Finite Group Databasebeta

← #37545202

Subgroups — #37545202

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×#19153841C211yesmaximal, characteristic, complemented
2C22×#15668434C211yesmaximal, characteristic, complemented
3D4×D724C211yesmaximal, characteristic, complemented
4#26041669C211yesmaximal, characteristic, complemented
5#26041530C211yesmaximal, characteristic, complemented
6#26041510C211yesmaximal, characteristic, complemented
7#26041491C211yesmaximal, characteristic, complemented
8#26041474C211yesmaximal, characteristic, complemented
9#26041429C211yesmaximal, characteristic, complemented
10#26041424C211yesmaximal, characteristic, complemented
11#26041413C211yesmaximal, characteristic, complemented
12#26041412C211yesmaximal, characteristic, complemented
13#26041389C211yesmaximal, characteristic, complemented
14#26041351C211yesmaximal, characteristic, complemented
15#546×C181C211yesmaximal, characteristic, complemented
16C22×D724C2211yescharacteristic
17C22×D724C2211yescharacteristic, complemented
18C2×D4×D181—21no—
19#19153841—24no—
20C2×#15668436C2211yescharacteristic, complemented
21C2×#15668436C2211yescharacteristic, complemented
22C2×#15668436C2211yescharacteristic, complemented
23C2×#15668434—24no—
24C2×#15668434C2211yescharacteristic
25C2×#15668434C2211yescharacteristic
26C2×#15668434C2211yescharacteristic, complemented
27C2×#15668434C2211yescharacteristic, complemented
28C2×#15668434C2211yescharacteristic, complemented
29C2×#15668434C2211yescharacteristic, complemented
30C2×#15668433C2211yescharacteristic, complemented
31C22×#13089678C2211yescharacteristic, complemented
32#19153813C2211yescharacteristic, complemented
33#19153791—21no—
34#19153790C2211yescharacteristic
35#19153790C2211yescharacteristic, complemented
36#19153786—21no—
37#19153786—21no—
38C4×D724C2211yescharacteristic, complemented
39#19153747C2211yescharacteristic, complemented
40C4⋊#13089678C2211yescharacteristic, complemented
41#19153740C2211yescharacteristic, complemented
42#19153737C2211yescharacteristic, complemented
43#19153737C2211yescharacteristic, complemented
44#19153736C2211yescharacteristic, complemented
45#19153736C2211yescharacteristic, complemented
46#19153736C2211yescharacteristic, complemented
47C22⋊#13089678C2211yescharacteristic, complemented
48C22⋊#13089678C2211yescharacteristic, complemented
49#19153727C2211yescharacteristic, complemented
50C2×#41×C181C2211yescharacteristic, complemented
51C22×D4×C181C2211yescharacteristic, complemented
52#124×C181C2211yescharacteristic, complemented
53#123×C181—21no—
54C22⋊Q8×C181C2211yescharacteristic, complemented
55#121×C181—21no—
56#121×C181C2211yescharacteristic, complemented
57#120×C181—21no—
58C4×D4×C181C2211yescharacteristic
59C4×D4×C181C2211yescharacteristic, complemented
60C2×C22⋊C4×C181—21no—
61C23×D181—21no—
62C23×D181—21no—
63C2×D724—22no—
64C2×D724—22no—
65C2×D724—21no—
66C2×D724—21no—
67C2×D724—21no—
68C2×D724—21no—
69C2×D724—41no—
70C2×D724—41no—
71C2×D724C2311yescharacteristic
