Finite Group Databasebeta

← #35224656

Subgroups — #35224656

Order 20864.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782212C211yesmaximal, characteristic
2C4×Q8×D163C211yesmaximal, characteristic, complemented
3#23781980C211yesmaximal, characteristic, complemented
4#23781977C211yesmaximal, characteristic, complemented
5#23781959C211yesmaximal, characteristic, complemented
6#23781958C211yesmaximal, characteristic, complemented
7#23781864C211yesmaximal, characteristic, complemented
8#23781863C211yesmaximal, characteristic, complemented
9#23781805C211yesmaximal, characteristic, complemented
10#23781802C211yesmaximal, characteristic, complemented
11#23781767C211yesmaximal, characteristic, complemented
12#23781765C211yesmaximal, characteristic, complemented
13#23781763C211yesmaximal, characteristic, complemented
14#23781720C211yesmaximal, characteristic, complemented
15#440×C163C211yesmaximal, characteristic, complemented
16C2×#14579699—21no—
17C2×Q8×D163C2211yescharacteristic
18C2×#14579688C2211yescharacteristic
19C2×#14579687C2211yescharacteristic
20C2×#14579686C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104731C2211yescharacteristic
23#19104721—21no—
24#19104710C2211yescharacteristic
25#19104708—21no—
26#19104706C2211yescharacteristic
27Q8×Dic163C2211yescharacteristic
28C4×Dic326—21no—
29C4×Dic326C2211yescharacteristic
30#19104700—21no—
31#19104700C2211yescharacteristic
32C4×D652C2211yescharacteristic
33C4×C22⋊D163—21no—
34C4⋊C4×D163—21no—
35C4⋊C4×D163C2211yescharacteristic
36C42×D163—21no—
37C42×D163C2211yescharacteristic
38#19104669C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104668C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104655C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104646C2211yescharacteristic
52C2×SD16×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59#14579699—44no—
60C2×Dic326—21no—
61C2×Dic326—21no—
62C2×Dic326C2311yescharacteristic
63Q8×D163—22no—
64Q8×D163—22no—
65C2×D652—21no—
66C2×D652C2311yescharacteristic
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163D411yescharacteristic
74C2×C4×D163D411yescharacteristic
75C2×C4×D163C2311yescharacteristic
76#14579688—22no—
77#14579687—22no—
78#14579686—22no—
79C2×C163⋊C8C2311yescharacteristic
80C2×C163⋊C8C2311yescharacteristic
81C4⋊Dic163—21no—
82C4⋊Dic163C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88C22⋊Dic163—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163—21no—
94C4×Dic163D411yescharacteristic
95C4×Dic163D411yescharacteristic
96C4×Dic163C2311yescharacteristic
97C2×Q8×C163C2311yescharacteristic
98C2×D4×C163C2311yescharacteristic
99C22×C4×C163—21no—
100SD16×C163—22no—
101C2×C8×C163C2311yescharacteristic
102C2×C8×C163C2311yescharacteristic
103C4⋊C4×C163—21no—
104C4⋊C4×C163C2311yescharacteristic
105C4⋊C4×C163C2311yescharacteristic
106C22⋊C4×C163—21no—
107C42×C163—21no—
108C42×C163C2311yescharacteristic
109C22×D163—21no—
110C22×D163C2×D411yescharacteristic
111Dic326—24no—
112Dic326—21no—
113Dic326—42no—
114Dic326#4112yescomplemented
115D652—21no—
116D652—21no—
117D652—42no—
118C22⋊D163—42no—
119C22⋊D163—42no—
120C22⋊D163—42no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C4×D163—24no—
128C4×D163—22no—
129C4×D163—21no—
130C4×D163—21no—
131C4×D163—22no—
132C4×D163—42no—
133C4×D163—42no—
134C163⋊C8—21no—
135C163⋊C8—21no—
136C23×C163—21no—
137Q8×C163—21no—
138Q8×C163#4112yescomplemented
139D4×C163—21no—
140D4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—42no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C8×C163—21no—
149C8×C163—21no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163C22×D411yescharacteristic
154C2×D163—21no—
155C2×D163—21no—
156C2×D163—41no—
157C2×D163—41no—
158C2×D163—41no—
159C2×D163AGL(1,ℤ/8)12yes—
160C4×C163—22no—
161C4×C163—21no—
162C4×C163—21no—
163C4×C163—41no—
164C4×C163—41no—
165C4×C163C2×#4111yescharacteristic
166C4×C163C2×#4111yescharacteristic
167Dic163—22no—
168Dic163—21no—
169Dic163—21no—
170Dic163—41no—
171Dic163—41no—
172Dic163#13712yes—
173C2×C163—81no—
174C2×C163#52511yescharacteristic
175C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C163C2×#13711yescharacteristic
177D163—22no—
178D163—81no—
