Finite Group Databasebeta

← #37545536

Subgroups — #37545536

Order 23168.

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

normal subgroups only

This group has 494 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#26041934C211yesmaximal, characteristic, complemented
2D4×D724C211yesmaximal, characteristic, complemented
3#26041626C211yesmaximal, characteristic, complemented
4#26041594C212yesmaximal, complemented
5#26041559C211yesmaximal, characteristic, complemented
6#26041514C212yesmaximal, complemented
7#26041474C211yesmaximal, characteristic, complemented
8#26041411C212yesmaximal, complemented
9C8×D724C211yesmaximal, characteristic, complemented
10#102×D181C212yesmaximal, complemented
11C4×D8×C181C211yesmaximal, characteristic, complemented
12C2×#15668449—21no—
13C22×D724—21no—
14C2×D4×D181C2212yes—
15C2×#15668434C2212yescomplemented
16C2×C8×D181C2212yescomplemented
17#19153816C2211yescharacteristic, complemented
18#19153813C2211yescharacteristic
19#19153798—22no—
20#19153795—21no—
21#19153791C2212yes—
22#19153790—21no—
23#19153787C2212yescomplemented
24#19153786—22no—
25C4×D724C2211yescharacteristic
26C4⋊C4×D181C2212yescomplemented
27#19153749C2212yescomplemented
28C4⋊#13089678C2211yescharacteristic, complemented
29#19153738C2212yescomplemented
30#19153738C2212yescomplemented
31#19153737C2212yes—
32#19153736C2212yescomplemented
33#19153728C2212yescomplemented
34C2×D8×C181C2211yescharacteristic
35C4×D4×C181C2211yescharacteristic
36C4×D4×C181C2211yescharacteristic, complemented
37#107×C181C2211yescharacteristic, complemented
38#102×C181C2211yescharacteristic
39#102×C181C2211yescharacteristic, complemented
40C4×C8×C181C2211yescharacteristic, complemented
41C23×D181—22no—
42#15668449—44no—
43C2×Dic362—21no—
44C2×D724—22no—
45C2×D724—21no—
46C2×D724—21no—
47C2×D724—44no—
48C2×D724D411yescharacteristic
49C2×D724C2312yes—
50C2×C22⋊D181—22no—
51C2×C22⋊D181—22no—
52D4×D181—24no—
53D4×D181—42no—
54C2×C4×D181—22no—
55C2×C4×D181C2312yes—
56#15668434—42no—
57C2×#13089678C2312yes—
58C8×D181—42no—
59C4⋊Dic181—21no—
60C4⋊Dic181D411yescharacteristic
61C4⋊Dic181C2312yes—
62#15668428—22no—
63C22⋊Dic181—22no—
64#15668426—22no—
65#15668426D412yes—
66C2×D4×C181C2311yescharacteristic
67C2×D4×C181C2311yescharacteristic
68C22×C4×C181—21no—
69C22×C4×C181—21no—
70D8×C181—22no—
71C2×C8×C181C2311yescharacteristic
72C2×C8×C181C2311yescharacteristic
73C4⋊C4×C181C2311yescharacteristic
74C4⋊C4×C181C2311yescharacteristic
75C22⋊C4×C181—21no—
76C22⋊C4×C181—21no—
77C42×C181C2311yescharacteristic
78C22×D181—22no—
79C22×D181—44no—
80C22×D181—44no—
81C22×D181—82no—
82C22×D181C2×D412yes—
83Dic362—42no—
84D724—22no—
85D724—22no—
86D724—44no—
87D724—42no—
88D724—81no—
89D724D814yescomplemented
90C22⋊D181—44no—
91C22⋊D181—44no—
92C2×Dic181—22no—
93C2×Dic181C2×D412yes—
94C4×D181—22no—
95C4×D181—22no—
96C4×D181—44no—
97#13089678—22no—
98C23×C181—21no—
99C23×C181—21no—
100D4×C181—21no—
101D4×C181—21no—
102D4×C181—21no—
103D4×C181#4112yescomplemented
104C2×C4×C181—21no—
105C2×C4×C181—21no—
106C2×C4×C181—42no—
107C2×C4×C181—42no—
108C2×C4×C181C2×D411yescharacteristic
109C2×C4×C181C2×D411yescharacteristic
110C2×C4×C181C2411yescharacteristic
111C8×C181—21no—
112C8×C181—21no—
113C22×C181—22no—
114C22×C181—41no—
115C22×C181—41no—
116C22×C181—41no—
117C22×C181—41no—
118C22×C181—41no—
119C22×C181C22×D411yescharacteristic
120C2×D181—22no—
121C2×D181—22no—
122C2×D181—42no—
123C2×D181—42no—
124C2×D181—42no—
125C2×D181—88no—
126C2×D181C2×D814yes—
127C4×C181—21no—
128C4×C181—41no—
129C4×C181—41no—
130C4×C181C2×D812yes—
131C4×C181C2×#4111yescharacteristic
132C4×C181H(5,𝔽2)11yescharacteristic
133Dic181—22no—
134Dic181—42no—
135C2×C181—42no—
136C2×C181—81no—
137C2×C181#54611yescharacteristic
138C2×C181C22×D811yescharacteristic
139C2×C181#57711yescharacteristic
140D181—24no—
141D181—82no—
142C181#329411yescharacteristic, complemented, Sylow
143#3294—1811nomaximal, Sylow
144#547—1811no—
145D42—1811no—
