Finite Group Databasebeta

← #37545488

Subgroups — #37545488

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#26042091C211yesmaximal, characteristic, complemented
2#26042086C211yesmaximal, characteristic, complemented
3#26042084C211yesmaximal, characteristic, complemented
4#26042081C211yesmaximal, characteristic, complemented
5#26041668C211yesmaximal, characteristic, complemented
6#26041550C211yesmaximal, characteristic, complemented
7#26041549C211yesmaximal, characteristic, complemented
8#26041543C211yesmaximal, characteristic, complemented
9#26041542C211yesmaximal, characteristic, complemented
10#26041458C211yesmaximal, characteristic, complemented
11#26041453C211yesmaximal, characteristic, complemented
12#26041448C211yesmaximal, characteristic, complemented
13#26041448C211yesmaximal, characteristic, complemented
14#26041402C211yesmaximal, characteristic, complemented
15#444×C181C211yesmaximal, characteristic, complemented
16#19153863C2211yescharacteristic, complemented
17#19153862C2211yescharacteristic
18#19153862C2211yescharacteristic, complemented
19#19153862C2211yescharacteristic, complemented
20#19153849—21no—
21#19153849C2211yescharacteristic, complemented
22#19153848—21no—
23#19153846C2211yescharacteristic, complemented
24#19153844—21no—
25#19153844—21no—
26#19153844—21no—
27#19153844C2211yescharacteristic
28#19153844C2211yescharacteristic, complemented
29#19153844C2211yescharacteristic, complemented
30#19153843—21no—
31#19153843—21no—
32#19153841C2211yescharacteristic
33#19153841C2211yescharacteristic, complemented
34#19153841C2211yescharacteristic, complemented
35#19153840—21no—
36#19153839—21no—
37#19153838—21no—
38#19153838—21no—
39#19153838C2211yescharacteristic
40#19153837—21no—
41C2×#15668438—21no—
42C2×#15668438C2211yescharacteristic, complemented
43C2×#15668438C2211yescharacteristic, complemented
44SD16×D181—21no—
45C2×D1448—21no—
46C2×D1448C2211yescharacteristic, complemented
47C2×D1448C2211yescharacteristic, complemented
48D8×D181—21no—
49#19153822C2211yescharacteristic, complemented
50#19153822C2211yescharacteristic, complemented
51#19153813C2211yescharacteristic, complemented
52#19153804C2211yescharacteristic, complemented
53#19153752C2211yescharacteristic, complemented
54#19153744C2211yescharacteristic, complemented
55#19153744C2211yescharacteristic, complemented
56#19153734C2211yescharacteristic, complemented
57#19153732C2211yescharacteristic, complemented
58#19153732C2211yescharacteristic, complemented
59#19153732C2211yescharacteristic, complemented
60#135×C181C2211yescharacteristic
61#131×C181C2211yescharacteristic, complemented
62#131×C181C2211yescharacteristic, complemented
63#108×C181C2211yescharacteristic, complemented
64#104×C181C2211yescharacteristic, complemented
65#104×C181C2211yescharacteristic, complemented
66#97×C181C2211yescharacteristic, complemented
67#15668450—21no—
68#15668450—21no—
69#15668450—21no—
70#15668449—21no—
71#15668449—21no—
72#15668449—21no—
73#15668449—21no—
74#15668449C2311yescharacteristic
75#15668449C2311yescharacteristic
76#15668448—21no—
77#15668448—41no—
78C2×Dic362—21no—
79C2×Dic362C2311yescharacteristic
80Q8×D181—21no—
81C2×D724—21no—
82C2×D724—21no—
83C2×D724—21no—
84C2×D724—41no—
85C2×D724C2311yescharacteristic
86C2×D724C2311yescharacteristic
87C2×D724C2311yescharacteristic
88D4×D181—21no—
89D4×D181—21no—
90D4×D181—21no—
91D4×D181—41no—
92D4×D181—41no—
93D4×D181—41no—
94Dic724—21no—
95Dic724D411yescharacteristic, complemented
96#15668439—21no—
97#15668438—21no—
98#15668438—21no—
99#15668438—21no—
100#15668438—21no—
101#15668438—21no—
102#15668438—21no—
103#15668438—21no—
104#15668438—21no—
105#15668438D411yescharacteristic, complemented
106#15668438D411yescharacteristic, complemented
107#15668437—21no—
108#15668436—21no—
109#15668436—21no—
110#15668436—21no—
111D1448—21no—
112D1448—21no—
113D1448—21no—
114D1448—21no—
115D1448—21no—
116D1448—21no—
117D1448—21no—
118D1448D411yescharacteristic, complemented
119#15668434—21no—
