Finite Group Databasebeta

← #37545482

Subgroups — #37545482

Order 23168.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042090C211yesmaximal, characteristic, complemented
2#26042089C211yesmaximal, characteristic, complemented
3#26042083C211yesmaximal, characteristic, complemented
4#26042082C211yesmaximal, characteristic, complemented
5#26041666C211yesmaximal, characteristic, complemented
6#26041548C211yesmaximal, characteristic, complemented
7#26041547C211yesmaximal, characteristic, complemented
8#26041541C211yesmaximal, characteristic, complemented
9#26041540C211yesmaximal, characteristic, complemented
10#26041458C211yesmaximal, characteristic, complemented
11#26041453C211yesmaximal, characteristic, complemented
12#26041446C211yesmaximal, characteristic, complemented
13#26041398C211yesmaximal, characteristic, complemented
14#26041398C211yesmaximal, characteristic, complemented
15#453×C181C211yesmaximal, characteristic, complemented
16#19153863C2211yescharacteristic, complemented
17#19153862C2211yescharacteristic
18#19153861C2211yescharacteristic, complemented
19#19153849C2211yescharacteristic, complemented
20#19153846—21no—
21#19153846C2211yescharacteristic, complemented
22#19153845—21no—
23#19153844C2211yescharacteristic, complemented
24#19153843—21no—
25#19153842—21no—
26#19153842—21no—
27#19153842C2211yescharacteristic
28#19153842C2211yescharacteristic, complemented
29#19153841—21no—
30#19153841C2211yescharacteristic, complemented
31#19153840—21no—
32#19153840—21no—
33#19153837—21no—
34#19153836—21no—
35#19153836—21no—
36#19153836C2211yescharacteristic
37#19153836C2211yescharacteristic
38#19153835—21no—
39C2×#15668437—21no—
40C2×#15668437C2211yescharacteristic, complemented
41C2×#15668437C2211yescharacteristic, complemented
42SD16×D181—21no—
43C2×#15668434—21no—
44C2×#15668434C2211yescharacteristic, complemented
45C2×#15668434C2211yescharacteristic, complemented
46D8×D181—21no—
47#19153824C2211yescharacteristic, complemented
48#19153824C2211yescharacteristic, complemented
49#19153823C2211yescharacteristic, complemented
50#19153823C2211yescharacteristic, complemented
51#19153804C2211yescharacteristic, complemented
52#19153751C2211yescharacteristic, complemented
53#19153751C2211yescharacteristic, complemented
54#19153744C2211yescharacteristic, complemented
55#19153744C2211yescharacteristic, complemented
56#19153734C2211yescharacteristic, complemented
57#19153733C2211yescharacteristic, complemented
58#19153732C2211yescharacteristic, complemented
59#19153727C2211yescharacteristic, complemented
60H(5,𝔽2)×C181C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C181C2211yescharacteristic
62AGL(1,ℤ/8)×C181C2211yescharacteristic, complemented
63#127×C181C2211yescharacteristic, complemented
64#104×C181C2211yescharacteristic, complemented
65#104×C181C2211yescharacteristic, complemented
66#100×C181C2211yescharacteristic, complemented
67#15668450—21no—
68#15668449—21no—
69#15668449—21no—
70#15668449C2311yescharacteristic
71#15668449C2311yescharacteristic
72#15668448—21no—
73#15668448—21no—
74#15668448—21no—
75#15668448—41no—
76C2×Dic362—21no—
77C2×Dic362C2311yescharacteristic