146#486—1811no—
147#480—1812no—
148#474—1811no—
149#466—1812no—
150#459—1811no—
151#447—1812no—
152C4×D8D18111yescharacteristic, complemented
153C8×D4—1811no—
154C2×#102—1812no—
155C2×#41—3621no—
156C22×D4—3621no—
157C22×D4—1812no—
158C2×D8—1812no—
159C2×D8C2×D18111yescharacteristic
160C22×C8—1812no—
161C4⋊Q8—1811no—
162#127—1811no—
163#123—3622no—
164C22⋊Q8—3621no—
165#121—3621no—
166#121—1812no—
167#121—1812no—
168#120—3622no—
169C4×D4—1811no—
170C4×D4C2×D18111yescharacteristic
171C4×D4C2×D18111yescharacteristic, complemented
172C2×C4⋊C4—1812no—
173#107—1812no—
174#107C2×D18111yescharacteristic, complemented
175C4⋊C8—1811no—
176#102—1812no—
177#102—1812no—
178#102—1812no—
179#102—1812no—
180#102C2×D18111yescharacteristic
181#102C2×D18111yescharacteristic, complemented
182C22⋊C8—1812no—
183C4×C8C2×D18111yescharacteristic, complemented
184C24—3622no—
185#41—7244no—
186C2×Q8—3621no—
187C2×D4—3622no—
188C2×D4—3624no—
189C2×D4—3622no—
190C2×D4—3622no—
191C2×D4—3621no—
192C2×D4—3621no—
193C2×D4—1811no—
194C2×D4—1812no—
195C2×D4—7244no—
196C2×D4—7242no—
197C2×D4C22×D18111yescharacteristic
198C2×D4C22×D18111yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—3622no—
202C22×C4—1812no—
203D8—22no—
204D8—7242no—
205C2×C8—1812no—
206C2×C8—7242no—
207C2×C8C22×D18111yescharacteristic
208C2×C8C22×D18111yescharacteristic
209C4⋊C4—3622no—
210C4⋊C4—3621no—
211C4⋊C4—1812no—
212C4⋊C4—1811no—
213C4⋊C4C22×D18111yescharacteristic
214C4⋊C4C22×D18111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—3622no—
218C22⋊C4—3622no—
219C22⋊C4—1812no—
220C42C22×D18111yescharacteristic
221C23—21no—
222C23—21no—
223C23—3622no—
224C23—14482no—
225C23—1812no—
226C23—7244no—
227C23—7244no—
228Q8—7242no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—3622no—
233D4—3622no—
234D4—14481no—
235D4—1814no—
236D4—7244no—
237D4—7244no—
238D4—7244no—
239D4—7242no—
240D4#1566844912yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—3622no—
246C2×C4—3622no—
247C2×C4—3622no—
248C2×C4—1812no—
249C2×C4—7244no—
250C2×C4D4×D18111yescharacteristic
251C2×C4D4×D18111yescharacteristic
252C2×C4C23×D18111yescharacteristic
253C8—21no—
254C8—21no—
255C8—3622no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—3622no—
263C22—3622no—
264C22—14488no—
265C22—1814no—
266C22—7242no—
267C22—7242no—
268C22—7242no—
269C22C2×D4×D18111yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—3622no—
274C4—7242no—
275C4D8×D18112yes—
276C4C2×#1566844911yescharacteristic
277C4#1915386211yescharacteristic
278C2—42no—
279C2—81no—
280C2—3624no—
281C2—14482no—
282C2#2604192511yescharacteristic
283C2C2×D8×D18111yescharacteristic
284C2#2604208611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC4×D8×C181C2
socleC22×C181C22×D4
hypercenterC4×D8D181
hypocenterC181#3294
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #26041934C2semidirect
2. D4×D724C2semidirect
3. #26041626C2semidirect
4. #26041594C2semidirect
5. #26041559C2semidirect
6. #26041514C2semidirect
7. #26041474C2semidirect
8. #26041411C2semidirect
9. C8×D724C2semidirect
10. #102×D181C2semidirect
11. C4×D8×C181C2semidirect
15. C2×#15668434C22semidirect
16. C2×C8×D181C22semidirect
17. #19153816C22semidirect
23. #19153787C22semidirect
26. C4⋊C4×D181C22semidirect
27. #19153749C22semidirect
28. C4⋊#13089678C22semidirect
29. #19153738C22semidirect
30. #19153738C22semidirect
32. #19153736C22semidirect
33. #19153728C22semidirect
36. C4×D4×C181C22semidirect
37. #107×C181C22semidirect
39. #102×C181C22semidirect
40. C4×C8×C181C22semidirect
89. D724D8semidirect
103. D4×C181#41semidirect
142. C181#3294semidirect
152. C4×D8D181semidirect
171. C4×D4C2×D181semidirect
174. #107C2×D181semidirect
181. #102C2×D181semidirect
183. C4×C8C2×D181semidirect
240. D4#15668449semidirect

Maximal subgroups

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

subgroupindex
#260419342
D4×D7242
#260416262
#260415942
#260415592
#260415142
#260414742
#260414112
C8×D7242
#102×D1812
C4×D8×C1812
#3294181