Finite Group Databasebeta

← #37543212

Subgroups — #37543212

Order 23168.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153848C211yesmaximal, characteristic, complemented
2C2×#19153843C211yesmaximal, characteristic
3#124×D181C211yesmaximal, characteristic, complemented
4C4×#15668450C211yesmaximal, characteristic, complemented
5#26041503C211yesmaximal, characteristic, complemented
6#26041501C211yesmaximal, characteristic, complemented
7#26041499C211yesmaximal, characteristic, complemented
8#26041497C211yesmaximal, characteristic, complemented
9#26041495C211yesmaximal, characteristic, complemented
10#26041493C211yesmaximal, characteristic, complemented
11#26041491C211yesmaximal, characteristic, complemented
12#26041301C211yesmaximal, characteristic, complemented
13#26041286C211yesmaximal, characteristic, complemented
14#26041282C211yesmaximal, characteristic, complemented
15#464×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic
19#19153848—24no—
20#19153843—24no—
21C2×#15668439C2211yescharacteristic, complemented
22C2×#15668438C2211yescharacteristic, complemented
23C2×#15668436C2211yescharacteristic
24C2×#15668436C2211yescharacteristic, complemented
25C2×D1448C2211yescharacteristic
26C2×#15668434C2211yescharacteristic
27C2×#15668432C2211yescharacteristic
28C2×#15668432C2211yescharacteristic
29#19153805C2211yescharacteristic
30#19153804C2211yescharacteristic
31#19153803C2211yescharacteristic
32#19153802—21no—
33#19153802—21no—
34Q8×Dic181C2211yescharacteristic, complemented
35C4×D724—21no—
36C4×D724C2211yescharacteristic, complemented
37#19153778—21no—
38#19153778C2211yescharacteristic, complemented
39#19153774—21no—
40#19153774C2211yescharacteristic
41C22⋊C4×D181—21no—
42C22⋊C4×D181—21no—
43C42×D181—21no—
44C42×D181C2211yescharacteristic
45C4⋊#13089678C2211yescharacteristic, complemented
46#19153745C2211yescharacteristic, complemented
47#19153745C2211yescharacteristic, complemented
48#19153741C2211yescharacteristic, complemented
49#19153740C2211yescharacteristic, complemented
50#19153739C2211yescharacteristic, complemented
51#19153738C2211yescharacteristic
52#19153737C2211yescharacteristic
53#19153736C2211yescharacteristic
54C2×Q16×C181C2211yescharacteristic, complemented
55C2×SD16×C181C2211yescharacteristic
56#124×C181C2211yescharacteristic
57C4×Q8×C181C2211yescharacteristic, complemented
58C4⋊C8×C181C2211yescharacteristic, complemented
59#103×C181C2211yescharacteristic, complemented
60#102×C181C2211yescharacteristic
61C23×D181—21no—
62#15668450—22no—
63#15668450D414yes—
64C2×Dic362—21no—
65C2×Dic362C2311yescharacteristic
66Q8×D181—22no—
67Q8×D181—22no—
68C2×D724—21no—
69C2×D724C2311yescharacteristic
70C2×D724C2311yescharacteristic
71C2×C22⋊D181—21no—
72D4×D181—22no—
73D4×D181—22no—
74C2×C4×D181—21no—
75C2×C4×D181—21no—
76C2×C4×D181—21no—
77C2×C4×D181—21no—
78C2×C4×D181D411yescharacteristic
79C2×C4×D181D411yescharacteristic
80C2×C4×D181C2311yescharacteristic
81#15668439—22no—
82#15668438—22no—
83#15668436—22no—
84#15668436—22no—
85D1448—22no—
86#15668434—22no—
87#15668432—22no—
88#15668432—22no—
89C2×#13089678C2311yescharacteristic
90C2×#13089678C2311yescharacteristic
91C4⋊Dic181—21no—
92C4⋊Dic181C2311yescharacteristic
93C22⋊Dic181—21no—
94C22⋊Dic181—21no—
95#15668426—21no—
96#15668426—21no—
97#15668426—21no—
98#15668426—21no—
99#15668426—21no—
100#15668426—21no—
101#15668426—21no—
102C4×Dic181—21no—
103C4×Dic181—21no—
104C4×Dic181D411yescharacteristic
105C4×Dic181D411yescharacteristic
106C4×Dic181C2311yescharacteristic
107C2×Q8×C181C2311yescharacteristic