72C2×D724C2311yescharacteristic
73C2×D724C2311yescharacteristic
74C2×C22⋊D181—21no—
75C2×C22⋊D181—21no—
76D4×D181—44no—
77C2×C4×D181—21no—
78#15668436—22no—
79#15668436—22no—
80#15668436—41no—
81#15668434—22no—
82#15668434—22no—
83#15668434—22no—
84#15668434—22no—
85#15668434—41no—
86#15668434D414yescomplemented
87#15668433—22no—
88C2×#13089678—22no—
89C2×#13089678C2311yescharacteristic
90C2×#13089678C2311yescharacteristic
91C2×#13089678C2311yescharacteristic
92C2×#13089678C2311yescharacteristic
93C4⋊Dic181C2311yescharacteristic
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97C24×C181—21no—
98#41×C181—22no—
99#41×C181—41no—
100C2×Q8×C181C2311yescharacteristic
101C2×D4×C181—22no—
102C2×D4×C181—21no—
103C2×D4×C181—21no—
104C2×D4×C181—21no—
105C2×D4×C181—41no—
106C2×D4×C181D411yescharacteristic
107C2×D4×C181C2311yescharacteristic
108C2×D4×C181C2311yescharacteristic
109C22×C4×C181—21no—
110C22×C4×C181—21no—
111C22×C4×C181C2311yescharacteristic
112C22×C4×C181C2311yescharacteristic
113C4⋊C4×C181—21no—
114C4⋊C4×C181D411yescharacteristic
115C4⋊C4×C181C2311yescharacteristic
116C22⋊C4×C181—21no—
117C22⋊C4×C181—21no—
118C22⋊C4×C181—21no—
119C22⋊C4×C181—21no—
120C22⋊C4×C181—21no—
121C22⋊C4×C181D411yescharacteristic
122C22⋊C4×C181D411yescharacteristic
123C42×C181C2311yescharacteristic
124C22×D181—21no—
125C22×D181—21no—
126C22×D181—21no—
127C22×D181—42no—
128C22×D181—42no—
129C22×D181—42no—
130C22×D181—42no—
131C22×D181—81no—
132C22×D181—81no—
133D724—21no—
134D724—21no—
135D724—21no—
136D724—21no—
137D724—21no—
138D724—42no—
139D724—42no—
140D724—42no—
141D724—81no—
142D724C2×D412yes—
143C22⋊D181—42no—
144C22⋊D181—42no—
145C2×Dic181—21no—
146C4×D181—42no—
147#13089678—21no—
148#13089678—21no—
149#13089678—21no—
150#13089678C2×D412yescomplemented
151C23×C181—21no—
152C23×C181—21no—
153C23×C181—42no—
154C23×C181—42no—
155C23×C181—81no—
156C23×C181C2×D411yescharacteristic
157C23×C181C2×D411yescharacteristic
158Q8×C181—21no—
159Q8×C181—21no—
160D4×C181—22no—
161D4×C181—21no—
162D4×C181—21no—
163D4×C181—21no—
164D4×C181—42no—
165D4×C181—42no—
166D4×C181—81no—
167D4×C181C2×D412yescomplemented
168C2×C4×C181—21no—
169C2×C4×C181—21no—
170C2×C4×C181—21no—
171C2×C4×C181—21no—
172C2×C4×C181—21no—
173C2×C4×C181—21no—
174C2×C4×C181—42no—
175C2×C4×C181—81no—
176C2×C4×C181C2×D411yescharacteristic
177C2×C4×C181C2×D411yescharacteristic
178C2×C4×C181C2×D411yescharacteristic
179C2×C4×C181C2×D411yescharacteristic
180C2×C4×C181C2411yescharacteristic
181C22×C181—21no—
182C22×C181—21no—
183C22×C181—21no—
184C22×C181—22no—
185C22×C181—21no—
186C22×C181—21no—
187C22×C181—41no—
188C22×C181—41no—
189C22×C181—41no—
190C22×C181—41no—
191C22×C181—41no—
192C22×C181—84no—
193C22×C181AGL(1,ℤ/8)12yescomplemented
194C22×C181C22×D411yescharacteristic
195C2×D181—22no—
196C2×D181—41no—
197C2×D181—41no—
198C2×D181—41no—
199C2×D181—41no—
200C2×D181—41no—
201C2×D181—41no—
202C2×D181—41no—
203C2×D181—41no—
204C2×D181—84no—
205C2×D181—84no—
206C4×C181—21no—
207C4×C181—21no—
208C4×C181—21no—
