Finite Group Databasebeta

← #37545201

Subgroups — #37545201

Order 23168.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153840C211yesmaximal, characteristic, complemented
2#26041923C211yesmaximal, characteristic, complemented
3D4×C22⋊D181C211yesmaximal, characteristic, complemented
4#26041667C211yesmaximal, characteristic, complemented
5#26041529C211yesmaximal, characteristic, complemented
6#26041508C211yesmaximal, characteristic, complemented
7#26041492C211yesmaximal, characteristic, complemented
8#26041472C211yesmaximal, characteristic, complemented
9#26041430C211yesmaximal, characteristic, complemented
10#26041421C211yesmaximal, characteristic, complemented
11#26041413C211yesmaximal, characteristic, complemented
12#26041411C211yesmaximal, characteristic, complemented
13#26041391C211yesmaximal, characteristic, complemented
14#26041350C211yesmaximal, characteristic, complemented
15C22×D8×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic, complemented
17C22×C22⋊D181—21no—
18C2×D4×D181C2211yescharacteristic, complemented
19#19153840—24no—
20C2×#15668438C2211yescharacteristic, complemented
21C2×#15668437C2211yescharacteristic, complemented
22C2×#15668437C2211yescharacteristic, complemented
23C2×#15668434C2211yescharacteristic, complemented
24C2×#15668434C2211yescharacteristic, complemented
25C2×#15668432C2211yescharacteristic, complemented
26#19153812C2211yescharacteristic, complemented
27#19153803C2211yescharacteristic, complemented
28#19153797—21no—
29#19153795C2211yescharacteristic, complemented
30#19153790C2211yescharacteristic
31#19153788—21no—
32#19153788—21no—
33#19153787C2211yescharacteristic, complemented
34#19153787C2211yescharacteristic, complemented
35#19153785—21no—
36#19153785—21no—
37#19153784—21no—
38D4×Dic181C2211yescharacteristic, complemented
39D4×Dic181C2211yescharacteristic, complemented
40C4×C22⋊D181C2211yescharacteristic
41C2×C22⋊Dic181—21no—
42#19153748C2211yescharacteristic, complemented
43#19153745C2211yescharacteristic, complemented
44#19153741C2211yescharacteristic, complemented
45#19153738C2211yescharacteristic, complemented
46#19153737C2211yescharacteristic, complemented
47#19153737C2211yescharacteristic, complemented
48#19153737C2211yescharacteristic, complemented
49#19153737C2211yescharacteristic, complemented
50C22⋊#13089678C2211yescharacteristic, complemented
51#19153728C2211yescharacteristic, complemented
52#19153726C2211yescharacteristic, complemented
53C22×D4×C181C2211yescharacteristic
54C22×D4×C181C2211yescharacteristic, complemented
55C2×D8×C181—24no—
56C2×D8×C181C2211yescharacteristic
57C2×D8×C181C2211yescharacteristic
58C2×D8×C181C2211yescharacteristic, complemented
59C2×D8×C181C2211yescharacteristic, complemented
60C22×C8×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668448—22no—
63#15668448—41no—
64C2×Dic362C2311yescharacteristic
65C2×D724C2311yescharacteristic
66C2×C22⋊D181—21no—
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—44no—
74C2×C22⋊D181D411yescharacteristic
75D4×D181—22no—
76D4×D181—41no—
77C22×Dic181—21no—
