Finite Group Databasebeta

← #37544845

Subgroups — #37544845

Order 23168.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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#26041827C211yesmaximal, characteristic
2C4⋊Q8×D181C211yesmaximal, characteristic, complemented
3#26041677C211yesmaximal, characteristic, complemented
4#26041675C211yesmaximal, characteristic, complemented
5#26041625C211yesmaximal, characteristic, complemented
6#26041596C214yesmaximal, complemented
7#26041567C214yesmaximal, complemented
8#26041330C211yesmaximal, characteristic, complemented
9#486×C181C211yesmaximal, characteristic, complemented
10C2×#15668448—21no—
11C2×Q8×D181—21no—
12#19153816C2211yescharacteristic
13#19153816C2211yescharacteristic
14#19153815—21no—
15#19153815C2212yes—
16#19153814—21no—
17#19153803—21no—
18#19153787—21no—
19#19153787C2212yes—
20D4×Dic181—21no—
21D4×Dic181C2212yes—
22C4⋊C4×D181—21no—
23C4⋊C4×D181C2212yes—
24C42×D181C2211yescharacteristic
25#19153748C2214yes—
26#19153747C2214yes—
27#19153745C2212yes—
28#19153737C2214yes—
29#19153728C2212yes—
30C4×#13089678C2211yescharacteristic
31C4⋊Q8×C181C2211yescharacteristic
32#127×C181C2211yescharacteristic
33#102×C181C2214yes—
34C4×C8×C181C2211yescharacteristic
35#15668448—44no—
36C2×Dic362—22no—
37C2×Dic362—21no—
38C2×Dic362—21no—
39Q8×D181—44no—
40C2×C22⋊D181—22no—
41C22×Dic181—22no—
42C2×C4×D181—22no—
43C2×C4×D181D411yescharacteristic
44C2×C4×D181D411yescharacteristic
45C2×C4×D181C2311yescharacteristic
46C2×#13089678C2312yes—
47C4⋊Dic181—21no—
48C4⋊Dic181—21no—
49C4⋊Dic181C2312yes—
50C4⋊Dic181C2312yes—
51#15668428—22no—
52#15668428—22no—
53C22⋊Dic181—22no—
54C22⋊Dic181—22no—
55C4×Dic181D411yescharacteristic
56C4×Dic181D411yescharacteristic
57C4×Dic181C2311yescharacteristic
58C2×Q8×C181—21no—
59C2×D4×C181—21no—
60C2×D4×C181—21no—
61C2×D4×C181C2312yes—
62C2×C8×C181C2312yes—
63C4⋊C4×C181—21no—
64C4⋊C4×C181C2312yes—
65C42×C181C2311yescharacteristic
66C22×D181C2×D411yescharacteristic
67Dic362—44no—
68Dic362—42no—
69Dic362—42no—
70C22⋊D181—44no—
71C2×Dic181—22no—
72C2×Dic181—22no—
73C2×Dic181—44no—
74C2×Dic181C2×D411yescharacteristic
75C2×Dic181C2×D411yescharacteristic
76C2×Dic181C2×D411yescharacteristic
77C4×D181—22no—
78C4×D181—22no—
79C4×D181—44no—
80C4×D181SD1614yes—
81#13089678—22no—
82C23×C181—22no—
83Q8×C181—42no—
84D4×C181—24no—
85D4×C181—42no—
86D4×C181—42no—
87C2×C4×C181—22no—
88C2×C4×C181C2×D411yescharacteristic
89C2×C4×C181C2×D411yescharacteristic
90C2×C4×C181C2411yescharacteristic
91C8×C181—22no—
92C22×C181—42no—
93C22×C181—42no—
94C22×C181—42no—
95C22×C181C22×D411yescharacteristic
96C2×D181—21no—
97C2×D181—21no—
98C2×D181C2×SD1612yes—
99C4×C181—42no—
100C4×C181C2×SD1612yes—
101C4×C181AGL(1,ℤ/8)12yes—
102C4×C1812−1+412yes—
103Dic181—21no—
104Dic181—21no—
105Dic181—42no—
106Dic181—42no—
107Dic181C2×SD1612yes—
108C2×C181—82no—
109C2×C181#53611yescharacteristic
