Finite Group Databasebeta

← #35224475

Subgroups — #35224475

Order 20864.

462 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×D163C211yesmaximal, characteristic
2#117×D163C211yesmaximal, characteristic, complemented
3#23781990C211yesmaximal, characteristic, complemented
4#23781989C211yesmaximal, characteristic, complemented
5#23781986C211yesmaximal, characteristic, complemented
6#23781985C211yesmaximal, characteristic, complemented
7#23781982C211yesmaximal, characteristic, complemented
8#23781980C211yesmaximal, characteristic, complemented
9#23781977C211yesmaximal, characteristic, complemented
10#23781706C211yesmaximal, characteristic, complemented
11#23781704C211yesmaximal, characteristic, complemented
12#23781668C211yesmaximal, characteristic, complemented
13#23781668C211yesmaximal, characteristic, complemented
14#23781627C211yesmaximal, characteristic, complemented
15#482×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163—21no—
17C2×D4×D163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19C2×#14579682C2211yescharacteristic
20C2×#14579682C2211yescharacteristic
21#19104710C2211yescharacteristic
22#19104709C2211yescharacteristic
23#19104708—21no—
24#19104707—21no—
25#19104706C2211yescharacteristic
26#19104694C2211yescharacteristic, complemented
27#19104693C2211yescharacteristic
28#19104693C2211yescharacteristic
29#19104692—21no—
30#19104691—21no—
31C4⋊C4×D163C2211yescharacteristic
32C4⋊C4×D163C2211yescharacteristic, complemented
33C4⋊C4×D163C2211yescharacteristic, complemented
34C22⋊C4×D163—21no—
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#19104657C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic, complemented
46#19104656C2211yescharacteristic, complemented
47#19104655C2211yescharacteristic, complemented
48#19104655C2211yescharacteristic, complemented
49C22⋊(C163⋊C8)C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51(C2×D163)⋊C8C2211yescharacteristic, complemented
52#121×C163C2211yescharacteristic
53#117×C163C2211yescharacteristic, complemented
54#107×C163C2211yescharacteristic, complemented
55#106×C163C2211yescharacteristic, complemented
56#102×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C23×D163D411yescharacteristic
61C2×D652—21no—
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66D4×D163—22no—
67D4×D163—22no—
68D4×D163—44no—
69C22×Dic163D411yescharacteristic
70C2×C4×D163—22no—
71C2×C4×D163—22no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163—21no—
75C2×C4×D163D411yescharacteristic
76C2×C4×D163D411yescharacteristic
77C2×C4×D163C2311yescharacteristic
78C2×C4×D163C2311yescharacteristic
79#14579682—22no—
80#14579682—22no—
81C2×C163⋊C8C2311yescharacteristic
82C2×C163⋊C8C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84C4⋊Dic163C2311yescharacteristic
85C4⋊Dic163C2311yescharacteristic
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95C4×Dic163—21no—
96C4×Dic163—21no—
97C4×Dic163—21no—
98C2×D4×C163—21no—
99C2×D4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C2×C8×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C4⋊C4×C163C2311yescharacteristic
106C22⋊C4×C163—21no—
107C22⋊C4×C163—21no—
108C42×C163—21no—
109C22×D163—22no—
110C22×D163—22no—
111C22×D163—21no—
112C22×D163—41no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—42no—