78C22×Dic181—21no—
79C22×Dic181—21no—
80C2×C4×D181C2311yescharacteristic
81#15668438—22no—
82#15668437—22no—
83#15668437—41no—
84#15668434—22no—
85#15668434—41no—
86#15668432—22no—
87C2×#13089678C2311yescharacteristic
88C2×#13089678C2311yescharacteristic
89C4⋊Dic181C2311yescharacteristic
90C4⋊Dic181C2311yescharacteristic
91#15668428—21no—
92#15668428D411yescharacteristic
93C22⋊Dic181—21no—
94C22⋊Dic181—21no—
95C22⋊Dic181—21no—
96C22⋊Dic181—21no—
97C22⋊Dic181—21no—
98C22⋊Dic181—21no—
99C22⋊Dic181—21no—
100C22⋊Dic181D411yescharacteristic
101#15668426—21no—
102#15668426D411yescharacteristic
103C4×Dic181C2311yescharacteristic
104C24×C181—21no—
105C24×C181—21no—
106C2×D4×C181—22no—
107C2×D4×C181—22no—
108C2×D4×C181—21no—
109C2×D4×C181—21no—
110C2×D4×C181—41no—
111C2×D4×C181—41no—
112C2×D4×C181C2311yescharacteristic
113C2×D4×C181C2311yescharacteristic
114C2×D4×C181C2311yescharacteristic
115C2×D4×C181C2311yescharacteristic
116C22×C4×C181C2311yescharacteristic
117D8×C181—22no—
118D8×C181—22no—
119D8×C181—22no—
120D8×C181D414yescomplemented
121C2×C8×C181—22no—
122C2×C8×C181C2311yescharacteristic
123C2×C8×C181C2311yescharacteristic
124C22×D181—21no—
125C22×D181—42no—
126C22×D181—42no—
127C22×D181—81no—
128C22×D181C2×D411yescharacteristic
129Dic362—21no—
130Dic362—21no—
131D724—21no—
132D724—21no—
133C22⋊D181—22no—
134C22⋊D181—42no—
135C22⋊D181—42no—
136C22⋊D181—42no—
137C22⋊D181—42no—
138C22⋊D181—42no—
139C22⋊D181—81no—
140C22⋊D181—81no—
141C2×Dic181—21no—
142C2×Dic181—21no—
143C2×Dic181—21no—
144C2×Dic181—42no—
145C2×Dic181—42no—
146C2×Dic181—81no—
147C2×Dic181C2×D411yescharacteristic
148C2×Dic181C2×D411yescharacteristic
149C2×Dic181C2×D411yescharacteristic
150C4×D181—21no—
151C4×D181—21no—
152#13089678—21no—
153#13089678—21no—
154C23×C181—21no—
155C23×C181—21no—
156C23×C181—21no—
157C23×C181—21no—
158C23×C181—42no—
159C23×C181—42no—
160C23×C181—42no—
161C23×C181—42no—
162C23×C181—81no—
163C23×C181—81no—
164C23×C181C2×D411yescharacteristic
165D4×C181—21no—
166D4×C181—21no—
167D4×C181—21no—
168D4×C181—21no—
169D4×C181—21no—
170D4×C181—21no—
171D4×C181—42no—
172D4×C181—42no—
173D4×C181C2×D412yes—
174D4×C181C2×D412yescomplemented
175C2×C4×C181—21no—
176C2×C4×C181—21no—
177C2×C4×C181C2×D411yescharacteristic
178C2×C4×C181C2411yescharacteristic
179C8×C181—21no—
180C8×C181C2×D412yescomplemented
181C22×C181—21no—
182C22×C181—21no—
183C22×C181—22no—
184C22×C181—22no—
185C22×C181—41no—
186C22×C181—41no—
187C22×C181—41no—
188C22×C181—41no—
189C22×C181—41no—
190C22×C181—41no—
191C22×C181—41no—
192C22×C181—41no—
193C22×C181—41no—
194C22×C181—41no—
195C22×C181—84no—
196C22×C181—84no—
197C22×C181AGL(1,ℤ/8)12yescomplemented
198C22×C181C22×D411yescharacteristic
199C2×D181—21no—
200C2×D181—21no—
201C2×D181—21no—
202C2×D181—41no—
203C2×D181—41no—
204C2×D181—41no—
205C2×D181—84no—
206C4×C181—21no—
207C4×C181C22×D411yescharacteristic
208C4×C181H(5,𝔽2)11yescharacteristic
209Dic181—21no—
210Dic181—21no—
211Dic181—21no—
212Dic181—41no—
213Dic181—41no—
214Dic181—41no—
215C2×C181—22no—
216C2×C181—42no—
217C2×C181—42no—