179C163#311211yescharacteristic, complemented, Sylow
180#3112—1631nomaximal, Sylow
181#532—1631no—
182C2×C4×Q8—1631no—
183#480—1631no—
184#480—1631no—
185#477—1631no—
186#477—1631no—
187#460—1631no—
188#460—1631no—
189#448—1631no—
190#448—1631no—
191#440—1631no—
192#440—1631no—
193#440—1631no—
194#440D16311yescharacteristic, complemented
195#432—1631no—
196C2×#41—3261no—
197C22×Q8—1631no—
198C2×SD16—1631no—
199C2×SD16—1631no—
200C2×SD16—1631no—
201C2×SD16C2×D16311yescharacteristic
202C4⋊Q8—1631no—
203#127—1631no—
204#124—3261no—
205#121—3261no—
206#121—1631no—
207#121—1631no—
208C4×Q8—3261no—
209C4×Q8—3261no—
210C4×Q8—1631no—
211C4×Q8—1631no—
212C4×Q8—1631no—
213C4×Q8C2×D16311yescharacteristic
214C4×D4—3261no—
215C4×D4—1631no—
216C4×D4C2×D16311yescharacteristic
217C2×C4⋊C4—3261no—
218C2×C4⋊C4—1631no—
219C2×C42—3261no—
220C2×C42—1631no—
221#107—1631no—
222#107—1631no—
223#107—1631no—
224#107C2×D16311yescharacteristic
225C4⋊C8—1631no—
226C4⋊C8—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103C2×D16311yescharacteristic
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#102C2×D16311yescharacteristic
235C22⋊C8—1631no—
236C22⋊C8—1631no—
237#97—1631no—
238#97C2×D16311yescharacteristic
239#41—6524no—
240C2×Q8—3261no—
241C2×Q8—3261no—
242C2×Q8—3262no—
243C2×Q8—3262no—
244C2×Q8—1631no—
245C2×Q8C22×D16311yescharacteristic
246C2×D4—3261no—
247C2×D4—3261no—
248C2×D4—3261no—
249C2×D4—3261no—
250C2×D4—1631no—
251C2×D4C22×D16311yescharacteristic
252C22×C4—21no—
253C22×C4—3261no—
254C22×C4—3261no—
255C22×C4—3261no—
256C22×C4—1631no—
257C22×C4—1631no—
258C22×C4—1631no—
259SD16—22no—
260SD16—3262no—
261SD16—3262no—
262SD16—3262no—
263C2×C8—1631no—
264C2×C8—1631no—
265C2×C8C22×D16311yescharacteristic
266C2×C8C22×D16311yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—3261no—
272C4⋊C4—3261no—
273C4⋊C4—1631no—
274C4⋊C4—1631no—
275C4⋊C4C22×D16311yescharacteristic
276C4⋊C4C22×D16311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C42—21no—
282C42—3261no—
283C42—3261no—
284C42—3261no—
285C42—1631no—
286C42—1631no—
287C42—1631no—
288C42C22×D16311yescharacteristic
289C23—21no—
290C23—3261no—
291C23—1631no—
292Q8—21no—
293Q8—3264no—
294Q8—3261no—
295Q8—6522no—
296Q8—1632no—
297Q8#1457969912yescomplemented
298D4—21no—
299D4—21no—
300D4—3261no—
301D4—3261no—
302D4—6522no—
303D4—6522no—
304D4—6522no—
305D4—6522no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—3264no—
311C2×C4—3262no—
312C2×C4—3262no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—6522no—
319C2×C4—6522no—
320C2×C4—1631no—
321C2×C4—1631no—
322C2×C4—1631no—
323C2×C4D4×D16311yescharacteristic
324C2×C4D4×D16311yescharacteristic
325C2×C4C23×D16311yescharacteristic
326C8—21no—
327C8—21no—
328C8—3261no—
329C8—3261no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—3261no—
334C22—3261no—
335C22—6521no—
336C22—6521no—
337C22—6521no—
338C22—1632no—
339C22C2×D4×D16311yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—3262no—
346C4—3261no—
347C4—3261no—
348C4—6521no—
349C4—6521no—
350C4—1632no—
351C4#41×D16311yescharacteristic
352C4C2×#1457969911yescharacteristic
353C2—81no—
354C2—13041no—
355C2—3262no—
356C2#2378217411yescharacteristic
357C2AGL(1,ℤ/8)×D16311yescharacteristic
358C2#137×D16311yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#440×C163C2
socleC22×C163C22×D4
prime radical#23782212C2
hypercenter#440D163
hypocenterC163#3112
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D163C2semidirect
3. #23781980C2semidirect
4. #23781977C2semidirect
5. #23781959C2semidirect
6. #23781958C2semidirect
7. #23781864C2semidirect
8. #23781863C2semidirect
9. #23781805C2semidirect
10. #23781802C2semidirect
11. #23781767C2semidirect
12. #23781765C2semidirect
13. #23781763C2semidirect
14. #23781720C2semidirect
15. #440×C163C2semidirect
114. Dic326#41semidirect
138. Q8×C163#41semidirect
179. C163#3112semidirect
194. #440D163semidirect
297. Q8#14579699semidirect

Maximal subgroups

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

subgroupindex
#237822122
C4×Q8×D1632
#237819802
#237819772
#237819592
#237819582
#237818642
#237818632
#237818052
#237818022
#237817672
#237817652
#237817632
#237817202
#440×C1632
#3112163