Finite Group Databasebeta

← #35227572

Subgroups — #35227572

Order 20864.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#23782390C211yesmaximal, characteristic, complemented
2#23782389C211yesmaximal, characteristic
3#23782058C212yesmaximal
4#23781889C212yesmaximal
5#23781873C212yesmaximal, complemented
6#23781872C211yesmaximal, characteristic, complemented
7#23781867C211yesmaximal, characteristic
8#23781795C211yesmaximal, characteristic, complemented
9#23781772C212yesmaximal, complemented
10C4×#14579688C211yesmaximal, characteristic
11#494×C163C211yesmaximal, characteristic, complemented
12C2×#14579689C2212yes—
13C2×#14579688C2211yescharacteristic
14C2×#14579686C2212yes—
15#19104733C2212yes—
16#19104730—22no—
17#19104726—21no—
18#19104725—22no—
19#19104724C2212yes—
20#19104715—21no—
21Q8×Dic163C2212yes—
22Q8×Dic163C2212yes—
23C4×Dic326C2211yescharacteristic
24C4×Dic326C2211yescharacteristic
25#19104700—21no—
26C4×D652C2211yescharacteristic
27#19104693—21no—
28#19104667C2211yescharacteristic
29C4⋊(C163⋊C8)C2211yescharacteristic
30Dic163⋊C8C2212yes—
31#19104660C2211yescharacteristic
32#19104659C2212yes—
33#19104659C2212yes—
34#19104657C2211yescharacteristic
35#19104645C2212yes—
36C2×Q16×C163C2212yes—
37C2×Q16×C163C2212yes—
38C4⋊Q8×C163C2211yescharacteristic
39C4⋊Q8×C163C2211yescharacteristic
40C4×C8×C163C2211yescharacteristic
41C2×Dic326D411yescharacteristic
42C2×Dic326C2311yescharacteristic
43C2×D652C2311yescharacteristic
44C2×C4×D163—21no—
45#14579689—42no—
46#14579688—22no—
47#14579686—42no—
48C2×C163⋊C8C2312yes—
49C4⋊Dic163—22no—
50C4⋊Dic163—22no—
51C4⋊Dic163D411yescharacteristic
52C4⋊Dic163C2311yescharacteristic
53C4⋊Dic163C2311yescharacteristic
54Dic163⋊C4—22no—
55Dic163⋊C4—22no—
56Dic163⋊C4—21no—
57Dic163⋊C4D412yes—
58(C2×D163)⋊C4—22no—
59(C2×D163)⋊C4—22no—
60(C2×D163)⋊C4—21no—
61C4×Dic163—22no—
62C4×Dic163—22no—
63C4×Dic163—21no—
64C4×Dic163C2312yes—
65C2×Q8×C163C2312yes—
66C2×Q8×C163C2312yes—
67Q16×C163—24no—
68Q16×C163—42no—
69C2×C8×C163C2311yescharacteristic
70C2×C8×C163C2311yescharacteristic
71C4⋊C4×C163—21no—
72C4⋊C4×C163—21no—
73C4⋊C4×C163—21no—
74C4⋊C4×C163—21no—
75C42×C163C2311yescharacteristic
76C22×D163—21no—
77Dic326—22no—
78Dic326—21no—
79Dic326—21no—
80D652—21no—
81D652—21no—
82C2×Dic163—21no—
83C2×Dic163—21no—
84C2×Dic163—21no—
85C2×Dic163C2×D412yes—
86C2×Dic163C2×D412yes—
87C4×D163—42no—
88C163⋊C8—22no—
89Q8×C163—22no—
90Q8×C163—22no—
91Q8×C163—22no—
92Q8×C163—22no—
93C2×C4×C163—22no—
94C2×C4×C163—22no—
95C2×C4×C163C2×D411yescharacteristic
96C2×C4×C163C2×D411yescharacteristic
97C2×C4×C163C2411yescharacteristic
98C8×C163—21no—
99C8×C163#4112yes—
100C22×C163C22×D411yescharacteristic
101C2×D163—41no—
102C2×D163—41no—
103C2×D163—41no—
104C4×C163—21no—
105C4×C163—42no—
106C4×C163—42no—
107C4×C163#13712yes—
108C4×C163C2×#4111yescharacteristic
109C4×C163H(5,𝔽2)11yescharacteristic
110Dic163—22no—
111Dic163—22no—
112Dic163—41no—
113Dic163—41no—
114Dic163—41no—
115C2×C163#55011yescharacteristic
