Finite Group Databasebeta

← #35224066

Subgroups — #35224066

Order 20864.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19104766C211yesmaximal, characteristic
2C22×#14579689C211yesmaximal, characteristic, complemented
3#23782241C211yesmaximal, characteristic
4#23781887C214yesmaximal, complemented
5#23781875C214yesmaximal, complemented
6#23781692C211yesmaximal, characteristic, complemented
7#23781673C211yesmaximal, characteristic, complemented
8#23781659C211yesmaximal, characteristic
9#519×C163C211yesmaximal, characteristic, complemented
10C2×#14579699C2211yescharacteristic
11C22×Dic326C2211yescharacteristic
12#19104766—24no—
13C2×#14579689—24no—
14C2×#14579689C2214yes—
15C2×#14579689C2212yes—
16C2×#14579686C2212yes—
17C2×#14579683C2211yescharacteristic
18C22×C163⋊C8C2211yescharacteristic
19#19104735C2212yes—
20#19104723C2212yes—
21#19104719—22no—
22#19104715—22no—
23C4⋊(C163⋊C8)C2214yes—
24#19104658C2212yes—
25#19104658C2214yes—
26#19104655C2212yes—
27C22×Q8×C163C2211yescharacteristic
28C4×Q8×C163—22no—
29C4×Q8×C163C2214yes—
30#117×C163—22no—
31#117×C163C2211yescharacteristic
32#117×C163C2211yescharacteristic
33#14579699—22no—
34#14579699—22no—
35C2×Dic326—21no—
36C2×Dic326—22no—
37C2×Dic326—22no—
38C2×Dic326C2312yes—
39C2×Dic326C2311yescharacteristic
40C2×D652C2311yescharacteristic
41C2×C22⋊D163—21no—
42C22×Dic163—21no—
43C2×C4×D163—21no—
44#14579689—28no—
45#14579689—24no—
46#14579686—24no—
47#14579683—22no—
48C2×C163⋊C8—22no—
49C2×C163⋊C8C2312yes—
50C2×C163⋊C8C2312yes—
51C4⋊Dic163—24no—
52(C2×D163)⋊C4—24no—
53C2×Q8×C163—21no—
54C2×Q8×C163—22no—
55C2×Q8×C163D414yes—
56C2×Q8×C163C2312yes—
57C22×C4×C163—21no—
58C22×C4×C163C2311yescharacteristic
59C4⋊C4×C163—24no—
60C4⋊C4×C163D412yes—
61C4⋊C4×C163C2312yes—
62C22⋊C4×C163—21no—
63C22⋊C4×C163D412yes—
64C42×C163—24no—
65C42×C163—21no—
66C42×C163C2312yes—
67C22×D163—21no—
68Dic326—22no—
69Dic326—21no—
70Dic326—22no—
71Dic326—21no—
72Dic326—42no—
73D652—21no—
74D652—21no—
75C22⋊D163—42no—
76C2×Dic163—22no—
77C2×Dic163—21no—
78C2×Dic163—42no—
79C4×D163—42no—
80C163⋊C8—22no—
81C163⋊C8—22no—
82C23×C163C2×D411yescharacteristic
83Q8×C163—24no—
84Q8×C163—22no—
85Q8×C163C2×D414yes—
86C2×C4×C163—24no—
87C2×C4×C163—22no—
88C2×C4×C163—22no—
89C2×C4×C163C2×D414yes—
90C2×C4×C163C2×D411yescharacteristic
91C2×C4×C163C2×D412yes—
92C2×C4×C163#4112yes—
93C2×C4×C163C2411yescharacteristic
94C22×C163—22no—
95C22×C163#12112yes—
96C22×C163C22×D411yescharacteristic
97C2×D163—41no—
98C2×D163—42no—
99C4×C163—24no—
100C4×C163—24no—
101C4×C163—42no—
102C4×C163#12112yes—
103C4×C163C22×D411yescharacteristic
104C4×C163C2×#4111yescharacteristic
105Dic163—41no—
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#519D16311yescharacteristic, complemented
117#464—1634no—
118#462—1634no—
119#424—1631no—
120#419—1631no—
121#416—1631no—
122C2×#41—1631no—
123C22×Q8—1631no—
124C22×Q8C2×D16311yescharacteristic
125#137—3264no—
126C2×Q16—3264no—
127C2×Q16—1634no—
128C2×Q16—1632no—
129C2×SD16—1632no—
130C2×M16—1631no—
131C22×C8—1631no—
132C4⋊Q8—1632no—
133#124—1632no—
134#123—3262no—
135C22⋊Q8—3262no—
136C4×Q8—22no—
137C4×Q8C2×D16314yes—
138#117—22no—
139#117C2×D16311yescharacteristic
140#117C2×D16311yescharacteristic
141C4⋊C8—1634no—
142#103—1632no—
143#103—1634no—
144#102—1632no—
145#41—3262no—
146#41—3262no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—3261no—
150C2×Q8—3262no—
151C2×Q8—3262no—
152C2×Q8—1632no—
153C2×Q8—1631no—
154C2×Q8D65214yes—
155C2×Q8C22×D16312yes—
156C2×D4—3261no—
157C2×D4—1631no—
158C22×C4—21no—
159C22×C4—3261no—
160C22×C4—3261no—
161C22×C4C22×D16311yescharacteristic
162Q16—3268no—
163Q16—3264no—
164SD16—3264no—
165M16—3262no—
166C2×C8—3262no—
167C2×C8—1632no—
168C2×C8—1632no—
169C4⋊C4—24no—
170C4⋊C4—3264no—
171C4⋊C4C22⋊D16312yes—
172C4⋊C4C22×D16312yes—
173C22⋊C4—21no—
174C22⋊C4—3264no—
175C22⋊C4C22⋊D16312yes—
176C42—24no—
177C42—21no—
178C42C22×D16312yes—
179C23—3261no—
180C23C2×C22⋊D16311yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—3262no—
184Q8—3261no—
185Q8—3262no—
186Q8—3261no—
187Q8—6522no—
188Q8C2×D65214yes—
189D4—3261no—
190D4—3261no—
191D4—6522no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—3262no—
196C2×C4—3261no—
197C2×C4—6522no—
198C2×C4—6522no—
199C2×C4C2×C22⋊D16314yes—
200C2×C4C2×C22⋊D16311yescharacteristic
201C2×C4C2×D65212yes—
202C2×C4#1457969912yes—
203C2×C4C23×D16311yescharacteristic
204C8—3262no—
205C8—3262no—
206C22—22no—
207C22—6521no—
208C22—6522no—
209C22#1910470912yes—
210C22C22×C22⋊D16311yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—6521no—
215C4—6522no—
216C4#1910470912yes—
217C4C22×D65211yescharacteristic
218C4C2×#1457969911yescharacteristic
219C2—22no—
220C2—13041no—
221C2C2×#1910470911yescharacteristic
222C2#2378248012yes—
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#519×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579699C22
hypercenter#519D163
hypocenterC163#3044
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. C22×#14579689C2semidirect
4. #23781887C2semidirect
5. #23781875C2semidirect
6. #23781692C2semidirect
7. #23781673C2semidirect
9. #519×C163C2semidirect
111. C163#3044semidirect
116. #519D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047662
C22×#145796892
#237822412
#237818872
#237818752
#237816922
#237816732
#237816592
#519×C1632
#3044163