209C4×C181—21no—
210C4×C181—41no—
211C4×C181—41no—
212C4×C181C22×D411yescharacteristic
213C4×C181H(5,𝔽2)11yescharacteristic
214Dic181—41no—
215C2×C181—22no—
216C2×C181—41no—
217C2×C181—42no—
218C2×C181—81no—
219C2×C181D4211yescharacteristic
220C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C181#57611yescharacteristic
222D181—42no—
223D181—81no—
224D181—81no—
225C181#326411yescharacteristic, complemented, Sylow
226#3264—1811nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1811no—
228C22×D8—1811no—
229#546D18111yescharacteristic, complemented
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×#41C2×D18111yescharacteristic, complemented
243C22×D4—3621no—
244C22×D4—1811no—
245C22×D4—1811no—
246C22×D4C2×D18111yescharacteristic, complemented
247AGL(1,ℤ/8)—3624no—
248C2×SD16—1811no—
249C2×SD16—1811no—
250C2×SD16—1811no—
251C2×D8—3624no—
252C2×D8—1811no—
253C2×D8—1811no—
254C2×D8—1811no—
255C2×D8—1811no—
256C2×D8—1811no—
257C2×D8—1811no—
258C2×M16—1811no—
259C22×C8—1811no—
260#127—1811no—
261#124C2×D18111yescharacteristic, complemented
262#123—21no—
263C22⋊Q8C2×D18111yescharacteristic, complemented
264#121—21no—
265#121—3621no—
266#121—1811no—
267#121—1811no—
268#121C2×D18111yescharacteristic, complemented
269#120—21no—
270#120—3621no—
271#120—3621no—
272C4×D4—1811no—
273C4×D4C2×D18111yescharacteristic
274C4×D4C2×D18111yescharacteristic, complemented
275C2×C22⋊C4—21no—
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—3621no—
289C24—3621no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D18111yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—3622no—
299C2×D4—3622no—
300C2×D4—3621no—
301C2×D4—3621no—
302C2×D4—3621no—
303C2×D4—3621no—
304C2×D4—3621no—
305C2×D4—3621no—
306C2×D4—1811no—
307C2×D4—1811no—
308C2×D4—1811no—
309C2×D4—7241no—
310C2×D4—7241no—
311C2×D4—7244no—
312C2×D4C22⋊D18111yescharacteristic
313C2×D4C22×D18111yescharacteristic
314C2×D4C22×D18111yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—3621no—
318C22×C4C22×D18111yescharacteristic
319C22×C4C22×D18111yescharacteristic
320SD16—3622no—
321SD16—3622no—
322SD16—7241no—
323D8—3622no—
324D8—3622no—
325D8—3622no—
326D8—3622no—
327D8—1814no—
328D8—7241no—
329M16—3622no—
330C2×C8—3622no—
331C2×C8—1811no—
332C2×C8—1811no—
333C2×C8—1811no—
334C2×C8—1811no—
335C4⋊C4—21no—
336C4⋊C4—1811no—
337C4⋊C4C22⋊D18111yescharacteristic
338C4⋊C4C22×D18111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—3621no—
345C22⋊C4—3621no—
346C22⋊C4—3621no—
347C22⋊C4C22⋊D18111yescharacteristic
348C22⋊C4C22⋊D18111yescharacteristic
349C42C22×D18111yescharacteristic
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—3621no—
356C23—3621no—
357C23—3621no—
358C23—14481no—
359C23—14481no—
360C23—7242no—
361C23—7242no—
362C23—7242no—
363C23—7242no—
364C23C2×C22⋊D18111yescharacteristic
365C23C2×C22⋊D18111yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—3621no—
376D4—3621no—
377D4—3621no—
378D4—3621no—
379D4—3621no—
380D4—14481no—
381D4—1812no—
382D4—7242no—
383D4—7242no—
384D4—7242no—
