Finite Group Databasebeta

← #37545535

Subgroups — #37545535

Order 23168.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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#26041933C211yesmaximal, characteristic, complemented
2D4×D724C211yesmaximal, characteristic, complemented
3#26041626C211yesmaximal, characteristic, complemented
4#26041593C212yesmaximal, complemented
5#26041560C211yesmaximal, characteristic, complemented
6#26041515C212yesmaximal, complemented
7#26041475C211yesmaximal, characteristic, complemented
8#26041415C212yesmaximal, complemented
9C4×D1448C211yesmaximal, characteristic, complemented
10C2×#19153738C212yesmaximal, complemented
11C8×D4×C181C211yesmaximal, characteristic, complemented
12C2×#15668448—21no—
13C22×D724C2212yes—
14C2×D4×D181—21no—
15C2×D1448C2211yescharacteristic
16C2×D1448C2212yescomplemented
17#19153816C2211yescharacteristic, complemented
18#19153813C2211yescharacteristic
19#19153800—22no—
20#19153796—21no—
21#19153791—21no—
22#19153790C2212yes—
23#19153790C2212yescomplemented
24#19153786—22no—
25C4×D724C2211yescharacteristic
26C4×D724C2211yescharacteristic, complemented
27C2×C4⋊Dic181C2212yescomplemented
28#19153750C2211yescharacteristic, complemented
29#19153750C2212yescomplemented
30#19153738C2211yescharacteristic
31#19153738C2212yes—
32#19153738C2211yescharacteristic, complemented
33#19153738C2212yescomplemented
34#19153738C2212yescomplemented
35#19153738C2212yescomplemented
36C22×C8×C181C2212yescomplemented
37C4×D4×C181C2211yescharacteristic
38C4⋊C8×C181C2211yescharacteristic, complemented
39C22⋊C8×C181C2212yescomplemented
40C4×C8×C181C2211yescharacteristic, complemented
41C23×D181—22no—
42#15668448—44no—
43C2×Dic362—21no—
44C2×D724—24no—
45C2×D724—22no—
46C2×D724—21no—
47C2×D724—21no—
48C2×D724—42no—
49C2×D724C2312yes—
50C2×D724C2311yescharacteristic
51C2×D724C2311yescharacteristic
52C2×C22⋊D181—22no—
53C2×C22⋊D181—22no—
54D4×D181—44no—
55C22×Dic181—22no—
56C2×C4×D181—21no—
57C2×C4×D181—21no—
58D1448—22no—
59D1448—42no—
60C4⋊Dic181—22no—
61C4⋊Dic181—21no—
62C4⋊Dic181C2311yescharacteristic
63C4⋊Dic181C2312yes—
64C4⋊Dic181C2311yescharacteristic
65#15668426—21no—
66#15668426—21no—
67#15668426—22no—
68#15668426—22no—
69C2×D4×C181D411yescharacteristic
70C22×C4×C181C2312yes—
71C2×C8×C181—42no—
72C2×C8×C181C2311yescharacteristic
73C2×C8×C181C2311yescharacteristic
74C2×C8×C181C2312yes—
75C4⋊C4×C181D411yescharacteristic
76C22⋊C4×C181D412yes—
77C42×C181C2311yescharacteristic
78C22×D181—21no—
79C22×D181—21no—
80C22×D181—22no—
81C22×D181—44no—
82C22×D181—44no—
83C22×D181—82no—
84Dic362—42no—
85D724—21no—
86D724—22no—
87D724—21no—
88D724—21no—
89D724—22no—
90D724—44no—
91D724—42no—
92D724—81no—
93D724#4112yescomplemented
94C22⋊D181—44no—
95C22⋊D181—44no—
96C2×Dic181—21no—
97C2×Dic181—21no—
98C2×Dic181—22no—
99C2×Dic181—44no—
100C4×D181—42no—
101C4×D181—42no—
102C23×C181C2×D412yes—
103D4×C181D814yescomplemented
104C2×C4×C181—22no—
105C2×C4×C181—22no—
106C2×C4×C181C2×D411yescharacteristic
107C2×C4×C181C2×D412yes—
108C2×C4×C181C2×D411yescharacteristic
109C2×C4×C181C2411yescharacteristic
110C8×C181—21no—
111C8×C181—21no—
112C8×C181—22no—
113C22×C181—22no—
114C22×C181—22no—
115C22×C181C2×D814yes—
116C22×C181C22×D411yescharacteristic
117C2×D181—22no—
118C2×D181—41no—
119C2×D181—41no—
120C2×D181—42no—
121C2×D181—41no—
122C2×D181—41no—
123C2×D181—42no—
124C2×D181—42no—
125C2×D181—41no—
126C2×D181—88no—
127C4×C181—22no—
128C4×C181—21no—
129C4×C181C2×D812yes—
130C4×C181C2×#4111yescharacteristic
131C4×C181H(5,𝔽2)11yescharacteristic
132Dic181—41no—
133Dic181—41no—
134Dic181—42no—
135C2×C181—24no—
136C2×C181#54611yescharacteristic
137C2×C181C22×D811yescharacteristic
138C2×C181#57711yescharacteristic
139D181—42no—
140D181—81no—