78Q8×D181—21no—
79C2×D724—21no—
80C2×D724C2311yescharacteristic
81C2×C22⋊D181—41no—
82D4×D181—21no—
83D4×D181—21no—
84D4×D181—21no—
85D4×D181—41no—
86Dic724—21no—
87#15668439—21no—
88#15668439D411yescharacteristic, complemented
89#15668439D411yescharacteristic, complemented
90#15668438—21no—
91#15668438—21no—
92#15668437—21no—
93#15668437—21no—
94#15668437—21no—
95#15668437—21no—
96#15668437—21no—
97#15668437—21no—
98#15668437—21no—
99#15668437D411yescharacteristic, complemented
100#15668437D411yescharacteristic, complemented
101#15668436—21no—
102#15668436D411yescharacteristic, complemented
103#15668436D411yescharacteristic, complemented
104D1448—21no—
105#15668434—21no—
106#15668434—21no—
107#15668434—21no—
108#15668434—21no—
109#15668434—21no—
110#15668434—21no—
111#15668434—21no—
112#15668434D411yescharacteristic, complemented
113#15668434D411yescharacteristic, complemented
114#15668433—21no—
115#15668433—21no—
116#15668433C2311yescharacteristic
117#15668433C2311yescharacteristic
118#15668432—21no—
119#15668432—21no—
120C2×#13089678—21no—
121C2×#13089678—21no—
122C2×#13089678C2311yescharacteristic
123C2×#13089678C2311yescharacteristic
124C8×D181—21no—
125C8×D181—21no—
126#15668426—41no—
127#41×C181—21no—
128#41×C181—21no—
129#41×C181C2311yescharacteristic
130#41×C181C2311yescharacteristic
131C2×D4×C181—21no—
132C2×D4×C181—21no—
133C2×D4×C181—41no—
134C2×D4×C181—41no—
135C2×D4×C181C2311yescharacteristic
136C2×D4×C181C2311yescharacteristic
137SD16×C181—21no—
138SD16×C181—21no—
139D8×C181—21no—
140D8×C181—21no—
141M16×C181C2311yescharacteristic
142M16×C181C2311yescharacteristic
143C42×C181C2311yescharacteristic
144C22×D181—21no—
145C22×D181—41no—
146C22×D181—41no—
147Dic362—21no—
148Dic362—21no—
149Dic362—41no—
150Dic362C2×D411yescharacteristic
151Dic362C2×D411yescharacteristic
152D724—21no—
153D724—21no—
154D724—41no—
155D724C2×D411yescharacteristic
156D724C2×D411yescharacteristic
157C22⋊D181—21no—
158C22⋊D181—21no—
159C22⋊D181—41no—
160C22⋊D181—41no—
161C22⋊D181—41no—
162C22⋊D181—41no—
163C2×Dic181—21no—
164C2×Dic181—41no—
165C2×Dic181—41no—
166C4×D181—21no—
167C4×D181—21no—
168C4×D181—41no—
169C4×D181—41no—
170#13089678—21no—
171#13089678—21no—
172#13089678C2×D411yescharacteristic
173#13089678C2×D411yescharacteristic
174#13089678C2×D411yescharacteristic
175#13089678C2×D411yescharacteristic
176C23×C181—21no—
177C23×C181—21no—
178C23×C181—41no—
179Q8×C181C2×D411yescharacteristic
180Q8×C181C2×D411yescharacteristic
181D4×C181—21no—
182D4×C181—21no—
183D4×C181—21no—
184D4×C181—21no—
185D4×C181—21no—
186D4×C181—21no—
187D4×C181—41no—
188D4×C181—41no—
189D4×C181—82no—
190D4×C181C2×D411yescharacteristic
191D4×C181C2×D411yescharacteristic
192C2×C4×C181—21no—
193C2×C4×C181—21no—
194C2×C4×C181—21no—
195C2×C4×C181—41no—
196C2×C4×C181C2411yescharacteristic
197C8×C181—21no—
198C8×C181—21no—
199C22×C181—21no—
200C22×C181—21no—
201C22×C181—41no—
202C22×C181—41no—
203C22×C181—81no—
204C22×C181—81no—
205C22×C181—161no—
206C22×C181H(5,𝔽2)11yescharacteristic
207C2×D181—21no—
208C2×D181—21no—
209C2×D181—41no—
210C2×D181—81no—
211C2×D181—161no—
212C2×D181—161no—
213C4×C181—21no—
214C4×C181—21no—
