Finite Group Databasebeta

← #37545342

Subgroups — #37545342

Order 23168.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19153843C212yesmaximal, complemented
2#26041951C211yesmaximal, characteristic, complemented
3D4×D724C211yesmaximal, characteristic, complemented
4#26041656C211yesmaximal, characteristic, complemented
5#26041533C212yesmaximal, complemented
6#26041479C211yesmaximal, characteristic, complemented
7#26041474C211yesmaximal, characteristic, complemented
8#26041427C212yesmaximal, complemented
9#26041414C212yesmaximal, complemented
10#26041358C211yesmaximal, characteristic, complemented
11C4×SD16×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2212yescomplemented
13C22×D724—21no—
14C2×D4×D181C2212yes—
15#19153843—28no—
16C2×#15668436C2212yescomplemented
17C2×D1448C2212yescomplemented
18C2×D1448C2212yescomplemented
19C2×#15668434C2212yes—
20C2×#15668432C2212yescomplemented
21#19153813—21no—
22#19153813C2211yescharacteristic
23#19153813C2211yescharacteristic, complemented
24#19153791—22no—
25#19153791C2212yes—
26#19153791C2212yescomplemented
27#19153790—21no—
28#19153786—22no—
29C4×D724—21no—
30C4×D724C2211yescharacteristic
31C4⋊#13089678C2211yescharacteristic, complemented
32#19153736C2212yescomplemented
33#19153736C2212yescomplemented
34#19153728C2212yescomplemented
35C2×SD16×C181C2211yescharacteristic
36C4×Q8×C181C2211yescharacteristic, complemented
37C4×D4×C181C2211yescharacteristic
38#106×C181C2211yescharacteristic, complemented
39#103×C181C2211yescharacteristic, complemented
40#102×C181C2211yescharacteristic
41C4×C8×C181C2211yescharacteristic, complemented
42C23×D181—22no—
43#15668450—24no—
44#15668450—42no—
45C2×D724—22no—
46C2×D724—22no—
47C2×D724—22no—
48C2×D724—21no—
49C2×D724—21no—
50C2×D724—21no—
51C2×D724—21no—
52C2×D724—44no—
53C2×D724D411yescharacteristic
54C2×D724C2312yes—
55C2×D724C2312yes—
56C2×C22⋊D181—22no—
57D4×D181—24no—
58D4×D181—42no—
59C2×C4×D181—22no—
60C2×C4×D181C2312yes—
61#15668436—24no—
62D1448—24no—
63D1448—42no—
64#15668434—24no—
65#15668432—24no—
66C2×#13089678C2312yes—
67C4⋊Dic181D411yescharacteristic
68#15668426—22no—
69#15668426—22no—
70#15668426D412yes—
71C2×Q8×C181C2311yescharacteristic
72C2×D4×C181C2311yescharacteristic
73C22×C4×C181—21no—
74SD16×C181D414yescomplemented
75C2×C8×C181C2311yescharacteristic
76C2×C8×C181C2311yescharacteristic
77C4⋊C4×C181—21no—
78C4⋊C4×C181C2311yescharacteristic
79C4⋊C4×C181C2311yescharacteristic
80C22⋊C4×C181—21no—
81C42×C181—21no—
82C42×C181C2311yescharacteristic
83C22×D181—22no—
84C22×D181—22no—
85C22×D181—44no—
86C22×D181—44no—
87C22×D181—82no—
88C22×D181C2×D412yes—
89D724—22no—
90D724—22no—
91D724—22no—
92D724—22no—
93D724—22no—
94D724—44no—
95D724—44no—
96D724—42no—
97D724—42no—
98D724—81no—
99D724—81no—
100D724—82no—
101C22⋊D181—44no—
102C2×Dic181C2×D412yes—
103C4×D181—22no—
104C4×D181—22no—
105C4×D181—44no—
106#13089678—22no—
107C23×C181—21no—
108Q8×C181—21no—
109Q8×C181C2×D412yescomplemented
110D4×C181—21no—
111D4×C181C2×D412yes—
112C2×C4×C181—21no—
113C2×C4×C181—21no—
114C2×C4×C181—21no—
115C2×C4×C181—42no—
116C2×C4×C181C2×D411yescharacteristic
117C2×C4×C181C2×D411yescharacteristic
118C2×C4×C181C2411yescharacteristic
119C8×C181—21no—
120C8×C181C2×D412yescomplemented
121C22×C181—22no—
122C22×C181—41no—
123C22×C181—41no—
124C22×C181C22×D411yescharacteristic
125C2×D181—22no—
126C2×D181—22no—
127C2×D181—22no—
128C2×D181—42no—
129C2×D181—42no—
130C2×D181—42no—
131C2×D181—42no—
132C2×D181—42no—
133C2×D181—42no—
134C2×D181—88no—
135C4×C181—22no—
136C4×C181—21no—
137C4×C181—41no—
138C4×C181—41no—
139C4×C181AGL(1,ℤ/8)12yescomplemented
140C4×C181C22×D411yescharacteristic
141C4×C181H(5,𝔽2)11yescharacteristic
142Dic181—22no—
143C2×C181—42no—
144C2×C181D4211yescharacteristic
145C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C181#57611yescharacteristic
147D181—42no—
148D181—82no—
149D181—82no—
150C181#327411yescharacteristic, complemented, Sylow
151#3274—1811nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1812no—
153#550—1811no—
154D42—1811no—
155#493—1811no—
156#469—1812no—
157#460—1811no—
158#459—1811no—
159#449—1812no—
160#447—1812no—
161C4×SD16D18111yescharacteristic, complemented