108C2×Q8×C181C2311yescharacteristic
109C2×D4×C181C2311yescharacteristic
110Q16×C181—22no—
111SD16×C181—22no—
112C2×C8×C181C2311yescharacteristic
113C2×C8×C181C2311yescharacteristic
114C4⋊C4×C181—21no—
115C4⋊C4×C181C2311yescharacteristic
116C22⋊C4×C181—21no—
117C22⋊C4×C181—21no—
118C42×C181—21no—
119C42×C181C2311yescharacteristic
120C22×D181—21no—
121C22×D181—21no—
122C22×D181—42no—
123C22×D181—42no—
124C22×D181—42no—
125C22×D181C2×D411yescharacteristic
126Dic362—21no—
127Dic362—21no—
128Dic362—42no—
129D724—24no—
130D724—21no—
131D724—21no—
132D724—21no—
133D724C2×D412yes—
134C22⋊D181—42no—
135C2×Dic181—21no—
136C2×Dic181—21no—
137C2×Dic181C2×D411yescharacteristic
138C2×Dic181C2×D411yescharacteristic
139C2×Dic181C2×D411yescharacteristic
140C4×D181—24no—
141C4×D181—41no—
142C4×D181—41no—
143C4×D181—42no—
144C4×D181—42no—
145C4×D181—42no—
146C4×D181C2×D412yes—
147C4×D181#4112yes—
148#13089678—21no—
149#13089678—21no—
150C23×C181—21no—
151Q8×C181—21no—
152Q8×C181—21no—
153Q8×C181—21no—
154Q8×C181C2×D412yes—
155D4×C181—21no—
156D4×C181—21no—
157C2×C4×C181—21no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181C2×D411yescharacteristic
161C2×C4×C181C2×D411yescharacteristic
162C2×C4×C181C2411yescharacteristic
163C8×C181—21no—
164C8×C181—21no—
165C22×C181—41no—
166C22×C181—41no—
167C22×C181—41no—
168C22×C181C22×D411yescharacteristic
169C2×D181—21no—
170C2×D181—21no—
171C2×D181—22no—
172C2×D181—41no—
173C2×D181—41no—
174C2×D181—41no—
175C2×D181—41no—
176C2×D181—41no—
177C2×D181—84no—
178C2×D181#12112yes—
179C4×C181—22no—
180C4×C181—21no—
181C4×C181—21no—
182C4×C181—41no—
183C4×C181—41no—
184C4×C181C22×D411yescharacteristic
185C4×C181C2×#4111yescharacteristic
186Dic181—21no—
187Dic181—21no—
188Dic181—41no—
189Dic181—41no—
190Dic181#12112yes—
191C2×C181—81no—
192C2×C181C2×#12111yescharacteristic
193C2×C181#57511yescharacteristic
194C2×C181#57511yescharacteristic
195D181—22no—
196D181—42no—
197D181—81no—
198C181#304011yescharacteristic, complemented, Sylow
199#3040—1811nomaximal, Sylow
200C2×#137—1811no—
201C2×AGL(1,ℤ/8)—1811no—
202C2×#124—1811no—
203C4×#41—1811no—
204#464—1811no—
205#464—1811no—
206#464—1811no—
207#464D18111yescharacteristic, complemented
208#463—1811no—
209#463—1811no—
210#463—1811no—
211#463—1811no—
212#423—1811no—
213#419—1811no—
214#418—1811no—
215C2×#41—1811no—
216C22×Q8—1811no—
217C22×D4—1811no—
218#137—3624no—
219AGL(1,ℤ/8)—3624no—
220C2×Q16—1811no—
221C2×Q16C2×D18111yescharacteristic, complemented
222C2×SD16—1811no—
223C2×SD16—1811no—
224C2×SD16—1811no—
225C2×SD16C2×D18111yescharacteristic
226C2×D8—1811no—
227C2×D8—1811no—
228C2×M16—1811no—
229C2×M16—1811no—
230#124—3621no—
231#124—3621no—
232#124—1811no—
233#124—1811no—
234#124—1811no—
235#124C2×D18111yescharacteristic
236C4×Q8—1811no—
237C4×Q8C2×D18111yescharacteristic, complemented
238C4×D4—3621no—
239C4×D4—3621no—
240C4×D4—1811no—
241C4×D4—1811no—
242#117—3621no—
243#117—1811no—
244C2×C22⋊C4—3621no—
245C2×C22⋊C4—3621no—
246C2×C42—3621no—
247C2×C42—1811no—
248C4⋊C8—1811no—
249C4⋊C8—1811no—
250C4⋊C8—1811no—
251C4⋊C8C2×D18111yescharacteristic, complemented
252#103—1811no—
253#103—1811no—
254#103—1811no—
255#103C2×D18111yescharacteristic, complemented
256#102—1811no—
257#102—1811no—
258#102—1811no—
259#102C2×D18111yescharacteristic
260C24—3621no—
261#41—3622no—
262#41—1814no—
263C2×Q8—3621no—
264C2×Q8—3622no—
265C2×Q8—3622no—
266C2×Q8—1811no—
267C2×Q8C22×D18111yescharacteristic
268C2×Q8C22×D18111yescharacteristic
269C2×D4—3621no—
270C2×D4—3622no—
271C2×D4—3622no—
