Finite Group Databasebeta

← #37545395

Subgroups — #37545395

Order 23168.

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

normal subgroups only

This group has 620 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
1C22×D1448C211yesmaximal, characteristic, complemented
2C2×D8×D181C211yesmaximal, characteristic, complemented
3#26041913C212yesmaximal, complemented
4#26041654C211yesmaximal, characteristic, complemented
5#26041516C211yesmaximal, characteristic, complemented
6#26041473C212yesmaximal, complemented
7#26041414C212yesmaximal, complemented
8#26041412C212yesmaximal, complemented
9#26041361C211yesmaximal, characteristic, complemented
10C8×C22⋊D181C211yesmaximal, characteristic, complemented
11#466×C181C211yesmaximal, characteristic, complemented
12C22×D724C2212yescomplemented
13C2×D4×D181—22no—
14C2×D4×D181C2212yescomplemented
15C2×D1448—24no—
16C2×D1448C2211yescharacteristic
17C2×D1448C2211yescharacteristic, complemented
18C2×D1448C2212yescomplemented
19C2×#15668434C2212yescomplemented
20C2×#15668434C2212yescomplemented
21D8×D181—24no—
22C2×C8×D181C2211yescharacteristic, complemented
23#19153812C2212yescomplemented
24#19153791C2212yescomplemented
25#19153789—22no—
26#19153786—22no—
27#19153784—22no—
28#19153778C2212yescomplemented
29C4×C22⋊D181C2211yescharacteristic
30#19153745C2211yescharacteristic, complemented
31#19153736C2212yescomplemented
32#19153736C2212yescomplemented
33C22⋊#13089678C2211yescharacteristic, complemented
34#19153728C2211yescharacteristic, complemented
35C8×Dic181C2211yescharacteristic, complemented
36C2×D8×C181C2211yescharacteristic, complemented
37C22×C8×C181C2211yescharacteristic, complemented
38#121×C181C2212yescomplemented
39#107×C181C2211yescharacteristic, complemented
40#102×C181C2212yescomplemented
41C23×D181—22no—
42C23×D181—22no—
43C2×D724—24no—
44C2×D724—22no—
45C2×D724—22no—
46C2×D724—42no—
47C2×D724C2312yes—
48C2×D724C2312yes—
49C2×C22⋊D181—22no—
50C2×C22⋊D181—22no—
51C2×C22⋊D181—22no—
52C2×C22⋊D181D411yescharacteristic
53D4×D181—24no—
54D4×D181—42no—
55D4×D181—48no—
56C2×C4×D181—22no—
57C2×C4×D181C2311yescharacteristic
58D1448—24no—
59D1448—22no—
60D1448D414yescomplemented
61#15668434—24no—
62#15668434—42no—
63C2×#13089678C2311yescharacteristic
64C2×#13089678C2311yescharacteristic
65C8×D181—22no—
66#15668428D411yescharacteristic
67C22⋊Dic181D411yescharacteristic
68#15668426—22no—
69#15668426—22no—
70#15668426D411yescharacteristic
71C4×Dic181C2311yescharacteristic
72C2×D4×C181—22no—
73C2×D4×C181C2312yes—
74C22×C4×C181C2311yescharacteristic
75D8×C181—22no—
76C2×C8×C181—22no—
77C2×C8×C181C2311yescharacteristic
78C2×C8×C181C2311yescharacteristic
79C4⋊C4×C181C2312yes—
80C22⋊C4×C181—22no—
81C22×D181—22no—
82C22×D181—22no—
83C22×D181—44no—
84C22×D181—44no—
85C22×D181—44no—
86C22×D181—44no—
87C22×D181—82no—
88C22×D181—82no—
89C22×D181C2×D411yescharacteristic
90D724—22no—
91D724—22no—
92D724—22no—
93D724—44no—
94D724—44no—
