Finite Group Databasebeta

← #35224680

Subgroups — #35224680

Order 20864.

430 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#23782210C211yesmaximal, characteristic, complemented
2#41×Dic163C211yesmaximal, characteristic, complemented
3#23781990C211yesmaximal, characteristic, complemented
4#23781986C211yesmaximal, characteristic, complemented
5#23781948C211yesmaximal, characteristic, complemented
6#23781943C211yesmaximal, characteristic, complemented
7#23781863C211yesmaximal, characteristic, complemented
8#23781858C211yesmaximal, characteristic, complemented
9#23781810C211yesmaximal, characteristic, complemented
10#23781777C211yesmaximal, characteristic, complemented
11#23781774C211yesmaximal, characteristic, complemented
12#23781768C211yesmaximal, characteristic, complemented
13#23781766C211yesmaximal, characteristic, complemented
14#23781719C211yesmaximal, characteristic, complemented
15#449×C163C211yesmaximal, characteristic, complemented
16C2×#14579698—21no—
17C2×#14579686C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19#19104734C2211yescharacteristic
20#19104731C2211yescharacteristic
21#19104721—21no—
22#19104709C2211yescharacteristic
23#19104707—21no—
24Q8×Dic163C2211yescharacteristic
25Q8×Dic163C2211yescharacteristic
26D4×Dic163—21no—
27D4×Dic163—21no—
28D4×Dic163C2211yescharacteristic
29D4×Dic163C2211yescharacteristic
30D4×Dic163C2211yescharacteristic
31#19104697C2211yescharacteristic
32(C22⋊D163)⋊C4—21no—
33#19104694—21no—
34#19104694C2211yescharacteristic
35C2×C4×Dic163—21no—
36C2×C4×Dic163C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104667C2211yescharacteristic
40#19104666C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104659C2211yescharacteristic
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50#19104645C2211yescharacteristic
51#19104645C2211yescharacteristic
52C2×#41×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54C2×D8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59#14579698—44no—
60C2×Dic326—21no—
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C22×Dic163—21no—
66C22×Dic163—21no—
67C22×Dic163—21no—
68C22×Dic163D411yescharacteristic
69C22×Dic163D411yescharacteristic
70C2×C4×D163—21no—
71#14579686—22no—
72#14579684—22no—
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80Dic163⋊C4—21no—
81C22⋊Dic163—21no—
82C22⋊Dic163—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87C4×Dic163—21no—
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163D411yescharacteristic
91C4×Dic163D411yescharacteristic
92C4×Dic163C2311yescharacteristic
93C4×Dic163C2311yescharacteristic
94#41×C163—22no—
95#41×C163—22no—
96C2×Q8×C163C2311yescharacteristic
97C2×D4×C163—21no—
98C2×D4×C163—21no—
99C2×D4×C163C2311yescharacteristic
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163—21no—
102C22×C4×C163C2311yescharacteristic
103SD16×C163—22no—
104D8×C163—22no—
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163C2311yescharacteristic
108C22⋊C4×C163—21no—
109C22×D163—21no—
110Dic326—42no—
111D652—21no—
112D652—21no—
113C22⋊D163—42no—
114C22⋊D163—42no—
115C22⋊D163—42no—
116C2×Dic163—22no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—41no—
121C2×Dic163—42no—
122C2×Dic163—42no—
123C2×Dic163—42no—
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C4×D163—42no—
129C163⋊C8—21no—
130C163⋊C8—21no—
131C23×C163—21no—
132C23×C163—21no—
133C23×C163C2×D411yescharacteristic
134Q8×C163—21no—
135Q8×C163#4112yescomplemented
136D4×C163—24no—
137D4×C163—21no—
138D4×C163—21no—
139D4×C163—21no—
140D4×C163—42no—
141D4×C163#4112yescomplemented
142C2×C4×C163—24no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2411yescharacteristic
149C8×C163—21no—
150C8×C163—21no—
151C22×C163—21no—
152C22×C163—22no—
153C22×C163—21no—
154C22×C163—21no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163C22×D411yescharacteristic
161C2×D163—41no—
162C2×D163—41no—
163C2×D163—41no—
164C4×C163—21no—
165C4×C163—22no—
166C4×C163—41no—
167C4×C163C2×#4111yescharacteristic
168C4×C163C2×#4111yescharacteristic
169Dic163—21no—
170Dic163—21no—
171Dic163—21no—
172Dic163—41no—
173Dic163—41no—
174Dic163—41no—
175Dic163AGL(1,ℤ/8)12yes—
176C2×C163—41no—
177C2×C163—42no—
178C2×C163—81no—
179C2×C163#52511yescharacteristic
