Finite Group Databasebeta

← #35227620

Subgroups — #35227620

Order 20864.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782400C211yesmaximal, characteristic, complemented
2#23782055C211yesmaximal, characteristic, complemented
3#23782052C211yesmaximal, characteristic, complemented
4#23782047C211yesmaximal, characteristic, complemented
5#23781887C211yesmaximal, characteristic, complemented
6#23781884C211yesmaximal, characteristic, complemented
7#23781880C211yesmaximal, characteristic, complemented
8#23781877C211yesmaximal, characteristic, complemented
9#23781860C211yesmaximal, characteristic, complemented
10#23781857C211yesmaximal, characteristic, complemented
11#23781796C211yesmaximal, characteristic, complemented
12#23781765C211yesmaximal, characteristic, complemented
13#23781763C211yesmaximal, characteristic, complemented
14C4×#14579684C211yesmaximal, characteristic, complemented
15#559×C163C211yesmaximal, characteristic, complemented
16C2×#14579689C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20C2×#14579686C2211yescharacteristic
21C2×#14579684C2211yescharacteristic
22C2×#14579684C2211yescharacteristic
23C2×#14579684C2211yescharacteristic
24C2×#14579684C2211yescharacteristic
25#19104732C2211yescharacteristic
26#19104724—21no—
27#19104723C2211yescharacteristic
28#19104723C2211yescharacteristic
29#19104720—21no—
30#19104720—21no—
31C4×Dic326C2211yescharacteristic
32C4×D652C2211yescharacteristic
33C4×D652C2211yescharacteristic
34#19104697—21no—
35#19104668C2211yescharacteristic
36C4⋊(C163⋊C8)C2211yescharacteristic
37C4⋊(C163⋊C8)C2211yescharacteristic
38C4⋊(C163⋊C8)C2211yescharacteristic
39#19104659C2211yescharacteristic
40#19104658C2211yescharacteristic
41#19104656C2211yescharacteristic
42#19104656C2211yescharacteristic
43#19104655C2211yescharacteristic
44#19104655C2211yescharacteristic
45#19104646C2211yescharacteristic
46#19104646C2211yescharacteristic
47C4×C163⋊C8C2211yescharacteristic
48#127×C163C2211yescharacteristic
49#124×C163—21no—
50#124×C163C2211yescharacteristic
51#124×C163C2211yescharacteristic
52#123×C163—21no—
53#123×C163—21no—
54C4×Q8×C163C2211yescharacteristic
55C4×Q8×C163C2211yescharacteristic
56C4×D4×C163—21no—
57C4×D4×C163C2211yescharacteristic
58C4×D4×C163C2211yescharacteristic
59C2×Dic326C2311yescharacteristic
60C2×D652—21no—
61C2×D652—21no—
62C2×D652C2311yescharacteristic
63C2×D652C2311yescharacteristic
64C2×C4×D163—21no—
65C2×C4×D163—21no—
66#14579689—41no—
67#14579687—22no—
68#14579687—41no—
69#14579686—22no—
70#14579686—41no—
71#14579684—22no—
72#14579684—41no—
73#14579684—41no—
74#14579684—41no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80Dic163⋊C4—21no—
81(C2×D163)⋊C4—21no—
82(C2×D163)⋊C4—21no—
83(C2×D163)⋊C4—21no—
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87C4×Dic163—21no—
88C2×Q8×C163D411yescharacteristic
89C2×Q8×C163C2311yescharacteristic
90C2×D4×C163—21no—
91C2×D4×C163—21no—
92C2×D4×C163C2311yescharacteristic
93C2×D4×C163C2311yescharacteristic
94C22×C4×C163—21no—
95C22×C4×C163—21no—
96C4⋊C4×C163—21no—
97C4⋊C4×C163D411yescharacteristic
98C4⋊C4×C163D411yescharacteristic
99C4⋊C4×C163D411yescharacteristic
100C4⋊C4×C163C2311yescharacteristic
101C22⋊C4×C163—21no—
102C22⋊C4×C163—21no—
103C22⋊C4×C163—21no—
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C42×C163—21no—
108C42×C163C2311yescharacteristic
109C42×C163C2311yescharacteristic
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112C22×D163—21no—
113Dic326—21no—
114Dic326—21no—
115D652—21no—
116D652—21no—
117D652—21no—
118D652—21no—
119D652—42no—
120D652—81no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C4×D163—81no—
124C4×D163—81no—
125C163⋊C8—21no—
126C163⋊C8—21no—
127C163⋊C8—21no—
128C163⋊C8#4112yes—
129C23×C163—21no—
130C23×C163—21no—
131Q8×C163—21no—
132Q8×C163—21no—
133Q8×C163—41no—
134D4×C163—21no—
135D4×C163—21no—
136D4×C163—21no—
137D4×C163—21no—
138D4×C163—42no—
139D4×C163—81no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—81no—
143C2×C4×C163—81no—
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2×D411yescharacteristic
150C2×C4×C163C2411yescharacteristic
151C22×C163—41no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163C22×D411yescharacteristic
158C2×D163—41no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—41no—
162C2×D163—41no—
163C2×D163—41no—
164C4×C163—21no—
165C4×C163—21no—
166C4×C163—21no—
167C4×C163—21no—
168C4×C163—21no—
169C4×C163—21no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163C2×#4111yescharacteristic
