Finite Group Databasebeta

← #37542946

Subgroups — #37542946

Order 23168.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D4×D181C211yesmaximal, characteristic, complemented
2C2×#19153841C211yesmaximal, characteristic, complemented
3#26041786C211yesmaximal, characteristic, complemented
4#26041428C214yesmaximal, complemented
5#26041411C214yesmaximal, complemented
6#26041285C211yesmaximal, characteristic, complemented
7#26041284C211yesmaximal, characteristic, complemented
8#26041242C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C181C211yesmaximal, characteristic, complemented
10C24×D181—21no—
11C22×D724C2211yescharacteristic
12C22×C22⋊D181—21no—
13C2×D4×D181—24no—
14C2×D4×D181—22no—
15C2×D4×D181—24no—
16C2×D4×D181C2214yes—
17C22×C4×D181C2211yescharacteristic
18#19153841—24no—
19C2×#15668436C2212yescomplemented
20C2×#15668434C2212yescomplemented
21C2×#15668433C2211yescharacteristic, complemented
22#19153797—21no—
23#19153792C2212yescomplemented
24#19153787C2212yescomplemented
25#19153784—21no—
26C4×C22⋊D181—22no—
27#19153775C2211yescharacteristic, complemented
28#19153738C2214yescomplemented
29#19153737C2214yescomplemented
30#19153728C2214yescomplemented
31C2×#41×C181C2211yescharacteristic, complemented
32C22×D4×C181C2211yescharacteristic
33AGL(1,ℤ/8)×C181—24no—
34C2×SD16×C181C2212yescomplemented
35C2×D8×C181C2212yescomplemented
36C2×M16×C181C2211yescharacteristic, complemented
37C23×D181—28no—
38C23×D181—24no—
39C23×D181—21no—
40C23×D181—48no—
41C23×D181D411yescharacteristic
42C2×D724—21no—
43C2×D724—22no—
44C2×D724D414yescomplemented
45C2×D724C2312yes—
46C2×C22⋊D181—28no—
47C2×C22⋊D181—22no—
48C2×C22⋊D181—24no—
49D4×D181—216no—
50D4×D181—28no—
51D4×D181—28no—
52C22×Dic181D411yescharacteristic
53C2×C4×D181—22no—
54C2×C4×D181—22no—
55C2×C4×D181D412yes—
56C2×C4×D181C2312yes—
57#15668436—24no—
58#15668434—24no—
59#15668433—22no—
60C2×#13089678C2312yescomplemented
61C4⋊Dic181C2312yescomplemented
62#15668428—22no—
63C22⋊Dic181—22no—
64C22⋊Dic181—21no—
65#15668426—22no—
66C4×Dic181—21no—
67C24×C181—21no—
68#41×C181—22no—
69#41×C181—22no—
70C2×Q8×C181C2311yescharacteristic, complemented
71C2×D4×C181—21no—
72C2×D4×C181—21no—
73C2×D4×C181—22no—
74C2×D4×C181D414yescomplemented
75C2×D4×C181C2312yes—
76C2×D4×C181C2311yescharacteristic, complemented
77C22×C4×C181—21no—
78C22×C4×C181C2311yescharacteristic
79SD16×C181—24no—
80D8×C181—24no—
81M16×C181—22no—
82C2×C8×C181C2312yescomplemented
83C22×D181—22no—
84C22×D181—24no—
85C22×D181—216no—
86C22×D181—24no—
87C22×D181—22no—
88C22×D181—432no—
89C22×D181—48no—
90C22×D181—44no—
91C22×D181—48no—
92C22×D181C2×D412yes—
93D724—24no—
94D724—22no—
95D724C2×D414yes—
96C22⋊D181—216no—
97C22⋊D181—44no—
98C2×Dic181—22no—
99C2×Dic181—22no—
100C2×Dic181C2×D412yes—
101C4×D181—24no—
102C4×D181—22no—
103C4×D181—22no—
104#13089678—22no—
105C23×C181—24no—
106C23×C181—22no—
107C23×C181—21no—
108C23×C181—44no—
109C23×C181C2×D411yescharacteristic
110Q8×C181—21no—
111Q8×C181—21no—
112D4×C181—24no—
113D4×C181—22no—
114D4×C181—21no—
115D4×C181—21no—
116D4×C181—42no—
117D4×C181C2×D414yes—
118C2×C4×C181—21no—
119C2×C4×C181—42no—
120C2×C4×C181C2×D411yescharacteristic
121C2×C4×C181C2×D412yes—
122C2×C4×C181C2×D412yes—
123C2×C4×C181C2411yescharacteristic
124C8×C181—22no—
125C22×C181—22no—
126C22×C181—24no—
127C22×C181—48no—
128C22×C181—44no—
129C22×C181—41no—
130C22×C181—42no—
131C22×C181#12012yes—
132C22×C181C22×D411yescharacteristic
133C2×D181—28no—
134C2×D181—22no—
135C2×D181—22no—
136C2×D181—24no—
137C2×D181—432no—
138C2×D181—48no—
139C2×D181—44no—
140C2×D181AGL(1,ℤ/8)14yescomplemented
141C4×C181—41no—
142C4×C181#12012yes—
143C4×C181C22×D411yescharacteristic
144C4×C181C22×D411yescharacteristic
145Dic181—22no—
146Dic181—42no—
147C2×C181—22no—
148C2×C181—44no—
149C2×C181—81no—
150C2×C181C2×#12011yescharacteristic
151C2×C181C2×AGL(1,ℤ/8)12yes—
152D181—24no—
153D181—44no—
154C181#300311yescharacteristic, complemented, Sylow
155#3003—1811nomaximal, Sylow
156C23×D4—1811no—
157C2×AGL(1,ℤ/8)—1811no—
158C2×AGL(1,ℤ/8)D18111yescharacteristic, complemented
159#525—1811no—
160#450—1814no—
161#447—1814no—
162#418—1811no—
163#418—1811no—
