Finite Group Databasebeta

← #35226307

Subgroups — #35226307

Order 20864.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782480C211yesmaximal, characteristic, complemented
2#23782473C211yesmaximal, characteristic, complemented
3#23782472C211yesmaximal, characteristic, complemented
4#23782058C211yesmaximal, characteristic, complemented
5#23781938C211yesmaximal, characteristic, complemented
6#23781937C211yesmaximal, characteristic, complemented
7#23781930C211yesmaximal, characteristic, complemented
8#23781928C211yesmaximal, characteristic, complemented
9#23781848C211yesmaximal, characteristic, complemented
10#23781846C211yesmaximal, characteristic, complemented
11#23781841C211yesmaximal, characteristic, complemented
12#23781838C211yesmaximal, characteristic, complemented
13#23781787C211yesmaximal, characteristic, complemented
14#23781785C211yesmaximal, characteristic, complemented
15#578×C163C211yesmaximal, characteristic, complemented
16#19104782C2211yescharacteristic
17#19104781C2211yescharacteristic
18#19104767—21no—
19#19104766—21no—
20#19104766—21no—
21#19104766C2211yescharacteristic
22#19104766C2211yescharacteristic
23#19104765—21no—
24#19104765C2211yescharacteristic
25#19104765C2211yescharacteristic, complemented
26#19104760—21no—
27#19104760C2211yescharacteristic
28#19104760C2211yescharacteristic
29#19104759—21no—
30#19104756—21no—
31#19104755—21no—
32#19104755—21no—
33#19104755C2211yescharacteristic
34C2×#14579689—21no—
35C2×#14579689C2211yescharacteristic, complemented
36C2×#14579688C2211yescharacteristic
37C2×#14579686—21no—
38C2×#14579686C2211yescharacteristic
39SD16×D163—21no—
40#19104742C2211yescharacteristic
41#19104742C2211yescharacteristic, complemented
42#19104741C2211yescharacteristic
43#19104733C2211yescharacteristic, complemented
44#19104724C2211yescharacteristic
45#19104671C2211yescharacteristic, complemented
46#19104670C2211yescharacteristic, complemented
47#19104662C2211yescharacteristic
48#19104662C2211yescharacteristic, complemented
49#19104662C2211yescharacteristic, complemented
50#19104661C2211yescharacteristic
51#19104654C2211yescharacteristic, complemented
52#19104654C2211yescharacteristic, complemented
53#19104653C2211yescharacteristic, complemented
54#19104651C2211yescharacteristic
55#19104645C2211yescharacteristic, complemented
562−1+4×C163C2211yescharacteristic
572−1+4×C163C2211yescharacteristic, complemented
58#137×C163—21no—
59#137×C163—21no—
60#137×C163C2211yescharacteristic
61#137×C163C2211yescharacteristic, complemented
62#135×C163—21no—
63#135×C163C2211yescharacteristic
64C2×Q16×C163—21no—
65C2×Q16×C163C2211yescharacteristic, complemented
66#131×C163C2211yescharacteristic, complemented
67#14579700—21no—
68#14579699—21no—
69#14579699—21no—
70#14579699C2311yescharacteristic
71#14579698—21no—
72#14579698—41no—
73C2×Dic326—21no—
74C2×Dic326C2311yescharacteristic
75Q8×D163—21no—
76C2×D652—21no—
77C2×D652C2311yescharacteristic
78D4×D163—21no—
79D4×D163—41no—
80#14579689—21no—
81#14579689—21no—
82#14579689—21no—
83#14579689—21no—
84#14579689—21no—
85#14579689—21no—
86#14579689D411yescharacteristic
87#14579688—21no—
88#14579688—21no—
89#14579687—21no—
90#14579687—21no—
91#14579687D411yescharacteristic
92#14579686—21no—
93#14579686—21no—
94#14579686—21no—
95#14579686—21no—
96#14579686—21no—
97#14579686—21no—
98#14579686D411yescharacteristic
99#14579684—21no—
100#14579684—21no—
101#14579684D411yescharacteristic
102#14579683—21no—
103#14579683—21no—
104#14579683C2311yescharacteristic
105#14579683C2311yescharacteristic
106#14579683C2311yescharacteristic
107#14579682—21no—
108C2×C163⋊C8—21no—
109C2×C163⋊C8—21no—
110C2×C163⋊C8C2311yescharacteristic
111C8×D163—21no—
112Dic163⋊C4—41no—
113(C2×D163)⋊C4—41no—
114C4×Dic163C2311yescharacteristic
115#41×C163—21no—
116#41×C163—21no—
117#41×C163—21no—
118#41×C163—21no—
