Finite Group Databasebeta

← #35223367

Subgroups — #35223367

Order 20864.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104762C211yesmaximal, characteristic
2#23782147C211yesmaximal, characteristic, complemented
3#23782137C211yesmaximal, characteristic, complemented
4#23781778C211yesmaximal, characteristic, complemented
5#23781774C211yesmaximal, characteristic, complemented
6#23781768C211yesmaximal, characteristic, complemented
7SD16×Dic163C211yesmaximal, characteristic, complemented
8C4×#14579686C211yesmaximal, characteristic, complemented
9#23781747C211yesmaximal, characteristic, complemented
10C4×#14579684C211yesmaximal, characteristic, complemented
11#23781706C211yesmaximal, characteristic, complemented
12#23781660C211yesmaximal, characteristic, complemented
13#102×D163C211yesmaximal, characteristic, complemented
14#23781622C211yesmaximal, characteristic, complemented
15#440×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18#19104762C418yescomplemented
19C2×#14579686C2211yescharacteristic
20C2×D1304C2211yescharacteristic
21C2×#14579684C2211yescharacteristic
22C2×#14579682C2211yescharacteristic
23C2×C8×D163C2211yescharacteristic
24Q8×Dic163C2211yescharacteristic, complemented
25C4×D652—21no—
26C4×D652C2211yescharacteristic
27C4×D652C2211yescharacteristic, complemented
28D4×Dic163C2211yescharacteristic
29#19104697—21no—
30#19104697C2211yescharacteristic
31#19104697C2211yescharacteristic, complemented
32C4×C22⋊D163—21no—
33(C22⋊D163)⋊C4—21no—
34#19104693C2211yescharacteristic
35#19104692—21no—
36#19104692C2211yescharacteristic
37C4⋊C4×D163—21no—
38C4⋊C4×D163C2211yescharacteristic
39C2×(C2×D163)⋊C4—21no—
40C22⋊C4×D163—21no—
41#19104668C2211yescharacteristic, complemented
42#19104667C2211yescharacteristic, complemented
43#19104666C2211yescharacteristic, complemented
44#19104659C2211yescharacteristic, complemented
45#19104657C2211yescharacteristic
46#19104657C2211yescharacteristic, complemented
47#19104656C2211yescharacteristic
48#19104655C2211yescharacteristic
49#19104655C2211yescharacteristic, complemented
50#19104645C2211yescharacteristic, complemented
51C4×C163⋊C8C2211yescharacteristic, complemented
52C8×Dic163C2211yescharacteristic, complemented
53C2×SD16×C163C2211yescharacteristic
54C4×Q8×C163C2211yescharacteristic, complemented
55C4×D4×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic
59#97×C163C2211yescharacteristic, complemented
60C23×D163—21no—
61#14579700—22no—
62#14579700C2×C414yescomplemented
63C2×D652—21no—
64C2×D652C2311yescharacteristic
65C2×D652C2311yescharacteristic
66C2×C22⋊D163—21no—
67D4×D163—22no—
68D4×D163C2×C414yescomplemented
69C22×Dic163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163—21no—
75C2×C4×D163—21no—
76C2×C4×D163C2311yescharacteristic
77#14579686C2×C414yescomplemented
78D1304C2×C414yescomplemented
79#14579684C2×C414yescomplemented
80#14579682C2×C414yescomplemented
81C2×C163⋊C8C2311yescharacteristic
82C2×C163⋊C8C2311yescharacteristic
83C8×D163—22no—
84C4⋊Dic163—21no—
85C4⋊Dic163C2311yescharacteristic
86C4⋊Dic163C2311yescharacteristic
87Dic163⋊C4—21no—
88Dic163⋊C4D411yescharacteristic
89Dic163⋊C4D411yescharacteristic
90C22⋊Dic163—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4D411yescharacteristic
97(C2×D163)⋊C4D411yescharacteristic
98C4×Dic163—21no—
99C4×Dic163—21no—
100C4×Dic163C2311yescharacteristic
101C2×Q8×C163C2311yescharacteristic
102C2×D4×C163C2311yescharacteristic
103C22×C4×C163—21no—
104SD16×C163C2×C414yescomplemented