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×D8—1811no—
153C8×D4D18111yescharacteristic, complemented
154C2×#102—1812no—
155C2×#41—3621no—
156C22×D4—3621no—
157C22×D4—1812no—
158C2×D8—1811no—
159C2×D8—1812no—
160C22×C8C2×D18112yescomplemented
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×D4—1811no—
171C4×D4C2×D18111yescharacteristic
172C2×C4⋊C4—1812no—
173#107—1811no—
174#107—1812no—
175C4⋊C8C2×D18111yescharacteristic, complemented
176#102—1811no—
177#102—1812no—
178#102—1812no—
179#102—1812no—
180#102—1811no—
181#102—1812no—
182C22⋊C8C2×D18112yescomplemented
183C4×C8C2×D18111yescharacteristic, complemented
184C24—3622no—
185#41—7244no—
186C2×Q8—3621no—
187C2×D4—3624no—
188C2×D4—3622no—
189C2×D4—3622no—
190C2×D4—3621no—
191C2×D4—3622no—
192C2×D4—3621no—
193C2×D4—1812no—
194C2×D4—1811no—
195C2×D4—1811no—
196C2×D4—7242no—
197C2×D4—7244no—
198C2×D4D72411yescharacteristic
199C22×C4—3622no—
200C22×C4—3621no—
201C22×C4—3621no—
202C22×C4C22×D18112yes—
203D8—3622no—
204D8—7242no—
205C2×C8—42no—
206C2×C8C22×D18111yescharacteristic
207C2×C8C22×D18111yescharacteristic
208C2×C8C22×D18112yes—
209C4⋊C4—3622no—
210C4⋊C4—3621no—
211C4⋊C4—1811no—
212C4⋊C4—1812no—
213C4⋊C4—1811no—
214C4⋊C4D72411yescharacteristic
215C22⋊C4—3621no—
216C22⋊C4—3621no—
217C22⋊C4—3622no—
218C22⋊C4—3622no—
219C22⋊C4D72412yes—
220C42C22×D18111yescharacteristic
221C23—3621no—
222C23—3621no—
223C23—3622no—
224C23—14482no—
225C23—7244no—
226C23—7244no—
227C23C2×D72412yes—
228Q8—7242no—
229D4—3621no—
230D4—3622no—
231D4—3621no—
232D4—3621no—
233D4—3622no—
234D4—14481no—
235D4—1812no—
236D4—7244no—
237D4—7244no—
238D4—7244no—
239D4—7242no—
240D4D144814yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—3621no—
244C2×C4—3621no—
245C2×C4—3622no—
246C2×C4—7244no—
247C2×C4—7242no—
248C2×C4—7242no—
249C2×C4C2×D72411yescharacteristic
250C2×C4C2×D72412yes—
251C2×C4C2×D72411yescharacteristic
252C2×C4C23×D18111yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—3622no—
259C22—14488no—
260C22—7242no—
261C22—7241no—
262C22—7241no—
263C22—7242no—
264C22—7242no—
265C22—7241no—
266C22—7241no—
267C22—7241no—
268C22C2×D144814yes—
269C22C22×D72411yescharacteristic
270C4—22no—
271C4—21no—
272C4—7241no—
273C4—7241no—
274C4—7242no—
275C4C2×D144812yes—
276C4#41×D18111yescharacteristic
277C4#1915386111yescharacteristic
278C2—24no—
279C2—14482no—
280C2—14481no—
281C2—7242no—
282C2#2604192411yescharacteristic
283C2C22×D144811yescharacteristic
284C2#2604209111yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC8×D4×C181C2
socleC22×C181C22×D4
hypercenterC8×D4D181
hypocenterC181#3294
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #26041933C2semidirect
2. D4×D724C2semidirect
3. #26041626C2semidirect
4. #26041593C2semidirect
5. #26041560C2semidirect
6. #26041515C2semidirect
7. #26041475C2semidirect
8. #26041415C2semidirect
9. C4×D1448C2semidirect
10. C2×#19153738C2semidirect
11. C8×D4×C181C2semidirect
16. C2×D1448C22semidirect
17. #19153816C22semidirect
23. #19153790C22semidirect
26. C4×D724C22semidirect
27. C2×C4⋊Dic181C22semidirect
28. #19153750C22semidirect
29. #19153750C22semidirect
32. #19153738C22semidirect
33. #19153738C22semidirect
34. #19153738C22semidirect
35. #19153738C22semidirect
36. C22×C8×C181C22semidirect
38. C4⋊C8×C181C22semidirect
39. C22⋊C8×C181C22semidirect
40. C4×C8×C181C22semidirect
93. D724#41semidirect
103. D4×C181D8semidirect
142. C181#3294semidirect
153. C8×D4D181semidirect
160. C22×C8C2×D181semidirect
175. C4⋊C8C2×D181semidirect
182. C22⋊C8C2×D181semidirect
183. C4×C8C2×D181semidirect
240. D4D1448semidirect

Maximal subgroups

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

subgroupindex
#260419332
D4×D7242
#260416262
#260415932
#260415602
#260415152
#260414752
#260414152
C4×D14482
C2×#191537382
C8×D4×C1812
#3294181