Finite Group Databasebeta

← #37543210

Subgroups — #37543210

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×#19153845C211yesmaximal, characteristic, complemented
2C2×#19153840C211yesmaximal, characteristic
3#124×D181C211yesmaximal, characteristic, complemented
4C4×#15668448C211yesmaximal, characteristic, complemented
5#26041504C211yesmaximal, characteristic, complemented
6#26041502C211yesmaximal, characteristic, complemented
7#26041500C211yesmaximal, characteristic, complemented
8#26041498C211yesmaximal, characteristic, complemented
9#26041496C211yesmaximal, characteristic, complemented
10#26041494C211yesmaximal, characteristic, complemented
11#26041492C211yesmaximal, characteristic, complemented
12#26041301C211yesmaximal, characteristic, complemented
13#26041286C211yesmaximal, characteristic, complemented
14#26041282C211yesmaximal, characteristic, complemented
15#463×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic
19#19153845—24no—
20#19153840—24no—
21C2×Dic724C2211yescharacteristic, complemented
22C2×#15668439C2211yescharacteristic, complemented
23C2×#15668438C2211yescharacteristic
24C2×#15668437C2211yescharacteristic
25C2×#15668437C2211yescharacteristic, complemented
26C2×#15668434C2211yescharacteristic
27C2×#15668432C2211yescharacteristic
28C2×#15668432C2211yescharacteristic
29#19153805C2211yescharacteristic
30#19153804C2211yescharacteristic
31#19153803C2211yescharacteristic
32#19153802—21no—
33#19153802—21no—
34C4×Dic362C2211yescharacteristic, complemented
35#19153781C2211yescharacteristic, complemented
36D4×Dic181C2211yescharacteristic, complemented
37C4×C22⋊D181—21no—
38#19153776—21no—
39#19153774C2211yescharacteristic
40#19153772—21no—
41C22⋊C4×D181—21no—
42C22⋊C4×D181—21no—
43C2×C4×Dic181—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×SD16×C181C2211yescharacteristic, complemented
55C2×D8×C181C2211yescharacteristic
56#124×C181C2211yescharacteristic
57C4×D4×C181C2211yescharacteristic, complemented
58C4⋊C8×C181C2211yescharacteristic, complemented
59#103×C181C2211yescharacteristic, complemented
60#102×C181C2211yescharacteristic
61C23×D181—21no—
62#15668448—22no—
63#15668448D414yes—
64C2×Dic362—21no—
65C2×Dic362C2311yescharacteristic
66C2×Dic362C2311yescharacteristic
67Q8×D181—22no—
68Q8×D181—22no—
69C2×D724C2311yescharacteristic
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72D4×D181—22no—
73D4×D181—22no—
74C22×Dic181—21no—
75C22×Dic181—21no—
76C2×C4×D181—21no—
77C2×C4×D181D411yescharacteristic
78C2×C4×D181D411yescharacteristic
79C2×C4×D181C2311yescharacteristic
80Dic724—22no—
81#15668439—22no—
82#15668438—22no—
83#15668437—22no—
84#15668437—22no—
85#15668434—22no—
86#15668432—22no—
87#15668432—22no—
88C2×#13089678C2311yescharacteristic
89C2×#13089678C2311yescharacteristic
90C4⋊Dic181C2311yescharacteristic
91#15668428—21no—
92#15668428—21no—
93C22⋊Dic181—21no—
94C22⋊Dic181—21no—
95C22⋊Dic181—21no—
96#15668426—21no—
97#15668426—21no—
98#15668426—21no—
99#15668426—21no—
100#15668426—21no—
101C4×Dic181—21no—
102C4×Dic181—21no—
103C4×Dic181—21no—
104C4×Dic181D411yescharacteristic