116C22×D163C2×D411yescharacteristic
117C22×D163C2×D411yescharacteristic
118D652—21no—
119D652—21no—
120D652—42no—
121C22⋊D163—42no—
122C22⋊D163—42no—
123C22⋊D163—42no—
124C2×Dic163—22no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C4×D163—22no—
131C4×D163—21no—
132C4×D163—21no—
133C4×D163—22no—
134C4×D163—42no—
135C4×D163—42no—
136C4×D163—42no—
137C4×D163#4112yes—
138C4×D163#4112yes—
139C163⋊C8—21no—
140C163⋊C8—21no—
141C23×C163—21no—
142C23×C163C2×D411yescharacteristic
143D4×C163—21no—
144D4×C163—21no—
145D4×C163—42no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2411yescharacteristic
153C8×C163—21no—
154C8×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163—21no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163C22×D411yescharacteristic
162C2×D163—21no—
163C2×D163—21no—
164C2×D163—21no—
165C2×D163—21no—
166C2×D163—44no—
167C2×D163—41no—
168C2×D163—41no—
169C2×D163—41no—
170C2×D163—84no—
171C2×D163#12312yes—
172C2×D163AGL(1,ℤ/8)12yes—
173C4×C163—21no—
174C4×C163—41no—
175C4×C163—41no—
176C4×C163—41no—
177C4×C163C2×#4111yescharacteristic
178C4×C163C2×#4111yescharacteristic
179Dic163—21no—
180Dic163—41no—
181Dic163—41no—
182Dic163—41no—
183Dic163#12312yes—
184C2×C163—41no—
185C2×C163—81no—
186C2×C163C2×#12311yescharacteristic
187C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C163#57511yescharacteristic
189D163—22no—
190D163—22no—
191D163—81no—
192C163#309411yescharacteristic, complemented, Sylow
193#3094—1631nomaximal, Sylow
194C2×#121—1631no—
195C2×#117—1631no—
196#482—1631no—
197#482—1631no—
198#482—1631no—
199#482D16311yescharacteristic, complemented
200#481—1631no—
201#481—1631no—
202#480—1631no—
203#480—1631no—
204#428—1631no—
205#428—1631no—
206#418—1631no—
207#418—1631no—
208#407—1631no—
209C22×D4—3261no—
210C22×D4—1631no—
211C23×C4—1631no—
212C2×M16—1631no—
213C2×M16—1631no—
214#121—3261no—
215#121—3261no—
216#121—1631no—
217#121—1631no—
218#121—1631no—
219#121C2×D16311yescharacteristic
220#117—3261no—
221#117—3261no—
222#117—1631no—
223#117—1631no—
224#117—1631no—
225#117C2×D16311yescharacteristic, complemented
226C2×C4⋊C4—1631no—
227C2×C4⋊C4—1631no—
228C2×C4⋊C4—1631no—
229C2×C22⋊C4—3261no—
230C2×C22⋊C4—3261no—
231C2×C42—3261no—
232#107—1631no—
233#107—1631no—
234#107—1631no—
235#107C2×D16311yescharacteristic, complemented
236#106—1631no—
237#106—1631no—
238#106—1631no—
239#106C2×D16311yescharacteristic, complemented
240#102—1631no—
241#102—1631no—
242#102—1631no—
243#102—1631no—
244#102—1631no—
245#102—1631no—
246#102C2×D16311yescharacteristic, complemented
247#102C2×D16311yescharacteristic, complemented
248C22⋊C8—1631no—
249C22⋊C8—1631no—
250C22⋊C8—1631no—
251C22⋊C8C2×D16311yescharacteristic, complemented
252C24—3261no—
253C24—1631no—
254C2×D4—21no—
255C2×D4—3261no—
256C2×D4—3261no—
257C2×D4—3261no—
258C2×D4—3262no—
259C2×D4—3262no—
260C2×D4—3261no—
261C2×D4—6524no—
262C2×D4—1631no—
263C2×D4C22×D16311yescharacteristic
264C22×C4—3262no—
265C22×C4—3262no—
266C22×C4—3261no—
267C22×C4—3261no—
268C22×C4—3261no—
269C22×C4—1631no—
270C22×C4—1631no—
271C22×C4—1631no—
272C22×C4—1631no—
273C22×C4—1631no—
274C22×C4C22×D16311yescharacteristic
275M16—3262no—
276M16—3262no—
277C2×C8—1631no—
278C2×C8—1631no—
279C2×C8C22×D16311yescharacteristic
280C2×C8C22×D16311yescharacteristic