218C2×C181—81no—
219C2×C181—81no—
220C2×C181D4211yescharacteristic
221C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C181#57611yescharacteristic
223D181—41no—
224D181—81no—
225C181#326411yescharacteristic, complemented, Sylow
226#3264—1811nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1811no—
228C22×D8D18111yescharacteristic, complemented
229#546—1811no—
230D42—1811no—
231#496—1811no—
232#469—1811no—
233#465—1811no—
234#463—1811no—
235#459—1811no—
236#450—1811no—
237#449—1811no—
238#447—1811no—
239#447—1811no—
240#442—1811no—
241#435—1811no—
242C2×#41—1811no—
243C22×D4—3621no—
244C22×D4—1811no—
245C22×D4C2×D18111yescharacteristic
246C22×D4C2×D18111yescharacteristic, complemented
247AGL(1,ℤ/8)—3624no—
248C2×SD16—1811no—
249C2×SD16—1811no—
250C2×SD16—1811no—
251C2×D8—24no—
252C2×D8—1811no—
253C2×D8—1811no—
254C2×D8C2×D18111yescharacteristic
255C2×D8C2×D18111yescharacteristic
256C2×D8C2×D18111yescharacteristic, complemented
257C2×D8C2×D18111yescharacteristic, complemented
258C2×M16—1811no—
259C22×C8C2×D18111yescharacteristic, complemented
260#127—1811no—
261#124—1811no—
262#123—3621no—
263C22⋊Q8—1811no—
264#121—3621no—
265#121—3621no—
266#121—1811no—
267#121—1811no—
268#121—1811no—
269#120—3621no—
270#120—3621no—
271#120—3621no—
272C4×D4—1811no—
273C4×D4—1811no—
274C4×D4—1811no—
275C2×C22⋊C4—3621no—
276#106—1811no—
277C4⋊C8—1811no—
278#103—1811no—
279#102—1811no—
280#102—1811no—
281#102—1811no—
282#102—1811no—
283#102—1811no—
284C22⋊C8—1811no—
285C22⋊C8—1811no—
286#97—1811no—
287C24—21no—
288C24—21no—
289C24—3621no—
290#41—3622no—
291#41—7241no—
292C2×Q8—1811no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—3622no—
300C2×D4—3621no—
301C2×D4—3621no—
302C2×D4—3621no—
303C2×D4—3621no—
304C2×D4—3621no—
305C2×D4—3621no—
306C2×D4—3621no—
307C2×D4—1811no—
308C2×D4—1811no—
309C2×D4—7241no—
310C2×D4—7244no—
311C2×D4C22×D18111yescharacteristic
312C2×D4C22×D18111yescharacteristic
313C2×D4C22×D18111yescharacteristic
314C2×D4C22×D18111yescharacteristic
315C22×C4—3621no—
316C22×C4—3621no—
317C22×C4—3621no—
318C22×C4—1811no—
319C22×C4C22×D18111yescharacteristic
320SD16—3622no—
321SD16—3622no—
322SD16—7241no—
323D8—22no—
324D8—22no—
325D8—22no—
326D8—3622no—
327D8—7241no—
328D8C22⋊D18114yescomplemented
329M16—3622no—
330C2×C8—22no—
331C2×C8—1811no—
332C2×C8—1811no—
333C2×C8C22×D18111yescharacteristic
334C2×C8C22×D18111yescharacteristic
335C4⋊C4—3621no—
336C4⋊C4—1811no—
337C4⋊C4—1811no—
338C4⋊C4—1811no—
339C22⋊C4—3621no—
340C22⋊C4—3621no—
341C22⋊C4—3621no—
342C22⋊C4—3621no—
343C22⋊C4—3621no—
344C22⋊C4—3621no—
345C22⋊C4—3621no—
346C22⋊C4—3621no—
347C22⋊C4—1811no—
348C22⋊C4—1811no—
349C42—1811no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—3621no—
361C23—14481no—
362C23—1811no—
363C23—7242no—
364C23—7242no—
365C23D4×D18111yescharacteristic
366Q8—3621no—
367Q8—3621no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—3622no—
377D4—3621no—
378D4—3621no—
379D4—14481no—
380D4—14481no—
381D4—7242no—
382D4—7242no—
383D4—7242no—
384D4—7242no—
385D4—7242no—
386D4C2×C22⋊D18112yes—
