Finite Group Databasebeta

← #35224713

Subgroups — #35224713

Order 20864.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782282C211yesmaximal, characteristic, complemented
2#23782140C211yesmaximal, characteristic, complemented
3#23782007C211yesmaximal, characteristic, complemented
4#23781986C211yesmaximal, characteristic, complemented
5#23781953C211yesmaximal, characteristic, complemented
6#23781943C211yesmaximal, characteristic, complemented
7#23781878C211yesmaximal, characteristic, complemented
8#23781878C211yesmaximal, characteristic, complemented
9#23781816C211yesmaximal, characteristic, complemented
10#23781777C211yesmaximal, characteristic, complemented
11#23781762C211yesmaximal, characteristic, complemented
12#23781756C211yesmaximal, characteristic, complemented
13D8×Dic163C211yesmaximal, characteristic, complemented
14#23781723C211yesmaximal, characteristic, complemented
15#447×C163C211yesmaximal, characteristic
16C2×#14579687C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18#19104734C2211yescharacteristic, complemented
19#19104728—21no—
20#19104722—21no—
21#19104722C2211yescharacteristic, complemented
22#19104716—21no—
23#19104714C2211yescharacteristic
24#19104712—21no—
25#19104700C2211yescharacteristic, complemented
26D4×Dic163—21no—
27D4×Dic163—21no—
28D4×Dic163C2211yescharacteristic
29D4×Dic163C2211yescharacteristic
30D4×Dic163C2211yescharacteristic
31D4×Dic163C2211yescharacteristic
32D4×Dic163C2211yescharacteristic, complemented
33#19104694C2211yescharacteristic
34#19104694C2211yescharacteristic
35C2×C22⋊Dic163—21no—
36C2×C22⋊Dic163—21no—
37#19104669C2211yescharacteristic, complemented
38#19104668C2211yescharacteristic, complemented
39#19104667C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42Dic163⋊C8C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic, complemented
48#19104656C2211yescharacteristic, complemented
49C22⋊(C163⋊C8)C2211yescharacteristic
50#19104645C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C2×D8×C163C2211yescharacteristic
54C2×D8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61C22×Dic163—21no—
62C22×Dic163—21no—
63C22×Dic163—21no—
64C22×Dic163—21no—
65C22×Dic163—21no—
66#14579687—22no—
67#14579687—22no—
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C4⋊Dic163D411yescharacteristic
71C4⋊Dic163D411yescharacteristic
72C4⋊Dic163C2311yescharacteristic
73C4⋊Dic163C2311yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78C22⋊Dic163—21no—
79C22⋊Dic163—21no—
80C22⋊Dic163—21no—
81C22⋊Dic163—21no—
82C22⋊Dic163—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163D411yescharacteristic
88C22⋊Dic163D411yescharacteristic
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163C2311yescharacteristic
92C4×Dic163C2311yescharacteristic
93C24×C163—21no—
94C2×D4×C163—22no—
95C2×D4×C163—22no—
96C2×D4×C163—21no—
97C2×D4×C163—21no—
98C2×D4×C163C2311yescharacteristic
99C2×D4×C163C2311yescharacteristic
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163C2311yescharacteristic
102D8×C163—22no—
103D8×C163—22no—
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108Dic326—21no—
109Dic326—21no—
110Dic326—81no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—81no—
116C2×Dic163—81no—
117C2×Dic163—81no—
118C2×Dic163—81no—
119C2×Dic163—81no—
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C163⋊C8—21no—
125C163⋊C8—21no—
126C23×C163—24no—
127C23×C163—21no—
128C23×C163—21no—
129C23×C163—21no—
130C23×C163—81no—
131C23×C163—81no—
132C23×C163C2×D411yescharacteristic
133D4×C163—24no—
134D4×C163—21no—
135D4×C163—21no—
136D4×C163—21no—
137D4×C163—21no—
138D4×C163—81no—
139D4×C163#4112yes—
140D4×C163#4112yes—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2411yescharacteristic
146C8×C163—21no—
147C8×C163—21no—
148C22×C163—21no—
149C22×C163—22no—
150C22×C163—22no—
151C22×C163—21no—
152C22×C163—21no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—84no—
161C22×C163C22×D411yescharacteristic
162C4×C163—21no—
163C4×C163—41no—
164C4×C163C2×#4111yescharacteristic
165C4×C163C2×#4111yescharacteristic
166Dic163—21no—
167Dic163—21no—
168Dic163—21no—
169Dic163—21no—
170Dic163—41no—
171Dic163—41no—
172Dic163—41no—
173Dic163—41no—
174C2×C163—41no—
175C2×C163—42no—
176C2×C163—42no—
177C2×C163—81no—
178C2×C163#52511yescharacteristic
179C2×C163#57611yescharacteristic
