented
4#22549674C212yesmaximal, complemented
5#456×C127C211yesmaximal, characteristic
6#16861239—22no—
7#16861208—22no—
8#16861207—21no—
9#16861207C2212yes—
10C4×Dic254—21no—
11#16861174—22no—
12#16861136—22no—
13#16861128C2212yes—
142−1+4×C127C2211yescharacteristic
15#137×C127—21no—
16C4⋊Q8×C127C2211yescharacteristic
17#104×C127—21no—
18#101×C127C2211yescharacteristic
19C2×Dic254—21no—
20C2×Dic254—21no—
21C2×Dic254D412yes—
22#14523238—42no—
23#14523236—42no—
24#14523232—22no—
25C4⋊Dic127—21no—
26Dic127⋊C4—22no—
27Dic127⋊C4—22no—
28Dic127⋊C4—22no—
29C4×Dic127—21no—
30C4×Dic127—22no—
31C4×Dic127D412yes—
32#41×C127—22no—
33#41×C127—21no—
34#41×C127—41no—
35C2×Q8×C127—41no—
36C2×Q8×C127D411yescharacteristic
37C2×Q8×C127C2311yescharacteristic
38Q16×C127—41no—
39SD16×C127—41no—
40M16×C127—21no—
41C4⋊C4×C127—22no—
42C42×C127D411yescharacteristic
43Dic254—24no—
44Dic254—42no—
45Dic254—81no—
46C2×Dic127—22no—
47C2×Dic127—22no—
48C127⋊C8—42no—
49Q8×C127—21no—
50Q8×C127—22no—
51Q8×C127—41no—
52Q8×C127—41no—
53D4×C127—21no—
54D4×C127—22no—
55D4×C127—41no—
56C2×C4×C127—21no—
57C2×C4×C127—21no—
58C2×C4×C127—41no—
59C2×C4×C127—42no—
60C2×C4×C127C2×D412yes—
61C2×C4×C127C2×D411yescharacteristic
62C8×C127—41no—
63C22×C127—41no—
64C22×C127#12011yescharacteristic
65C4×C127—22no—
66C4×C127—21no—
67C4×C127—22no—
68C4×C127—41no—
69C4×C127—41no—
70Dic127—24no—
71Dic127—42no—
72C2×C127—21no—
73C2×C127—81no—
74C2×C127B(4,𝔽2)11yescharacteristic
75#2205—1271nomaximal, Sylow
76C127#220511yescharacteristic, complemented, Sylow
77Q82—1271no—
78#456—1272no—
79#456D12711yescharacteristic
80#356—1271no—
81#356—1272no—
822−1+4C2×D12711yescharacteristic
83#137—21no—
84#137—2542no—
85C4⋊Q8—1272no—
86C4⋊Q8—2542no—
87C4⋊Q8—2541no—
88C4⋊Q8C2×D12711yescharacteristic
89C4×Q8—2542no—
90C4×Q8—2541no—
91#104—21no—
92#104—2542no—
93#101—1272no—
94#101C2×D12711yescharacteristic
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—1272no—
100C2×Q8—2541no—
101C2×Q8—2541no—
102C2×Q8C22⋊D12711yescharacteristic
103C2×Q8C22×D12711yescharacteristic
104Q16—41no—
105Q16—5082no—
106SD16—41no—
107SD16—5082no—
108M16—21no—
109M16—2542no—
110C4⋊C4—22no—
111C4⋊C4—2542no—
112C4⋊C4—2541no—
113C4⋊C4—2542no—
114C4⋊C4—2542no—
115C42—1272no—
116C42—2542no—
117C42—2541no—
118C42C22⋊D12711yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—10161no—
124Q8—5082no—
125Q8—2544no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—2542no—
134C2×C4—2542no—
135C2×C4D4×D12712yes—
136C2×C4C2×C22⋊D12711yescharacteristic
137C8—41no—
138C8—5082no—
139C22—41no—
140C22#1686117711yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—5082no—
147C4—2544no—
148C2—21no—
149C2—81no—
150C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×Q8×C127C23
FrattiniC2×Q8C22×D127
Fitting#456×C127C2
socleC2×C127B(4,𝔽2)
prime radical2−1+4×C127C22
hypercenter#456D127
hypocenterC127#2205
LCSG ▷ C2×Q8×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #22550606C2semidirect
3. #22549675C2semidirect
4. #22549674C2semidirect
76. C127#2205semidirect

Maximal subgroups

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

subgroupindex
#225506062
#225500592
#225496752
#225496742
#456×C1272
#2205127