387D4C2×C22⋊D18112yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—3621no—
391C2×C4—3621no—
392C2×C4—3621no—
393C2×C4—3621no—
394C2×C4—3621no—
395C2×C4—14481no—
396C2×C4—1811no—
397C2×C4—1811no—
398C2×C4—1811no—
399C2×C4—7242no—
400C2×C4—7242no—
401C2×C4D4×D18111yescharacteristic
402C2×C4C23×D18111yescharacteristic
403C8—21no—
404C8—3621no—
405C8—3621no—
406C8C2×C22⋊D18112yescomplemented
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—3621no—
424C22—3621no—
425C22—3621no—
426C22—14484no—
427C22—7241no—
428C22—7241no—
429C22—7241no—
430C22#1915384012yescomplemented
431C22C2×D4×D18111yescharacteristic
432C4—21no—
433C4—3621no—
434C4—3621no—
435C4—3621no—
436C4—7241no—
437C4—7241no—
438C4—7241no—
439C4C22×C22⋊D18111yescharacteristic
440C4#1915386111yescharacteristic
441C2—22no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—14481no—
447C2—7241no—
448C2D4×C22⋊D18111yescharacteristic
449C2C2×#1915384011yescharacteristic
450C2#2604208011yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC22×D8×C181C2
socleC22×C181C22×D4
hypercenterC22×D8D181
hypocenterC181#3264
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19153840C2semidirect
2. #26041923C2semidirect
3. D4×C22⋊D181C2semidirect
4. #26041667C2semidirect
5. #26041529C2semidirect
6. #26041508C2semidirect
7. #26041492C2semidirect
8. #26041472C2semidirect
9. #26041430C2semidirect
10. #26041421C2semidirect
11. #26041413C2semidirect
12. #26041411C2semidirect
13. #26041391C2semidirect
14. #26041350C2semidirect
15. C22×D8×C181C2semidirect
16. C2×#15668448C22semidirect
18. C2×D4×D181C22semidirect
20. C2×#15668438C22semidirect
21. C2×#15668437C22semidirect
22. C2×#15668437C22semidirect
23. C2×#15668434C22semidirect
24. C2×#15668434C22semidirect
25. C2×#15668432C22semidirect
26. #19153812C22semidirect
27. #19153803C22semidirect
29. #19153795C22semidirect
33. #19153787C22semidirect
34. #19153787C22semidirect
38. D4×Dic181C22semidirect
39. D4×Dic181C22semidirect
42. #19153748C22semidirect
43. #19153745C22semidirect
44. #19153741C22semidirect
45. #19153738C22semidirect
46. #19153737C22semidirect
47. #19153737C22semidirect
48. #19153737C22semidirect
49. #19153737C22semidirect
50. C22⋊#13089678C22semidirect
51. #19153728C22semidirect
52. #19153726C22semidirect
54. C22×D4×C181C22semidirect
58. C2×D8×C181C22semidirect
59. C2×D8×C181C22semidirect
60. C22×C8×C181C22semidirect
120. D8×C181D4semidirect
174. D4×C181C2×D4semidirect
180. C8×C181C2×D4semidirect
197. C22×C181AGL(1,ℤ/8)semidirect
225. C181#3264semidirect
228. C22×D8D181semidirect
246. C22×D4C2×D181semidirect
256. C2×D8C2×D181semidirect
257. C2×D8C2×D181semidirect
259. C22×C8C2×D181semidirect
328. D8C22⋊D181semidirect
387. D4C2×C22⋊D181semidirect
406. C8C2×C22⋊D181semidirect
430. C22#19153840semidirect

Maximal subgroups

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

subgroupindex
C2×#191538402
#260419232
D4×C22⋊D1812
#260416672
#260415292
#260415082
#260414922
#260414722
#260414302
#260414212
#260414132
#260414112
#260413912
#260413502
C22×D8×C1812
#3264181