385D4—7242no—
386D4—7242no—
387D4D4×D18112yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—3621no—
397C2×C4—7242no—
398C2×C4C2×C22⋊D18111yescharacteristic
399C2×C4C2×C22⋊D18111yescharacteristic
400C2×C4C2×C22⋊D18111yescharacteristic
401C2×C4C2×C22⋊D18111yescharacteristic
402C2×C4C23×D18111yescharacteristic
403C8—3621no—
404C8—3621no—
405C8—3621no—
406C8—1812no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—3622no—
420C22—14484no—
421C22—14484no—
422C22—7241no—
423C22—7241no—
424C22—7241no—
425C22—7241no—
426C22—7241no—
427C22—7241no—
428C22—7241no—
429C22—7241no—
430C22#1915384112yescomplemented
431C22C22×C22⋊D18111yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—7241no—
439C4C2×D4×D18111yescharacteristic
440C4#1915386111yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—14481no—
446C2—14481no—
447C2—7242no—
448C2D4×C22⋊D18111yescharacteristic
449C2C2×#1915384111yescharacteristic
450C2#2604208211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#546×C181C2
socleC22×C181C22×D4
hypercenter#546D181
hypocenterC181#3264
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#19153841C2semidirect
2. C22×#15668434C2semidirect
3. D4×D724C2semidirect
4. #26041669C2semidirect
5. #26041530C2semidirect
6. #26041510C2semidirect
7. #26041491C2semidirect
8. #26041474C2semidirect
9. #26041429C2semidirect
10. #26041424C2semidirect
11. #26041413C2semidirect
12. #26041412C2semidirect
13. #26041389C2semidirect
14. #26041351C2semidirect
15. #546×C181C2semidirect
17. C22×D724C22semidirect
20. C2×#15668436C22semidirect
21. C2×#15668436C22semidirect
22. C2×#15668436C22semidirect
26. C2×#15668434C22semidirect
27. C2×#15668434C22semidirect
28. C2×#15668434C22semidirect
29. C2×#15668434C22semidirect
30. C2×#15668433C22semidirect
31. C22×#13089678C22semidirect
32. #19153813C22semidirect
35. #19153790C22semidirect
38. C4×D724C22semidirect
39. #19153747C22semidirect
40. C4⋊#13089678C22semidirect
41. #19153740C22semidirect
42. #19153737C22semidirect
43. #19153737C22semidirect
44. #19153736C22semidirect
45. #19153736C22semidirect
46. #19153736C22semidirect
47. C22⋊#13089678C22semidirect
48. C22⋊#13089678C22semidirect
49. #19153727C22semidirect
50. C2×#41×C181C22semidirect
51. C22×D4×C181C22semidirect
52. #124×C181C22semidirect
54. C22⋊Q8×C181C22semidirect
56. #121×C181C22semidirect
59. C4×D4×C181C22semidirect
86. #15668434D4semidirect
150. #13089678C2×D4semidirect
167. D4×C181C2×D4semidirect
193. C22×C181AGL(1,ℤ/8)semidirect
225. C181#3264semidirect
229. #546D181semidirect
242. C2×#41C2×D181semidirect
246. C22×D4C2×D181semidirect
261. #124C2×D181semidirect
263. C22⋊Q8C2×D181semidirect
268. #121C2×D181semidirect
274. C4×D4C2×D181semidirect
387. D4D4×D181semidirect
430. C22#19153841semidirect

Maximal subgroups

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

subgroupindex
C2×#191538412
C22×#156684342
D4×D7242
#260416692
#260415302
#260415102
#260414912
#260414742
#260414292
#260414242
#260414132
#260414122
#260413892
#260413512
#546×C1812
#3264181