Finite Group Databasebeta

← #35224067

Subgroups — #35224067

Order 20864.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#19104768C211yesmaximal, characteristic
2C22×Dic652C211yesmaximal, characteristic, complemented
3#23782241C211yesmaximal, characteristic
4#23782142C211yesmaximal, characteristic, complemented
5#23781888C214yesmaximal, complemented
6#23781876C214yesmaximal, complemented
7#23781675C211yesmaximal, characteristic, complemented
8#23781662C211yesmaximal, characteristic
9#424×C163C211yesmaximal, characteristic, complemented
10C2×#14579699C2211yescharacteristic
11C22×Dic326C2211yescharacteristic
12C22×Dic326C2211yescharacteristic
13#19104768—24no—
14C2×Dic652—24no—
15C2×Dic652C2214yes—
16C2×Dic652C2212yes—
17C2×#14579688C2212yes—
18#19104735C2212yes—
19#19104723C2212yes—
20#19104719—22no—
21#19104715—22no—
22C4×Dic326C2214yes—
23#19104700—22no—
24#19104694C2211yescharacteristic
25#19104691—22no—
26#19104660C2214yes—
27#19104660C2212yes—
28#19104657C2212yes—
29C2×M16×C163C2211yescharacteristic
30C22×C8×C163C2211yescharacteristic
31#117×C163C2211yescharacteristic
32C4⋊C8×C163C2214yes—
33#14579699—22no—
34#14579699—22no—
35C2×Dic326—21no—
36C2×Dic326—21no—
37C2×Dic326—22no—
38C2×Dic326—22no—
39C2×Dic326—22no—
40C2×Dic326D414yes—
41C2×Dic326C2312yes—
42C2×Dic326C2311yescharacteristic
43C2×Dic326C2312yes—
44C2×D652C2311yescharacteristic
45C2×C22⋊D163—21no—
46C22×Dic163—21no—
47C22×Dic163—21no—
48C2×C4×D163—21no—
49Dic652—24no—
50Dic652—28no—
51#14579688—24no—
52C4⋊Dic163—24no—
53C4⋊Dic163C2312yes—
54Dic163⋊C4—24no—
55C22⋊Dic163—21no—
56(C2×D163)⋊C4—24no—
57C4×Dic163—24no—
58C4×Dic163—21no—
59C22×C4×C163C2311yescharacteristic
60M16×C163—22no—
61C2×C8×C163—22no—
62C2×C8×C163C2312yes—
63C2×C8×C163C2312yes—
64C4⋊C4×C163D412yes—
65C22⋊C4×C163D412yes—
66C42×C163C2312yes—
67C22×D163—21no—
68Dic326—24no—
69Dic326—22no—
70Dic326—22no—
71Dic326—21no—
72Dic326—22no—
73Dic326—21no—
74Dic326—42no—
75Dic326C2×D414yes—
76D652—21no—
77D652—21no—
78C22⋊D163—42no—
79C2×Dic163—24no—
80C2×Dic163—22no—
81C2×Dic163—22no—
82C2×Dic163—22no—
83C2×Dic163—21no—
84C2×Dic163—42no—
85C4×D163—42no—
86C23×C163C2×D411yescharacteristic
87C2×C4×C163C2×D412yes—
88C2×C4×C163C2×D414yes—
89C2×C4×C163C2×D411yescharacteristic
90C2×C4×C163#4112yes—
91C2×C4×C163C2411yescharacteristic
92C8×C163—22no—
93C8×C163—22no—
94C22×C163—22no—
95C22×C163#12112yes—
96C22×C163C22×D411yescharacteristic
97C2×D163—41no—
98C2×D163—42no—
99C4×C163—24no—
100C4×C163#12112yes—
101C4×C163C22×D411yescharacteristic
102C4×C163C2×#4111yescharacteristic
103Dic163—24no—
104Dic163—41no—
105Dic163—42no—
106Dic163—42no—
107C2×C163—22no—
108C2×C163C2×#12111yescharacteristic
109C2×C163#57812yes—
110D163—81no—
