Finite Group Databasebeta

← #35224464

Subgroups — #35224464

Order 20864.

414 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782175C211yesmaximal, characteristic
2#117×D163C211yesmaximal, characteristic, complemented
3#23782009C211yesmaximal, characteristic, complemented
4#23782008C211yesmaximal, characteristic, complemented
5#23782007C211yesmaximal, characteristic, complemented
6#23782006C211yesmaximal, characteristic, complemented
7#23781988C211yesmaximal, characteristic, complemented
8#23781987C211yesmaximal, characteristic, complemented
9#23781986C211yesmaximal, characteristic, complemented
10#23781706C211yesmaximal, characteristic, complemented
11#23781704C211yesmaximal, characteristic, complemented
12#23781663C211yesmaximal, characteristic, complemented
13#23781663C211yesmaximal, characteristic, complemented
14#23781627C211yesmaximal, characteristic, complemented
15#482×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C22×C4×D163C2211yescharacteristic
18C2×#14579682C2211yescharacteristic
19C2×#14579682C2211yescharacteristic
20#19104717—21no—
21#19104715C2211yescharacteristic
22#19104714C2211yescharacteristic
23#19104706C2211yescharacteristic
24#19104703—21no—
25D4×Dic163—21no—
26(C22⋊D163)⋊C4—21no—
27#19104694C2211yescharacteristic
28#19104693C2211yescharacteristic
29#19104693C2211yescharacteristic
30#19104693C2211yescharacteristic
31#19104692—21no—
32#19104691—21no—
33C4⋊C4×D163C2211yescharacteristic
34C4⋊C4×D163C2211yescharacteristic
35C22⋊C4×D163—21no—
36C42×D163—21no—
37#19104669C2211yescharacteristic, complemented
38#19104668C2211yescharacteristic, complemented
39#19104668C2211yescharacteristic, complemented
40#19104667C2211yescharacteristic, complemented
41#19104666C2211yescharacteristic, complemented
42#19104666C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104658C2211yescharacteristic, complemented
46#19104658C2211yescharacteristic, complemented
47#19104656C2211yescharacteristic, complemented
48#19104656C2211yescharacteristic, complemented
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic
52#121×C163C2211yescharacteristic
53#117×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic, complemented
55#106×C163C2211yescharacteristic, complemented
56#102×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic
59C23×D163D411yescharacteristic
60#14579698—22no—
61#14579698—22no—
62C2×Dic326C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C22×Dic163—21no—
66C22×Dic163D411yescharacteristic
67C2×C4×D163—22no—
68C2×C4×D163—22no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163D411yescharacteristic
72C2×C4×D163D411yescharacteristic
73C2×C4×D163C2311yescharacteristic
74C2×C4×D163C2311yescharacteristic
75#14579682—22no—
76#14579682—22no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C4⋊Dic163—21no—
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82C4⋊Dic163C2311yescharacteristic
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93C4×Dic163—21no—
94C4×Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163—21no—
97C2×D4×C163—21no—
98C2×D4×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C42×C163—21no—
108C22×D163—22no—
109C22×D163—41no—
110C22×D163—41no—
111C22×D163C2×D411yescharacteristic
112C22×D163C2×D411yescharacteristic
113Dic326—21no—
114Dic326—21no—
115C22⋊D163—42no—
116C22⋊D163—42no—
117C2×Dic163—22no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—42no—
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C4×D163—22no—
126C4×D163—21no—
127C4×D163—21no—
128C4×D163—22no—
129C4×D163—42no—
130C4×D163—42no—
131C4×D163#4112yes—
132C4×D163#4112yes—
133C163⋊C8—21no—
134C163⋊C8—21no—
135C23×C163—21no—
136C23×C163C2×D411yescharacteristic
137D4×C163—21no—
138D4×C163—21no—
139D4×C163—42no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2411yescharacteristic
147C8×C163—21no—
148C8×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163C22×D411yescharacteristic
156C2×D163—21no—
157C2×D163—21no—
158C2×D163—21no—
159C2×D163—21no—
160C2×D163—21no—
161C2×D163—44no—
162C2×D163#12312yes—
163C4×C163—21no—
164C4×C163—41no—
165C4×C163—41no—
166C4×C163—41no—
167C4×C163C2×#4111yescharacteristic
168C4×C163C2×#4111yescharacteristic
169Dic163—21no—
170Dic163—41no—
171Dic163—41no—
172Dic163—41no—
173Dic163—41no—
174Dic163#12312yes—
175C2×C163—41no—
176C2×C163—81no—
177C2×C163C2×#12311yescharacteristic
178C2×C163#57511yescharacteristic
179C2×C163#57511yescharacteristic
180D163—22no—
181D163—41no—
182C163#309311yescharacteristic, complemented, Sylow