119#41×C163—41no—
120#41×C163—41no—
121#41×C163C2311yescharacteristic
122#41×C163C2311yescharacteristic
123#41×C163C2311yescharacteristic
124C2×Q8×C163—21no—
125C2×Q8×C163—21no—
126C2×Q8×C163—21no—
127C2×Q8×C163—21no—
128C2×Q8×C163C2311yescharacteristic
129C2×Q8×C163C2311yescharacteristic
130Q16×C163—21no—
131Q16×C163—21no—
132Q16×C163—21no—
133Q16×C163—21no—
134Q16×C163D411yescharacteristic
135SD16×C163—21no—
136SD16×C163—21no—
137SD16×C163D411yescharacteristic
138SD16×C163D411yescharacteristic
139D8×C163D411yescharacteristic
140M16×C163—21no—
141M16×C163C2311yescharacteristic
142C2×C8×C163—21no—
143C2×C8×C163C2311yescharacteristic
144C22×D163—21no—
145C22×D163—41no—
146Dic326—21no—
147Dic326—21no—
148Dic326—41no—
149Dic326C2×D411yescharacteristic
150D652—21no—
151D652—21no—
152D652—41no—
153D652C2×D411yescharacteristic
154C22⋊D163—21no—
155C22⋊D163—41no—
156C22⋊D163—41no—
157C2×Dic163—21no—
158C2×Dic163—21no—
159C2×Dic163—41no—
160C4×D163—21no—
161C4×D163—41no—
162C4×D163—41no—
163C163⋊C8—21no—
164C163⋊C8—21no—
165C163⋊C8—21no—
166C163⋊C8C2×D411yescharacteristic
167C163⋊C8C2×D411yescharacteristic
168Q8×C163—21no—
169Q8×C163—21no—
170Q8×C163—21no—
171Q8×C163—21no—
172Q8×C163—41no—
173Q8×C163—41no—
174Q8×C163C2×D411yescharacteristic
175Q8×C163C2×D411yescharacteristic
176Q8×C163C2×D411yescharacteristic
177D4×C163—21no—
178D4×C163—21no—
179D4×C163—21no—
180D4×C163—41no—
181D4×C163—41no—
182D4×C163C2×D411yescharacteristic
183D4×C163C2×D411yescharacteristic
184D4×C163C2×D411yescharacteristic
185C2×C4×C163—21no—
186C2×C4×C163—21no—
187C2×C4×C163—21no—
188C2×C4×C163—21no—
189C2×C4×C163—21no—
190C2×C4×C163—41no—
191C2×C4×C163—41no—
192C2×C4×C163—41no—
193C2×C4×C163—41no—
194C2×C4×C163C2411yescharacteristic
195C8×C163—21no—
196C8×C163C2×D411yescharacteristic
197C8×C163C2×D411yescharacteristic
198C22×C163—21no—
199C22×C163—21no—
200C22×C163—21no—
201C22×C163H(5,𝔽2)11yescharacteristic
202C2×D163—21no—
203C2×D163—41no—
204C2×D163—81no—
205C2×D163—161no—
206C4×C163—21no—
207C4×C163—21no—
208C4×C163—21no—
209C4×C163—41no—
210C4×C163—41no—
211C4×C163C22×D411yescharacteristic
212C4×C163C22×D411yescharacteristic
213Dic163—21no—
214Dic163—41no—
215Dic163—41no—
216C2×C163—21no—
217C2×C163—41no—
218C2×C163—41no—
219C2×C163—41no—
220C2×C163D4211yescharacteristic
221D163—41no—
222D163—81no—
223C163#328911yescharacteristic, complemented, Sylow
224#3289—1631nomaximal, Sylow
225#578—1631no—
226#578D16311yescharacteristic, complemented
227#577—1631no—
228#577—1631no—
229#497—1631no—
230#473—1631no—
231#473—1631no—
232#471—1631no—
233#471—1631no—
234#456—1631no—
235#456—1631no—
236#455—1631no—
237#454—1631no—
238#444—1631no—
239#444—1631no—
2402−1+4—1631no—
2412−1+4C2×D16311yescharacteristic
2422−1+4C2×D16311yescharacteristic, complemented
243H(5,𝔽2)—1631no—
244#137—21no—
245#137—21no—
246#137—3261no—
247#137—3261no—
248#137—3261no—
249#137—3261no—
250#137—1631no—
251#137—1631no—
252#137—1631no—
253#137—1631no—
254#137C2×D16311yescharacteristic
255#137C2×D16311yescharacteristic, complemented
256AGL(1,ℤ/8)—3261no—
257AGL(1,ℤ/8)—3261no—
258AGL(1,ℤ/8)—1631no—
259AGL(1,ℤ/8)—1631no—
260#135—21no—
261#135—3261no—
262#135—3261no—
263#135—3261no—
264#135—1631no—
265#135C2×D16311yescharacteristic
266C2×Q16—21no—
267C2×Q16—3261no—
268C2×Q16—1631no—
269C2×Q16C2×D16311yescharacteristic, complemented
270C2×SD16—3261no—
271C2×SD16—3261no—
272C2×SD16—1631no—
273C2×SD16—1631no—
274#131—1631no—
275#131—1631no—
276#131—1631no—
277#131C2×D16311yescharacteristic, complemented
278C4⋊Q8—1631no—
279#124—1631no—
280#108—1631no—
281#108—1631no—
282#104—1631no—
283#104—1631no—
284#104—1631no—
285#104—1631no—
286#101—1631no—
287#101—1631no—
288#101—1631no—
289#100—1631no—
290#97—1631no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295#41—41no—
296#41—41no—
297#41—3261no—
