Finite Group Databasebeta

← #35224728

Subgroups — #35224728

Order 20864.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782268C211yesmaximal, characteristic, complemented
2#23782148C211yesmaximal, characteristic, complemented
3#23781981C211yesmaximal, characteristic, complemented
4#23781979C211yesmaximal, characteristic, complemented
5#23781974C211yesmaximal, characteristic, complemented
6#23781963C211yesmaximal, characteristic, complemented
7#23781863C211yesmaximal, characteristic, complemented
8#23781858C211yesmaximal, characteristic, complemented
9#23781810C211yesmaximal, characteristic, complemented
10#23781774C211yesmaximal, characteristic, complemented
11#23781766C211yesmaximal, characteristic, complemented
12#23781750C211yesmaximal, characteristic, complemented
13D8×Dic163C211yesmaximal, characteristic, complemented
14#23781723C211yesmaximal, characteristic, complemented
15#449×C163C211yesmaximal, characteristic, complemented
16C2×#14579686C2211yescharacteristic
17C2×#14579684C2211yescharacteristic
18#19104731C2211yescharacteristic
19#19104726C2211yescharacteristic
20#19104719—21no—
21#19104717—21no—
22#19104709C2211yescharacteristic
23#19104707—21no—
24#19104706—21no—
25Q8×Dic163C2211yescharacteristic
26Q8×Dic163C2211yescharacteristic
27D4×Dic163—21no—
28D4×Dic163—21no—
29D4×Dic163C2211yescharacteristic
30D4×Dic163C2211yescharacteristic
31D4×Dic163C2211yescharacteristic
32#19104697C2211yescharacteristic
33#19104694—21no—
34#19104694C2211yescharacteristic
35C2×C4⋊Dic163—21no—
36C2×C4⋊Dic163C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104669C2211yescharacteristic
39#19104666C2211yescharacteristic
40#19104666C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104659C2211yescharacteristic
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×#41×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54C2×D8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59C2×D652C2311yescharacteristic
60C2×C22⋊D163—21no—
61C2×C22⋊D163—21no—
62C2×C22⋊D163—21no—
63C22×Dic163—21no—
64C22×Dic163—21no—
65C22×Dic163—21no—
66C22×Dic163—21no—
67C2×C4×D163—21no—
68#14579686—22no—
69#14579684—22no—
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C4⋊Dic163—21no—
73C4⋊Dic163—21no—
74C4⋊Dic163—21no—
75C4⋊Dic163—21no—
76C4⋊Dic163D411yescharacteristic
77C4⋊Dic163D411yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81Dic163⋊C4—21no—
82C22⋊Dic163—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163D411yescharacteristic
86C22⋊Dic163D411yescharacteristic
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163C2311yescharacteristic
92C4×Dic163C2311yescharacteristic
93#41×C163—22no—
94#41×C163—22no—
95C2×Q8×C163C2311yescharacteristic
96C2×D4×C163—21no—
97C2×D4×C163—21no—
98C2×D4×C163C2311yescharacteristic
99C2×D4×C163C2311yescharacteristic
100C22×C4×C163—21no—
101C22×C4×C163C2311yescharacteristic
102SD16×C163—22no—
103D8×C163—22no—
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108C22×D163—21no—
109D652—21no—
110D652—21no—
111C22⋊D163—81no—
112C22⋊D163—81no—
113C22⋊D163—81no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—81no—
118C2×Dic163—81no—
119C2×Dic163—81no—
120C2×Dic163—81no—
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C4×D163—81no—
126C163⋊C8—21no—
127C163⋊C8—21no—
128C23×C163—21no—
129C23×C163—21no—
130C23×C163C2×D411yescharacteristic
131Q8×C163—21no—
132Q8×C163#4112yes—
133D4×C163—24no—
134D4×C163—21no—
135D4×C163—21no—
136D4×C163—21no—
137D4×C163—81no—
138D4×C163#4112yes—
139C2×C4×C163—24no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2411yescharacteristic
146C8×C163—21no—
147C8×C163—21no—
148C22×C163—21no—
149C22×C163—22no—
150C22×C163—21no—
151C22×C163—21no—
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—
161C4×C163—21no—
162C4×C163—22no—
163C4×C163—41no—
164C4×C163C2×#4111yescharacteristic
165C4×C163C2×#4111yescharacteristic
166Dic163—21no—
167Dic163—21no—
168Dic163—21no—
169Dic163—21no—
170Dic163—41no—
171Dic163—41no—
172Dic163—41no—
173C2×C163—41no—
174C2×C163—42no—
175C2×C163—81no—