183#3093—1631nomaximal, Sylow
184#525—1631no—
185C2×#117—1631no—
186#485—1631no—
187#485—1631no—
188#485—1631no—
189#485—1631no—
190#482—1631no—
191#482—1631no—
192#482—1631no—
193#482D16311yescharacteristic, complemented
194#428—1631no—
195#428—1631no—
196#417—1631no—
197#417—1631no—
198#407—1631no—
199C2×#41—1631no—
200C23×C4—1631no—
201C2×M16—1631no—
202C2×M16—1631no—
203#123—3261no—
204C22⋊Q8—1631no—
205C22⋊Q8—1631no—
206#121—1631no—
207#121C2×D16311yescharacteristic
208#120—3261no—
209C4×D4—3261no—
210C4×D4—3261no—
211#117—3261no—
212#117—3261no—
213#117—1631no—
214#117—1631no—
215#117—1631no—
216#117—1631no—
217#117C2×D16311yescharacteristic
218C2×C4⋊C4—1631no—
219C2×C4⋊C4—1631no—
220C2×C22⋊C4—3261no—
221C2×C42—3261no—
222#107—1631no—
223#107—1631no—
224#107—1631no—
225#107C2×D16311yescharacteristic, complemented
226#106—1631no—
227#106—1631no—
228#106—1631no—
229#106C2×D16311yescharacteristic, complemented
230#103—1631no—
231#103—1631no—
232#103—1631no—
233#103—1631no—
234#102—1631no—
235#102—1631no—
236#102C2×D16311yescharacteristic, complemented
237#102C2×D16311yescharacteristic, complemented
238C22⋊C8—1631no—
239C22⋊C8—1631no—
240C22⋊C8—1631no—
241C22⋊C8C2×D16311yescharacteristic
242C24—1631no—
243#41—3262no—
244#41—3262no—
245C2×Q8—1631no—
246C2×D4—21no—
247C2×D4—3261no—
248C2×D4—3261no—
249C2×D4C22×D16311yescharacteristic
250C22×C4—3262no—
251C22×C4—3262no—
252C22×C4—3261no—
253C22×C4—3261no—
254C22×C4—3261no—
255C22×C4—1631no—
256C22×C4—1631no—
257C22×C4—1631no—
258C22×C4—1631no—
259C22×C4—1631no—
260C22×C4C22×D16311yescharacteristic
261M16—3262no—
262M16—3262no—
263C2×C8—1631no—
264C2×C8—1631no—
265C2×C8C22×D16311yescharacteristic
266C2×C8C22×D16311yescharacteristic
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4—1631no—
274C4⋊C4C22×D16311yescharacteristic
275C4⋊C4C22×D16311yescharacteristic
276C4⋊C4C22×D16311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C42—21no—
287C42—3261no—
288C42—3261no—
289C42—3261no—
290C42—3261no—
291C23—21no—
292C23—3262no—
293C23—6521no—
294C23—6521no—
295C23—1631no—
296C23—1631no—
297C23D4×D16311yescharacteristic
298Q8—3261no—
299Q8—3261no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—6522no—
304D4—6522no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—3262no—
311C2×C4—3262no—
312C2×C4—3262no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—6522no—
320C2×C4—6522no—
321C2×C4—6522no—
322C2×C4—1631no—
323C2×C4—1631no—
324C2×C4—1632no—
325C2×C4—1632no—
326C2×C4D4×D16311yescharacteristic
327C2×C4C23×D16311yescharacteristic
328C8—21no—
329C8—21no—
330C8—3261no—
331C8—3261no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—3261no—
339C22—3261no—
340C22—3261no—
341C22—3261no—
342C22—3261no—
343C22—6524no—
344C22—1632no—
345C22C2×D4×D16311yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—3261no—
351C4—6521no—
352C4—6521no—
353C4—6521no—
354C4—6521no—
355C4—1632no—
356C4#41×D16311yescharacteristic
357C4#41×D16311yescharacteristic
358C2—41no—
359C2—81no—
360C2—3262no—
361C2—6521no—
362C2#123×D16311yescharacteristic
363C2#2378246111yescharacteristic
364C2#2378246211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#482×C163C2
socleC22×C163C22×D4
prime radical#23782175C2
hypercenter#482D163
hypocenterC163#3093
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #117×D163C2semidirect
3. #23782009C2semidirect
4. #23782008C2semidirect
5. #23782007C2semidirect
6. #23782006C2semidirect
7. #23781988C2semidirect
8. #23781987C2semidirect
9. #23781986C2semidirect
10. #23781706C2semidirect
11. #23781704C2semidirect
12. #23781663C2semidirect
13. #23781663C2semidirect
14. #23781627C2semidirect
15. #482×C163C2semidirect
37. #19104669C22semidirect
38. #19104668C22semidirect
39. #19104668C22semidirect
40. #19104667C22semidirect
41. #19104666C22semidirect
42. #19104666C22semidirect
43. #19104660C22semidirect
44. #19104660C22semidirect
45. #19104658C22semidirect
46. #19104658C22semidirect
47. #19104656C22semidirect
48. #19104656C22semidirect
54. #107×C163C22semidirect
55. #106×C163C22semidirect
56. #102×C163C22semidirect
57. #102×C163C22semidirect
182. C163#3093semidirect
193. #482D163semidirect
225. #107C2×D163semidirect
229. #106C2×D163semidirect
236. #102C2×D163semidirect
237. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237821752
#117×D1632
#237820092
#237820082
#237820072
#237820062
#237819882
#237819872
#237819862
#237817062
#237817042
#237816632
#237816272
#482×C1632
#3093163