Finite Group Databasebeta

← #36365327

Subgroups — #36365327

Order 22144.

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#26012302C211yesmaximal, characteristic, complemented
2#26012298C211yesmaximal, characteristic, complemented
3#26012294C211yesmaximal, characteristic, complemented
4#26011879C211yesmaximal, characteristic, complemented
5#26011759C211yesmaximal, characteristic, complemented
6#26011758C211yesmaximal, characteristic, complemented
7#26011754C211yesmaximal, characteristic, complemented
8#26011752C211yesmaximal, characteristic, complemented
9#26011669C211yesmaximal, characteristic, complemented
10#26011662C211yesmaximal, characteristic, complemented
11#26011659C211yesmaximal, characteristic, complemented
12#26011657C211yesmaximal, characteristic, complemented
13#26011613C211yesmaximal, characteristic, complemented
14#26011611C211yesmaximal, characteristic, complemented
15#576×C173C211yesmaximal, characteristic, complemented
16#19149726C2211yescharacteristic, complemented
17#19149725C2211yescharacteristic, complemented
18#19149711—21no—
19#19149709—21no—
20#19149709C2211yescharacteristic, complemented
21#19149709C2211yescharacteristic, complemented
22#19149705—21no—
23#19149705—21no—
24#19149705C2211yescharacteristic, complemented
25#19149705C2211yescharacteristic, complemented
26#19149704—21no—
27#19149704C2211yescharacteristic
28#19149704C2211yescharacteristic, complemented
29#19149703—21no—
30#19149700—21no—
31#19149699—21no—
32#19149699—21no—
33#19149699C2211yescharacteristic
34C2×#14583815C2211yescharacteristic, complemented
35C2×#14583814—21no—
36C2×#14583814C2211yescharacteristic, complemented
37SD16×D173—21no—
38C2×#14583811—21no—
39C2×#14583811C2211yescharacteristic, complemented
40#19149686C2211yescharacteristic, complemented
41#19149686C2211yescharacteristic, complemented
42#19149685C2211yescharacteristic, complemented
43#19149675C2211yescharacteristic, complemented
44#19149666C2211yescharacteristic, complemented
45#19149615C2211yescharacteristic, complemented
46#19149614C2211yescharacteristic, complemented
47#19149606C2211yescharacteristic, complemented
48#19149606C2211yescharacteristic, complemented
49#19149606C2211yescharacteristic, complemented
50#19149605C2211yescharacteristic, complemented
51#19149597C2211yescharacteristic, complemented
52#19149596C2211yescharacteristic, complemented
53#19149596C2211yescharacteristic, complemented
54#19149595C2211yescharacteristic, complemented
55#19149589C2211yescharacteristic, complemented
56H(5,𝔽2)×C173C2211yescharacteristic
57H(5,𝔽2)×C173C2211yescharacteristic, complemented
58AGL(1,ℤ/8)×C173—21no—
59AGL(1,ℤ/8)×C173—21no—
60AGL(1,ℤ/8)×C173C2211yescharacteristic
61AGL(1,ℤ/8)×C173C2211yescharacteristic, complemented
62#135×C173—21no—
63#135×C173C2211yescharacteristic
64C2×D8×C173—21no—
65C2×D8×C173C2211yescharacteristic, complemented
66#131×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583826—21no—
69#14583826—21no—
70#14583826C2311yescharacteristic
71#14583825—21no—
72#14583825—41no—
73C2×Dic346—21no—
74C2×Dic346C2311yescharacteristic
75Q8×D173—21no—
76C2×D692—21no—
77C2×D692C2311yescharacteristic
78C2×C22⋊D173—41no—
