Finite Group Databasebeta

← #35213702

Subgroups — #35213702

Order 20864.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#23782330C211yesmaximal, characteristic
2#23782017C211yesmaximal, characteristic, complemented
3#23781961C211yesmaximal, characteristic, complemented
4#23781956C211yesmaximal, characteristic
5#23781395C211yesmaximal, characteristic, complemented
6#23781392C211yesmaximal, characteristic
7#332×C163C211yesmaximal, characteristic, complemented
8C2×Q8×D163—21no—
9#19104735C2211yescharacteristic
10#19104734—21no—
11#19104713—22no—
12#19104711—21no—
13C4×Dic326C2211yescharacteristic
14C4×D652C2211yescharacteristic
15#19104697—21no—
16#19104669—21no—
17#19104667—21no—
18#19104660—21no—
19#19104658—22no—
20#19104657—21no—
21C4×C163⋊C8C2211yescharacteristic
22C4⋊Q8×C163C2211yescharacteristic
23C4⋊C8×C163C2211yescharacteristic
24C4⋊C8×C163C2211yescharacteristic
25C2×Dic326—22no—
26C2×Dic326—21no—
27C2×Dic326D411yescharacteristic
28Q8×D163—42no—
29Q8×D163—42no—
30C2×D652D411yescharacteristic
31C2×C4×D163—22no—
32C2×C4×D163—21no—
33C2×C163⋊C8—21no—
34C4⋊Dic163—21no—
35C4⋊Dic163—21no—
36C4⋊Dic163D411yescharacteristic
37C4⋊Dic163D411yescharacteristic
38Dic163⋊C4—22no—
39Dic163⋊C4—21no—
40(C2×D163)⋊C4—22no—
41(C2×D163)⋊C4—21no—
42C4×Dic163—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—
50Dic326—21no—
51Dic326—44no—
52Dic326—42no—
53Dic326SD1612yes—
54D652—21no—
55D652Q1612yes—
56C2×Dic163—21no—
57C2×Dic163—21no—
58C2×Dic163—21no—
59C2×Dic163—41no—
60C4×D163—44no—
61C4×D163—41no—
62C4×D163—41no—
63C163⋊C8—41no—
64Q8×C163—41no—
65Q8×C163—41no—
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—22no—
74C2×D163—41no—
75C2×D163—41no—
76C4×C163—21no—
77C4×C163—42no—
78C4×C163C2×SD1611yescharacteristic
79C4×C163C2×Q1611yescharacteristic
80C4×C163AGL(1,ℤ/8)11yescharacteristic
81C4×C163#13711yescharacteristic
82Dic163—22no—
83Dic163—41no—
84Dic163—41no—
85Dic163—81no—
86C2×C163#45011yescharacteristic
87C2×C163#45111yescharacteristic
88C2×C163#45411yescharacteristic
89D163—42no—
90C163#163711yescharacteristic, complemented, Sylow
91#1637—1631nomaximal, Sylow
92D4×Q8—1631no—
93#487—1631no—
94#477—1631no—
95#476—1631no—
96#332D16311yescharacteristic, complemented
97Q8⋊C8—1631no—
98D4⋊C8—1631no—
99C22×Q8—3261no—
100C4⋊Q8—3261no—
101C4⋊Q8—1631no—
102C4⋊Q8C2×D16311yescharacteristic
103C22⋊Q8—3261no—
104C22⋊Q8—3262no—
105C4×Q8—1631no—
106C4×D4—3261no—
107C4×D4—1631no—
108#107—3261no—
109#106—3261no—
110C4⋊C8C2×D16311yescharacteristic
111C4⋊C8C2×D16311yescharacteristic
112#103—3261no—
113#103—3262no—
114#102—3261no—
115C4×C8—1631no—
116C2×Q8—21no—
117C2×Q8—3262no—
118C2×Q8—3261no—
119C2×Q8—6522no—
120C2×Q8—6522no—
121C2×Q8—1631no—
122C2×D4—1631no—
123C22×C4—3262no—
124C22×C4—3261no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—3261no—
128C4⋊C4—21no—
129C4⋊C4—3262no—
130C4⋊C4—3261no—
131C4⋊C4—3261no—
132C4⋊C4—3261no—
133C4⋊C4—1631no—
134C4⋊C4—1631no—
135C4⋊C4D65212yes—
136C22⋊C4—3262no—
137C22⋊C4—3261no—
138C42—3261no—
139C42C22×D16311yescharacteristic
140C23—3261no—
141Q8—41no—
142Q8—41no—
143Q8—3261no—
144Q8—6522no—
145Q8—6524no—
146Q8—1632no—
147D4—3261no—
148D4—1632no—
149C2×C4—22no—
150C2×C4—3261no—
151C2×C4—3261no—
152C2×C4—3261no—
153C2×C4—6524no—
154C2×C4—6521no—
155C2×C4—6521no—
156C2×C4—6521no—
157C2×C4D4×D16311yescharacteristic
158C2×C4D4×D16311yescharacteristic
159C2×C4C2×D65211yescharacteristic
160C8—41no—
161C8—41no—
162C8—6521no—
163C22—3262no—
164C22—6521no—
165C22—6521no—
166C22#1910470511yescharacteristic
167C4—21no—
168C4—42no—
169C4—13041no—
170C4—3262no—
171C4—6521no—
172C4—6521no—
173C4SD16×D16311yescharacteristic
174C4Q16×D16311yescharacteristic
175C4#1910475911yescharacteristic
176C4#1910476411yescharacteristic
177C2—6522no—
178C2#2378181711yescharacteristic
179C2#2378182211yescharacteristic
180C2#2378184011yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#332×C163C2
socleC22×C163#120
prime radicalC2×D652D4
hypercenter#332D163
hypocenterC163#1637
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. #23781961C2semidirect
5. #23781395C2semidirect
7. #332×C163C2semidirect
90. C163#1637semidirect
96. #332D163semidirect

Maximal subgroups

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

subgroupindex
#237823302
#237820172
#237819612
#237819562
#237813952
#237813922
#332×C1632
#1637163