162#436—1811no—
163C2×#41—1812no—
164C22×D4—3621no—
165C22×D4—1812no—
166AGL(1,ℤ/8)—3628no—
167C2×SD16—1812no—
168C2×SD16C2×D18111yescharacteristic
169C2×D8—1812no—
170C2×D8—1812no—
171C2×D8—1812no—
172C2×M16—1812no—
173#127—3621no—
174#127—1811no—
175#127—1811no—
176#121—3621no—
177#121—3622no—
178#121—1812no—
179#121—1812no—
180#120—3622no—
181C4×Q8C2×D18111yescharacteristic, complemented
182C4×D4—3621no—
183C4×D4—1811no—
184C4×D4C2×D18111yescharacteristic
185#106C2×D18111yescharacteristic, complemented
186C4⋊C8—1811no—
187#103C2×D18111yescharacteristic, complemented
188#102—1812no—
189#102—1812no—
190#102C2×D18111yescharacteristic
191C22⋊C8—1812no—
192C4×C8C2×D18111yescharacteristic, complemented
193C24—3622no—
194#41—3624no—
195#41—7242no—
196C2×Q8C22×D18111yescharacteristic
197C2×D4—3622no—
198C2×D4—3622no—
199C2×D4—3622no—
200C2×D4—3624no—
201C2×D4—3622no—
202C2×D4—3621no—
203C2×D4—3621no—
204C2×D4—3621no—
205C2×D4—3621no—
206C2×D4—1811no—
207C2×D4—1812no—
208C2×D4—1812no—
209C2×D4—7244no—
210C2×D4—7242no—
211C2×D4C22×D18111yescharacteristic
212C22×C4—21no—
213C22×C4—3622no—
214C22×C4—1812no—
215SD16—3624no—
216SD16D72414yescomplemented
217D8—3624no—
218D8—3624no—
219D8—7242no—
220M16—3624no—
221C2×C8—1812no—
222C2×C8C22×D18111yescharacteristic
223C2×C8C22×D18111yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—1811no—
226C4⋊C4C22×D18111yescharacteristic
227C4⋊C4C22×D18111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—3622no—
230C22⋊C4—3622no—
231C22⋊C4—1812no—
232C42—21no—
233C42C22×D18111yescharacteristic
234C23—21no—
235C23—3622no—
236C23—3622no—
237C23—14482no—
238C23—1812no—
239C23—7244no—
240C23—7244no—
241Q8—21no—
242Q8C2×D72412yescomplemented
243D4—21no—
244D4—3622no—
245D4—3622no—
246D4—3622no—
247D4—3622no—
248D4—3622no—
249D4—14481no—
250D4—14481no—
251D4—14482no—
252D4—7244no—
253D4—7244no—
254D4—7244no—
255D4—7242no—
256D4—7242no—
257D4C2×D72412yes—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—3622no—
263C2×C4—3622no—
264C2×C4—1812no—
265C2×C4—7244no—
266C2×C4D4×D18111yescharacteristic
267C2×C4D4×D18111yescharacteristic
268C2×C4C23×D18111yescharacteristic
269C8—21no—
270C8—3622no—
271C8C2×D72412yescomplemented
272C22—22no—
273C22—41no—
274C22—41no—
275C22—3622no—
276C22—3622no—
277C22—3622no—
278C22—14488no—
279C22—7242no—
280C22—7242no—
281C22—7242no—
282C22—7242no—
283C22—7242no—
284C22—7242no—
285C22C2×D4×D18111yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—3622no—
291C4#1915384312yescomplemented
292C4C22×D72411yescharacteristic
293C4#1915386211yescharacteristic
294C2—42no—
295C2—14482no—
296C2—14482no—
297C2—7242no—
298C2D4×D72411yescharacteristic
299C2C2×#1915384311yescharacteristic
300C2#2604208111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC4×SD16×C181C2
socleC22×C181C22×D4
hypercenterC4×SD16D181
hypocenterC181#3274
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19153843C2semidirect
2. #26041951C2semidirect
3. D4×D724C2semidirect
4. #26041656C2semidirect
5. #26041533C2semidirect
6. #26041479C2semidirect
7. #26041474C2semidirect
8. #26041427C2semidirect
9. #26041414C2semidirect
10. #26041358C2semidirect
11. C4×SD16×C181C2semidirect
12. C2×#15668450C22semidirect
16. C2×#15668436C22semidirect
17. C2×D1448C22semidirect
18. C2×D1448C22semidirect
20. C2×#15668432C22semidirect
23. #19153813C22semidirect
26. #19153791C22semidirect
31. C4⋊#13089678C22semidirect
32. #19153736C22semidirect
33. #19153736C22semidirect
34. #19153728C22semidirect
36. C4×Q8×C181C22semidirect
38. #106×C181C22semidirect
39. #103×C181C22semidirect
41. C4×C8×C181C22semidirect
74. SD16×C181D4semidirect
109. Q8×C181C2×D4semidirect
120. C8×C181C2×D4semidirect
139. C4×C181AGL(1,ℤ/8)semidirect
150. C181#3274semidirect
161. C4×SD16D181semidirect
181. C4×Q8C2×D181semidirect
185. #106C2×D181semidirect
187. #103C2×D181semidirect
192. C4×C8C2×D181semidirect
216. SD16D724semidirect
242. Q8C2×D724semidirect
271. C8C2×D724semidirect
291. C4#19153843semidirect

Maximal subgroups

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

subgroupindex
C2×#191538432
#260419512
D4×D7242
#260416562
#260415332
#260414792
#260414742
#260414272
#260414142
#260413582
C4×SD16×C1812
#3274181