79D4×D173—21no—
80D4×D173—41no—
81#14583816—21no—
82#14583816—21no—
83#14583816D411yescharacteristic, complemented
84#14583815—21no—
85#14583815—21no—
86#14583814—21no—
87#14583814—21no—
88#14583814—21no—
89#14583814—21no—
90#14583814—21no—
91#14583814—21no—
92#14583814D411yescharacteristic, complemented
93#14583813—21no—
94#14583813—21no—
95#14583813D411yescharacteristic, complemented
96#14583811—21no—
97#14583811—21no—
98#14583811—21no—
99#14583811—21no—
100#14583811—21no—
101#14583811—21no—
102#14583811D411yescharacteristic, complemented
103#14583810—21no—
104#14583810—21no—
105#14583810C2311yescharacteristic
106#14583810C2311yescharacteristic
107#14583810C2311yescharacteristic
108#14583809—21no—
109C2×#13063436—21no—
110C2×#13063436—21no—
111C2×#13063436C2311yescharacteristic
112C8×D173—21no—
113C22⋊Dic173—41no—
114C4×Dic173C2311yescharacteristic
115#41×C173—21no—
116#41×C173—21no—
117#41×C173—21no—
118#41×C173—21no—
119#41×C173C2311yescharacteristic
120#41×C173C2311yescharacteristic
121#41×C173C2311yescharacteristic
122C2×D4×C173—21no—
123C2×D4×C173—21no—
124C2×D4×C173—21no—
125C2×D4×C173—21no—
126C2×D4×C173—41no—
127C2×D4×C173—41no—
128C2×D4×C173C2311yescharacteristic
129C2×D4×C173C2311yescharacteristic
130Q16×C173D411yescharacteristic, complemented
131SD16×C173—21no—
132SD16×C173—21no—
133SD16×C173D411yescharacteristic, complemented
134SD16×C173D411yescharacteristic, complemented
135D8×C173—21no—
136D8×C173—21no—
137D8×C173—21no—
138D8×C173—21no—
139D8×C173D411yescharacteristic, complemented
140M16×C173—21no—
141M16×C173C2311yescharacteristic
142C2×C8×C173—21no—
143C2×C8×C173C2311yescharacteristic
144C22×D173—21no—
145C22×D173—41no—
146Dic346—21no—
147Dic346—21no—
148Dic346—41no—
149Dic346C2×D411yescharacteristic
150D692—21no—
151D692—21no—
152D692—41no—
153D692C2×D411yescharacteristic
154C22⋊D173—21no—
155C22⋊D173—41no—
156C22⋊D173—41no—
157C22⋊D173—82no—
158C2×Dic173—21no—
159C2×Dic173—21no—
160C2×Dic173—41no—
161C4×D173—21no—
162C4×D173—41no—
163C4×D173—41no—
164#13063436—21no—
165#13063436—21no—
166#13063436—21no—
167#13063436C2×D411yescharacteristic
168#13063436C2×D411yescharacteristic
169C23×C173—21no—
170C23×C173—21no—
171C23×C173—41no—
172C23×C173—41no—
173Q8×C173C2×D411yescharacteristic
174Q8×C173C2×D411yescharacteristic
175Q8×C173C2×D411yescharacteristic
176D4×C173—21no—
177D4×C173—21no—
178D4×C173—21no—
179D4×C173—21no—
180D4×C173—21no—
181D4×C173—21no—
182D4×C173—21no—
183D4×C173—41no—
184D4×C173—41no—
185D4×C173—41no—
186D4×C173—41no—
187D4×C173C2×D411yescharacteristic
188D4×C173C2×D411yescharacteristic
189D4×C173C2×D411yescharacteristic
190C2×C4×C173—21no—
191C2×C4×C173—21no—
192C2×C4×C173—21no—
193C2×C4×C173—41no—
194C2×C4×C173—41no—
195C2×C4×C173C2411yescharacteristic
196C8×C173—21no—
197C8×C173C2×D411yescharacteristic
198C8×C173C2×D411yescharacteristic
199C22×C173—21no—
200C22×C173—21no—
201C22×C173—21no—
202C22×C173—41no—
203C22×C173—41no—
204C22×C173—81no—
205C22×C173—81no—
206C22×C173—161no—
207C22×C173—161no—
208C22×C173H(5,𝔽2)11yescharacteristic
209C2×D173—21no—
210C2×D173—41no—
211C2×D173—81no—
212C2×D173—161no—
213C4×C173—21no—
214C4×C173—21no—
215C4×C173—21no—
