Finite Group Databasebeta

← #35226356

Subgroups — #35226356

Order 20864.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782370C211yesmaximal, characteristic, complemented
2D4×C22⋊D163C211yesmaximal, characteristic
3#23782010C211yesmaximal, characteristic, complemented
4#23781983C211yesmaximal, characteristic, complemented
5#23781982C211yesmaximal, characteristic, complemented
6#23781953C211yesmaximal, characteristic, complemented
7#23781894C211yesmaximal, characteristic, complemented
8#23781893C211yesmaximal, characteristic, complemented
9#23781864C211yesmaximal, characteristic, complemented
10#23781817C211yesmaximal, characteristic, complemented
11#23781816C211yesmaximal, characteristic, complemented
12#23781756C211yesmaximal, characteristic, complemented
13C8×C22⋊D163C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15C2×#102×C163C211yesmaximal, characteristic, complemented
16C22×C22⋊D163—21no—
17C2×D4×D163C2211yescharacteristic
18C2×#14579688C2211yescharacteristic, complemented
19C2×#14579687C2211yescharacteristic
20C2×#14579687C2211yescharacteristic, complemented
21C2×C8×D163C2211yescharacteristic, complemented
22#19104734C2211yescharacteristic, complemented
23#19104731C2211yescharacteristic
24#19104725—21no—
25#19104715C2211yescharacteristic, complemented
26#19104714—21no—
27#19104714C2211yescharacteristic, complemented
28#19104713—21no—
29#19104709C2211yescharacteristic
30#19104707—21no—
31#19104706C2211yescharacteristic
32#19104704—21no—
33#19104703—21no—
34#19104700C2211yescharacteristic, complemented
35D4×Dic163C2211yescharacteristic
36C4×C22⋊D163—21no—
37C4×C22⋊D163C2211yescharacteristic
38C4⋊C4×D163C2211yescharacteristic, complemented
39#19104667C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41#19104666C2211yescharacteristic, complemented
42Dic163⋊C8C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic, complemented
47#19104655C2211yescharacteristic, complemented
48#19104655C2211yescharacteristic, complemented
49C22⋊(C163⋊C8)C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic, complemented
54C2×C4⋊C4×C163C2211yescharacteristic, complemented
55#102×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C2×Dic326—21no—
61C2×Dic326C2311yescharacteristic
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C22⋊D163—44no—
69C2×C22⋊D163D411yescharacteristic
70D4×D163—22no—
71D4×D163—41no—
72C22×Dic163—21no—
73C2×C4×D163—21no—
74C2×C4×D163C2311yescharacteristic
75#14579688—41no—
76#14579687—22no—
77#14579687—41no—
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C8×D163—41no—
81C4⋊Dic163—21no—
82C4⋊Dic163C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4D411yescharacteristic
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92C22⋊Dic163D411yescharacteristic
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4D411yescharacteristic
96C4×Dic163—21no—
97C4×Dic163C2311yescharacteristic
98C24×C163—21no—
99C2×D4×C163—22no—
100C2×D4×C163—21no—
101C2×D4×C163—41no—
102C2×D4×C163C2311yescharacteristic
103C2×D4×C163C2311yescharacteristic
104C22×C4×C163—21no—
105C22×C4×C163C2311yescharacteristic
106C2×C8×C163—41no—
107C2×C8×C163C2311yescharacteristic
108C2×C8×C163C2311yescharacteristic
109C4⋊C4×C163—21no—
110C4⋊C4×C163C2311yescharacteristic
111C4⋊C4×C163C2311yescharacteristic
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116C22×D163C2×D411yescharacteristic
117Dic326—21no—
118Dic326—21no—
119Dic326—81no—
120D652—21no—
