Finite Group Databasebeta

← #35213609

Subgroups — #35213609

Order 20864.

236 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 55 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#23782376C211yesmaximal, characteristic, complemented
2#23782039C211yesmaximal, characteristic, complemented
3#23781866C211yesmaximal, characteristic, complemented
4#23781861C211yesmaximal, characteristic
5#23781395C211yesmaximal, characteristic, complemented
6#23781392C211yesmaximal, characteristic, complemented
7#333×C163C211yesmaximal, characteristic, complemented
8C2×#14579688—21no—
9C2×#14579686—22no—
10C2×D1304—21no—
11#19104732C2211yescharacteristic
12#19104726—21no—
13#19104715—21no—
14#19104711—22no—
15C4×Dic326C2211yescharacteristic
16C4×D652C2211yescharacteristic
17#19104697—21no—
18C4⋊C4×D163—21no—
19#19104657—21no—
20#19104657—21no—
21C4×C163⋊C8C2211yescharacteristic
22C4⋊Q8×C163C2211yescharacteristic
23C4⋊C8×C163C2211yescharacteristic
24C4⋊C8×C163C2211yescharacteristic
25C2×Dic326D411yescharacteristic
26C2×D652—21no—
27C2×D652—21no—
28C2×D652—21no—
29C2×D652D411yescharacteristic
30C2×C4×D163—22no—
31C2×C4×D163—21no—
32#14579688—42no—
33#14579686—44no—
34D1304—42no—
35C2×C163⋊C8—21no—
36C4⋊Dic163—22no—
37C4⋊Dic163D411yescharacteristic
38C4⋊Dic163D411yescharacteristic
39Dic163⋊C4—22no—
40Dic163⋊C4—21no—
41(C2×D163)⋊C4—22no—
42(C2×D163)⋊C4—21no—
43C4×Dic163—21no—
44C2×Q8×C163D412yes—
45C2×C8×C163—21no—
46C2×C8×C163—21no—
47C4⋊C4×C163—21no—
48C4⋊C4×C163—21no—
49C42×C163C2311yescharacteristic
50C22×D163—21no—
51C22×D163—41no—
52Dic326—21no—
53Dic326—21no—
54D652—21no—
55D652—42no—
56D652—41no—
57D652—41no—
58D652—41no—
59D652—41no—
60D652SD1612yescomplemented
61C2×Dic163—21no—
62C2×Dic163—21no—
63C2×Dic163—21no—
64C4×D163—41no—
65C4×D163—41no—
66C4×D163—82no—
67C163⋊C8—22no—
68Q8×C163—42no—
69C2×C4×C163—22no—
70C2×C4×C163C2×D411yescharacteristic
71C2×C4×C163C2×D411yescharacteristic
72C2×C4×C163C2×D411yescharacteristic
73C8×C163—41no—
74C8×C163—41no—
75C22×C163#12011yescharacteristic
76C2×D163—22no—
77C2×D163—41no—
78C2×D163—41no—
79C2×D163—81no—
80C2×D163—81no—
81C2×D163—81no—
82C4×C163—21no—
83C4×C163—42no—
84C4×C163C2×SD1611yescharacteristic
85C4×C163#13511yescharacteristic
86C4×C163AGL(1,ℤ/8)11yescharacteristic
87C4×C163#13711yescharacteristic
88Dic163—41no—
89Dic163—41no—
90Dic163—41no—
91C2×C163#44811yescharacteristic
92C2×C163#44911yescharacteristic
93C2×C163#45311yescharacteristic
94D163—42no—
95D163—161no—
96C163#162211yescharacteristic, complemented, Sylow
97#1622—1631nomaximal, Sylow
98#554—1631no—
99#492—1631no—
100#460—1631no—
101#459—1631no—
102#333D16311yescharacteristic, complemented
103Q8⋊C8—1631no—
104D4⋊C8—1631no—
105C2×SD16—3261no—
106C2×SD16—3262no—
107C2×D8—3261no—
108C4⋊Q8C2×D16311yescharacteristic
109#127—1631no—
110#125—3261no—
111C22⋊Q8—3261no—
112C22⋊Q8—3262no—
113C4×Q8—1631no—
114C4×D4—3261no—
115C4×D4—1631no—
116C2×C4⋊C4—3261no—
117C4⋊C8C2×D16311yescharacteristic
118C4⋊C8C2×D16311yescharacteristic
119#102—3261no—
120#102—3261no—
121C4×C8—1631no—
122C2×Q8—1631no—
123C2×Q8D65212yes—
124C2×D4—3261no—
125C2×D4—3261no—
126C2×D4—3261no—
127C2×D4—1631no—
128C22×C4—3262no—
129C22×C4—3261no—
130SD16—6522no—
131SD16—6524no—
132D8—6522no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—3261no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—3262no—
139C4⋊C4—3262no—
140C4⋊C4—3261no—
141C4⋊C4—1631no—
142C4⋊C4—1631no—
143C22⋊C4—3262no—
144C22⋊C4—3261no—
145C42—3261no—
146C42C22×D16311yescharacteristic
147C23—3261no—
148C23—6521no—
149Q8—42no—
150Q8—3261no—
151Q8—3261no—
152D4—3261no—
153D4—6522no—
154D4—6521no—
155D4—6521no—
156D4—6521no—
157D4—6521no—
158D4—1632no—
159C2×C4—22no—
160C2×C4—13042no—
161C2×C4—3261no—
162C2×C4—3261no—
163C2×C4—3261no—
164C2×C4—6521no—
165C2×C4—6521no—
166C2×C4D4×D16311yescharacteristic
167C2×C4D4×D16311yescharacteristic
168C2×C4C2×D65211yescharacteristic
169C8—41no—
170C8—41no—
171C8—3262no—
172C22—13041no—
173C22—13041no—
174C22—13041no—
175C22—3262no—
176C22—6521no—
177C22—6521no—
178C22#1910470511yescharacteristic
179C4—21no—
180C4—42no—
181C4—6521no—
182C4—6521no—
183C4—6521no—
184C4SD16×D16311yescharacteristic
185C4#1910475811yescharacteristic
186C4#1910476211yescharacteristic
187C4#1910476711yescharacteristic
188C2—26081no—
189C2—6522no—
190C2#2378180511yescharacteristic
191C2#2378181311yescharacteristic
192C2#2378183511yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#333×C163C2
socleC22×C163#120
prime radical#23781861C2
hypercenter#333D163
hypocenterC163#1622
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #23782376C2semidirect
2. #23782039C2semidirect
3. #23781866C2semidirect
5. #23781395C2semidirect
6. #23781392C2semidirect
7. #333×C163C2semidirect
60. D652SD16semidirect
96. C163#1622semidirect
102. #333D163semidirect

Maximal subgroups

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

subgroupindex
#237823762
#237820392
#237818662
#237818612
#237813952
#237813922
#333×C1632
#1622163