173C4×C163H(5,𝔽2)11yescharacteristic
174Dic163—41no—
175Dic163—41no—
176C2×C163—81no—
177C2×C163—81no—
178C2×C163#55011yescharacteristic
179C2×C163#57511yescharacteristic
180C2×C163#57611yescharacteristic
181D163—81no—
182D163—81no—
183C163#340311yescharacteristic, complemented, Sylow
184#3403—1631nomaximal, Sylow
185#559—1631no—
186#559D16311yescharacteristic, complemented
187#496—1631no—
188#496—1631no—
189#495—1631no—
190#464—1631no—
191#464—1631no—
192#463—1631no—
193#463—1631no—
194#459—1631no—
195#459—1631no—
196#446—1631no—
197#440—1631no—
198#440—1631no—
199C4×D8—1631no—
200C2×Q16—1631no—
201C2×SD16—1631no—
202C2×SD16—1631no—
203C2×SD16—1631no—
204C2×SD16—1631no—
205C2×D8—1631no—
206C2×D8—1631no—
207C2×D8—1631no—
208C2×D8—1631no—
209#127—1631no—
210#127C2×D16311yescharacteristic
211#124—21no—
212#124—3261no—
213#124—1631no—
214#124—1631no—
215#124C2×D16311yescharacteristic
216#124C2×D16311yescharacteristic
217#123—21no—
218#123—21no—
219#123—3261no—
220#123—3261no—
221C4×Q8—1631no—
222C4×Q8C2×D16311yescharacteristic
223C4×Q8C2×D16311yescharacteristic
224C4×D4—21no—
225C4×D4—3261no—
226C4×D4—1631no—
227C4×D4—1631no—
228C4×D4C2×D16311yescharacteristic
229C4×D4C2×D16311yescharacteristic
230#107—1631no—
231C4⋊C8—1631no—
232C4⋊C8—1631no—
233C4⋊C8—1631no—
234#103—1631no—
235#103—1631no—
236#102—1631no—
237#102—1631no—
238#102—1631no—
239#102—1631no—
240#97—1631no—
241#97—1631no—
242C4×C8—1631no—
243C2×Q8—1631no—
244C2×Q8C22⋊D16311yescharacteristic
245C2×Q8C22×D16311yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—3261no—
249C2×D4—3261no—
250C2×D4—1631no—
251C2×D4—1631no—
252C2×D4C22×D16311yescharacteristic
253C2×D4C22×D16311yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—3261no—
257C22×C4—3261no—
258Q16—6521no—
259SD16—3262no—
260SD16—3262no—
261SD16—6521no—
262SD16—6521no—
263D8—3262no—
264D8—6521no—
265D8—6521no—
266D8—6521no—
267C2×C8—1631no—
268C2×C8—1631no—
269C2×C8—1631no—
270C2×C8—1631no—
271C4⋊C4—21no—
272C4⋊C4—3261no—
273C4⋊C4—1631no—
274C4⋊C4C22⋊D16311yescharacteristic
275C4⋊C4C22⋊D16311yescharacteristic
276C4⋊C4C22⋊D16311yescharacteristic
277C4⋊C4C22×D16311yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C22⋊C4—3261no—
287C22⋊C4—3261no—
288C22⋊C4—3261no—
289C22⋊C4—3261no—
290C42—21no—
291C42—3261no—
292C42C22×D16311yescharacteristic
293C42C22×D16311yescharacteristic
294C42C22×D16311yescharacteristic
295C23—21no—
296C23—21no—
297C23—3261no—
298C23—3261no—
299Q8—21no—
300Q8—21no—
301Q8—41no—
302Q8—3261no—
303Q8—3261no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—13041no—
311D4—3261no—
312D4—3261no—
313D4—3261no—
314D4—3261no—
315D4—6522no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—13041no—
321C2×C4—13041no—
322C2×C4—3261no—
323C2×C4—3261no—
324C2×C4C2×C22⋊D16311yescharacteristic
325C2×C4C2×C22⋊D16311yescharacteristic
326C2×C4C2×C22⋊D16311yescharacteristic
327C2×C4C2×C22⋊D16311yescharacteristic
328C2×C4C2×C22⋊D16311yescharacteristic
329C2×C4C2×C22⋊D16311yescharacteristic
330C2×C4C23×D16311yescharacteristic
331C8—3261no—
332C8—3261no—
333C8—3261no—
334C8—1632no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—6521no—
342C22—6521no—
343C22—6521no—
344C22—6521no—
345C22—6521no—
346C22—6521no—
347C22C22×C22⋊D16311yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—6521no—
357C4—6521no—
358C4#41×D16311yescharacteristic
359C4#1910478111yescharacteristic
360C2—81no—
361C2—81no—
362C2—13041no—
363C2—13041no—
364C2#2378233411yescharacteristic
365C2#2378246311yescharacteristic
366C2#2378246811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#559×C163C2
socleC22×C163C22×D4
hypercenter#559D163
hypocenterC163#3403
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #23782400C2semidirect
2. #23782055C2semidirect
3. #23782052C2semidirect
4. #23782047C2semidirect
5. #23781887C2semidirect
6. #23781884C2semidirect
7. #23781880C2semidirect
8. #23781877C2semidirect
9. #23781860C2semidirect
10. #23781857C2semidirect
11. #23781796C2semidirect
12. #23781765C2semidirect
13. #23781763C2semidirect
14. C4×#14579684C2semidirect
15. #559×C163C2semidirect
183. C163#3403semidirect
186. #559D163semidirect

Maximal subgroups

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

subgroupindex
#237824002
#237820552
#237820522
#237820472
#237818872
#237818842
#237818802
#237818772
#237818602
#237818572
#237817962
#237817652
#237817632
C4×#145796842
#559×C1632
#3403163