180C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C163#57511yescharacteristic
182D163—81no—
183C163#311511yescharacteristic, complemented, Sylow
184#3115—1631nomaximal, Sylow
185#532—1631no—
186C4×#41—1631no—
187#482—1631no—
188#482—1631no—
189#475—1631no—
190#474—1631no—
191#460—1631no—
192#459—1631no—
193#449—1631no—
194#449D16311yescharacteristic, complemented
195#442—1631no—
196#442—1631no—
197#440—1631no—
198#440—1631no—
199#432—1631no—
200C2×#41—3261no—
201C2×#41C2×D16311yescharacteristic
202C2×SD16—1631no—
203C2×SD16C2×D16311yescharacteristic
204C2×D8—1631no—
205C2×D8C2×D16311yescharacteristic
206C4⋊Q8—1631no—
207#127—1631no—
208#124—3261no—
209#121—3261no—
210#121—1631no—
211#121C2×D16311yescharacteristic
212C4×Q8—1631no—
213C4×Q8—1631no—
214C4×D4—3261no—
215C4×D4—3261no—
216C4×D4—3261no—
217C4×D4—1631no—
218C4×D4—1631no—
219C4×D4—1631no—
220C4×D4—1631no—
221#117—3261no—
222#117—1631no—
223C2×C42—3261no—
224C2×C42—1631no—
225#107—1631no—
226#107—1631no—
227#106—1631no—
228#106—1631no—
229C4⋊C8—1631no—
230C4⋊C8—1631no—
231#103—1631no—
232#103C2×D16311yescharacteristic
233#102—1631no—
234#102—1631no—
235#102—1631no—
236#102—1631no—
237#102—1631no—
238#102C2×D16311yescharacteristic
239C22⋊C8—1631no—
240C22⋊C8C2×D16311yescharacteristic
241#97—1631no—
242#97—1631no—
243#41—22no—
244#41—22no—
245#41—6524no—
246C2×Q8—3261no—
247C2×Q8C22×D16311yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—3261no—
251C2×D4—3261no—
252C2×D4—3261no—
253C2×D4—1631no—
254C2×D4C22×D16311yescharacteristic
255C2×D4C22×D16311yescharacteristic
256C22×C4—21no—
257C22×C4—3261no—
258C22×C4—3261no—
259C22×C4—3261no—
260C22×C4—3261no—
261C22×C4—1631no—
262C22×C4—1631no—
263C22×C4C22×D16311yescharacteristic
264SD16—22no—
265SD16—3262no—
266D8—22no—
267D8—3262no—
268C2×C8—1631no—
269C2×C8—1631no—
270C2×C8C22×D16311yescharacteristic
271C2×C8C22×D16311yescharacteristic
272C4⋊C4—3261no—
273C4⋊C4—3261no—
274C4⋊C4—3261no—
275C4⋊C4—1631no—
276C4⋊C4—1631no—
277C4⋊C4—1631no—
278C4⋊C4C22×D16311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C42—3261no—
287C42—3261no—
288C42—3261no—
289C42—1631no—
290C42—1631no—
291C42—1631no—
292C42—1631no—
293C23—21no—
294C23—21no—
295C23—3261no—
296C23D4×D16311yescharacteristic
297Q8—21no—
298Q8—6522no—
299Q8#1457969812yescomplemented
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—3261no—
306D4—3261no—
307D4—6522no—
308D4—6522no—
309D4—6522no—
310D4#1457969812yescomplemented
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—3262no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—6522no—
321C2×C4—6521no—
322C2×C4—6522no—
323C2×C4—6522no—
324C2×C4—6522no—
325C2×C4—1631no—
326C2×C4—1631no—
327C2×C4—1631no—
328C2×C4—1631no—
329C2×C4D4×D16311yescharacteristic
330C2×C4C23×D16311yescharacteristic
331C8—21no—
332C8—21no—
333C8—3261no—
334C8—3261no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—6521no—
345C22—6521no—
346C22—6521no—
347C22C2×D4×D16311yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—3261no—
352C4—3261no—
353C4—3261no—
354C4—6521no—
355C4—6521no—
356C4—6521no—
357C4—1632no—
358C4C2×#1457969811yescharacteristic
359C4C2×#1457969811yescharacteristic
360C2—41no—
361C2—42no—
362C2—81no—
363C2—13041no—
364C2#2378217111yescharacteristic
365C2AGL(1,ℤ/8)×D16311yescharacteristic
366C2#2378246211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#449×C163C2
socleC22×C163C22×D4
hypercenter#449D163
hypocenterC163#3115
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #23782210C2semidirect
2. #41×Dic163C2semidirect
3. #23781990C2semidirect
4. #23781986C2semidirect
5. #23781948C2semidirect
6. #23781943C2semidirect
7. #23781863C2semidirect
8. #23781858C2semidirect
9. #23781810C2semidirect
10. #23781777C2semidirect
11. #23781774C2semidirect
12. #23781768C2semidirect
13. #23781766C2semidirect
14. #23781719C2semidirect
15. #449×C163C2semidirect
135. Q8×C163#41semidirect
141. D4×C163#41semidirect
183. C163#3115semidirect
194. #449D163semidirect
299. Q8#14579698semidirect
310. D4#14579698semidirect

Maximal subgroups

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

subgroupindex
#237822102
#41×Dic1632
#237819902
#237819862
#237819482
#237819432
#237818632
#237818582
#237818102
#237817772
#237817742
#237817682
#237817662
#237817192
#449×C1632
#3115163