215C4×C181—41no—
216C4×C181C22×D411yescharacteristic
217C4×C181C22×D411yescharacteristic
218Dic181—21no—
219Dic181—21no—
220Dic181—41no—
221C2×C181—21no—
222C2×C181—41no—
223C2×C181—41no—
224C2×C181—81no—
225C2×C181—81no—
226C2×C181D4211yescharacteristic
227D181—41no—
228D181—41no—
229D181—81no—
230C181#328811yescharacteristic, complemented, Sylow
231#3288—1811nomaximal, Sylow
232#577—1811no—
233#577—1811no—
234#576—1811no—
235#576—1811no—
236#496—1811no—
237#472—1811no—
238#472—1811no—
239#471—1811no—
240#471—1811no—
241#455—1811no—
242#454—1811no—
243#453—1811no—
244#453D18111yescharacteristic, complemented
245#444—1811no—
246#444—1811no—
2472−1+4—1811no—
248H(5,𝔽2)—1811no—
249H(5,𝔽2)—1811no—
250H(5,𝔽2)C2×D18111yescharacteristic, complemented
251#137—3621no—
252#137—3621no—
253#137—1811no—
254#137—1811no—
255AGL(1,ℤ/8)—3621no—
256AGL(1,ℤ/8)—3621no—
257AGL(1,ℤ/8)—3621no—
258AGL(1,ℤ/8)—3621no—
259AGL(1,ℤ/8)—3621no—
260AGL(1,ℤ/8)—3621no—
261AGL(1,ℤ/8)—1811no—
262AGL(1,ℤ/8)—1811no—
263AGL(1,ℤ/8)—1811no—
264AGL(1,ℤ/8)—1811no—
265AGL(1,ℤ/8)C2×D18111yescharacteristic
266AGL(1,ℤ/8)C2×D18111yescharacteristic, complemented
267#135—3621no—
268#135—3621no—
269#135—3621no—
270#135—3621no—
271#135—1811no—
272#135—1811no—
273C2×SD16—3621no—
274C2×SD16—3621no—
275C2×SD16—1811no—
276C2×SD16—1811no—
277C2×D8—3621no—
278C2×D8—3621no—
279C2×D8—1811no—
280C2×D8—1811no—
281#131—1811no—
282#131—1811no—
283#131—1811no—
284#131—1811no—
285#127C2×D18111yescharacteristic, complemented
286#124—1811no—
287#108—1811no—
288#108—1811no—
289#104—1811no—
290#104—1811no—
291#104C2×D18111yescharacteristic, complemented
292#104C2×D18111yescharacteristic, complemented
293#101—1811no—
294#100—1811no—
295#100—1811no—
296#100C2×D18111yescharacteristic, complemented
297#97—1811no—
298#41—21no—
299#41—21no—
300#41—3621no—
301#41—3621no—
302#41—3621no—
303#41—3621no—
304#41—3621no—
305#41—3621no—
306#41—1811no—
307#41—1811no—
308#41—7241no—
309#41C22×D18111yescharacteristic
310#41C22×D18111yescharacteristic
311C2×Q8—3621no—
312C2×Q8—3621no—
313C2×Q8—1811no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—3621no—
319C2×D4—3621no—
320C2×D4—3621no—
321C2×D4—3621no—
322C2×D4—1811no—
323C2×D4—7241no—
324C2×D4—7241no—
325C2×D4C22×D18111yescharacteristic
326C2×D4C22×D18111yescharacteristic
327Q16—3621no—
328Q16—3621no—
329Q16—1811no—
330Q16—1811no—
331SD16—21no—
332SD16—21no—
333SD16—3621no—
334SD16—3621no—
335SD16—3621no—
336SD16—3621no—
337SD16—3621no—
338SD16—3621no—
339SD16—3621no—
340SD16—3621no—
341SD16—3621no—
342SD16—3621no—
343SD16—1811no—
344SD16—1811no—
345SD16—1811no—
346SD16—1811no—
347D8—21no—
348D8—21no—
349D8—3621no—
350D8—3621no—
351D8—3621no—
352D8—3621no—
353D8—3621no—
354D8—3621no—
355D8—3621no—
356D8—3621no—
357D8—1811no—
358D8—1811no—
359M16—3621no—
360M16—3621no—
361M16—3621no—
362M16—3621no—
363M16—1811no—
364M16—1811no—
365M16C22×D18111yescharacteristic
366M16C22×D18111yescharacteristic
367C2×C8—3621no—
368C2×C8—3621no—
369C2×C8—3621no—
370C2×C8—3621no—
371C2×C8—1811no—
372C2×C8—1811no—
373C22⋊C4—7241no—
374C42C22×D18111yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—3621no—
379C23—7241no—
380C23—7241no—