216C4×C173C22×D411yescharacteristic
217C4×C173C22×D411yescharacteristic
218Dic173—21no—
219Dic173—41no—
220Dic173—41no—
221C2×C173—21no—
222C2×C173—41no—
223C2×C173—41no—
224C2×C173—41no—
225C2×C173—81no—
226C2×C173—81no—
227C2×C173D4211yescharacteristic
228D173—41no—
229D173—81no—
230C173#328811yescharacteristic, complemented, Sylow
231#3288—1731nomaximal, Sylow
232#577—1731no—
233#577—1731no—
234#576—1731no—
235#576D17311yescharacteristic, complemented
236#496—1731no—
237#472—1731no—
238#472—1731no—
239#471—1731no—
240#471—1731no—
241#455—1731no—
242#454—1731no—
243#453—1731no—
244#453—1731no—
245#444—1731no—
246#444—1731no—
2472−1+4—1731no—
248H(5,𝔽2)—1731no—
249H(5,𝔽2)C2×D17311yescharacteristic
250H(5,𝔽2)C2×D17311yescharacteristic, complemented
251#137—3461no—
252#137—3461no—
253#137—1731no—
254#137—1731no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—3461no—
258AGL(1,ℤ/8)—3461no—
259AGL(1,ℤ/8)—3461no—
260AGL(1,ℤ/8)—3461no—
261AGL(1,ℤ/8)—1731no—
262AGL(1,ℤ/8)—1731no—
263AGL(1,ℤ/8)—1731no—
264AGL(1,ℤ/8)—1731no—
265AGL(1,ℤ/8)C2×D17311yescharacteristic
266AGL(1,ℤ/8)C2×D17311yescharacteristic, complemented
267#135—21no—
268#135—3461no—
269#135—3461no—
270#135—3461no—
271#135—1731no—
272#135C2×D17311yescharacteristic
273C2×SD16—3461no—
274C2×SD16—3461no—
275C2×SD16—1731no—
276C2×SD16—1731no—
277C2×D8—21no—
278C2×D8—3461no—
279C2×D8—1731no—
280C2×D8C2×D17311yescharacteristic, complemented
281#131—1731no—
282#131—1731no—
283#131—1731no—
284#131C2×D17311yescharacteristic, complemented
285#127—1731no—
286#124—1731no—
287#108—1731no—
288#108—1731no—
289#104—1731no—
290#104—1731no—
291#104—1731no—
292#104—1731no—
293#101—1731no—
294#100—1731no—
295#100—1731no—
296#100—1731no—
297#97—1731no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—3461no—
303#41—3461no—
304#41—3461no—
305#41—3461no—
306#41—1731no—
307#41—6921no—
308#41C22×D17311yescharacteristic
309#41C22×D17311yescharacteristic
310#41C22×D17311yescharacteristic
311C2×Q8—3461no—
312C2×Q8—3461no—
313C2×Q8—1731no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—3461no—
321C2×D4—3461no—
322C2×D4—1731no—
323C2×D4—6921no—
324C2×D4—6921no—
325C2×D4C22×D17311yescharacteristic
326C2×D4C22×D17311yescharacteristic
327Q16—3461no—
328Q16—3461no—
329Q16—1731no—
330Q16C22⋊D17311yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—3461no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338SD16—3461no—
339SD16—3461no—
340SD16—3461no—
341SD16—3461no—
342SD16—3461no—
343SD16—1731no—
344SD16—1731no—
345SD16C22⋊D17311yescharacteristic, complemented
346SD16C22⋊D17311yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—3461no—
352D8—3461no—
353D8—3461no—
354D8—3461no—
355D8—3461no—
356D8—3461no—
357D8—1731no—
358D8C22⋊D17311yescharacteristic, complemented
359M16—21no—
360M16—3461no—
361M16—3461no—
362M16—3461no—
363M16—1731no—
364M16—1731no—
365M16—1731no—
366M16C22×D17311yescharacteristic
367C2×C8—21no—
368C2×C8—3461no—
369C2×C8—3461no—
370C2×C8—3461no—
371C2×C8—1731no—
372C2×C8C22×D17311yescharacteristic
373C22⋊C4—6921no—
374C42—1731no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—3461no—
380C23—6921no—
381Q8—3461no—
382Q8—3461no—
