Finite Group Databasebeta

← #35213786

Subgroups — #35213786

Order 20864.

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

normal subgroups only

This group has 234 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#23782068C211yesmaximal, characteristic, complemented
3#23781946C211yesmaximal, characteristic
4#23781876C211yesmaximal, characteristic, complemented
5#23781395C211yesmaximal, characteristic, complemented
6#23781392C211yesmaximal, characteristic
7#332×C163C211yesmaximal, characteristic, complemented
8C2×Q8×D163—21no—
9C2×Dic652—21no—
10#19104735C2211yescharacteristic
11#19104734—21no—
12#19104713—22no—
13#19104711—21no—
14C4×Dic326C2211yescharacteristic
15C4×D652C2211yescharacteristic
16#19104697—21no—
17#19104669—21no—
18#19104668—22no—
19#19104660—21no—
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—
33Dic652—42no—
34C2×C163⋊C8—21no—
35C4⋊Dic163—21no—
36C4⋊Dic163—21no—
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×C163—21no—
45C2×C8×C163—21no—
46C2×C8×C163—21no—
47C4⋊C4×C163—21no—
48C4⋊C4×C163D412yes—
49C42×C163C2311yescharacteristic
50C22×D163—21no—
51Dic326—21no—
52Dic326—44no—
53Dic326—41no—
54Dic326—41no—
55Dic326D812yes—
56D652—21no—
57D652Q1612yescomplemented
58C2×Dic163—21no—
59C2×Dic163—21no—
60C2×Dic163—21no—
61C2×Dic163—41no—
62C4×D163—44no—
63C4×D163—41no—
64C4×D163—41no—
65C163⋊C8—22no—
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—
78C4×C163—21no—
79C4×C163—42no—
80C4×C163C2×D811yescharacteristic
81C4×C163C2×Q1611yescharacteristic
82C4×C163AGL(1,ℤ/8)11yescharacteristic
83C4×C163#13711yescharacteristic
84Dic163—22no—
85Dic163—41no—
86Dic163—41no—
87Dic163—81no—
88C2×C163#44711yescharacteristic
89C2×C163#45111yescharacteristic
90C2×C163#45611yescharacteristic
91D163—42no—
92C163#164911yescharacteristic, complemented, Sylow
93#1649—1631nomaximal, Sylow
94D4×Q8—1631no—
95#500—1631no—
96#474—1631no—
97#462—1631no—
98#332D16311yescharacteristic, complemented
99Q8⋊C8—1631no—
100D4⋊C8—1631no—
101C22×Q8—3261no—
102C2×Q16—3261no—
103C4⋊Q8—3261no—
104C4⋊Q8—1631no—
105C4⋊Q8C2×D16311yescharacteristic
106C22⋊Q8—3261no—
107C22⋊Q8—3262no—
108C4×Q8—1631no—
109C4×D4—3261no—
110C4×D4—1631no—
111#107—3261no—
112#107—3262no—
113C4⋊C8C2×D16311yescharacteristic
114C4⋊C8C2×D16311yescharacteristic
115#103—3261no—
116#102—3261no—
117C4×C8—1631no—
118C2×Q8—21no—
119C2×Q8—3262no—
120C2×Q8—3261no—
121C2×Q8—6522no—
122C2×Q8—6522no—
123C2×Q8—1631no—
124C2×D4—1631no—
125C22×C4—3262no—
126C22×C4—3261no—
127Q16—6522no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—3261no—
131C4⋊C4—21no—
132C4⋊C4—3262no—
133C4⋊C4—3261no—
134C4⋊C4—3261no—
135C4⋊C4—3261no—
136C4⋊C4—1631no—
137C4⋊C4—1631no—
138C4⋊C4D65212yes—
139C22⋊C4—3262no—
140C22⋊C4—3261no—
141C42—3261no—
142C42C22×D16311yescharacteristic
143C23—3261no—
144Q8—41no—
145Q8—41no—
146Q8—3261no—
147Q8—6524no—
148Q8—6521no—
149Q8—6521no—
150Q8—1632no—
151D4—3261no—
152D4—1632no—
153C2×C4—22no—
154C2×C4—3261no—
155C2×C4—3261no—
156C2×C4—3261no—
157C2×C4—6524no—
158C2×C4—6521no—
159C2×C4—6521no—
160C2×C4—6521no—
161C2×C4D4×D16311yescharacteristic
162C2×C4D4×D16311yescharacteristic
163C2×C4C2×D65211yescharacteristic
164C8—41no—
165C8—41no—
166C8—3262no—
167C22—3262no—
168C22—6521no—
169C22—6521no—
170C22#1910470511yescharacteristic
171C4—21no—
172C4—42no—
173C4—13041no—
174C4—3262no—
175C4—6521no—
176C4—6521no—
177C4D8×D16311yescharacteristic
178C4Q16×D16311yescharacteristic
179C4#1910476211yescharacteristic
180C4#1910476411yescharacteristic
181C2—6522no—
182C2#2378180011yescharacteristic
183C2#2378182211yescharacteristic
184C2#2378185011yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#332×C163C2
socleC22×C163#120
prime radicalC2×D652D4
hypercenter#332D163
hypocenterC163#1649
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. #23782068C2semidirect
4. #23781876C2semidirect
5. #23781395C2semidirect
7. #332×C163C2semidirect
57. D652Q16semidirect
92. C163#1649semidirect
98. #332D163semidirect

Maximal subgroups

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

subgroupindex
#237823302
#237820682
#237819462
#237818762
#237813952
#237813922
#332×C1632
#1649163