121D652—21no—
122C22⋊D163—42no—
123C22⋊D163—42no—
124C22⋊D163—42no—
125C22⋊D163—42no—
126C22⋊D163—81no—
127C22⋊D163SD1614yescomplemented
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—21no—
131C2×Dic163—42no—
132C2×Dic163C2×D411yescharacteristic
133C2×Dic163C2×D411yescharacteristic
134C2×Dic163C2×D411yescharacteristic
135C4×D163—21no—
136C4×D163—21no—
137C4×D163—81no—
138C163⋊C8—21no—
139C163⋊C8—21no—
140C23×C163—21no—
141C23×C163—21no—
142C23×C163—42no—
143C23×C163—42no—
144C23×C163—81no—
145C23×C163C2×D411yescharacteristic
146D4×C163—21no—
147D4×C163—21no—
148D4×C163—21no—
149D4×C163—42no—
150D4×C163#4112yes—
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—21no—
154C2×C4×C163—21no—
155C2×C4×C163—81no—
156C2×C4×C163C2×D411yescharacteristic
157C2×C4×C163C2411yescharacteristic
158C8×C163—21no—
159C8×C163—21no—
160C22×C163—21no—
161C22×C163—21no—
162C22×C163—22no—
163C22×C163—41no—
164C22×C163—41no—
165C22×C163—41no—
166C22×C163—41no—
167C22×C163—41no—
168C22×C163—84no—
169C22×C163C2×SD1612yes—
170C22×C163C22×D411yescharacteristic
171C2×D163—21no—
172C2×D163—21no—
173C2×D163—41no—
174C2×D163—41no—
175C2×D163—41no—
176C2×D163—84no—
177C2×D163C2×SD1612yes—
178C4×C163—21no—
179C4×C163—41no—
180C4×C163—41no—
181C4×C163C2×#4111yescharacteristic
182C4×C163H(5,𝔽2)11yescharacteristic
183Dic163—21no—
184Dic163—21no—
185Dic163—41no—
186Dic163—41no—
187Dic163—41no—
188Dic163C2×SD1612yes—
189C2×C163—22no—
190C2×C163—42no—
191C2×C163—81no—
192C2×C163#54611yescharacteristic
193C2×C163C22×SD1611yescharacteristic
194C2×C163#57611yescharacteristic
195D163—22no—
196D163—81no—
197C163#329511yescharacteristic, complemented, Sylow
198#3295—1631nomaximal, Sylow
199#554—1631no—
200D42—1631no—
201#486—1631no—
202#481—1631no—
203#481—1631no—
204#476—1631no—
205#465—1631no—
206#465—1631no—
207#460—1631no—
208#450—1631no—
209#450—1631no—
210C4×SD16—1631no—
211C8×D4—1631no—
212C2×#102—1631no—
213C2×#102D16311yescharacteristic, complemented
214C22×D4—3261no—
215C22×D4—1631no—
216C22×D4C2×D16311yescharacteristic
217C2×SD16—1631no—
218C2×SD16—1631no—
219C2×SD16—1631no—
220C22×C8—1631no—
221C22×C8C2×D16311yescharacteristic, complemented
222C4⋊Q8—1631no—
223#127—1631no—
224#125—3261no—
225C22⋊Q8—3261no—
226C22⋊Q8—3261no—
227C22⋊Q8—1631no—
228C22⋊Q8—1631no—
229#121—3261no—
230#121—1631no—
231#121—1631no—
232#120—3261no—
233#120—3261no—
234C4×Q8—1631no—
235C4×D4—3261no—
236C4×D4—1631no—
237C4×D4—1631no—
238C2×C4⋊C4—1631no—
239C2×C4⋊C4C2×D16311yescharacteristic, complemented
240#106—1631no—
241#106—1631no—
242#106—1631no—
243C4⋊C8—1631no—
244#103—1631no—
245#102—1631no—
246#102—1631no—
247#102—1631no—
248#102—1631no—
249#102—1631no—
250#102C2×D16311yescharacteristic
251#102C2×D16311yescharacteristic
252#102C2×D16311yescharacteristic, complemented
253#102C2×D16311yescharacteristic, complemented
254C22⋊C8—1631no—
255C22⋊C8—1631no—
256C4×C8—1631no—
257C24—21no—
258C24—3261no—
259C2×Q8—3261no—
260C2×Q8—1631no—
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—3262no—
265C2×D4—3261no—
266C2×D4—3261no—
267C2×D4—3261no—
268C2×D4—3261no—
269C2×D4—3261no—
270C2×D4—6521no—
271C2×D4—6524no—
272C2×D4—1631no—
273C2×D4—1631no—
274C2×D4C22×D16311yescharacteristic
275C2×D4C22×D16311yescharacteristic
276C22×C4—21no—
277C22×C4—3261no—
278C22×C4—3261no—
279C22×C4—1631no—
280C22×C4C22×D16311yescharacteristic
281SD16—3262no—
282SD16—6521no—
283SD16—6521no—
284C2×C8—41no—
285C2×C8—6521no—