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—3261no—
284C4⋊C4—1631no—
285C4⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4C22×D16311yescharacteristic
288C4⋊C4C22×D16311yescharacteristic
289C4⋊C4C22×D16311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—3261no—
296C22⋊C4—3261no—
297C22⋊C4—3261no—
298C42—21no—
299C42—3261no—
300C42—3261no—
301C42—3261no—
302C23—21no—
303C23—3262no—
304C23—3262no—
305C23—3261no—
306C23—6521no—
307C23—6522no—
308C23—6522no—
309C23—6522no—
310C23—1631no—
311C23—1631no—
312C23D4×D16311yescharacteristic
313D4—21no—
314D4—21no—
315D4—42no—
316D4—3261no—
317D4—3261no—
318D4—6522no—
319D4—6522no—
320D4—6522no—
321D4—6522no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—3262no—
328C2×C4—3262no—
329C2×C4—3262no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—3261no—
333C2×C4—3261no—
334C2×C4—3261no—
335C2×C4—6522no—
336C2×C4—6522no—
337C2×C4—6522no—
338C2×C4—1631no—
339C2×C4—1631no—
340C2×C4—1632no—
341C2×C4—1632no—
342C2×C4D4×D16311yescharacteristic
343C2×C4C23×D16311yescharacteristic
344C8—21no—
345C8—21no—
346C8—3261no—
347C8—3261no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—13044no—
355C22—3261no—
356C22—3261no—
357C22—3261no—
358C22—3261no—
359C22—6524no—
360C22—6521no—
361C22—6521no—
362C22—6521no—
363C22—1632no—
364C22—1632no—
365C22C2×D4×D16311yescharacteristic
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—3261no—
371C4—6521no—
372C4—6521no—
373C4—6521no—
374C4—1632no—
375C4#41×D16311yescharacteristic
376C4#41×D16311yescharacteristic
377C2—41no—
378C2—81no—
379C2—13041no—
380C2—3262no—
381C2—3262no—
382C2#123×D16311yescharacteristic
383C2AGL(1,ℤ/8)×D16311yescharacteristic
384C2#2378246211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#482×C163C2
socleC22×C163C22×D4
prime radical#121×D163C2
hypercenter#482D163
hypocenterC163#3094
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. #23781990C2semidirect
4. #23781989C2semidirect
5. #23781986C2semidirect
6. #23781985C2semidirect
7. #23781982C2semidirect
8. #23781980C2semidirect
9. #23781977C2semidirect
10. #23781706C2semidirect
11. #23781704C2semidirect
12. #23781668C2semidirect
13. #23781668C2semidirect
14. #23781627C2semidirect
15. #482×C163C2semidirect
26. #19104694C22semidirect
32. C4⋊C4×D163C22semidirect
33. C4⋊C4×D163C22semidirect
37. #19104669C22semidirect
38. #19104668C22semidirect
39. #19104668C22semidirect
40. #19104667C22semidirect
41. #19104666C22semidirect
42. #19104666C22semidirect
43. #19104657C22semidirect
44. #19104657C22semidirect
45. #19104656C22semidirect
46. #19104656C22semidirect
47. #19104655C22semidirect
48. #19104655C22semidirect
49. C22⋊(C163⋊C8)C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. (C2×D163)⋊C8C22semidirect
53. #117×C163C22semidirect
54. #107×C163C22semidirect
55. #106×C163C22semidirect
56. #102×C163C22semidirect
57. #102×C163C22semidirect
58. C22⋊C8×C163C22semidirect
192. C163#3094semidirect
199. #482D163semidirect
225. #117C2×D163semidirect
235. #107C2×D163semidirect
239. #106C2×D163semidirect
246. #102C2×D163semidirect
247. #102C2×D163semidirect
251. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#121×D1632
#117×D1632
#237819902
#237819892
#237819862
#237819852
#237819822
#237819802
#237819772
#237817062
#237817042
#237816682
#237816272
#482×C1632
#3094163