Finite Group Databasebeta

← #37545338

Subgroups — #37545338

Order 23168.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153844C211yesmaximal, characteristic, complemented
2C2×#19153840C211yesmaximal, characteristic, complemented
3#26041947C211yesmaximal, characteristic, complemented
4#26041913C211yesmaximal, characteristic, complemented
5#26041654C211yesmaximal, characteristic, complemented
6#26041533C211yesmaximal, characteristic, complemented
7#26041478C211yesmaximal, characteristic, complemented
8#26041473C211yesmaximal, characteristic, complemented
9#26041425C211yesmaximal, characteristic, complemented
10#26041422C211yesmaximal, characteristic, complemented
11#26041414C211yesmaximal, characteristic, complemented
12#26041412C211yesmaximal, characteristic, complemented
13#26041366C211yesmaximal, characteristic, complemented
14#26041355C211yesmaximal, characteristic, complemented
15#469×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic, complemented
17C2×#15668448C2211yescharacteristic, complemented
18C22×D724C2211yescharacteristic
19C2×D4×D181—21no—
20C2×D4×D181C2211yescharacteristic
21#19153844—24no—
22#19153840—24no—
23C2×#15668438C2211yescharacteristic
24C2×#15668438C2211yescharacteristic, complemented
25C2×#15668437C2211yescharacteristic, complemented
26C2×D1448C2211yescharacteristic
27C2×D1448C2211yescharacteristic, complemented
28C2×#15668434C2211yescharacteristic
29C2×#15668434C2211yescharacteristic, complemented
30C2×#15668434C2211yescharacteristic, complemented
31C2×#15668432C2211yescharacteristic, complemented
32#19153812—21no—
33#19153812C2211yescharacteristic
34#19153812C2211yescharacteristic, complemented
35#19153791C2211yescharacteristic
36#19153791C2211yescharacteristic, complemented
37#19153789—21no—
38#19153789—21no—
39#19153788—21no—
40#19153786—21no—
41#19153784—21no—
42#19153781C2211yescharacteristic, complemented
43#19153778C2211yescharacteristic
44C4×C22⋊D181C2211yescharacteristic
45#19153776—21no—
46#19153745C2211yescharacteristic, complemented
47#19153739C2211yescharacteristic, complemented
48#19153736C2211yescharacteristic
49#19153736C2211yescharacteristic, complemented
50#19153736C2211yescharacteristic, complemented
51C22⋊#13089678C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic, complemented
53C8×Dic181C2211yescharacteristic, complemented
54C2×D8×C181C2211yescharacteristic, complemented
55C2×M16×C181C2211yescharacteristic, complemented
56#121×C181C2211yescharacteristic
57#121×C181C2211yescharacteristic, complemented
58#106×C181C2211yescharacteristic, complemented
59#102×C181C2211yescharacteristic, complemented
60#102×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62C23×D181—21no—
63#15668449—22no—
64#15668449—41no—
65#15668448—22no—
66#15668448—41no—
67C2×Dic362C2311yescharacteristic
68C2×D724—22no—
69C2×D724—21no—
70C2×D724—21no—
71C2×D724—21no—
72C2×D724—41no—
73C2×D724C2311yescharacteristic
74C2×D724C2311yescharacteristic
75C2×D724C2311yescharacteristic
76C2×C22⋊D181—21no—
77C2×C22⋊D181—21no—
78C2×C22⋊D181—21no—
79C2×C22⋊D181—21no—
80C2×C22⋊D181—21no—
81C2×C22⋊D181—21no—
82C2×C22⋊D181—21no—
83C2×C22⋊D181D411yescharacteristic
84D4×D181—22no—
85D4×D181—41no—
86D4×D181—44no—
87C22×Dic181—21no—
88C2×C4×D181—21no—
89C2×C4×D181—21no—
90C2×C4×D181C2311yescharacteristic
91#15668438—22no—
92#15668438D414yescomplemented
93#15668437—22no—
94D1448—22no—
95D1448—22no—
96#15668434—22no—
97#15668434—41no—
98#15668434—41no—
99#15668432—22no—
100C2×#13089678C2311yescharacteristic
101C2×#13089678C2311yescharacteristic
102#15668428—21no—
103#15668428D411yescharacteristic
104C22⋊Dic181D411yescharacteristic
105#15668426—21no—
106#15668426—21no—
107#15668426—21no—
108#15668426D411yescharacteristic
109C4×Dic181—21no—
110C4×Dic181C2311yescharacteristic
111C2×D4×C181—21no—
112C2×D4×C181—21no—
113C2×D4×C181C2311yescharacteristic
114C2×D4×C181C2311yescharacteristic
