Finite Group Databasebeta

← #35216559

Subgroups — #35216559

Order 20864.

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

normal subgroups only

This group has 216 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#23782415C211yesmaximal, characteristic, complemented
2#23781850C212yesmaximal, complemented
3#23781837C211yesmaximal, characteristic
4#23781463C212yesmaximal, complemented
5#355×C163C211yesmaximal, characteristic, complemented
6#19104783C2211yescharacteristic
7#19104768—22no—
8#19104761—21no—
9#19104733C2212yes—
10#19104728—22no—
11#19104727—21no—
12#19104716—21no—
13#19104712—22no—
14#19104663—21no—
15#19104661—22no—
16#19104654C2211yescharacteristic
17#124×C163C2211yescharacteristic
18#101×C163C2212yes—
19#14579700—41no—
20#14579699—22no—
21#14579699—21no—
22C2×Dic326D412yes—
23Q8×D163—41no—
24C22×Dic163—21no—
25Dic652—42no—
26#14579688—42no—
27#14579687—41no—
28#14579684—41no—
29#14579683—21no—
30C4⋊Dic163—21no—
31C4⋊Dic163—21no—
32Dic163⋊C4—22no—
33Dic163⋊C4—22no—
34Dic163⋊C4—22no—
35C22⋊Dic163—22no—
36C22⋊Dic163—21no—
37C4×Dic163D412yes—
38C2×Q8×C163C2311yescharacteristic
39C2×D4×C163D411yescharacteristic
40M16×C163—22no—
41C22⋊C4×C163—22no—
42C42×C163D411yescharacteristic
43Dic326—21no—
44Dic326—22no—
45Dic326—42no—
46D652—21no—
47D652—22no—
48C22⋊D163—41no—
49C2×Dic163—22no—
50C2×Dic163—22no—
51C2×Dic163—81no—
52C4×D163—42no—
53C4×D163—41no—
54C163⋊C8—41no—
55C23×C163—21no—
56Q8×C163—41no—
57D4×C163—41no—
58C2×C4×C163—21no—
59C2×C4×C163C2×D412yes—
60C2×C4×C163C2×D411yescharacteristic
61C8×C163—42no—
62C22×C163—42no—
63C22×C163—41no—
64C22×C163#12011yescharacteristic
65C2×D163—41no—
66C4×C163—22no—
67C4×C163—21no—
68C4×C163—41no—
69Dic163—42no—
70Dic163—41no—
71Dic163—42no—
72C2×C163—21no—
73C2×C163—81no—
74C2×C163B(4,𝔽2)11yescharacteristic
75D163—81no—
76C163#220311yescharacteristic, complemented, Sylow
77#2203—1631nomaximal, Sylow
78#563—1631no—
79#456—1632no—
80#454—1631no—
81#356—1632no—
82#355D16311yescharacteristic, complemented
832−1+4—1631no—
84#137—3262no—
85AGL(1,ℤ/8)—3261no—
86C4⋊Q8—1632no—
87#126—3262no—
88#125—3261no—
89#124C2×D16311yescharacteristic
90#123—3261no—
91C22⋊Q8—3262no—
92#104—3262no—
93#104—3261no—
94#101—1631no—
95#101C2×D16312yes—
96#41—3262no—
97#41—3261no—
98#41—6521no—
99C2×Q8—6521no—
100C2×Q8—1632no—
101C2×Q8C22×D16311yescharacteristic
102C2×D4D65211yescharacteristic
103C22×C4—3261no—
104Q16—6522no—
105SD16—6522no—
106SD16—6521no—
107D8—6521no—
108M16—22no—
109M16—3261no—
110C4⋊C4—3261no—
111C4⋊C4—3262no—
112C4⋊C4—3261no—
113C4⋊C4—3262no—
114C4⋊C4—3262no—
115C22⋊C4—22no—
116C22⋊C4—3262no—
117C22⋊C4—3261no—
118C42—1632no—
119C42D65211yescharacteristic
120C23—21no—
121Q8—41no—
122Q8—3261no—
123Q8—3262no—
124Q8—6522no—
125D4—41no—
126D4—3261no—
127D4—3262no—
128D4—6521no—
129C2×C4—21no—
130C2×C4—13041no—
131C2×C4—3262no—
132C2×C4—3262no—
133C2×C4—6522no—
134C2×C4—6521no—
135C2×C4D4×D16312yes—
136C2×C4C2×D65211yescharacteristic
137C8—42no—
138C8—6521no—
139C22—42no—
140C22—41no—
141C22—6521no—
142C22#1910470511yescharacteristic
143C4—22no—
144C4—21no—
145C4—41no—
146C4—6522no—
147C4—6522no—
148C4—6521no—
149C2—21no—
150C2—81no—
151C2—13041no—
152C2#2378185311yescharacteristic
subgroupG/H
centerC2#23781853
commutator, cosocleC2×Q8×C163C23
FrattiniC2×Q8C22×D163
Fitting#355×C163C2
socleC2×C163B(4,𝔽2)
prime radical#23781837C2
hypercenter#355D163
hypocenterC163#2203
LCSG ▷ C2×Q8×C163 ▷ C22×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #23782415C2semidirect
2. #23781850C2semidirect
4. #23781463C2semidirect
5. #355×C163C2semidirect
76. C163#2203semidirect
82. #355D163semidirect

Maximal subgroups

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

subgroupindex
#237824152
#237818502
#237818372
#237814632
#355×C1632
#2203163