Finite Group Databasebeta

← #35213826

Subgroups — #35213826

Order 20864.

262 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 45 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#23782407C211yesmaximal, characteristic, complemented
2#23782055C211yesmaximal, characteristic, complemented
3#23782054C212yesmaximal, complemented
4#23781404C212yesmaximal, complemented
5#329×C163C211yesmaximal, characteristic, complemented
6C2×#14579688—22no—
7C2×#14579686—21no—
8C2×D1304—22no—
9C2×#14579684—21no—
10#19104732C2211yescharacteristic, complemented
11#19104724—21no—
12#19104723C2212yes—
13#19104721—22no—
14#19104705—22no—
15#19104703—21no—
16#19104646C2211yescharacteristic, complemented
17#124×C163C2211yescharacteristic
18#97×C163C2212yescomplemented
19C23×D163—21no—
20C2×Dic326D412yes—
21C2×D652—21no—
22C2×D652—21no—
23C2×D652—22no—
24C2×D652D412yes—
25C2×C22⋊D163—22no—
26#14579688—44no—
27#14579686—42no—
28D1304—44no—
29#14579684—42no—
30C2×C163⋊C8—21no—
31C22⋊Dic163—21no—
32(C2×D163)⋊C4—22no—
33(C2×D163)⋊C4—22no—
34(C2×D163)⋊C4—22no—
35C4×Dic163—21no—
36C2×Q8×C163D411yescharacteristic
37C2×D4×C163D411yescharacteristic
38C2×C8×C163—22no—
39C22⋊C4×C163—22no—
40C42×C163C2311yescharacteristic
41C22×D163—22no—
42C22×D163—41no—
43C22×D163—82no—
44C22×D163—81no—
45Dic326—42no—
46D652—42no—
47D652—41no—
48D652—42no—
49D652—41no—
50D652—42no—
51D652—81no—
52C22⋊D163—82no—
53C2×Dic163—22no—
54C163⋊C8—22no—
55C23×C163—21no—
56Q8×C163—41no—
57D4×C163—41no—
58C2×C4×C163—21no—
59C2×C4×C163C2×D412yes—
60C2×C4×C163C2×D411yescharacteristic
61C8×C163—24no—
62C22×C163—41no—
63C22×C163—42no—
64C22×C163#12011yescharacteristic
65C2×D163—42no—
66C2×D163—42no—
67C2×D163—42no—
68C2×D163—84no—
69C2×D163—82no—
70C2×D163—81no—
71C4×C163—22no—
72C4×C163—21no—
73C4×C163—41no—
74Dic163—42no—
75C2×C163—81no—
76C2×C163#45312yes—
77C2×C163#45311yescharacteristic
78D163—82no—
79D163—161no—
80C163#165511yescharacteristic, complemented, Sylow
81#1655—1631nomaximal, Sylow
82H(3,𝔽4)—1631no—
83#496—1631no—
84#496—1632no—
85#329—1632no—
86#329D16311yescharacteristic, complemented
87C2×SD16—3262no—
88C2×SD16—3261no—
89C2×D8—3262no—
90C2×D8—3261no—
91#127—1631no—
92#124—3262no—
93#124—3261no—
94#124—1632no—
95#124C2×D16311yescharacteristic
96#120—3262no—
97#120—3261no—
98#97—1631no—
99#97C2×D16312yescomplemented
100C24—3261no—
101C2×Q8—1632no—
102C2×Q8D65211yescharacteristic
103C2×D4—3261no—
104C2×D4—3262no—
105C2×D4—3262no—
106C2×D4—3261no—
107C2×D4—1632no—
108C2×D4D65211yescharacteristic
109SD16—6524no—
110SD16—6522no—
111D8—6524no—
112D8—6522no—
113C2×C8—22no—
114C2×C8—3261no—
115C22⋊C4—22no—
116C22⋊C4—3262no—
117C22⋊C4—3261no—
118C22⋊C4—3262no—
119C22⋊C4—3262no—
120C42—3261no—
121C42C22×D16311yescharacteristic
122C23—21no—
123C23—13042no—
124C23—13041no—
125C23—3262no—
126C23—6521no—
127Q8—41no—
128Q8—6522no—
129D4—41no—
130D4—13041no—
131D4—13042no—
132D4—6522no—
133D4—6521no—
134D4—6522no—
135D4—6521no—
136D4—6522no—
137C2×C4—21no—
138C2×C4—3262no—
139C2×C4D4×D16312yes—
140C2×C4C2×D65211yescharacteristic
141C8—24no—
142C8—3262no—
143C22—42no—
144C22—41no—
145C22—13044no—
146C22—13042no—
147C22—13041no—
148C22—6522no—
149C22—6522no—
150C22—6522no—
151C22#1910470511yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—6522no—
156C2—81no—
157C2—13042no—
158C2—26081no—
159C2#2378183412yes—
160C2#2378183511yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#329×C163C2
socleC22×C163#120
hypercenter#329D163
hypocenterC163#1655
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #23782407C2semidirect
2. #23782055C2semidirect
3. #23782054C2semidirect
4. #23781404C2semidirect
5. #329×C163C2semidirect
10. #19104732C22semidirect
16. #19104646C22semidirect
18. #97×C163C22semidirect
80. C163#1655semidirect
86. #329D163semidirect
99. #97C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824072
#237820552
#237820542
#237814042
#329×C1632
#1655163