Finite Group Databasebeta

← #35213765

Subgroups — #35213765

Order 20864.

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

normal subgroups only

This group has 212 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#23782381C211yesmaximal, characteristic
2#23782015C211yesmaximal, characteristic, complemented
3#23781946C211yesmaximal, characteristic, complemented
4#23781871C211yesmaximal, characteristic, complemented
5#23781392C211yesmaximal, characteristic, complemented
6#23781392C211yesmaximal, characteristic, complemented
7#333×C163C211yesmaximal, characteristic, complemented
8C2×#14579688—21no—
9#19104735C2211yescharacteristic
10#19104724—22no—
11#19104710—21no—
12#19104710—21no—
13C4×D652C2211yescharacteristic
14C4×D652C2211yescharacteristic
15#19104693—21no—
16#19104693—21no—
17#19104669—21no—
18#19104660—21no—
19#19104659—22no—
20#19104657—21no—
21C4×C163⋊C8C2211yescharacteristic
22C4⋊Q8×C163C2211yescharacteristic
23C4⋊C8×C163C2211yescharacteristic
24C4⋊C8×C163C2211yescharacteristic
25C2×Dic326—21no—
26C2×D652—22no—
27C2×D652D411yescharacteristic
28C2×D652D411yescharacteristic
29C2×C4×D163—21no—
30C2×C4×D163—21no—
31#14579688—42no—
32C2×C163⋊C8—21no—
33C4⋊Dic163—21no—
34C4⋊Dic163—21no—
35C4⋊Dic163D411yescharacteristic
36C4⋊Dic163D411yescharacteristic
37(C2×D163)⋊C4—22no—
38(C2×D163)⋊C4—22no—
39(C2×D163)⋊C4—21no—
40(C2×D163)⋊C4—21no—
41C4×Dic163—22no—
42C2×Q8×C163D412yes—
43C2×C8×C163—21no—
44C2×C8×C163—21no—
45C4⋊C4×C163—21no—
46C4⋊C4×C163—21no—
47C42×C163C2311yescharacteristic
48C22×D163—21no—
49C22×D163—21no—
50Dic326—41no—
51Dic326—41no—
52D652—21no—
53D652—21no—
54D652—21no—
55D652—21no—
56D652—82no—
57C2×Dic163—21no—
58C2×Dic163—21no—
59C2×Dic163—41no—
60C4×D163—41no—
61C4×D163—41no—
62C4×D163—41no—
63C4×D163—41no—
64C163⋊C8—41no—
65Q8×C163—42no—
66C2×C4×C163—22no—
67C2×C4×C163C2×D411yescharacteristic
68C2×C4×C163C2×D411yescharacteristic
69C2×C4×C163C2×D411yescharacteristic
70C8×C163—41no—
71C8×C163—41no—
72C22×C163#12011yescharacteristic
73C2×D163—41no—
74C2×D163—41no—
75C2×D163—41no—
76C2×D163—41no—
77C2×D163—41no—
78C2×D163—41no—
79C4×C163—21no—
80C4×C163—42no—
81C4×C163#13511yescharacteristic
82C4×C163#13511yescharacteristic
83C4×C163AGL(1,ℤ/8)11yescharacteristic
84C4×C163#13711yescharacteristic
85Dic163—41no—
86Dic163—41no—
87Dic163—81no—
88C2×C163#44911yescharacteristic
89C2×C163#45211yescharacteristic
90C2×C163#45511yescharacteristic
91D163—81no—
92D163—81no—
93C163#164611yescharacteristic, complemented, Sylow
94#1646—1631nomaximal, Sylow
95#555—1631no—
96#487—1631no—
97#474—1631no—
98#461—1631no—
99#333D16311yescharacteristic, complemented
100D4⋊C8—1631no—
101D4⋊C8—1631no—
102C2×SD16—3261no—
103C4⋊Q8—1631no—
104C4⋊Q8C2×D16311yescharacteristic
105#124—3262no—
106#121—3261no—
107#121—3261no—
108C4×D4—1631no—
109C4×D4—1631no—
110#117—3261no—
111#117—3261no—
112#107—3261no—
113C4⋊C8C2×D16311yescharacteristic
114C4⋊C8C2×D16311yescharacteristic
115#103—3262no—
116#103—3261no—
117#102—3261no—
118C4×C8—1631no—
119C2×Q8—3261no—
120C2×Q8D65212yes—
121C2×D4—3262no—
122C2×D4—1631no—
123C2×D4—1631no—
124C22×C4—3261no—
125C22×C4—3261no—
126SD16—6522no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—3261no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—3261no—
133C4⋊C4—3261no—
134C4⋊C4—1631no—
135C4⋊C4—1631no—
136C22⋊C4—3262no—
137C22⋊C4—3262no—
138C22⋊C4—3261no—
139C22⋊C4—3261no—
140C42—3262no—
141C42C22×D16311yescharacteristic
142C23—3261no—
143C23—3261no—
144Q8—42no—
145Q8—6521no—
146Q8—6521no—
147D4—13042no—
148D4—3261no—
149D4—3261no—
150D4—3261no—
151D4—3261no—
152C2×C4—22no—
153C2×C4—3261no—
154C2×C4—3261no—
155C2×C4—6521no—
156C2×C4—6521no—
157C2×C4—6521no—
158C2×C4—6521no—
159C2×C4—6521no—
160C2×C4D4×D16311yescharacteristic
161C2×C4D4×D16311yescharacteristic
162C2×C4C2×D65211yescharacteristic
163C8—41no—
164C8—41no—
165C8—6521no—
166C22—6521no—
167C22—6521no—
168C22—6521no—
169C22—6521no—
170C22—6521no—
171C22—6521no—
172C22#1910470511yescharacteristic
173C4—21no—
174C4—42no—
175C4—13041no—
176C4—6521no—
177C4—6521no—
178C4#1910475611yescharacteristic
179C4#1910475811yescharacteristic
180C4#1910476211yescharacteristic
181C4#1910476711yescharacteristic
182C2—13041no—
183C2—13041no—
184C2#2378181311yescharacteristic
185C2#2378183011yescharacteristic
186C2#2378184511yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#333×C163C2
socleC22×C163#120
prime radical#23782381C2
hypercenter#333D163
hypocenterC163#1646
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #23782015C2semidirect
3. #23781946C2semidirect
4. #23781871C2semidirect
5. #23781392C2semidirect
6. #23781392C2semidirect
7. #333×C163C2semidirect
93. C163#1646semidirect
99. #333D163semidirect

Maximal subgroups

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

subgroupindex
#237823812
#237820152
#237819462
#237818712
#237813922
#333×C1632
#1646163