176C2×C163#52511yescharacteristic
177C2×C163#57611yescharacteristic
178C2×C163#57711yescharacteristic
179D163—81no—
180C163#311911yescharacteristic, complemented, Sylow
181#3119—1631nomaximal, Sylow
182#540—1631no—
183#520—1631no—
184#481—1631no—
185#480—1631no—
186#479—1631no—
187#478—1631no—
188#460—1631no—
189#459—1631no—
190#449—1631no—
191#449D16311yescharacteristic, complemented
192#442—1631no—
193#440—1631no—
194C4×SD16—1631no—
195C4×D8—1631no—
196#433—1631no—
197C2×#41C2×D16311yescharacteristic
198C2×SD16—1631no—
199C2×SD16C2×D16311yescharacteristic
200C2×D8—1631no—
201C2×D8C2×D16311yescharacteristic
202#127—1631no—
203#125—1631no—
204#123—3261no—
205#123—3261no—
206#121—3261no—
207#121—3261no—
208#121—1631no—
209#121C2×D16311yescharacteristic
210C4×Q8—1631no—
211C4×Q8—1631no—
212C4×D4—3261no—
213C4×D4—3261no—
214C4×D4—1631no—
215C4×D4—1631no—
216C4×D4—1631no—
217C4×D4—1631no—
218#117—3261no—
219#117—1631no—
220C2×C4⋊C4—3261no—
221C2×C4⋊C4—1631no—
222#107—1631no—
223#107—1631no—
224#106—1631no—
225#106—1631no—
226C4⋊C8—1631no—
227C4⋊C8—1631no—
228#103—1631no—
229#103C2×D16311yescharacteristic
230#102—1631no—
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#102—1631no—
235#102C2×D16311yescharacteristic
236C22⋊C8—1631no—
237C22⋊C8C2×D16311yescharacteristic
238#97—1631no—
239C4×C8—1631no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D16311yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—3261no—
246C2×D4—3261no—
247C2×D4—3261no—
248C2×D4—1631no—
249C2×D4C22×D16311yescharacteristic
250C2×D4C22×D16311yescharacteristic
251C22×C4—21no—
252C22×C4—3261no—
253C22×C4—3261no—
254C22×C4—3261no—
255C22×C4—3261no—
256C22×C4—3261no—
257C22×C4C22×D16311yescharacteristic
258SD16—22no—
259SD16—3262no—
260D8—22no—
261D8—3262no—
262C2×C8—1631no—
263C2×C8—1631no—
264C2×C8C22×D16311yescharacteristic
265C2×C8C22×D16311yescharacteristic
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
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—
283C22⋊C4—1631no—
284C22⋊C4—1631no—
285C42—3261no—
286C42—3261no—
287C42—1631no—
288C42—1631no—
289C23—21no—
290C23—21no—
291C23—3261no—
292C23D4×D16311yescharacteristic
293Q8—21no—
294Q8#1457969812yes—
295D4—24no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—13041no—
301D4—13041no—
302D4—13041no—
303D4—3261no—
304D4—3261no—
305D4#1457969812yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—13041no—
312C2×C4—13041no—
313C2×C4—13041no—
314C2×C4—13041no—
315C2×C4—13041no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—1631no—
320C2×C4—1631no—
321C2×C4—1631no—
322C2×C4—1631no—
323C2×C4D4×D16311yescharacteristic
324C2×C4C23×D16311yescharacteristic
325C8—21no—
326C8—21no—
327C8—3261no—
328C8—3261no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—6521no—
339C22—6521no—
340C22—6521no—
341C22C2×D4×D16311yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—3261no—
346C4—3261no—
347C4—3261no—
348C4—3261no—
349C4—6521no—
350C4—6521no—
351C4—6521no—
352C4C2×#1457969811yescharacteristic
353C4C2×#1457969811yescharacteristic
354C2—41no—
355C2—42no—
356C2—81no—
357C2—13041no—
358C2#2378217111yescharacteristic
359C2#2378246711yescharacteristic
360C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#449×C163C2
socleC22×C163C22×D4
hypercenter#449D163
hypocenterC163#3119
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. #23782268C2semidirect
2. #23782148C2semidirect
3. #23781981C2semidirect
4. #23781979C2semidirect
5. #23781974C2semidirect
6. #23781963C2semidirect
7. #23781863C2semidirect
8. #23781858C2semidirect
9. #23781810C2semidirect
10. #23781774C2semidirect
11. #23781766C2semidirect
12. #23781750C2semidirect
13. D8×Dic163C2semidirect
14. #23781723C2semidirect
15. #449×C163C2semidirect
180. C163#3119semidirect
191. #449D163semidirect

Maximal subgroups

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

subgroupindex
#237822682
#237821482
#237819812
#237819792
#237819742
#237819632
#237818632
#237818582
#237818102
#237817742
#237817662
#237817502
D8×Dic1632
#237817232
#449×C1632
#3119163