Finite Group Databasebeta

← #35227238

Subgroups — #35227238

Order 20864.

396 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#23782374C211yesmaximal, characteristic, complemented
2#23782335C211yesmaximal, characteristic, complemented
3#23782029C211yesmaximal, characteristic, complemented
4#23782028C211yesmaximal, characteristic, complemented
5#23782010C211yesmaximal, characteristic, complemented
6#23781966C211yesmaximal, characteristic, complemented
7#23781963C211yesmaximal, characteristic, complemented
8#23781953C211yesmaximal, characteristic, complemented
9#23781858C211yesmaximal, characteristic, complemented
10#23781858C211yesmaximal, characteristic, complemented
11#23781794C211yesmaximal, characteristic, complemented
12#23781777C211yesmaximal, characteristic, complemented
13#23781775C211yesmaximal, characteristic, complemented
14D8×Dic163C211yesmaximal, characteristic, complemented
15#459×C163C211yesmaximal, characteristic, complemented
16C2×#14579698—21no—
17C2×#14579684C2211yescharacteristic
18#19104734C2211yescharacteristic
19#19104731C2211yescharacteristic
20#19104731C2211yescharacteristic
21#19104727C2211yescharacteristic
22#19104726C2211yescharacteristic
23#19104714—21no—
24#19104712—21no—
25#19104712—21no—
26#19104707—21no—
27#19104707—21no—
28#19104706—21no—
29C4×Dic326C2211yescharacteristic
30C4×D652C2211yescharacteristic
31D4×Dic163C2211yescharacteristic
32D4×Dic163C2211yescharacteristic
33D4×Dic163C2211yescharacteristic
34D4×Dic163C2211yescharacteristic
35C2×Dic163⋊C4—21no—
36#19104669C2211yescharacteristic
37#19104667C2211yescharacteristic
38#19104666C2211yescharacteristic
39Dic163⋊C8C2211yescharacteristic
40Dic163⋊C8C2211yescharacteristic
41#19104657C2211yescharacteristic
42#19104657C2211yescharacteristic
43#19104656C2211yescharacteristic
44#19104656C2211yescharacteristic
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104655C2211yescharacteristic
49#19104646C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×D8×C163C2211yescharacteristic
53C2×D8×C163C2211yescharacteristic
54#127×C163C2211yescharacteristic
55C4×D4×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#14579698—44no—
60C2×Dic326—21no—
61C2×Dic326D411yescharacteristic
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C22×Dic163—21no—
68C22×Dic163—21no—
69C22×Dic163—21no—
70C22×Dic163—21no—
71C2×C4×D163—21no—
72#14579684—22no—
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C4⋊Dic163—21no—
76C4⋊Dic163D411yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4D411yescharacteristic
85Dic163⋊C4D411yescharacteristic
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90(C2×D163)⋊C4—21no—
91C4×Dic163C2311yescharacteristic
92C4×Dic163C2311yescharacteristic
93C2×D4×C163—21no—
94C2×D4×C163—21no—
95C2×D4×C163C2311yescharacteristic
96C2×D4×C163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98C22×C4×C163—21no—
99D8×C163—22no—
100D8×C163—22no—
101C2×C8×C163C2311yescharacteristic
102C2×C8×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C22⋊C4×C163—21no—
105C42×C163C2311yescharacteristic
106C22×D163—21no—
107Dic326—22no—
108Dic326—81no—
109D652—21no—
110D652—21no—
111C22⋊D163—42no—
112C22⋊D163—42no—
113C22⋊D163—81no—
114C22⋊D163—81no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—42no—
119C2×Dic163—42no—
120C2×Dic163—42no—
121C2×Dic163—81no—
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C4×D163—42no—
127C163⋊C8—21no—
128C163⋊C8—21no—
129C23×C163—21no—
130C23×C163—21no—
131C23×C163—21no—
132D4×C163—21no—
133D4×C163—21no—
134D4×C163—21no—
135D4×C163—21no—
136D4×C163—21no—
137D4×C163—42no—
138D4×C163—81no—
139D4×C163#4112yescomplemented
140C2×C4×C163—21no—
141C2×C4×C163—81no—
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2411yescharacteristic
145C8×C163—21no—
146C8×C163—21no—
147C22×C163—22no—
148C22×C163—41no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163—41no—
156C22×C163C22×D411yescharacteristic
157C2×D163—41no—
158C2×D163—41no—
159C2×D163—41no—
160C4×C163—21no—
161C4×C163—21no—
162C4×C163—41no—
163C4×C163C2×#4111yescharacteristic
164C4×C1632−1+411yescharacteristic
165Dic163—21no—
166Dic163—21no—
167Dic163—21no—
168Dic163—41no—
169Dic163—41no—
170Dic163—41no—
171Dic163AGL(1,ℤ/8)12yes—
172C2×C163—42no—