120#15668434—21no—
121#15668434—21no—
122#15668433C2311yescharacteristic
123#15668433C2311yescharacteristic
124#15668432—21no—
125#15668432—21no—
126C8×D181—21no—
127C8×D181—21no—
128#15668426—41no—
129#41×C181C2311yescharacteristic
130#41×C181C2311yescharacteristic
131Q16×C181D411yescharacteristic, complemented
132SD16×C181D411yescharacteristic, complemented
133SD16×C181D411yescharacteristic, complemented
134D8×C181D411yescharacteristic, complemented
135M16×C181—21no—
136M16×C181—21no—
137M16×C181C2311yescharacteristic
138M16×C181C2311yescharacteristic
139C2×C8×C181—21no—
140C2×C8×C181—21no—
141C2×C8×C181C2311yescharacteristic
142C2×C8×C181C2311yescharacteristic
143C42×C181C2311yescharacteristic
144C22×D181—21no—
145C22×D181—21no—
146C22×D181—21no—
147C22×D181—41no—
148C22×D181—41no—
149C22×D181—41no—
150Dic362—21no—
151Dic362—21no—
152Dic362—41no—
153Dic362C2×D411yescharacteristic
154Dic362C2×D411yescharacteristic
155D724—21no—
156D724—21no—
157D724—21no—
158D724—21no—
159D724—21no—
160D724—21no—
161D724—41no—
162D724—41no—
163D724—41no—
164D724—82no—
165D724C2×D411yescharacteristic
166D724C2×D411yescharacteristic
167C22⋊D181—21no—
168C22⋊D181—21no—
169C22⋊D181—41no—
170C22⋊D181—41no—
171C22⋊D181—41no—
172C22⋊D181—41no—
173C2×Dic181—21no—
174C2×Dic181—41no—
175C4×D181—21no—
176C4×D181—21no—
177C4×D181—41no—
178C4×D181—41no—
179C4×D181—41no—
180C4×D181—41no—
181#13089678—21no—
182#13089678—21no—
183Q8×C181C2×D411yescharacteristic
184Q8×C181C2×D411yescharacteristic
185D4×C181—21no—
186D4×C181—21no—
187D4×C181C2×D411yescharacteristic
188D4×C181C2×D411yescharacteristic
189C2×C4×C181—21no—
190C2×C4×C181—21no—
191C2×C4×C181—21no—
192C2×C4×C181C2411yescharacteristic
193C8×C181—21no—
194C8×C181—21no—
195C8×C181C2×D411yescharacteristic
196C8×C181C2×D411yescharacteristic
197C8×C181C2×D411yescharacteristic
198C8×C181C2×D411yescharacteristic
199C22×C181—21no—
200C22×C181—21no—
201C22×C181H(5,𝔽2)11yescharacteristic
202C2×D181—21no—
203C2×D181—21no—
204C2×D181—41no—
205C2×D181—41no—
206C2×D181—41no—
207C2×D181—81no—
208C2×D181—81no—
209C2×D181—81no—
210C2×D181—161no—
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×C181D4211yescharacteristic
225D181—41no—
226D181—41no—
227D181—81no—
228D181—81no—
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#453—1811no—
245#444—1811no—
246#444D18111yescharacteristic, complemented
2472−1+4—1811no—
248H(5,𝔽2)—1811no—
249H(5,𝔽2)—1811no—
250H(5,𝔽2)—1811no—
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)—1811no—
266AGL(1,ℤ/8)—1811no—
267#135—3621no—
268#135—3621no—
269#135—3621no—
270#135—3621no—
271#135—1811no—
272#135C2×D18111yescharacteristic
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#131C2×D18111yescharacteristic, complemented
284#131C2×D18111yescharacteristic, complemented
285#127—1811no—
286#124—1811no—
287#108—1811no—
288#108C2×D18111yescharacteristic, complemented
289#104—1811no—
290#104—1811no—
291#104C2×D18111yescharacteristic, complemented
292#104C2×D18111yescharacteristic, complemented
293#101—1811no—
294#100—1811no—
295#100—1811no—
296#100—1811no—
297#97C2×D18111yescharacteristic, complemented
298#41—3621no—
299#41—3621no—
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—3621no—
315C2×D4—3621no—
316C2×D4—3621no—
317C2×D4—3621no—
318C2×D4—3621no—
319C2×D4—3621no—
320C2×D4—1811no—
321C2×D4—1811no—
322C2×D4—1811no—
323C2×D4—7241no—
324C2×D4—7241no—
325C2×D4—7241no—
326C2×D4—7241no—
327Q16—3621no—
328Q16—3621no—
329Q16—1811no—
330Q16D72411yescharacteristic, complemented
331SD16—3621no—
332SD16—3621no—
333SD16—3621no—
334SD16—3621no—
335SD16—3621no—
336SD16—3621no—
337SD16—3621no—
338SD16—3621no—
339SD16—3621no—
340SD16—3621no—
341SD16—3621no—
342SD16—3621no—
343SD16—1811no—
344SD16—1811no—
345SD16D72411yescharacteristic, complemented
346SD16D72411yescharacteristic, complemented
347D8—3621no—
348D8—3621no—
349D8—3621no—
350D8—3621no—
351D8—3621no—
352D8—3621no—
353D8—3621no—
354D8—3621no—
355D8—3621no—
356D8—3621no—
357D8—1811no—