111C163#304411yescharacteristic, complemented, Sylow
112#3044—1631nomaximal, Sylow
113C2×#137—1631no—
114C22×Q16—1631no—
115#536—1631no—
116#519—1631no—
117#464—1634no—
118#462—1634no—
119#424D16311yescharacteristic, complemented
120#419—1631no—
121#416—1631no—
122C2×#41—1631no—
123C22×Q8—1631no—
124C22×Q8—1631no—
125#137—3264no—
126C2×Q16—3264no—
127C2×Q16—1634no—
128C2×Q16—1632no—
129C2×SD16—1632no—
130C2×M16C2×D16311yescharacteristic
131C22×C8C2×D16311yescharacteristic
132C4⋊Q8—1632no—
133#124—1632no—
134#123—3262no—
135C22⋊Q8—3262no—
136C4×Q8—3262no—
137C4×Q8—1634no—
138#117—3262no—
139#117—1631no—
140#117C2×D16311yescharacteristic
141C4⋊C8C2×D16314yes—
142#103—1632no—
143#103—1634no—
144#102—1632no—
145#41—3262no—
146#41—3262no—
147C2×Q8—3261no—
148C2×Q8—3261no—
149C2×Q8—3262no—
150C2×Q8—3262no—
151C2×Q8—3262no—
152C2×Q8—1634no—
153C2×Q8—1632no—
154C2×Q8—1631no—
155C2×Q8—1632no—
156C2×D4—3261no—
157C2×D4—1631no—
158C22×C4—3261no—
159C22×C4—3261no—
160C22×C4—3261no—
161C22×C4C22×D16311yescharacteristic
162Q16—3268no—
163Q16—3264no—
164SD16—3264no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D16312yes—
168C2×C8C22×D16312yes—
169C4⋊C4—3264no—
170C4⋊C4—3264no—
171C4⋊C4—1632no—
172C4⋊C4D65212yes—
173C22⋊C4—3261no—
174C22⋊C4—3264no—
175C22⋊C4D65212yes—
176C42—3264no—
177C42—3261no—
178C42C22×D16312yes—
179C23—3261no—
180C23C2×D65211yescharacteristic
181Q8—3262no—
182Q8—3264no—
183Q8—3262no—
184Q8—3261no—
185Q8—3262no—
186Q8—3261no—
187Q8—6522no—
188Q8—1634no—
189D4—3261no—
190D4—3261no—
191D4—6522no—
192C2×C4—3262no—
193C2×C4—3264no—
194C2×C4—3262no—
195C2×C4—3262no—
196C2×C4—3261no—
197C2×C4—6522no—
198C2×C4—6522no—
199C2×C4D4×D16312yes—
200C2×C4C2×D65214yes—
201C2×C4C2×D65211yescharacteristic
202C2×C4#1457970012yes—
203C2×C4C23×D16311yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—6521no—
208C22—6522no—
209C22#1910471012yes—
210C22C22×D65211yescharacteristic
211C4—24no—
212C4—3264no—
213C4—6521no—
214C4—6522no—
215C4—6522no—
216C4#1910471012yes—
217C4C2×D4×D16311yescharacteristic
218C4C2×#1457970011yescharacteristic
219C2—22no—
220C2—13041no—
221C2C2×#1910471011yescharacteristic
222C2#2378248212yes—
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#424×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579699C22
hypercenter#424D163
hypocenterC163#3044
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic652C2semidirect
4. #23782142C2semidirect
5. #23781888C2semidirect
6. #23781876C2semidirect
7. #23781675C2semidirect
9. #424×C163C2semidirect
111. C163#3044semidirect
119. #424D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047682
C22×Dic6522
#237822412
#237821422
#237818882
#237818762
#237816752
#237816622
#424×C1632
#3044163