95D724C2×D414yescomplemented
96C22⋊D181—44no—
97C22⋊D181—44no—
98C22⋊D181—82no—
99C22⋊D181D814yescomplemented
100C2×Dic181C2×D411yescharacteristic
101C2×Dic181C2×D411yescharacteristic
102C2×Dic181C2×D411yescharacteristic
103C4×D181—21no—
104C4×D181—21no—
105C4×D181—44no—
106#13089678—21no—
107#13089678—21no—
108C23×C181—22no—
109C23×C181C2×D411yescharacteristic
110D4×C181—22no—
111D4×C181—22no—
112D4×C181—44no—
113C2×C4×C181—21no—
114C2×C4×C181—21no—
115C2×C4×C181—22no—
116C2×C4×C181C2×D411yescharacteristic
117C2×C4×C181C2411yescharacteristic
118C8×C181—21no—
119C8×C181C2×D412yescomplemented
120C22×C181—21no—
121C22×C181—21no—
122C22×C181—42no—
123C22×C181—42no—
124C22×C181—42no—
125C22×C181C2×D812yescomplemented
126C22×C181C22×D411yescharacteristic
127C2×D181—21no—
128C2×D181—21no—
129C2×D181—24no—
130C2×D181—42no—
131C2×D181—42no—
132C2×D181—42no—
133C2×D181—42no—
134C2×D181—42no—
135C2×D181—88no—
136C2×D181—88no—
137C2×D181C2×D812yescomplemented
138C4×C181—21no—
139C4×C181—42no—
140C4×C181C22×D411yescharacteristic
141C4×C181H(5,𝔽2)11yescharacteristic
142Dic181—21no—
143Dic181—21no—
144Dic181C2×D812yescomplemented
145C2×C181—22no—
146C2×C181—82no—
147C2×C181D4211yescharacteristic
148C2×C181C22×D811yescharacteristic
149C2×C181#57611yescharacteristic
150D181—22no—
151D181—44no—
152D181—82no—
153C181D4×D811yescharacteristic, complemented, Sylow
154D4×D8—1811nomaximal, Sylow
155C22×D8—1811no—
156C22×D8—1811no—
157D42—1812no—
158#493—1811no—
159#466—1811no—
160#466D18111yescharacteristic, complemented
161#459—1812no—
162#447—1812no—
163#447—1812no—
164C4×D8—1811no—
165C8×D4—1811no—
166C22×D4—3622no—
167C22×D4—1812no—
168C22×D4—1812no—
169C2×D8—3624no—
170C2×D8—3624no—
171C2×D8—1812no—
172C2×D8—1812no—
173C2×D8—1812no—
174C2×D8—1811no—
175C2×D8—1811no—
176C2×D8C2×D18111yescharacteristic, complemented
177C22×C8—1811no—
178C22×C8C2×D18111yescharacteristic, complemented
179#127—1812no—
180#121—3622no—
181#121—1812no—
182#121C2×D18112yescomplemented
183#120—3622no—
184#120—3622no—
185C4×D4—1812no—
186C4×D4—1811no—
187#107C2×D18111yescharacteristic, complemented
188C4⋊C8—1811no—
189#102—1812no—
190#102—1812no—
191#102C2×D18112yescomplemented
192C22⋊C8—1811no—
193C22⋊C8—1811no—
194C4×C8—1811no—
195C24—3622no—
196C24—3622no—
197C2×D4—22no—
198C2×D4—3624no—
199C2×D4—3622no—
200C2×D4—3624no—
201C2×D4—3622no—
202C2×D4—3622no—
203C2×D4—3622no—
204C2×D4—3622no—
205C2×D4—1811no—
206C2×D4—1812no—
207C2×D4—1812no—
208C2×D4—7242no—
209C2×D4—7242no—
210C2×D4—7248no—
211C2×D4C22×D18112yes—
212C22×C4—3622no—
213C22×C4—1811no—
214C22×C4C22×D18111yescharacteristic
215D8—22no—
216D8—3624no—
217D8—3624no—
218D8—3622no—
219D8—1814no—
220D8—7242no—
221C2×C8—22no—
222C2×C8—3622no—
223C2×C8—1811no—
224C2×C8—1811no—
225C2×C8C22×D18111yescharacteristic
226C2×C8C22×D18111yescharacteristic
227C4⋊C4—1811no—
228C4⋊C4C22×D18112yes—
229C22⋊C4—22no—
230C22⋊C4—3622no—
231C22⋊C4—3622no—
232C22⋊C4—1811no—