180C2×C163#57711yescharacteristic
181C163#311811yescharacteristic, complemented, Sylow
182#3118—1631nomaximal, Sylow
183#542—1631no—
184#518—1631no—
185#485—1631no—
186#482—1631no—
187#476—1631no—
188#474—1631no—
189#463—1631no—
190#463—1631no—
191#450—1631no—
192#447D16311yescharacteristic
193#442—1631no—
194#440—1631no—
195C4×SD16—1631no—
196C4×D8—1631no—
197#433—1631no—
198C22×D4C2×D16311yescharacteristic
199C2×SD16—1631no—
200C2×SD16—1631no—
201C2×D8C2×D16311yescharacteristic
202C2×D8C2×D16311yescharacteristic
203C4⋊Q8—1631no—
204#126—3261no—
205#124—3261no—
206#124—1631no—
207#123—3261no—
208C22⋊Q8—3261no—
209C22⋊Q8—1631no—
210#121C2×D16311yescharacteristic
211C4×Q8—1631no—
212C4×D4—3261no—
213C4×D4—3261no—
214C4×D4—1631no—
215C4×D4—1631no—
216C4×D4—1631no—
217C4×D4—1631no—
218C4×D4—1631no—
219#117—1631no—
220#117—1631no—
221C2×C22⋊C4—3261no—
222C2×C22⋊C4—3261no—
223#107—1631no—
224#107—1631no—
225#106—1631no—
226#106—1631no—
227C4⋊C8—1631no—
228C4⋊C8—1631no—
229#103—1631no—
230#103—1631no—
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#102—1631no—
235#102C2×D16311yescharacteristic
236#102C2×D16311yescharacteristic
237C22⋊C8—1631no—
238C22⋊C8C2×D16311yescharacteristic
239#97—1631no—
240C4×C8—1631no—
241C24—21no—
242C2×Q8—3261no—
243C2×Q8—1631no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22×D16311yescharacteristic
249C2×D4C22×D16311yescharacteristic
250C2×D4C22×D16311yescharacteristic
251C22×C4—3261no—
252C22×C4—3261no—
253C22×C4—3261no—
254C22×C4—3261no—
255C22×C4—3261no—
256C22×C4C22×D16311yescharacteristic
257SD16—3262no—
258SD16—3262no—
259D8—22no—
260D8—22no—
261C2×C8—1631no—
262C2×C8—1631no—
263C2×C8C22×D16311yescharacteristic
264C2×C8C22×D16311yescharacteristic
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—1631no—
269C4⋊C4—1631no—
270C4⋊C4—1631no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4C22×D16311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—1631no—
285C22⋊C4—1631no—
286C42—3261no—
287C42—3261no—
288C42—1631no—
289C42—1631no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—81no—
295C23—81no—
296C23D4×D16311yescharacteristic
297Q8—13041no—
298Q8—3261no—
299Q8—3261no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—21no—
305D4—81no—
306D4#1457969812yes—
307D4#1457969812yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—13041no—
312C2×C4—13041no—
313C2×C4—13041no—
314C2×C4—13041no—
315C2×C4—13041no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—1631no—
321C2×C4—1631no—
322C2×C4—1631no—
323C2×C4—1631no—
324C2×C4D4×D16311yescharacteristic
325C2×C4C23×D16311yescharacteristic
326C8—21no—
327C8—21no—
328C8—3261no—
329C8—3261no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—84no—
343C22C2×D4×D16311yescharacteristic
344C4—21no—
345C4—41no—
346C4—3261no—
347C4—3261no—
348C4—3261no—
349C4—3261no—
350C4—6521no—
351C4—6521no—
352C4—6521no—
353C4—6521no—
354C4C2×#1457969811yescharacteristic
355C4C2×#1457969811yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—81no—
360C2#2378217111yescharacteristic
361C2#2378246611yescharacteristic
362C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting, prime radical#447×C163C2
socleC22×C163C22×D4
hypercenter#447D163
hypocenterC163#3118
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #23782282C2semidirect
2. #23782140C2semidirect
3. #23782007C2semidirect
4. #23781986C2semidirect
5. #23781953C2semidirect
6. #23781943C2semidirect
7. #23781878C2semidirect
8. #23781878C2semidirect
9. #23781816C2semidirect
10. #23781777C2semidirect
11. #23781762C2semidirect
12. #23781756C2semidirect
13. D8×Dic163C2semidirect
14. #23781723C2semidirect
18. #19104734C22semidirect
21. #19104722C22semidirect
25. #19104700C22semidirect
32. D4×Dic163C22semidirect
37. #19104669C22semidirect
38. #19104668C22semidirect
39. #19104667C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. Dic163⋊C8C22semidirect
43. #19104660C22semidirect
44. #19104660C22semidirect
47. #19104656C22semidirect
48. #19104656C22semidirect
50. #19104645C22semidirect
51. C8×Dic163C22semidirect
181. C163#3118semidirect

Maximal subgroups

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

subgroupindex
#237822822
#237821402
#237820072
#237819862
#237819532
#237819432
#237818782
#237818162
#237817772
#237817622
#237817562
D8×Dic1632
#237817232
#447×C1632
#3118163