Finite Group Databasebeta

← #35213688

Subgroups — #35213688

Order 20864.

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

normal subgroups only

This group has 234 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#23782318C211yesmaximal, characteristic
2#23782017C211yesmaximal, characteristic, complemented
3#23781956C211yesmaximal, characteristic, complemented
4#23781956C211yesmaximal, characteristic, complemented
5#23781392C211yesmaximal, characteristic, complemented
6#23781392C211yesmaximal, characteristic, complemented
7#332×C163C211yesmaximal, characteristic, complemented
8C2×#14579700—21no—
9#19104735C2211yescharacteristic
10#19104720—22no—
11#19104711—21no—
12#19104710—21no—
13C4×D652C2211yescharacteristic
14C4×D652C2211yescharacteristic
15C4⋊C4×D163—21no—
16#19104667—21no—
17#19104667—21no—
18#19104658—22no—
19#19104657—21no—
20#19104657—21no—
21C4×C163⋊C8C2211yescharacteristic
22C4⋊Q8×C163C2211yescharacteristic
23C4⋊C8×C163C2211yescharacteristic
24C4⋊C8×C163C2211yescharacteristic
25#14579700—42no—
26#14579700—42no—
27C2×Dic326—21no—
28C2×D652—22no—
29C2×D652D411yescharacteristic
30C2×D652D411yescharacteristic
31C2×C4×D163—22no—
32C2×C4×D163—21no—
33C2×C4×D163—21no—
34C2×C163⋊C8—21no—
35C4⋊Dic163—21no—
36C4⋊Dic163—21no—
37C4⋊Dic163D411yescharacteristic
38C4⋊Dic163D411yescharacteristic
39Dic163⋊C4—22no—
40(C2×D163)⋊C4—22no—
41(C2×D163)⋊C4—21no—
42(C2×D163)⋊C4—21no—
43C2×Q8×C163—21no—
44C2×C8×C163—21no—
45C2×C8×C163—21no—
46C4⋊C4×C163—21no—
47C4⋊C4×C163D412yes—
48C42×C163C2311yescharacteristic
49C22×D163—21no—
50C22×D163—21no—
51Dic326—42no—
52D652—21no—
53D652—21no—
54D652—21no—
55D652—44no—
56D652SD1612yes—
57C2×Dic163—21no—
58C2×Dic163—21no—
59C2×Dic163—41no—
60C4×D163—44no—
61C4×D163—41no—
62C4×D163—41no—
63C4×D163—41no—
64C4×D163—41no—
65C163⋊C8—41no—
66Q8×C163—41no—
67Q8×C163—41no—
68C2×C4×C163—22no—
69C2×C4×C163C2×D411yescharacteristic
70C2×C4×C163C2×D411yescharacteristic
71C2×C4×C163C2×D411yescharacteristic
72C8×C163—41no—
73C8×C163—41no—
74C22×C163#12011yescharacteristic
75C2×D163—22no—
76C2×D163—41no—
77C2×D163—41no—
78C2×D163—41no—
79C2×D163—41no—
80C2×D163—41no—
81C4×C163—21no—
82C4×C163—42no—
83C4×C163C2×SD1611yescharacteristic
84C4×C163#13511yescharacteristic
85C4×C163AGL(1,ℤ/8)11yescharacteristic
86C4×C163#13711yescharacteristic
87Dic163—41no—
88Dic163—41no—
89Dic163—81no—
90C2×C163#45011yescharacteristic
91C2×C163#45211yescharacteristic
92C2×C163#45411yescharacteristic
93D163—42no—
94D163—81no—
95C163#163511yescharacteristic, complemented, Sylow
96#1635—1631nomaximal, Sylow
97#547—1631no—
98#487—1631no—
99#476—1631no—
100#476—1631no—
101#332D16311yescharacteristic, complemented
102D4⋊C8—1631no—
103D4⋊C8—1631no—
104C2×#41—3261no—
105C4⋊Q8—1631no—
106C4⋊Q8C2×D16311yescharacteristic
107#123—3262no—
108C22⋊Q8—3261no—
109#121—3261no—
110C4×D4—1631no—
111C4×D4—1631no—
112C2×C4⋊C4—3261no—
113#106—3261no—
114#106—3261no—
115C4⋊C8C2×D16311yescharacteristic
116C4⋊C8C2×D16311yescharacteristic
117#103—3262no—
118#102—3261no—
119#102—3261no—
120C4×C8—1631no—
121#41—6522no—
122#41—6522no—
123C2×Q8—21no—
124C2×Q8—3261no—
125C2×D4—3262no—
126C2×D4—1631no—
127C2×D4—1631no—
128C22×C4—3262no—
129C22×C4—3261no—
130C22×C4—3261no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—3261no—
134C4⋊C4—21no—
135C4⋊C4—3262no—
136C4⋊C4—3261no—
137C4⋊C4—3261no—
138C4⋊C4—1631no—
139C4⋊C4—1631no—
140C4⋊C4D65212yes—
141C22⋊C4—3262no—
142C22⋊C4—3261no—
143C22⋊C4—3261no—
144C42C22×D16311yescharacteristic
145C23—3261no—
146C23—3261no—
147Q8—41no—
148Q8—41no—
149Q8—6522no—
150D4—3261no—
151D4—3261no—
152D4—3261no—
153D4—6524no—
154D4—1632no—
155C2×C4—22no—
156C2×C4—3261no—
157C2×C4—3261no—
158C2×C4—6524no—
159C2×C4—6521no—
160C2×C4—6521no—
161C2×C4—6521no—
162C2×C4—6521no—
163C2×C4—6521no—
164C2×C4D4×D16311yescharacteristic
165C2×C4D4×D16311yescharacteristic
166C2×C4C2×D65211yescharacteristic
167C8—41no—
168C8—41no—
169C8—6521no—
170C22—3262no—
171C22—6521no—
172C22—6521no—
173C22—6521no—
174C22—6521no—
175C22—6521no—
176C22#1910470511yescharacteristic
177C4—21no—
178C4—42no—
179C4—13041no—
180C4—6521no—
181C4—6521no—
182C4SD16×D16311yescharacteristic
183C4#1910475611yescharacteristic
184C4#1910475911yescharacteristic
185C4#1910476711yescharacteristic
186C2—13041no—
187C2—6522no—
188C2#2378181711yescharacteristic
189C2#2378183011yescharacteristic
190C2#2378184011yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#332×C163C2
socleC22×C163#120
prime radical#23782318C2
hypercenter#332D163
hypocenterC163#1635
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #23782017C2semidirect
3. #23781956C2semidirect
4. #23781956C2semidirect
5. #23781392C2semidirect
6. #23781392C2semidirect
7. #332×C163C2semidirect
95. C163#1635semidirect
101. #332D163semidirect

Maximal subgroups

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

subgroupindex
#237823182
#237820172
#237819562
#237813922
#332×C1632
#1635163