233C22⋊C4—1811no—
234C42—1811no—
235C23—22no—
236C23—3622no—
237C23—3622no—
238C23—14482no—
239C23—14482no—
240C23—1811no—
241C23—7244no—
242C23—7244no—
243C23—7244no—
244C23—7244no—
245C23D4×D18111yescharacteristic
246D4—22no—
247D4—22no—
248D4—44no—
249D4—3622no—
250D4—3622no—
251D4—3622no—
252D4—14482no—
253D4—1814no—
254D4—1814no—
255D4—7244no—
256D4—7244no—
257D4—7244no—
258D4—7244no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—3621no—
263C2×C4—3621no—
264C2×C4—1811no—
265C2×C4—1811no—
266C2×C4—1811no—
267C2×C4—7244no—
268C2×C4D4×D18111yescharacteristic
269C2×C4C23×D18111yescharacteristic
270C8—21no—
271C8—3621no—
272C8—3621no—
273C8D4×D18112yescomplemented
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—3621no—
280C22—3621no—
281C22—3624no—
282C22—14488no—
283C22—14488no—
284C22—1812no—
285C22—7242no—
286C22—7242no—
287C22—7242no—
288C22—7242no—
289C22—7242no—
290C22D8×D18112yescomplemented
291C22C2×D4×D18111yescharacteristic
292C4—21no—
293C4—42no—
294C4—3621no—
295C4—3621no—
296C4—1812no—
297C4C2×D4×D18111yescharacteristic
298C4#1915386211yescharacteristic
299C2—22no—
300C2—82no—
301C2—3622no—
302C2—14482no—
303C2—7244no—
304C2#2604191311yescharacteristic
305C2C2×D8×D18111yescharacteristic
306C2#2604208111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#466×C181C2
socleC22×C181C22×D4
hypercenter#466D181
hypocenterC181D4×D8
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C22×D1448C2semidirect
2. C2×D8×D181C2semidirect
3. #26041913C2semidirect
4. #26041654C2semidirect
5. #26041516C2semidirect
6. #26041473C2semidirect
7. #26041414C2semidirect
8. #26041412C2semidirect
9. #26041361C2semidirect
10. C8×C22⋊D181C2semidirect
11. #466×C181C2semidirect
12. C22×D724C22semidirect
14. C2×D4×D181C22semidirect
17. C2×D1448C22semidirect
18. C2×D1448C22semidirect
19. C2×#15668434C22semidirect
20. C2×#15668434C22semidirect
22. C2×C8×D181C22semidirect
23. #19153812C22semidirect
24. #19153791C22semidirect
28. #19153778C22semidirect
30. #19153745C22semidirect
31. #19153736C22semidirect
32. #19153736C22semidirect
33. C22⋊#13089678C22semidirect
34. #19153728C22semidirect
35. C8×Dic181C22semidirect
36. C2×D8×C181C22semidirect
37. C22×C8×C181C22semidirect
38. #121×C181C22semidirect
39. #107×C181C22semidirect
40. #102×C181C22semidirect
60. D1448D4semidirect
95. D724C2×D4semidirect
99. C22⋊D181D8semidirect
119. C8×C181C2×D4semidirect
125. C22×C181C2×D8semidirect
137. C2×D181C2×D8semidirect
144. Dic181C2×D8semidirect
153. C181D4×D8semidirect
160. #466D181semidirect
176. C2×D8C2×D181semidirect
178. C22×C8C2×D181semidirect
182. #121C2×D181semidirect
187. #107C2×D181semidirect
191. #102C2×D181semidirect
273. C8D4×D181semidirect
290. C22D8×D181semidirect

Maximal subgroups

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

subgroupindex
C22×D14482
C2×D8×D1812
#260419132
#260416542
#260415162
#260414732
#260414142
#260414122
#260413612
C8×C22⋊D1812
#466×C1812
D4×D8181