298#41—3261no—
299#41—3261no—
300#41—3261no—
301#41—6521no—
302#41—1631no—
303#41C22×D16311yescharacteristic
304#41C22×D16311yescharacteristic
305#41C22×D16311yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—3261no—
311C2×Q8—3261no—
312C2×Q8—1631no—
313C2×Q8C22×D16311yescharacteristic
314C2×Q8C22×D16311yescharacteristic
315C2×D4—3261no—
316C2×D4—3261no—
317C2×D4—6521no—
318C2×D4—1631no—
319Q16—21no—
320Q16—21no—
321Q16—21no—
322Q16—21no—
323Q16—3261no—
324Q16—3261no—
325Q16—3261no—
326Q16—3261no—
327Q16—3261no—
328Q16—3261no—
329Q16—1631no—
330Q16C22⋊D16311yescharacteristic
331SD16—21no—
332SD16—21no—
333SD16—3261no—
334SD16—3261no—
335SD16—3261no—
336SD16—3261no—
337SD16—3261no—
338SD16—3261no—
339SD16—3261no—
340SD16—3261no—
341SD16—3261no—
342SD16—3261no—
343SD16—1631no—
344SD16—1631no—
345SD16C22⋊D16311yescharacteristic
346SD16C22⋊D16311yescharacteristic
347D8—3261no—
348D8—3261no—
349D8—1631no—
350D8C22⋊D16311yescharacteristic
351M16—21no—
352M16—3261no—
353M16—3261no—
354M16—3261no—
355M16—1631no—
356M16—1631no—
357M16—1631no—
358M16C22×D16311yescharacteristic
359C2×C8—21no—
360C2×C8—3261no—
361C2×C8—3261no—
362C2×C8—3261no—
363C2×C8—1631no—
364C2×C8C22×D16311yescharacteristic
365C4⋊C4—6521no—
366C22⋊C4—6521no—
367C42—1631no—
368C23—3261no—
369C23—6521no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—41no—
376Q8—3261no—
377Q8—3261no—
378Q8—6521no—
379Q8—1631no—
380Q8D4×D16311yescharacteristic
381Q8C2×C22⋊D16311yescharacteristic
382Q8C2×C22⋊D16311yescharacteristic
383D4—21no—
384D4—21no—
385D4—21no—
386D4—41no—
387D4—41no—
388D4—3261no—
389D4—3261no—
390D4—3261no—
391D4—6521no—
392D4—6521no—
393D4—6521no—
394D4—1631no—
395D4D4×D16311yescharacteristic
396D4C2×C22⋊D16311yescharacteristic
397D4C2×C22⋊D16311yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—3261no—
408C2×C4—3261no—
409C2×C4—3261no—
410C2×C4—6521no—
411C2×C4—6521no—
412C2×C4—6521no—
413C2×C4C23×D16311yescharacteristic
414C8—21no—
415C8—3261no—
416C8—3261no—
417C8—3261no—
418C8—1631no—
419C8—1631no—
420C8C2×C22⋊D16311yescharacteristic
421C8C2×C22⋊D16311yescharacteristic
422C22—21no—
423C22—21no—
424C22—21no—
425C22—13041no—
426C22—26081no—
427C22—3261no—
428C22—6521no—
429C22#1910478011yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—3261no—
436C4—6521no—
437C4—6521no—
438C4C2×D4×D16311yescharacteristic
439C4C22×C22⋊D16311yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—13041no—
445C2—6521no—
446C2D4×C22⋊D16311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#578×C163C2
socleC2×C163D42
hypercenter#578D163
hypocenterC163#3289
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #23782480C2semidirect
2. #23782473C2semidirect
3. #23782472C2semidirect
4. #23782058C2semidirect
5. #23781938C2semidirect
6. #23781937C2semidirect
7. #23781930C2semidirect
8. #23781928C2semidirect
9. #23781848C2semidirect
10. #23781846C2semidirect
11. #23781841C2semidirect
12. #23781838C2semidirect
13. #23781787C2semidirect
14. #23781785C2semidirect
15. #578×C163C2semidirect
25. #19104765C22semidirect
35. C2×#14579689C22semidirect
41. #19104742C22semidirect
43. #19104733C22semidirect
45. #19104671C22semidirect
46. #19104670C22semidirect
48. #19104662C22semidirect
49. #19104662C22semidirect
51. #19104654C22semidirect
52. #19104654C22semidirect
53. #19104653C22semidirect
55. #19104645C22semidirect
57. 2−1+4×C163C22semidirect
61. #137×C163C22semidirect
65. C2×Q16×C163C22semidirect
66. #131×C163C22semidirect
223. C163#3289semidirect
226. #578D163semidirect
242. 2−1+4C2×D163semidirect
255. #137C2×D163semidirect
269. C2×Q16C2×D163semidirect
277. #131C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824802
#237824732
#237824722
#237820582
#237819382
#237819372
#237819302
#237819282
#237818482
#237818462
#237818412
#237818382
#237817872
#237817852
#578×C1632
#3289163