173C2×C163—81no—
174C2×C163—81no—
175C2×C163#54811yescharacteristic
176C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C163#57611yescharacteristic
178D163—81no—
179C163#336611yescharacteristic, complemented, Sylow
180#3366—1631nomaximal, Sylow
181#554—1631no—
182#550—1631no—
183#490—1631no—
184#490—1631no—
185#486—1631no—
186#478—1631no—
187#478—1631no—
188#476—1631no—
189#459—1631no—
190#459—1631no—
191#459D16311yescharacteristic, complemented
192#446—1631no—
193#442—1631no—
194#442—1631no—
195C4×D8—1631no—
196C2×#41—3261no—
197C2×D8—1631no—
198C2×D8C2×D16311yescharacteristic
199C2×D8C2×D16311yescharacteristic
200C4⋊Q8—1631no—
201#127—1631no—
202#127—1631no—
203#127C2×D16311yescharacteristic
204#125—1631no—
205#125—1631no—
206C22⋊Q8—3261no—
207C22⋊Q8—3261no—
208C22⋊Q8—3261no—
209#121—3261no—
210#121—3261no—
211#121—3261no—
212C4×Q8—1631no—
213C4×D4—1631no—
214C4×D4—1631no—
215C4×D4—1631no—
216C4×D4—1631no—
217C4×D4—1631no—
218C4×D4C2×D16311yescharacteristic
219C2×C4⋊C4—3261no—
220#107—1631no—
221#106—1631no—
222#106—1631no—
223C4⋊C8—1631no—
224C4⋊C8—1631no—
225C4⋊C8C2×D16311yescharacteristic
226#102—1631no—
227#102—1631no—
228#102—1631no—
229#102—1631no—
230#102—1631no—
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#102C2×D16311yescharacteristic
235#102C2×D16311yescharacteristic
236#97—1631no—
237#97—1631no—
238C4×C8—1631no—
239#41—6524no—
240C2×Q8—3261no—
241C2×Q8—1631no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—3261no—
245C2×D4—3261no—
246C2×D4—3261no—
247C2×D4—3261no—
248C2×D4—1631no—
249C2×D4C22×D16311yescharacteristic
250C2×D4C22×D16311yescharacteristic
251C2×D4C22×D16311yescharacteristic
252C22×C4—21no—
253C22×C4—3261no—
254C22×C4—3261no—
255C22×C4—3261no—
256C22×C4—3261no—
257C22×C4—3261no—
258D8—22no—
259D8—22no—
260D8—3262no—
261C2×C8—1631no—
262C2×C8—1631no—
263C2×C8C22×D16311yescharacteristic
264C2×C8C22×D16311yescharacteristic
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—1631no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4—1631no—
274C4⋊C4—1631no—
275C4⋊C4—1631no—
276C4⋊C4C22×D16311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C42—1631no—
284C42—1631no—
285C42C22×D16311yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—3261no—
290Q8—13041no—
291Q8—3262no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—81no—
299D4—13041no—
300D4—13041no—
301D4—3261no—
302D4—3261no—
303D4—6522no—
304D4—6522no—
305D4#1457969812yescomplemented
306C2×C4—21no—
307C2×C4—81no—
308C2×C4—13041no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—6522no—
313C2×C4—6522no—
314C2×C4—6522no—
315C2×C4—6522no—
316C2×C4—1631no—
317C2×C4—1631no—
318C2×C4—1631no—
319C2×C4—1631no—
320C2×C4D4×D16311yescharacteristic
321C2×C4D4×D16311yescharacteristic
322C2×C4C23×D16311yescharacteristic
323C8—21no—
324C8—21no—
325C8—3261no—
326C8—3261no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—6521no—
337C22—6521no—
338C22—6521no—
339C22C2×D4×D16311yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—3261no—
344C4—3261no—
345C4—3261no—
346C4—6521no—
347C4—6521no—
348C4—6521no—
349C4—1632no—
350C4C2×#1457969811yescharacteristic
351C4#1910478211yescharacteristic
352C2—42no—
353C2—81no—
354C2—81no—
355C2—13041no—
356C2#2378232111yescharacteristic
357C2AGL(1,ℤ/8)×D16311yescharacteristic
358C2#2378246611yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#459×C163C2
socleC22×C163C22×D4
hypercenter#459D163
hypocenterC163#3366
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #23782374C2semidirect
2. #23782335C2semidirect
3. #23782029C2semidirect
4. #23782028C2semidirect
5. #23782010C2semidirect
6. #23781966C2semidirect
7. #23781963C2semidirect
8. #23781953C2semidirect
9. #23781858C2semidirect
10. #23781858C2semidirect
11. #23781794C2semidirect
12. #23781777C2semidirect
13. #23781775C2semidirect
14. D8×Dic163C2semidirect
15. #459×C163C2semidirect
139. D4×C163#41semidirect
179. C163#3366semidirect
191. #459D163semidirect
305. D4#14579698semidirect

Maximal subgroups

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

subgroupindex
#237823742
#237823352
#237820292
#237820282
#237820102
#237819662
#237819632
#237819532
#237818582
#237817942
#237817772
#237817752
D8×Dic1632
#459×C1632
#3366163