Finite Group Databasebeta

← #30638241

Subgroups — #30638241

Order 16256.

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

normal subgroups only

This group has 166 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#22550279C211yesmaximal, characteristic, complemented
2#22550081C211yesmaximal, characteristic
3#22549954C211yesmaximal, characteristic
4#22549626C211yesmaximal, characteristic, complemented
5#22549623C211yesmaximal, characteristic, complemented
6#22549622C211yesmaximal, characteristic, complemented
7#461×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9C2×#14523232—21no—
10#16861209C2211yescharacteristic
11#16861142—21no—
12#16861142—21no—
13C4⋊(C127⋊C8)C2211yescharacteristic
14C4⋊(C127⋊C8)C2211yescharacteristic
15#16861132—21no—
16C22⋊(C127⋊C8)—21no—
17C4×C127⋊C8C2211yescharacteristic
18C2×SD16×C127—21no—
19C4⋊Q8×C127C2211yescharacteristic
20C4×D4×C127C2211yescharacteristic
21C4⋊C8×C127C2211yescharacteristic
22#103×C127—21no—
23C2×Dic254—21no—
24#14523236—42no—
25#14523232—42no—
26C2×C127⋊C8—21no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8D411yescharacteristic
29C2×C127⋊C8D411yescharacteristic
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C2×Q8×C127—21no—
33C2×D4×C127D411yescharacteristic
34C22×C4×C127—21no—
35SD16×C127—42no—
36C2×C8×C127—21no—
37C4⋊C4×C127—21no—
38C4⋊C4×C127—21no—
39C4⋊C4×C127D411yescharacteristic
40C22⋊C4×C127—21no—
41C42×C127C2311yescharacteristic
42Dic254—41no—
43Dic254—41no—
44C2×Dic127—41no—
45C127⋊C8—21no—
46C127⋊C8—41no—
47C127⋊C8—41no—
48C127⋊C8SD1612yescomplemented
49C23×C127—21no—
50Q8×C127—41no—
51Q8×C127—41no—
52D4×C127—21no—
53D4×C127—21no—
54C2×C4×C127—21no—
55C2×C4×C127—41no—
56C2×C4×C127—41no—
57C2×C4×C127—41no—
58C2×C4×C127C2×D411yescharacteristic
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127#4111yescharacteristic
61C8×C127—41no—
62C22×C127—41no—
63C22×C127—41no—
64C22×C127—41no—
65C22×C127#12111yescharacteristic
66C4×C127—21no—
67C4×C127—41no—
68C4×C127—81no—
69C4×C127C2×SD1611yescharacteristic
70C4×C127AGL(1,ℤ/8)11yescharacteristic
71C4×C127AGL(1,ℤ/8)11yescharacteristic
72C4×C127#13711yescharacteristic
73Dic127—81no—
74C2×C127—81no—
75C2×C127#46111yescharacteristic
76C2×C127#46911yescharacteristic
77C2×C127#47311yescharacteristic
78#1693—1271nomaximal, Sylow
79C127#169311yescharacteristic, complemented, Sylow
80#500—1271no—
81#461—1271no—
82#461D12711yescharacteristic
83#436—1271no—
84#334—1271no—
85#332—1271no—
86#332—1271no—
87C2×SD16—21no—
88C2×SD16—2541no—
89C2×M16—2541no—
90C4⋊Q8—1271no—
91C4⋊Q8C2×D12711yescharacteristic
92C4×D4C2×D12711yescharacteristic
93#107—2541no—
94#107—2541no—
95C4⋊C8—1271no—
96C4⋊C8—1271no—
97C4⋊C8C2×D12711yescharacteristic
98#103—21no—
99#103—2541no—
100C22⋊C8—2541no—
101C4×C8—1271no—
102C2×Q8—21no—
103C2×Q8—2541no—
104C2×D4C22⋊D12711yescharacteristic
105C22×C4—21no—
106SD16—42no—
107SD16—5082no—
108M16—5082no—
109C2×C8—21no—
110C2×C8—1271no—
111C2×C8—1271no—
112C2×C8—2541no—
113C2×C8—2541no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—2541no—
117C4⋊C4—2541no—
118C4⋊C4C22⋊D12711yescharacteristic
119C22⋊C4—21no—
120C42C22×D12711yescharacteristic
121C23—21no—
122Q8—41no—
123Q8—41no—
124Q8—5081no—
125Q8—5081no—
126D4—21no—
127D4—21no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—5081no—
133C2×C4D4×D12711yescharacteristic
134C2×C4C2×C22⋊D12711yescharacteristic
135C2×C4#1452324711yescharacteristic
136C8—41no—
137C8—1272no—
138C8—5081no—
139C8—5081no—
140C8—2541no—
141C22—41no—
142C22—41no—
143C22—41no—
144C22#1686118111yescharacteristic
145C4—21no—
146C4—41no—
147C4—81no—
148C4—10161no—
149C4SD16×D12711yescharacteristic
150C4#1686123411yescharacteristic
151C4#1686123511yescharacteristic
152C4#1686123811yescharacteristic
153C2—81no—
154C2#2255007911yescharacteristic
155C2#2255012711yescharacteristic
156C2#2255014911yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#461×C127C2
socleC22×C127#121
prime radicalC2×D4×C127D4
hypercenter#461D127
hypocenterC127#1693
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #22550279C2semidirect
4. #22549626C2semidirect
5. #22549623C2semidirect
6. #22549622C2semidirect
48. C127⋊C8SD16semidirect
79. C127#1693semidirect

Maximal subgroups

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

subgroupindex
#225502792
#225500812
#225499542
#225496262
#225496232
#225496222
#461×C1272
#1693127