358D8D72411yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—3621no—
362M16—3621no—
363M16—1811no—
364M16—1811no—
365M16C22×D18111yescharacteristic
366M16C22×D18111yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—3621no—
370C2×C8—3621no—
371C2×C8C22×D18111yescharacteristic
372C2×C8C22×D18111yescharacteristic
373C22⋊C4—7241no—
374C42C22×D18111yescharacteristic
375C23—3621no—
376C23—3621no—
377C23—3621no—
378C23—7241no—
379C23—7241no—
380C23—7241no—
381Q8—3621no—
382Q8—3621no—
383Q8—1811no—
384Q8—1811no—
385Q8—7241no—
386Q8C2×D72411yescharacteristic
387Q8C2×D72411yescharacteristic
388D4—21no—
389D4—21no—
390D4—3621no—
391D4—3621no—
392D4—3621no—
393D4—3621no—
394D4—3621no—
395D4—3621no—
396D4—3621no—
397D4—3621no—
398D4—14482no—
399D4—1811no—
400D4—1811no—
401D4—7241no—
402D4—7241no—
403D4—7241no—
404D4—7241no—
405D4—7241no—
406D4—7241no—
407D4—7241no—
408D4C2×D72411yescharacteristic
409D4C2×D72411yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—3621no—
414C2×C4—3621no—
415C2×C4—3621no—
416C2×C4—7241no—
417C2×C4—7241no—
418C2×C4—7241no—
419C2×C4—7241no—
420C2×C4—7241no—
421C2×C4C23×D18111yescharacteristic
422C8—21no—
423C8—21no—
424C8—3621no—
425C8—3621no—
426C8D4×D18111yescharacteristic
427C8D4×D18111yescharacteristic
428C8C2×D72411yescharacteristic
429C8C2×D72411yescharacteristic
430C22—21no—
431C22—21no—
432C22—28961no—
433C22—28961no—
434C22—28961no—
435C22—3621no—
436C22—3621no—
437C22—14481no—
438C22—14481no—
439C22—14481no—
440C22—7241no—
441C22—7241no—
442C22—7241no—
443C22#1915386211yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3621no—
448C4—3621no—
449C4—7241no—
450C4C2×D4×D18111yescharacteristic
451C4C22×D72411yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—14481no—
456C2—14481no—
457C2—14481no—
458C2—7241no—
459C2—7241no—
460C2D4×D72411yescharacteristic
subgroupG/H
centerC2D4×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#444×C181C2
socleC2×C181D42
hypercenter#444D181
hypocenterC181#3288
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #26042091C2semidirect
2. #26042086C2semidirect
3. #26042084C2semidirect
4. #26042081C2semidirect
5. #26041668C2semidirect
6. #26041550C2semidirect
7. #26041549C2semidirect
8. #26041543C2semidirect
9. #26041542C2semidirect
10. #26041458C2semidirect
11. #26041453C2semidirect
12. #26041448C2semidirect
13. #26041448C2semidirect
14. #26041402C2semidirect
15. #444×C181C2semidirect
16. #19153863C22semidirect
18. #19153862C22semidirect
19. #19153862C22semidirect
21. #19153849C22semidirect
23. #19153846C22semidirect
28. #19153844C22semidirect
29. #19153844C22semidirect
33. #19153841C22semidirect
34. #19153841C22semidirect
42. C2×#15668438C22semidirect
43. C2×#15668438C22semidirect
46. C2×D1448C22semidirect
47. C2×D1448C22semidirect
49. #19153822C22semidirect
50. #19153822C22semidirect
51. #19153813C22semidirect
52. #19153804C22semidirect
53. #19153752C22semidirect
54. #19153744C22semidirect
55. #19153744C22semidirect
56. #19153734C22semidirect
57. #19153732C22semidirect
58. #19153732C22semidirect
59. #19153732C22semidirect
61. #131×C181C22semidirect
62. #131×C181C22semidirect
63. #108×C181C22semidirect
64. #104×C181C22semidirect
65. #104×C181C22semidirect
66. #97×C181C22semidirect
95. Dic724D4semidirect
105. #15668438D4semidirect
106. #15668438D4semidirect
118. D1448D4semidirect
131. Q16×C181D4semidirect
132. SD16×C181D4semidirect
133. SD16×C181D4semidirect
134. D8×C181D4semidirect
230. C181#3288semidirect
246. #444D181semidirect
283. #131C2×D181semidirect
284. #131C2×D181semidirect
288. #108C2×D181semidirect
291. #104C2×D181semidirect
292. #104C2×D181semidirect
297. #97C2×D181semidirect
330. Q16D724semidirect
345. SD16D724semidirect
346. SD16D724semidirect
358. D8D724semidirect

Maximal subgroups

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

subgroupindex
#260420912
#260420862
#260420842
#260420812
#260416682
#260415502
#260415492
#260415432
#260415422
#260414582
#260414532
#260414482
#260414022
#444×C1812
#3288181