Finite Group Databasebeta

← #35225124

Subgroups — #35225124

Order 20864.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782210C211yesmaximal, characteristic, complemented
2#23782201C211yesmaximal, characteristic, complemented
3#23782073C211yesmaximal, characteristic, complemented
4#23782051C211yesmaximal, characteristic, complemented
5#23782028C211yesmaximal, characteristic, complemented
6#23782026C211yesmaximal, characteristic, complemented
7#23781918C211yesmaximal, characteristic, complemented
8#23781916C211yesmaximal, characteristic, complemented
9#23781913C211yesmaximal, characteristic, complemented
10SD16×Dic163C211yesmaximal, characteristic, complemented
11#23781750C211yesmaximal, characteristic, complemented
12#23781747C211yesmaximal, characteristic, complemented
13#23781745C211yesmaximal, characteristic, complemented
14M16×Dic163C211yesmaximal, characteristic, complemented
15#468×C163C211yesmaximal, characteristic, complemented
16C2×#14579698—21no—
17C2×#14579686C2211yescharacteristic
18C2×D1304C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20C2×#14579683C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104731C2211yescharacteristic
23#19104730—21no—
24#19104726C2211yescharacteristic
25#19104724C2211yescharacteristic
26#19104721—21no—
27#19104719—21no—
28#19104709C2211yescharacteristic
29#19104709C2211yescharacteristic
30#19104707—21no—
31Q8×Dic163C2211yescharacteristic
32D4×Dic163C2211yescharacteristic
33#19104697C2211yescharacteristic
34#19104697C2211yescharacteristic
35(C22⋊D163)⋊C4—21no—
36(C22⋊D163)⋊C4—21no—
37#19104693—21no—
38C2×C4×Dic163C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104667C2211yescharacteristic
41#19104666C2211yescharacteristic
42#19104659C2211yescharacteristic
43#19104657C2211yescharacteristic
44#19104657C2211yescharacteristic
45#19104656C2211yescharacteristic
46#19104655C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104645C2211yescharacteristic
49#19104645C2211yescharacteristic
50C8×Dic163C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×SD16×C163C2211yescharacteristic
53C2×M16×C163C2211yescharacteristic
54C22⋊Q8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#14579698—44no—
60C2×Dic326—21no—
61C2×D652C2311yescharacteristic
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C22×Dic163—21no—
68C22×Dic163D411yescharacteristic
69C22×Dic163D411yescharacteristic
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72#14579686—22no—
73D1304—22no—
74#14579684—22no—
75#14579683—22no—
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C4⋊Dic163—21no—
79C4⋊Dic163—21no—
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84C22⋊Dic163—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91C4×Dic163—21no—
92C4×Dic163D411yescharacteristic
93C4×Dic163D411yescharacteristic
94C4×Dic163C2311yescharacteristic
95C4×Dic163C2311yescharacteristic
96C2×Q8×C163C2311yescharacteristic
97C2×D4×C163—21no—
98C2×D4×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100SD16×C163—22no—
101M16×C163—22no—
102C2×C8×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C4⋊C4×C163—21no—
105C4⋊C4×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108C22⋊C4×C163—21no—
109C22×D163—21no—
110C22×D163—21no—
111Dic326—42no—
112D652—21no—
113D652—21no—
114D652—21no—
115D652—21no—
116C22⋊D163—42no—
117C22⋊D163—42no—
118C22⋊D163—42no—
119C22⋊D163—42no—
120C2×Dic163—22no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—41no—
124C2×Dic163—42no—
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—42no—
130C4×D163—42no—
131C163⋊C8—21no—
132C163⋊C8#4112yescomplemented
133C23×C163—21no—
134C23×C163C2×D411yescharacteristic
135Q8×C163—21no—
136Q8×C163—21no—
137D4×C163—21no—
138D4×C163—21no—
139D4×C163—42no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2411yescharacteristic
147C8×C163—21no—
148C8×C163#4112yescomplemented
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163C22×D411yescharacteristic
156C2×D163—41no—
157C2×D163—41no—
158C2×D163—41no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—41no—
162C4×C163—21no—
163C4×C163—41no—
164C4×C163—41no—
165C4×C163—41no—
166C4×C163C2×#4111yescharacteristic
167C4×C163C2×#4111yescharacteristic
168Dic163—21no—
169Dic163—21no—
170Dic163—21no—
171Dic163—41no—
172Dic163—41no—