381Q8—3621no—
382Q8—3621no—
383Q8—1811no—
384Q8—1811no—
385Q8—7241no—
386Q8D4×D18111yescharacteristic
387Q8D4×D18111yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—3621no—
398D4—3621no—
399D4—3621no—
400D4—3621no—
401D4—1811no—
402D4—1811no—
403D4—7241no—
404D4—7241no—
405D4—7241no—
406D4—7241no—
407D4—7241no—
408D4D4×D18111yescharacteristic
409D4D4×D18111yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—3621no—
415C2×C4—3621no—
416C2×C4—3621no—
417C2×C4—7241no—
418C2×C4—7241no—
419C2×C4—7241no—
420C2×C4—7241no—
421C2×C4C23×D18111yescharacteristic
422C8—21no—
423C8—21no—
424C8—3621no—
425C8—3621no—
426C8—1811no—
427C8—1811no—
428C8—1811no—
429C8—1811no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—28961no—
438C22—28961no—
439C22—3621no—
440C22—3621no—
441C22—14481no—
442C22—7241no—
443C22#1915386111yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3621no—
448C4—3621no—
449C4—7241no—
450C4C2×D4×D18111yescharacteristic
451C4C2×D4×D18111yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—14481no—
458C2—7241no—
459C2—7241no—
460C2#2604191111yescharacteristic
subgroupG/H
centerC2#26041911
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#453×C181C2
socleC2×C181D42
hypercenter#453D181
hypocenterC181#3288
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #26042090C2semidirect
2. #26042089C2semidirect
3. #26042083C2semidirect
4. #26042082C2semidirect
5. #26041666C2semidirect
6. #26041548C2semidirect
7. #26041547C2semidirect
8. #26041541C2semidirect
9. #26041540C2semidirect
10. #26041458C2semidirect
11. #26041453C2semidirect
12. #26041446C2semidirect
13. #26041398C2semidirect
14. #26041398C2semidirect
15. #453×C181C2semidirect
16. #19153863C22semidirect
18. #19153861C22semidirect
19. #19153849C22semidirect
21. #19153846C22semidirect
23. #19153844C22semidirect
28. #19153842C22semidirect
30. #19153841C22semidirect
40. C2×#15668437C22semidirect
41. C2×#15668437C22semidirect
44. C2×#15668434C22semidirect
45. C2×#15668434C22semidirect
47. #19153824C22semidirect
48. #19153824C22semidirect
49. #19153823C22semidirect
50. #19153823C22semidirect
51. #19153804C22semidirect
52. #19153751C22semidirect
53. #19153751C22semidirect
54. #19153744C22semidirect
55. #19153744C22semidirect
56. #19153734C22semidirect
57. #19153733C22semidirect
58. #19153732C22semidirect
59. #19153727C22semidirect
60. H(5,𝔽2)×C181C22semidirect
62. AGL(1,ℤ/8)×C181C22semidirect
63. #127×C181C22semidirect
64. #104×C181C22semidirect
65. #104×C181C22semidirect
66. #100×C181C22semidirect
88. #15668439D4semidirect
89. #15668439D4semidirect
99. #15668437D4semidirect
100. #15668437D4semidirect
102. #15668436D4semidirect
103. #15668436D4semidirect
112. #15668434D4semidirect
113. #15668434D4semidirect
230. C181#3288semidirect
244. #453D181semidirect
250. H(5,𝔽2)C2×D181semidirect
266. AGL(1,ℤ/8)C2×D181semidirect
285. #127C2×D181semidirect
291. #104C2×D181semidirect
292. #104C2×D181semidirect
296. #100C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420902
#260420892
#260420832
#260420822
#260416662
#260415482
#260415472
#260415412
#260415402
#260414582
#260414532
#260414462
#260413982
#453×C1812
#3288181