105C4×Dic181D411yescharacteristic
106C4×Dic181C2311yescharacteristic
107C2×Q8×C181C2311yescharacteristic
108C2×D4×C181C2311yescharacteristic
109C2×D4×C181C2311yescharacteristic
110C22×C4×C181—21no—
111SD16×C181—22no—
112D8×C181—22no—
113C2×C8×C181C2311yescharacteristic
114C2×C8×C181C2311yescharacteristic
115C4⋊C4×C181C2311yescharacteristic
116C22⋊C4×C181—21no—
117C22⋊C4×C181—21no—
118C22⋊C4×C181—21no—
119C42×C181C2311yescharacteristic
120C22×D181—21no—
121C22×D181—42no—
122C22×D181—42no—
123C22×D181—42no—
124C22×D181C2×D411yescharacteristic
125Dic362—21no—
126Dic362—21no—
127Dic362—21no—
128Dic362—42no—
129Dic362C2×D412yes—
130D724—21no—
131D724—21no—
132C22⋊D181—24no—
133C22⋊D181—42no—
134C2×Dic181—21no—
135C2×Dic181—24no—
136C2×Dic181—21no—
137C2×Dic181—21no—
138C2×Dic181—42no—
139C2×Dic181C2×D411yescharacteristic
140C2×Dic181C2×D411yescharacteristic
141C2×Dic181C2×D411yescharacteristic
142C4×D181—41no—
143C4×D181—41no—
144C4×D181—42no—
145C4×D181C2×D412yes—
146C4×D181#4112yes—
147#13089678—21no—
148#13089678—21no—
149C23×C181—21no—
150C23×C181—21no—
151Q8×C181—21no—
152Q8×C181—21no—
153D4×C181—21no—
154D4×C181—21no—
155D4×C181—21no—
156D4×C181C2×D412yes—
157C2×C4×C181—21no—
158C2×C4×C181—21no—
159C2×C4×C181—42no—
160C2×C4×C181C2×D411yescharacteristic
161C2×C4×C181C2×D411yescharacteristic
162C2×C4×C181C2411yescharacteristic
163C8×C181—21no—
164C8×C181—21no—
165C22×C181—22no—
166C22×C181—41no—
167C22×C181—41no—
168C22×C181—41no—
169C22×C181—41no—
170C22×C181—41no—
171C22×C181C22×D411yescharacteristic
172C2×D181—21no—
173C2×D181—21no—
174C2×D181—41no—
175C2×D181—41no—
176C2×D181—41no—
177C2×D181—84no—
178C2×D181#12112yes—
179C4×C181—21no—
180C4×C181—21no—
181C4×C181—41no—
182C4×C181—41no—
183C4×C181C22×D411yescharacteristic
184C4×C181C2×#4111yescharacteristic
185Dic181—22no—
186Dic181—21no—
187Dic181—21no—
188Dic181—41no—
189Dic181—41no—
190Dic181#12112yes—
191C2×C181—42no—
192C2×C181—81no—
193C2×C181C2×#12111yescharacteristic
194C2×C181#57511yescharacteristic
195C2×C181#57511yescharacteristic
196D181—22no—
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#464—1811no—
208#463—1811no—
209#463—1811no—
210#463—1811no—
211#463D18111yescharacteristic, complemented
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×Q16—1811no—
222C2×SD16—1811no—
223C2×SD16—1811no—
224C2×SD16—1811no—
225C2×SD16C2×D18111yescharacteristic, complemented
226C2×D8—1811no—
227C2×D8C2×D18111yescharacteristic
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×Q8—1811no—
238C4×D4—3621no—
239C4×D4—3621no—
240C4×D4—1811no—
241C4×D4C2×D18111yescharacteristic, complemented
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×Q8—1811no—
268C2×Q8C22×D18111yescharacteristic
269C2×D4—3621no—
270C2×D4—3622no—
271C2×D4—3622no—
272C2×D4—3621no—
273C2×D4—1811no—
274C2×D4C22×D18111yescharacteristic
275C2×D4C22×D18111yescharacteristic
276C22×C4—21no—
277C22×C4—3621no—
278C22×C4—3621no—
279C22×C4—3621no—
280C22×C4—1811no—
281C22×C4—1811no—
282C22×C4—1811no—
283Q16—3622no—
284Q16—3622no—
285SD16—22no—
286SD16—3622no—
287SD16—3622no—