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163—21no—
108C4⋊C4×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C42×C163—21no—
112C42×C163C2311yescharacteristic
113C22×D163—21no—
114C22×D163—21no—
115C22×D163—24no—
116C22×D163—81no—
117C22×D163—81no—
118C22×D163C2×D411yescharacteristic
119D652—24no—
120D652—21no—
121D652—21no—
122D652C22×C412yes—
123D652C22×C412yes—
124C22⋊D163—24no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—81no—
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C2×Dic163C2×D411yescharacteristic
131C4×D163—24no—
132C4×D163—81no—
133C4×D163—81no—
134C4×D163—81no—
135C4×D163—81no—
136C4×D163—81no—
137C4×D163C22×C412yes—
138C4×D163#4112yes—
139C163⋊C8—21no—
140C163⋊C8C22×C412yes—
141C23×C163—21no—
142Q8×C163—21no—
143Q8×C163C22×C412yes—
144D4×C163—21no—
145D4×C163C22×C412yes—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—81no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2411yescharacteristic
153C8×C163—21no—
154C8×C163C22×C412yes—
155C22×C163—22no—
156C22×C163—41no—
157C22×C163—41no—
158C22×C163C22×D411yescharacteristic
159C2×D163—21no—
160C2×D163—21no—
161C2×D163—22no—
162C2×D163—22no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—41no—
167C2×D163—84no—
168C2×D163C4×D412yes—
169C4×C163—22no—
170C4×C163—21no—
171C4×C163—21no—
172C4×C163—41no—
173C4×C163—41no—
174C4×C163C23×C411yescharacteristic
175C4×C163C2×#4111yescharacteristic
176Dic163—21no—
177Dic163—21no—
178Dic163—41no—
179Dic163—41no—
180Dic163C4×D412yes—
181C2×C163—42no—
182C2×C163C2×C4×D411yescharacteristic
183C2×C163#57611yescharacteristic
184C2×C163#57711yescharacteristic
185D163—22no—
186D163—42no—
187D163—42no—
188C163#294611yescharacteristic, complemented, Sylow
189#2946—1631nomaximal, Sylow
190C2×AGL(1,ℤ/8)—1631no—
191#520—1631no—
192#518—1631no—
193#442—1631no—
194#442—1631no—
195#440—1631no—
196#440D16311yescharacteristic, complemented
197C4×SD16—1631no—
198C4×SD16—1631no—
199C4×D8—1631no—
200C4×D8—1631no—
201#428—1631no—
202#416—1631no—
203C2×#102—1631no—
204#405—1631no—
205C2×#41—1631no—
206C22×D4—1631no—
207AGL(1,ℤ/8)—1638no—
208C2×SD16—1631no—
209C2×SD16C2×D16311yescharacteristic
210C2×D8—1631no—
211C2×D8—1631no—
212C2×M16—1631no—
213C22×C8—1631no—
214C4×Q8—1631no—
215C4×Q8C2×D16311yescharacteristic, complemented
216C4×D4—3261no—
217C4×D4—3261no—
218C4×D4—3261no—
219C4×D4—3261no—
220C4×D4—1631no—
221C4×D4—1631no—
222C4×D4—1631no—
223C4×D4—1631no—
224C4×D4—1631no—
225C4×D4C2×D16311yescharacteristic
226#117—3261no—
227#117—1631no—
228#117—1631no—
229C2×C4⋊C4—3261no—
230C2×C4⋊C4—1631no—
231C2×C22⋊C4—3261no—
232C2×C22⋊C4—3261no—
233#107—1631no—
234#107C2×D16311yescharacteristic, complemented
235#106—1631no—
236#106—1631no—
237#103—1631no—
238#103C2×D16311yescharacteristic, complemented
239#102—1631no—
240#102—1631no—
241#102—1631no—
242#102—1631no—
243#102—1631no—
244#102C2×D16311yescharacteristic
245#97—1631no—
246#97C2×D16311yescharacteristic, complemented
247C4×C8—1631no—
248C4×C8—1631no—
249C24—3261no—
250#41—3262no—
251#41—1634no—
252C2×Q8C22×D16311yescharacteristic
253C2×D4—3261no—
254C2×D4—3262no—
255C2×D4—3261no—
256C2×D4—1634no—
257C2×D4—1631no—
258C2×D4—1631no—
259C2×D4C22×D16311yescharacteristic
260C22×C4—21no—
261C22×C4—3261no—
262C22×C4—3261no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4—3261no—
266C22×C4—3261no—
267C22×C4—3261no—
268C22×C4—1631no—
269SD16—1634no—
270SD16C4×D16314yescomplemented
271D8—1634no—
272D8—1634no—
273M16—1634no—