116C2×C163C2×#13711yescharacteristic
117C2×C163#57811yescharacteristic
118D163—81no—
119C163#339811yescharacteristic, complemented, Sylow
120#3398—1631nomaximal, Sylow
121#557—1631no—
122#556—1631no—
123#497—1632no—
124#494D16311yescharacteristic, complemented
125#464—1632no—
126#462—1632no—
127#462—1631no—
128#461—1631no—
129#446—1631no—
130#441—1632no—
131C4×SD16—1631no—
132C2×Q16—1632no—
133C2×Q16C2×D16312yes—
134C2×Q16C2×D16312yes—
135C2×SD16—1632no—
136C2×SD16—1631no—
137C4⋊Q8—1632no—
138C4⋊Q8C2×D16311yescharacteristic
139C4⋊Q8C2×D16311yescharacteristic
140#126—3262no—
141#125—3262no—
142#125—3261no—
143#124—1632no—
144C22⋊Q8—3261no—
145C4×Q8—3261no—
146C4×Q8—1632no—
147C4×Q8—1632no—
148C4×Q8—1631no—
149C4×Q8—1631no—
150C4×D4—1631no—
151#117—3261no—
152#106—1631no—
153C4⋊C8—1632no—
154C4⋊C8—1631no—
155#103—1632no—
156#103—1632no—
157#103—1631no—
158#102—1631no—
159#97—1632no—
160C4×C8C2×D16311yescharacteristic
161C2×Q8—1631no—
162C2×Q8—1631no—
163C2×Q8C22×D16312yes—
164C2×Q8C22×D16312yes—
165C2×D4—1631no—
166C22×C4—3261no—
167Q16—24no—
168Q16—42no—
169Q16—6522no—
170SD16—3262no—
171SD16—6522no—
172C2×C8—1632no—
173C2×C8C22×D16311yescharacteristic
174C2×C8C22×D16311yescharacteristic
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—3262no—
180C4⋊C4—3262no—
181C4⋊C4—3262no—
182C4⋊C4—3262no—
183C4⋊C4—3261no—
184C4⋊C4—1631no—
185C4⋊C4—1631no—
186C4⋊C4—1631no—
187C4⋊C4—1632no—
188C22⋊C4—3262no—
189C22⋊C4—3262no—
190C22⋊C4—3261no—
191C42—3262no—
192C42—3262no—
193C42—3261no—
194C42—1632no—
195C42C22×D16311yescharacteristic
196C23—3261no—
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—22no—
201Q8—3262no—
202Q8—3261no—
203Q8—3261no—
204D4—3261no—
205D4—3261no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—3261no—
209C2×C4—3261no—
210C2×C4—3261no—
211C2×C4—6522no—
212C2×C4—1632no—
213C2×C4—1632no—
214C2×C4D4×D16311yescharacteristic
215C2×C4D4×D16311yescharacteristic
216C2×C4C23×D16311yescharacteristic
217C8—21no—
218C8—3262no—
219C8#1457969812yes—
220C22—6521no—
221C22—6521no—
222C22—6521no—
223C22C2×D4×D16311yescharacteristic
224C4—21no—
225C4—42no—
226C4—42no—
227C4—3262no—
228C4—3262no—
229C4—6521no—
230C4—6521no—
231C4—6521no—
232C4#1910476712yes—
233C4C2×#1457969811yescharacteristic
234C4#1910478011yescharacteristic
235C2—13041no—
236C2#2378233511yescharacteristic
237C2C2×#1910476711yescharacteristic
238C2#2378247811yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#494×C163C2
socleC22×C163C22×D4
prime radicalC2×D652C23
hypercenter#494D163
hypocenterC163#3398
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #23782390C2semidirect
5. #23781873C2semidirect
6. #23781872C2semidirect
8. #23781795C2semidirect
9. #23781772C2semidirect
11. #494×C163C2semidirect
119. C163#3398semidirect
124. #494D163semidirect

Maximal subgroups

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

subgroupindex
#237823902
#237823892
#237820582
#237818892
#237818732
#237818722
#237818672
#237817952
#237817722
C4×#145796882
#494×C1632
#3398163