272C2×D4—3621no—
273C2×D4—1811no—
274C2×D4—1811no—
275C2×D4C22×D18111yescharacteristic
276C22×C4—3621no—
277C22×C4—3621no—
278C22×C4—3621no—
279C22×C4—3621no—
280C22×C4—1811no—
281C22×C4—1811no—
282C22×C4—1811no—
283Q16—22no—
284Q16—3622no—
285SD16—22no—
286SD16—3622no—
287SD16—3622no—
288SD16—3622no—
289D8—3622no—
290D8—3622no—
291M16—3622no—
292M16—3622no—
293C2×C8—1811no—
294C2×C8—1811no—
295C2×C8C22×D18111yescharacteristic
296C2×C8C22×D18111yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—3621no—
299C4⋊C4—1811no—
300C4⋊C4C22×D18111yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—3621no—
304C22⋊C4—3621no—
305C22⋊C4—3621no—
306C22⋊C4—3621no—
307C22⋊C4—3621no—
308C22⋊C4—3621no—
309C22⋊C4—3621no—
310C22⋊C4—3621no—
311C22⋊C4—3621no—
312C42—21no—
313C42—3621no—
314C42—3621no—
315C42—1811no—
316C42—1811no—
317C42—1811no—
318C42C22×D18111yescharacteristic
319C23—21no—
320C23—3621no—
321C23—3621no—
322C23—1811no—
323C23—7242no—
324C23—7242no—
325C23—7242no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—3621no—
330Q8—3621no—
331Q8—7242no—
332Q8D4×D18112yes—
333D4—21no—
334D4—21no—
335D4—3624no—
336D4—3621no—
337D4—3621no—
338D4—3621no—
339D4—1812no—
340D4—7242no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—3624no—
345C2×C4—3621no—
346C2×C4—3621no—
347C2×C4—1811no—
348C2×C4—1811no—
349C2×C4—1811no—
350C2×C4—1812no—
351C2×C4—1812no—
352C2×C4—7241no—
353C2×C4—7241no—
354C2×C4—7242no—
355C2×C4—7242no—
356C2×C4—7242no—
357C2×C4D4×D18111yescharacteristic
358C2×C4D4×D18111yescharacteristic
359C2×C4C23×D18111yescharacteristic
360C8—21no—
361C8—21no—
362C8—3621no—
363C8—3621no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—3621no—
368C22—3621no—
369C22—3622no—
370C22—14484no—
371C22—1812no—
372C22—7241no—
373C22—7241no—
374C22—7241no—
375C22—7241no—
376C22—7241no—
377C22C2×D4×D18111yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—3621no—
384C4—3621no—
385C4—1812no—
386C4—7241no—
387C4—7241no—
388C4C2×D4×D18111yescharacteristic
389C4#41×D18111yescharacteristic
390C2—81no—
391C2—3622no—
392C2—14481no—
393C2—7242no—
394C2#121×D18111yescharacteristic
395C2#2604207611yescharacteristic
396C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#464×C181C2
socleC22×C181C22×D4
prime radicalC2×#19153843C2
hypercenter#464D181
hypocenterC181#3040
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#19153848C2semidirect
3. #124×D181C2semidirect
4. C4×#15668450C2semidirect
5. #26041503C2semidirect
6. #26041501C2semidirect
7. #26041499C2semidirect
8. #26041497C2semidirect
9. #26041495C2semidirect
10. #26041493C2semidirect
11. #26041491C2semidirect
12. #26041301C2semidirect
13. #26041286C2semidirect
14. #26041282C2semidirect
15. #464×C181C2semidirect
21. C2×#15668439C22semidirect
22. C2×#15668438C22semidirect
24. C2×#15668436C22semidirect
34. Q8×Dic181C22semidirect
36. C4×D724C22semidirect
38. #19153778C22semidirect
45. C4⋊#13089678C22semidirect
46. #19153745C22semidirect
47. #19153745C22semidirect
48. #19153741C22semidirect
49. #19153740C22semidirect
50. #19153739C22semidirect
54. C2×Q16×C181C22semidirect
57. C4×Q8×C181C22semidirect
58. C4⋊C8×C181C22semidirect
59. #103×C181C22semidirect
198. C181#3040semidirect
207. #464D181semidirect
221. C2×Q16C2×D181semidirect
237. C4×Q8C2×D181semidirect
251. C4⋊C8C2×D181semidirect
255. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538482
C2×#191538432
#124×D1812
C4×#156684502
#260415032
#260415012
#260414992
#260414972
#260414952
#260414932
#260414912
#260413012
#260412862
#260412822
#464×C1812
#3040181