286C2×C8—1631no—
287C2×C8—1631no—
288C2×C8C22×D16311yescharacteristic
289C2×C8C22×D16311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—3261no—
292C4⋊C4—3261no—
293C4⋊C4—3261no—
294C4⋊C4—3261no—
295C4⋊C4—3261no—
296C4⋊C4—1631no—
297C4⋊C4—1631no—
298C4⋊C4—1631no—
299C4⋊C4C22×D16311yescharacteristic
300C4⋊C4C22×D16311yescharacteristic
301C22⋊C4—3261no—
302C22⋊C4—3261no—
303C22⋊C4—3261no—
304C22⋊C4—3261no—
305C22⋊C4—3261no—
306C22⋊C4—1631no—
307C22⋊C4—1631no—
308C42—3261no—
309C42—1631no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—13041no—
316C23—3261no—
317C23—6522no—
318C23—6522no—
319C23—1631no—
320C23D4×D16311yescharacteristic
321Q8—13041no—
322Q8—3261no—
323Q8—3261no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—13041no—
329D4—3261no—
330D4—3261no—
331D4—6522no—
332D4—6522no—
333D4—6522no—
334D4—6522no—
335D4—1634no—
336D4#1457969912yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—81no—
342C2×C4—13041no—
343C2×C4—3261no—
344C2×C4—3261no—
345C2×C4—3261no—
346C2×C4—3261no—
347C2×C4—3261no—
348C2×C4—6522no—
349C2×C4—1631no—
350C2×C4—1631no—
351C2×C4—1631no—
352C2×C4D4×D16311yescharacteristic
353C2×C4C23×D16311yescharacteristic
354C8—21no—
355C8—21no—
356C8—3261no—
357C8—3261no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—13044no—
368C22—3261no—
369C22—3261no—
370C22—6521no—
371C22—6521no—
372C22—6521no—
373C22—1632no—
374C22SD16×D16312yes—
375C22C2×D4×D16311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—3261no—
380C4—3261no—
381C4—6521no—
382C4—6521no—
383C4—6521no—
384C4—1632no—
385C4C2×#1457969911yescharacteristic
386C4#1910478011yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—13041no—
391C2—3262no—
392C2#2378230511yescharacteristic
393C2C2×SD16×D16311yescharacteristic
394C2#2378246611yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#102×C163C2
socleC22×C163C22×D4
prime radicalD4×C22⋊D163C2
hypercenterC2×#102D163
hypocenterC163#3295
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #23782370C2semidirect
3. #23782010C2semidirect
4. #23781983C2semidirect
5. #23781982C2semidirect
6. #23781953C2semidirect
7. #23781894C2semidirect
8. #23781893C2semidirect
9. #23781864C2semidirect
10. #23781817C2semidirect
11. #23781816C2semidirect
12. #23781756C2semidirect
13. C8×C22⋊D163C2semidirect
14. #102×D163C2semidirect
15. C2×#102×C163C2semidirect
18. C2×#14579688C22semidirect
20. C2×#14579687C22semidirect
21. C2×C8×D163C22semidirect
22. #19104734C22semidirect
25. #19104715C22semidirect
27. #19104714C22semidirect
34. #19104700C22semidirect
38. C4⋊C4×D163C22semidirect
39. #19104667C22semidirect
40. #19104666C22semidirect
41. #19104666C22semidirect
42. Dic163⋊C8C22semidirect
43. #19104660C22semidirect
44. #19104657C22semidirect
46. #19104656C22semidirect
47. #19104655C22semidirect
48. #19104655C22semidirect
49. C22⋊(C163⋊C8)C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C8×Dic163C22semidirect
53. C22×C8×C163C22semidirect
54. C2×C4⋊C4×C163C22semidirect
57. #102×C163C22semidirect
58. #102×C163C22semidirect
127. C22⋊D163SD16semidirect
197. C163#3295semidirect
213. C2×#102D163semidirect
221. C22×C8C2×D163semidirect
239. C2×C4⋊C4C2×D163semidirect
252. #102C2×D163semidirect
253. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823702
D4×C22⋊D1632
#237820102
#237819832
#237819822
#237819532
#237818942
#237818932
#237818642
#237818172
#237818162
#237817562
C8×C22⋊D1632
#102×D1632
C2×#102×C1632
#3295163