274C2×C8—3262no—
275C2×C8—1631no—
276C2×C8—1631no—
277C2×C8C22×D16311yescharacteristic
278C2×C8C22×D16311yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—3261no—
281C4⋊C4—3261no—
282C4⋊C4—1631no—
283C4⋊C4—1631no—
284C4⋊C4—1631no—
285C4⋊C4—1631no—
286C4⋊C4C22×D16311yescharacteristic
287C4⋊C4C22×D16311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—1631no—
296C22⋊C4—1631no—
297C42—21no—
298C42—3261no—
299C42—3261no—
300C42—1631no—
301C42C22×D16311yescharacteristic
302C23—21no—
303C23—13041no—
304C23—13041no—
305C23—3264no—
306C23—3261no—
307C23—3261no—
308C23—1631no—
309Q8—21no—
310Q8C2×C4×D16312yes—
311D4—21no—
312D4—3264no—
313D4—3264no—
314D4—3261no—
315D4—3261no—
316D4—1632no—
317D4—1632no—
318D4C2×C4×D16312yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—13041no—
324C2×C4—13041no—
325C2×C4—13041no—
326C2×C4—13041no—
327C2×C4—13041no—
328C2×C4—13041no—
329C2×C4—3264no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—1631no—
333C2×C4—1631no—
334C2×C4—1631no—
335C2×C4—1632no—
336C2×C4—1632no—
337C2×C4D4×D16311yescharacteristic
338C2×C4D4×D16311yescharacteristic
339C2×C4C23×D16311yescharacteristic
340C8—21no—
341C8—3261no—
342C8—1632no—
343C8C2×C4×D16312yes—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—13044no—
348C22—3261no—
349C22—3261no—
350C22—3262no—
351C22—3262no—
352C22—6521no—
353C22—6521no—
354C22—6521no—
355C22—6521no—
356C22—1632no—
357C22C2×D4×D16311yescharacteristic
358C4—21no—
359C4—21no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—3261no—
364C4—3261no—
365C4—6521no—
366C4—6521no—
367C4—1632no—
368C4C22×C4×D16311yescharacteristic
369C4#41×D16311yescharacteristic
370C2—42no—
371C2—3262no—
372C2—6522no—
373C2—6522no—
374C2C4×D4×D16311yescharacteristic
375C2#2378246911yescharacteristic
376C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutatorC4×C163C23×C4
FrattiniC2×C4C23×D163
Fitting#440×C163C2
socleC22×C163C22×D4
prime radicalC2×#19104762C2
cosocleC2×C4×C163C24
hypercenter#440D163
hypocenterC163#2946
LCSG ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #23782147C2semidirect
3. #23782137C2semidirect
4. #23781778C2semidirect
5. #23781774C2semidirect
6. #23781768C2semidirect
7. SD16×Dic163C2semidirect
8. C4×#14579686C2semidirect
9. #23781747C2semidirect
10. C4×#14579684C2semidirect
11. #23781706C2semidirect
12. #23781660C2semidirect
13. #102×D163C2semidirect
14. #23781622C2semidirect
15. #440×C163C2semidirect
18. #19104762C4semidirect
24. Q8×Dic163C22semidirect
27. C4×D652C22semidirect
31. #19104697C22semidirect
41. #19104668C22semidirect
42. #19104667C22semidirect
43. #19104666C22semidirect
44. #19104659C22semidirect
46. #19104657C22semidirect
49. #19104655C22semidirect
50. #19104645C22semidirect
51. C4×C163⋊C8C22semidirect
52. C8×Dic163C22semidirect
54. C4×Q8×C163C22semidirect
56. #107×C163C22semidirect
57. #103×C163C22semidirect
59. #97×C163C22semidirect
62. #14579700C2×C4semidirect
68. D4×D163C2×C4semidirect
77. #14579686C2×C4semidirect
78. D1304C2×C4semidirect
79. #14579684C2×C4semidirect
80. #14579682C2×C4semidirect
104. SD16×C163C2×C4semidirect
188. C163#2946semidirect
196. #440D163semidirect
215. C4×Q8C2×D163semidirect
234. #107C2×D163semidirect
238. #103C2×D163semidirect
246. #97C2×D163semidirect
270. SD16C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047622
#237821472
#237821372
#237817782
#237817742
#237817682
SD16×Dic1632
C4×#145796862
#237817472
C4×#145796842
#237817062
#237816602
#102×D1632
#237816222
#440×C1632
#2946163