164#407—1811no—
165C25—3621no—
166C2×#41C2×D18111yescharacteristic, complemented
167C22×D4—3624no—
168C22×D4—3624no—
169C22×D4—3622no—
170C22×D4—3621no—
171C22×D4—1814no—
172C22×D4—1811no—
173C22×D4C2×D18111yescharacteristic
174C23×C4—1811no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—3624no—
177C2×SD16—1812no—
178C2×SD16C2×D18112yescomplemented
179C2×D8—1812no—
180C2×D8C2×D18112yescomplemented
181C2×M16—1811no—
182C2×M16C2×D18111yescharacteristic, complemented
183#123—3621no—
184C22⋊Q8—1812no—
185#121—1812no—
186#120—3621no—
187C4×D4—3622no—
188#117—1811no—
189#102—1814no—
190#102—1814no—
191C22⋊C8—1814no—
192C24—21no—
193C24—3628no—
194C24—3624no—
195C24—3621no—
196C24—1811no—
197C24—7248no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D18111yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—36216no—
205C2×D4—3628no—
206C2×D4—3628no—
207C2×D4—3628no—
208C2×D4—3622no—
209C2×D4—3624no—
210C2×D4—3621no—
211C2×D4—3622no—
212C2×D4—1814no—
213C2×D4—1812no—
214C2×D4C22⋊D18114yescomplemented
215C2×D4C22×D18112yes—
216C2×D4C22×D18111yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—3622no—
219C22×C4—3622no—
220C22×C4—1811no—
221C22×C4—1812no—
222C22×C4—1812no—
223C22×C4C22×D18111yescharacteristic
224SD16—24no—
225SD16—3624no—
226D8—24no—
227D8—3624no—
228M16—22no—
229M16—3622no—
230C2×C8—1812no—
231C2×C8C22×D18112yescomplemented
232C4⋊C4—3622no—
233C4⋊C4—1812no—
234C22⋊C4—3622no—
235C22⋊C4—3621no—
236C22⋊C4—3622no—
237C42—3621no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—3624no—
243C23—36216no—
244C23—3622no—
245C23—3624no—
246C23—3622no—
247C23—1812no—
248C23—7244no—
249C23—72432no—
250C23—7248no—
251C23—7248no—
252C23D4×D18111yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—24no—
256D4—22no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4—36216no—
261D4—3624no—
262D4—3622no—
263D4—1814no—
264D4—7244no—
265D4C2×C22⋊D18114yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—3622no—
269C2×C4—3624no—
270C2×C4—3622no—
271C2×C4—3622no—
272C2×C4—3622no—
273C2×C4—1812no—
274C2×C4D4×D18111yescharacteristic
275C2×C4D4×D18112yes—
276C2×C4C2×C22⋊D18112yes—
277C2×C4C23×D18111yescharacteristic
278C8—22no—
279C8—3622no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—3628no—
287C22—3622no—
288C22—3622no—
289C22—3624no—
290C22—1814no—
291C22—7248no—
292C22—72432no—
293C22—7244no—
294C22#1915378412yes—
295C22C2×D4×D18111yescharacteristic
296C4—41no—
297C4—3622no—
298C4—7242no—
299C4#1915378412yes—
300C4C2×D4×D18111yescharacteristic
301C4C22×C22⋊D18111yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—3624no—
306C2—7244no—
307C2C2×#1915378411yescharacteristic
308C2AGL(1,ℤ/8)×D18112yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×AGL(1,ℤ/8)×C181C2
socleC22×C181C22×D4
hypercenterC2×AGL(1,ℤ/8)D181
hypocenterC181#3003
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×D4×D181C2semidirect
2. C2×#19153841C2semidirect
3. #26041786C2semidirect
4. #26041428C2semidirect
5. #26041411C2semidirect
6. #26041285C2semidirect
7. #26041284C2semidirect
8. #26041242C2semidirect
9. C2×AGL(1,ℤ/8)×C181C2semidirect
19. C2×#15668436C22semidirect
20. C2×#15668434C22semidirect
21. C2×#15668433C22semidirect
23. #19153792C22semidirect
24. #19153787C22semidirect
27. #19153775C22semidirect
28. #19153738C22semidirect
29. #19153737C22semidirect
30. #19153728C22semidirect
31. C2×#41×C181C22semidirect
34. C2×SD16×C181C22semidirect
35. C2×D8×C181C22semidirect
36. C2×M16×C181C22semidirect
44. C2×D724D4semidirect
60. C2×#13089678C23semidirect
61. C4⋊Dic181C23semidirect
70. C2×Q8×C181C23semidirect
74. C2×D4×C181D4semidirect
76. C2×D4×C181C23semidirect
82. C2×C8×C181C23semidirect
140. C2×D181AGL(1,ℤ/8)semidirect
154. C181#3003semidirect
158. C2×AGL(1,ℤ/8)D181semidirect
166. C2×#41C2×D181semidirect
178. C2×SD16C2×D181semidirect
180. C2×D8C2×D181semidirect
182. C2×M16C2×D181semidirect
200. C2×Q8C22×D181semidirect
214. C2×D4C22⋊D181semidirect
216. C2×D4C22×D181semidirect
231. C2×C8C22×D181semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1812
C2×#191538412
#260417862
#260414282
#260414112
#260412852
#260412842
#260412422
C2×AGL(1,ℤ/8)×C1812
#3003181