115C22×C4×C181C2311yescharacteristic
116D8×C181—22no—
117M16×C181—22no—
118C2×C8×C181C2311yescharacteristic
119C2×C8×C181C2311yescharacteristic
120C4⋊C4×C181C2311yescharacteristic
121C4⋊C4×C181C2311yescharacteristic
122C22⋊C4×C181—21no—
123C22⋊C4×C181—21no—
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—
133C22×D181C2×D411yescharacteristic
134Dic362—21no—
135Dic362C2×D412yescomplemented
136D724—21no—
137D724—21no—
138D724—21no—
139D724—21no—
140D724—21no—
141D724—42no—
142D724—42no—
143D724—81no—
144D724C2×D412yes—
145C22⋊D181—22no—
146C22⋊D181—42no—
147C22⋊D181—42no—
148C22⋊D181—42no—
149C22⋊D181—42no—
150C22⋊D181—42no—
151C22⋊D181—81no—
152C22⋊D181—81no—
153C2×Dic181—21no—
154C2×Dic181—42no—
155C2×Dic181C2×D411yescharacteristic
156C2×Dic181C2×D411yescharacteristic
157C2×Dic181C2×D411yescharacteristic
158C4×D181—21no—
159C4×D181—21no—
160C4×D181—42no—
161C4×D181—42no—
162#13089678—21no—
163#13089678—21no—
164C23×C181—21no—
165C23×C181—21no—
166C23×C181C2×D411yescharacteristic
167D4×C181—21no—
168D4×C181—21no—
169D4×C181—21no—
170D4×C181—21no—
171D4×C181—42no—
172D4×C181—81no—
173C2×C4×C181—21no—
174C2×C4×C181—21no—
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—21no—
184C22×C181—41no—
185C22×C181—41no—
186C22×C181—41no—
187C22×C181—41no—
188C22×C181—41no—
189C22×C181—41no—
190C22×C181C22×D411yescharacteristic
191C2×D181—21no—
192C2×D181—22no—
193C2×D181—21no—
194C2×D181—21no—
195C2×D181—41no—
196C2×D181—41no—
197C2×D181—41no—
198C2×D181—41no—
199C2×D181—41no—
200C2×D181—41no—
201C2×D181—41no—
202C2×D181—41no—
203C2×D181—84no—
204C2×D181—84no—
205C4×C181—21no—
206C4×C181—41no—
207C4×C181—41no—
208C4×C181C22×D411yescharacteristic
209C4×C181H(5,𝔽2)11yescharacteristic
210Dic181—21no—
211Dic181—22no—
212Dic181—21no—
213Dic181AGL(1,ℤ/8)12yescomplemented
214C2×C181—41no—
215C2×C181—81no—
216C2×C181—81no—
217C2×C181D4211yescharacteristic
218C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C181#57611yescharacteristic
220D181—41no—
221D181—42no—
222D181—81no—
223D181—81no—
224C181#327411yescharacteristic, complemented, Sylow
225#3274—1811nomaximal, Sylow
226C2×AGL(1,ℤ/8)—1811no—
227C2×AGL(1,ℤ/8)—1811no—
228#550—1811no—
229D42—1811no—
230#493—1811no—
231#469—1811no—
232#469D18111yescharacteristic, complemented
233#460—1811no—
234#459—1811no—
235#449—1811no—
236#449—1811no—
237#447—1811no—
238#447—1811no—
239C4×SD16—1811no—
240#436—1811no—
241C2×#41—1811no—
242C2×#41—1811no—
243C22×D4—3621no—
244C22×D4—1811no—
245C22×D4—1811no—
246AGL(1,ℤ/8)—3624no—
247AGL(1,ℤ/8)—3624no—
248C2×SD16—1811no—
249C2×SD16—1811no—
250C2×SD16—1811no—
251C2×D8—1811no—
252C2×D8—1811no—
253C2×D8—1811no—
254C2×D8—1811no—
255C2×D8—1811no—
256C2×D8C2×D18111yescharacteristic, complemented
257C2×M16—1811no—
258C2×M16C2×D18111yescharacteristic, complemented
259#127—3621no—
260#127—1811no—
261#127—1811no—
262#121—3621no—
263#121—3621no—
264#121—3621no—
265#121—1811no—
266#121—1811no—
267#121C2×D18111yescharacteristic
268#121C2×D18111yescharacteristic, complemented
269#120—3621no—
270#120—3621no—
271C4×Q8—1811no—
272C4×D4—3621no—
273C4×D4—1811no—
274C4×D4—1811no—
275#106C2×D18111yescharacteristic, complemented
276C4⋊C8—1811no—
277#103—1811no—
278#102—1811no—
279#102—1811no—
280#102—1811no—
281#102C2×D18111yescharacteristic, complemented
282#102C2×D18111yescharacteristic, complemented
283C22⋊C8—1811no—
284C22⋊C8—1811no—
285C4×C8—1811no—
286C24—3621no—
287C24—3621no—
288#41—3622no—
289#41—3622no—
290#41—7241no—
291#41—7241no—
292C2×Q8—1811no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—3622no—
296C2×D4—3621no—
297C2×D4—3621no—
298C2×D4—3622no—
299C2×D4—3621no—
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—1811no—
310C2×D4—1811no—
311C2×D4—7241no—
312C2×D4—7241no—
313C2×D4—7244no—
314C2×D4C22×D18111yescharacteristic
315C2×D4C22×D18111yescharacteristic
316C22×C4—3621no—
317C22×C4—3621no—
318C22×C4—3621no—