383Q8—1731no—
384Q8—6921no—
385Q8D4×D17311yescharacteristic
386Q8C2×C22⋊D17311yescharacteristic
387Q8C2×C22⋊D17311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—3461no—
400D4—3461no—
401D4—3461no—
402D4—13842no—
403D4—1731no—
404D4—6921no—
405D4—6921no—
406D4—6921no—
407D4D4×D17311yescharacteristic
408D4C2×C22⋊D17311yescharacteristic
409D4C2×C22⋊D17311yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—3461no—
416C2×C4—3461no—
417C2×C4—3461no—
418C2×C4—6921no—
419C2×C4—6921no—
420C2×C4—6921no—
421C2×C4C23×D17311yescharacteristic
422C8—21no—
423C8—3461no—
424C8—3461no—
425C8—3461no—
426C8—1731no—
427C8—1731no—
428C8C2×C22⋊D17311yescharacteristic
429C8C2×C22⋊D17311yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—161no—
438C22—161no—
439C22—3461no—
440C22—13841no—
441C22—6921no—
442C22—27681no—
443C22#1914972411yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—3461no—
448C4—6921no—
449C4—6921no—
450C4C2×D4×D17311yescharacteristic
451C4C22×C22⋊D17311yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—13841no—
459C2—6921no—
460C2D4×C22⋊D17311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#576×C173C2
socleC2×C173D42
hypercenter#576D173
hypocenterC173#3288
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #26012302C2semidirect
2. #26012298C2semidirect
3. #26012294C2semidirect
4. #26011879C2semidirect
5. #26011759C2semidirect
6. #26011758C2semidirect
7. #26011754C2semidirect
8. #26011752C2semidirect
9. #26011669C2semidirect
10. #26011662C2semidirect
11. #26011659C2semidirect
12. #26011657C2semidirect
13. #26011613C2semidirect
14. #26011611C2semidirect
15. #576×C173C2semidirect
16. #19149726C22semidirect
17. #19149725C22semidirect
20. #19149709C22semidirect
21. #19149709C22semidirect
24. #19149705C22semidirect
25. #19149705C22semidirect
28. #19149704C22semidirect
34. C2×#14583815C22semidirect
36. C2×#14583814C22semidirect
39. C2×#14583811C22semidirect
40. #19149686C22semidirect
41. #19149686C22semidirect
42. #19149685C22semidirect
43. #19149675C22semidirect
44. #19149666C22semidirect
45. #19149615C22semidirect
46. #19149614C22semidirect
47. #19149606C22semidirect
48. #19149606C22semidirect
49. #19149606C22semidirect
50. #19149605C22semidirect
51. #19149597C22semidirect
52. #19149596C22semidirect
53. #19149596C22semidirect
54. #19149595C22semidirect
55. #19149589C22semidirect
57. H(5,𝔽2)×C173C22semidirect
61. AGL(1,ℤ/8)×C173C22semidirect
65. C2×D8×C173C22semidirect
66. #131×C173C22semidirect
83. #14583816D4semidirect
92. #14583814D4semidirect
95. #14583813D4semidirect
102. #14583811D4semidirect
130. Q16×C173D4semidirect
133. SD16×C173D4semidirect
134. SD16×C173D4semidirect
139. D8×C173D4semidirect
230. C173#3288semidirect
235. #576D173semidirect
250. H(5,𝔽2)C2×D173semidirect
266. AGL(1,ℤ/8)C2×D173semidirect
280. C2×D8C2×D173semidirect
284. #131C2×D173semidirect
330. Q16C22⋊D173semidirect
345. SD16C22⋊D173semidirect
346. SD16C22⋊D173semidirect
358. D8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260123022
#260122982
#260122942
#260118792
#260117592
#260117582
#260117542
#260117522
#260116692
#260116622
#260116592
#260116572
#260116132
#260116112
#576×C1732
#3288173