288SD16—3622no—
289D8—22no—
290D8—3622no—
291M16—3622no—
292M16—3622no—
293C2×C8—1811no—
294C2×C8—1811no—
295C2×C8C22×D18111yescharacteristic
296C2×C8C22×D18111yescharacteristic
297C4⋊C4—3621no—
298C4⋊C4—3621no—
299C4⋊C4—1811no—
300C4⋊C4C22×D18111yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
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—3621no—
313C42—3621no—
314C42—3621no—
315C42—1811no—
316C42—1811no—
317C42—1811no—
318C42C22×D18111yescharacteristic
319C23—21no—
320C23—21no—
321C23—3621no—
322C23—1811no—
323C23—7242no—
324C23—7242no—
325C23—7242no—
326Q8—21no—
327Q8—21no—
328Q8—3621no—
329Q8—3621no—
330Q8—3621no—
331Q8—1812no—
332Q8—7242no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—3624no—
337D4—3621no—
338D4—3621no—
339D4—7242no—
340D4D4×D18112yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—3621no—
345C2×C4—3624no—
346C2×C4—3621no—
347C2×C4—3621no—
348C2×C4—1811no—
349C2×C4—1811no—
350C2×C4—1811no—
351C2×C4—1812no—
352C2×C4—1812no—
353C2×C4—7241no—
354C2×C4—7241no—
355C2×C4—7242no—
356C2×C4—7242no—
357C2×C4D4×D18111yescharacteristic
358C2×C4D4×D18111yescharacteristic
359C2×C4C23×D18111yescharacteristic
360C8—21no—
361C8—21no—
362C8—3621no—
363C8—3621no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—3621no—
371C22—3621no—
372C22—14484no—
373C22—1812no—
374C22—7241no—
375C22—7241no—
376C22—7241no—
377C22C2×D4×D18111yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—3622no—
383C4—3621no—
384C4—3621no—
385C4—1812no—
386C4—7241no—
387C4—7241no—
388C4C2×D4×D18111yescharacteristic
389C4#41×D18111yescharacteristic
390C2—42no—
391C2—81no—
392C2—3622no—
393C2—14481no—
394C2#121×D18111yescharacteristic
395C2#2604207511yescharacteristic
396C2#2604207611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#463×C181C2
socleC22×C181C22×D4
prime radicalC2×#19153840C2
hypercenter#463D181
hypocenterC181#3040
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19153845C2semidirect
3. #124×D181C2semidirect
4. C4×#15668448C2semidirect
5. #26041504C2semidirect
6. #26041502C2semidirect
7. #26041500C2semidirect
8. #26041498C2semidirect
9. #26041496C2semidirect
10. #26041494C2semidirect
11. #26041492C2semidirect
12. #26041301C2semidirect
13. #26041286C2semidirect
14. #26041282C2semidirect
15. #463×C181C2semidirect
21. C2×Dic724C22semidirect
22. C2×#15668439C22semidirect
25. C2×#15668437C22semidirect
34. C4×Dic362C22semidirect
35. #19153781C22semidirect
36. D4×Dic181C22semidirect
45. C4⋊#13089678C22semidirect
46. #19153745C22semidirect
47. #19153745C22semidirect
48. #19153741C22semidirect
49. #19153740C22semidirect
50. #19153739C22semidirect
54. C2×SD16×C181C22semidirect
57. C4×D4×C181C22semidirect
58. C4⋊C8×C181C22semidirect
59. #103×C181C22semidirect
198. C181#3040semidirect
211. #463D181semidirect
225. C2×SD16C2×D181semidirect
241. C4×D4C2×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×#191538452
C2×#191538402
#124×D1812
C4×#156684482
#260415042
#260415022
#260415002
#260414982
#260414962
#260414942
#260414922
#260413012
#260412862
#260412822
#463×C1812
#3040181