319C22×C4—1811no—
320C22×C4C22×D18111yescharacteristic
321SD16—3622no—
322SD16—3622no—
323SD16—1814no—
324D8—22no—
325D8—3622no—
326D8—3622no—
327D8—3622no—
328D8—7241no—
329D8—7241no—
330M16—22no—
331M16—3622no—
332C2×C8—1811no—
333C2×C8—1811no—
334C2×C8C22×D18111yescharacteristic
335C2×C8C22×D18111yescharacteristic
336C4⋊C4—3621no—
337C4⋊C4—1811no—
338C4⋊C4C22×D18111yescharacteristic
339C4⋊C4C22×D18111yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—3621no—
343C22⋊C4—3621no—
344C22⋊C4—3621no—
345C22⋊C4—1811no—
346C22⋊C4—1811no—
347C42—3621no—
348C42—1811no—
349C23—21no—
350C23—21no—
351C23—3621no—
352C23—3621no—
353C23—3621no—
354C23—14481no—
355C23—14481no—
356C23—1811no—
357C23—7242no—
358C23—7242no—
359C23—7242no—
360C23—7242no—
361C23D4×D18111yescharacteristic
362Q8—3621no—
363Q8—1812no—
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—81no—
370D4—3622no—
371D4—3621no—
372D4—3621no—
373D4—3621no—
374D4—3621no—
375D4—3621no—
376D4—14481no—
377D4—14481no—
378D4—14481no—
379D4—1812no—
380D4—7242no—
381D4—7242no—
382D4—7242no—
383D4—7242no—
384D4—7242no—
385D4—7242no—
386D4—7242no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—3621no—
392C2×C4—3621no—
393C2×C4—3621no—
394C2×C4—1811no—
395C2×C4—1811no—
396C2×C4—1811no—
397C2×C4—7242no—
398C2×C4—7242no—
399C2×C4—7242no—
400C2×C4D4×D18111yescharacteristic
401C2×C4C23×D18111yescharacteristic
402C8—21no—
403C8—3621no—
404C8—3621no—
405C8D4×D18112yescomplemented
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—3621no—
416C22—3622no—
417C22—3621no—
418C22—3621no—
419C22—14484no—
420C22—14484no—
421C22—7241no—
422C22—7241no—
423C22—7241no—
424C22—7241no—
425C22—7241no—
426C22—7241no—
427C22—7241no—
428C22—7241no—
429C22C2×D4×D18111yescharacteristic
430C4—21no—
431C4—41no—
432C4—41no—
433C4—3621no—
434C4—3622no—
435C4—3621no—
436C4—1812no—
437C4C2×D4×D18111yescharacteristic
438C4#1915386211yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—14481no—
443C2—14481no—
444C2—7241no—
445C2—7242no—
446C2#2604191311yescharacteristic
447C2AGL(1,ℤ/8)×D18111yescharacteristic
448C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#469×C181C2
socleC22×C181C22×D4
hypercenter#469D181
hypocenterC181#3274
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#19153844C2semidirect
2. C2×#19153840C2semidirect
3. #26041947C2semidirect
4. #26041913C2semidirect
5. #26041654C2semidirect
6. #26041533C2semidirect
7. #26041478C2semidirect
8. #26041473C2semidirect
9. #26041425C2semidirect
10. #26041422C2semidirect
11. #26041414C2semidirect
12. #26041412C2semidirect
13. #26041366C2semidirect
14. #26041355C2semidirect
15. #469×C181C2semidirect
16. C2×#15668449C22semidirect
17. C2×#15668448C22semidirect
24. C2×#15668438C22semidirect
25. C2×#15668437C22semidirect
27. C2×D1448C22semidirect
29. C2×#15668434C22semidirect
30. C2×#15668434C22semidirect
31. C2×#15668432C22semidirect
34. #19153812C22semidirect
36. #19153791C22semidirect
42. #19153781C22semidirect
46. #19153745C22semidirect
47. #19153739C22semidirect
49. #19153736C22semidirect
50. #19153736C22semidirect
51. C22⋊#13089678C22semidirect
52. #19153728C22semidirect
53. C8×Dic181C22semidirect
54. C2×D8×C181C22semidirect
55. C2×M16×C181C22semidirect
57. #121×C181C22semidirect
58. #106×C181C22semidirect
59. #102×C181C22semidirect
60. #102×C181C22semidirect
92. #15668438D4semidirect
135. Dic362C2×D4semidirect
180. C8×C181C2×D4semidirect
213. Dic181AGL(1,ℤ/8)semidirect
224. C181#3274semidirect
232. #469D181semidirect
256. C2×D8C2×D181semidirect
258. C2×M16C2×D181semidirect
268. #121C2×D181semidirect
275. #106C2×D181semidirect
281. #102C2×D181semidirect
282. #102C2×D181semidirect
405. C8D4×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538442
C2×#191538402
#260419472
#260419132
#260416542
#260415332
#260414782
#260414732
#260414252
#260414222
#260414142
#260414122
#260413662
#260413552
#469×C1812
#3274181