173Dic163AGL(1,ℤ/8)12yes—
174C2×C163—41no—
175C2×C163—81no—
176C2×C163#53211yescharacteristic
177C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
178C2×C163#57511yescharacteristic
179D163—81no—
180D163—81no—
181C163#317111yescharacteristic, complemented, Sylow
182#3171—1631nomaximal, Sylow
183#532—1631no—
184#529—1631no—
185#501—1631no—
186#496—1631no—
187#490—1631no—
188#489—1631no—
189#469—1631no—
190#469—1631no—
191#468—1631no—
192#468D16311yescharacteristic, complemented
193C4×SD16—1631no—
194C4×SD16—1631no—
195C4×D8—1631no—
196C4×D8—1631no—
197C4×M16—1631no—
198C2×#41—3261no—
199C2×SD16—1631no—
200C2×SD16C2×D16311yescharacteristic
201C2×D8—1631no—
202C2×D8—1631no—
203C2×M16—1631no—
204C2×M16C2×D16311yescharacteristic
205C4⋊Q8—1631no—
206#127—1631no—
207#126—3261no—
208#125—1631no—
209#124—3261no—
210#124—1631no—
211#123—3261no—
212C22⋊Q8C2×D16311yescharacteristic
213#121—3261no—
214#121—1631no—
215#121—1631no—
216#121C2×D16311yescharacteristic
217C4×Q8—1631no—
218C4×D4—3261no—
219C4×D4—3261no—
220C4×D4—1631no—
221C4×D4—1631no—
222C4×D4—1631no—
223#117—3261no—
224C2×C42—1631no—
225#107—1631no—
226#107C2×D16311yescharacteristic
227#106—1631no—
228#106—1631no—
229#103—1631no—
230#103C2×D16311yescharacteristic
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#102—1631no—
235#102—1631no—
236#102C2×D16311yescharacteristic
237#97—1631no—
238#97—1631no—
239C4×C8—1631no—
240C4×C8—1631no—
241#41—6524no—
242C2×Q8—3261no—
243C2×Q8C22×D16311yescharacteristic
244C2×D4—21no—
245C2×D4—3261no—
246C2×D4—3261no—
247C2×D4—3261no—
248C2×D4—3261no—
249C2×D4—1631no—
250C2×D4—1631no—
251C2×D4C22×D16311yescharacteristic
252C22×C4—3261no—
253C22×C4—3261no—
254C22×C4—3261no—
255C22×C4—1631no—
256C22×C4—1631no—
257C22×C4C22×D16311yescharacteristic
258SD16—22no—
259SD16—3262no—
260D8—3262no—
261D8—3262no—
262M16—22no—
263M16—3262no—
264C2×C8—1631no—
265C2×C8—1631no—
266C2×C8C22×D16311yescharacteristic
267C2×C8C22×D16311yescharacteristic
268C4⋊C4—21no—
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—21no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C42—3261no—
287C42—1631no—
288C42—1631no—
289C42—1631no—
290C42—1631no—
291C23—21no—
292C23—3261no—
293C23—3261no—
294C23D4×D16311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—6522no—
298D4—21no—
299D4—21no—
300D4—42no—
301D4—3261no—
302D4—3261no—
303D4—3261no—
304D4—3261no—
305D4—6522no—
306D4—6522no—
307D4—6522no—
308D4—6522no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—3262no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—6522no—
318C2×C4—6522no—
319C2×C4—6521no—
320C2×C4—6522no—
321C2×C4—1631no—
322C2×C4—1631no—
323C2×C4—1631no—
324C2×C4—1631no—
325C2×C4D4×D16311yescharacteristic
326C2×C4C23×D16311yescharacteristic
327C8—21no—
328C8—3261no—
329C8—1632no—
330C8#1457969812yescomplemented
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—6521no—
338C22—6521no—
339C22—6521no—
340C22—6521no—
341C22—6521no—
342C22—6521no—
343C22C2×D4×D16311yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—3261no—
349C4—3261no—
350C4—3261no—
351C4—6521no—
352C4—6521no—
353C4—1632no—
354C4#41×D16311yescharacteristic
355C4C2×#1457969811yescharacteristic
356C2—41no—
357C2—81no—
358C2—13041no—
359C2—13041no—
360C2#2378221011yescharacteristic
361C2AGL(1,ℤ/8)×D16311yescharacteristic
362C2#2378246411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#468×C163C2
socleC22×C163C22×D4
hypercenter#468D163
hypocenterC163#3171
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #23782210C2semidirect
2. #23782201C2semidirect
3. #23782073C2semidirect
4. #23782051C2semidirect
5. #23782028C2semidirect
6. #23782026C2semidirect
7. #23781918C2semidirect
8. #23781916C2semidirect
9. #23781913C2semidirect
10. SD16×Dic163C2semidirect
11. #23781750C2semidirect
12. #23781747C2semidirect
13. #23781745C2semidirect
14. M16×Dic163C2semidirect
15. #468×C163C2semidirect
132. C163⋊C8#41semidirect
148. C8×C163#41semidirect
181. C163#3171semidirect
192. #468D163semidirect
330. C8#14579698semidirect

Maximal subgroups

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

subgroupindex
#237822102
#237822012
#237820732
#237820512
#237820282
#237820262
#237819182
#237819162
#237819132
SD16×Dic1632
#237817502
#237817472
#237817452
M16×Dic1632
#468×C1632
#3171163