110C2×C181C22×SD1611yescharacteristic
111C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
112D181—22no—
113C181#322711yescharacteristic, complemented, Sylow
114#3227—1811nomaximal, Sylow
115#532—1811no—
116C2×C4⋊Q8—1811no—
117#498—1811no—
118#498—1811no—
119#486—1811no—
120#486D18111yescharacteristic, complemented
121#481—1814no—
122#476—1814no—
123#431—1811no—
124C2×#41—3621no—
125C22×Q8—3621no—
126C4⋊Q8—3621no—
127C4⋊Q8—3621no—
128C4⋊Q8—1812no—
129C4⋊Q8—1811no—
130C4⋊Q8—1811no—
131C4⋊Q8C2×D18111yescharacteristic
132#127C2×D18111yescharacteristic
133#124—3621no—
134#121—3621no—
135#121—1812no—
136C4×D4—3621no—
137C4×D4—1812no—
138C2×C4⋊C4—3621no—
139C2×C4⋊C4—1812no—
140C2×C42—1811no—
141#106—1814no—
142#106—1814no—
143C4⋊C8—1812no—
144#102—1814no—
145#102C2×D18114yes—
146C22⋊C8—1812no—
147C4×C8—1811no—
148C4×C8C2×D18111yescharacteristic
149#41—7244no—
150C2×Q8—21no—
151C2×Q8—3622no—
152C2×Q8—3621no—
153C2×Q8—3621no—
154C2×Q8—7244no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—3622no—
158C2×D4C22×D18112yes—
159C22×C4—3622no—
160C22×C4—3622no—
161C22×C4—1811no—
162C22×C4—1811no—
163C22×C4—1811no—
164C2×C8—1812no—
165C2×C8C22×D18112yes—
166C4⋊C4—21no—
167C4⋊C4—3622no—
168C4⋊C4—3622no—
169C4⋊C4—3621no—
170C4⋊C4—3621no—
171C4⋊C4—1812no—
172C4⋊C4—1812no—
173C4⋊C4C22×D18112yes—
174C22⋊C4—3622no—
175C22⋊C4—3622no—
176C42—1811no—
177C42—1811no—
178C42—1811no—
179C42C22×D18111yescharacteristic
180C23—22no—
181C23—1811no—
182Q8—42no—
183Q8—7244no—
184Q8—7242no—
185Q8—7242no—
186D4—24no—
187D4—42no—
188D4—42no—
189D4—7244no—
190C2×C4—22no—
191C2×C4—3622no—
192C2×C4—3622no—
193C2×C4—3622no—
194C2×C4—3622no—
195C2×C4—1811no—
196C2×C4—1811no—
197C2×C4—1811no—
198C2×C4—1814no—
199C2×C4—7244no—
200C2×C4—7244no—
201C2×C4D4×D18111yescharacteristic
202C2×C4D4×D18111yescharacteristic
203C2×C4C23×D18111yescharacteristic
204C8—22no—
205C8—3622no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—3621no—
210C22—3621no—
211C22—1812no—
212C22C2×D4×D18111yescharacteristic
213C4—42no—
214C4—3621no—
215C4—3621no—
216C4—1812no—
217C4—7242no—
218C4—7242no—
219C4SD16×D18112yes—
220C4#1915384012yes—
221C4#1915386312yes—
222C2—82no—
223C2—3622no—
224C2#2604184911yescharacteristic
225C2C2×SD16×D18111yescharacteristic
226C2C2×#1915384011yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#486×C181C2
socleC22×C181C22×D4
prime radical#26041827C2
hypercenter#486D181
hypocenterC181#3227
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D181C2semidirect
3. #26041677C2semidirect
4. #26041675C2semidirect
5. #26041625C2semidirect
6. #26041596C2semidirect
7. #26041567C2semidirect
8. #26041330C2semidirect
9. #486×C181C2semidirect
113. C181#3227semidirect
120. #486D181semidirect

Maximal subgroups

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

subgroupindex
#260418272
C4⋊Q8×D1812
#260416772
#260416752
#260416252
#260415962
#260415672
#260413302
#486×C1812
#3227181