Finite Group Databasebeta

← #35224814

Subgroups — #35224814

Order 20864.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782282C211yesmaximal, characteristic, complemented
2#23782148C211yesmaximal, characteristic, complemented
3#23781996C211yesmaximal, characteristic, complemented
4#23781981C211yesmaximal, characteristic, complemented
5#23781953C211yesmaximal, characteristic, complemented
6#23781948C211yesmaximal, characteristic, complemented
7#23781885C211yesmaximal, characteristic, complemented
8#23781878C211yesmaximal, characteristic, complemented
9#23781827C211yesmaximal, characteristic, complemented
10#23781777C211yesmaximal, characteristic, complemented
11#23781769C211yesmaximal, characteristic, complemented
12#23781756C211yesmaximal, characteristic, complemented
13SD16×Dic163C211yesmaximal, characteristic, complemented
14#23781723C211yesmaximal, characteristic, complemented
15#449×C163C211yesmaximal, characteristic
16C2×#14579689C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18#19104734C2211yescharacteristic
19#19104728—21no—
20#19104722—21no—
21#19104722C2211yescharacteristic
22#19104716—21no—
23#19104714C2211yescharacteristic
24#19104712—21no—
25Q8×Dic163C2211yescharacteristic
26Q8×Dic163C2211yescharacteristic
27#19104700C2211yescharacteristic
28D4×Dic163—21no—
29D4×Dic163—21no—
30D4×Dic163C2211yescharacteristic
31D4×Dic163C2211yescharacteristic
32D4×Dic163C2211yescharacteristic
33#19104694—21no—
34#19104694C2211yescharacteristic
35C2×C4⋊Dic163—21no—
36C2×C4⋊Dic163C2211yescharacteristic
37#19104667C2211yescharacteristic
38#19104667C2211yescharacteristic
39#19104666C2211yescharacteristic
40#19104666C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
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×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61C22×Dic163—21no—
62C22×Dic163—21no—
63C22×Dic163—21no—
64C22×Dic163—21no—
65#14579689—22no—
66#14579687—22no—
67C2×C163⋊C8C2311yescharacteristic
68C2×C163⋊C8C2311yescharacteristic
69C4⋊Dic163—21no—
70C4⋊Dic163—21no—
71C4⋊Dic163—21no—
72C4⋊Dic163D411yescharacteristic
73C4⋊Dic163D411yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80C22⋊Dic163—21no—
81C22⋊Dic163—21no—
82C22⋊Dic163—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163D411yescharacteristic
87C22⋊Dic163D411yescharacteristic
88C4×Dic163—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—
108Dic326—21no—
109Dic326—21no—
110Dic326—81no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—81no—
116C2×Dic163—81no—
117C2×Dic163—81no—
118C2×Dic163—81no—
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C163⋊C8—21no—
124C163⋊C8—21no—
125C23×C163—21no—
126C23×C163—21no—
127C23×C163C2×D411yescharacteristic
128Q8×C163—21no—
129Q8×C163#4112yes—
130D4×C163—24no—
131D4×C163—21no—
132D4×C163—21no—
133D4×C163—21no—
134D4×C163—81no—
135D4×C163#4112yes—
136C2×C4×C163—24no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2411yescharacteristic
143C8×C163—21no—
144C8×C163—21no—
145C22×C163—21no—
146C22×C163—22no—
147C22×C163—21no—
148C22×C163—21no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163C22×D411yescharacteristic
155C4×C163—21no—
156C4×C163—22no—
157C4×C163—41no—
158C4×C163C2×#4111yescharacteristic
159C4×C163C2×#4111yescharacteristic
160Dic163—21no—
161Dic163—21no—
162Dic163—21no—
163Dic163—21no—
164Dic163—41no—
165Dic163—41no—
166Dic163—41no—
167Dic163—41no—
168C2×C163—41no—
169C2×C163—42no—
170C2×C163—81no—
171C2×C163#52511yescharacteristic
172C2×C163#57711yescharacteristic
173C2×C163#57711yescharacteristic
174C163#312511yescharacteristic, complemented, Sylow
175#3125—1631nomaximal, Sylow
176#542—1631no—
177#520—1631no—
178#483—1631no—
179#481—1631no—
180#476—1631no—
181#475—1631no—
182#464—1631no—
183#463—1631no—
184#452—1631no—
185#449D16311yescharacteristic
186#442—1631no—
187#441—1631no—
188C4×SD16—1631no—
189C4×SD16—1631no—
190#433—1631no—
191C2×#41C2×D16311yescharacteristic
192C2×Q16—1631no—
193C2×SD16—1631no—
194C2×SD16C2×D16311yescharacteristic
195C2×D8C2×D16311yescharacteristic
196C4⋊Q8—1631no—
197#126—3261no—
198#124—3261no—
199#124—1631no—
200#123—3261no—
201C22⋊Q8—3261no—
202C22⋊Q8—1631no—
203#121C2×D16311yescharacteristic
204C4×Q8—1631no—
205C4×Q8—1631no—
206C4×Q8—1631no—
207C4×D4—3261no—
208C4×D4—3261no—
209C4×D4—1631no—
210C4×D4—1631no—
211C4×D4—1631no—
212#117—3261no—
213#117—1631no—
214C2×C4⋊C4—3261no—
215C2×C4⋊C4—1631no—
216#106—1631no—
217#106—1631no—
218#106—1631no—
219#106—1631no—
220C4⋊C8—1631no—
221C4⋊C8—1631no—
222#103—1631no—
223#103—1631no—
224#103—1631no—
225#103C2×D16311yescharacteristic
226#102—1631no—
227#102—1631no—
228#102—1631no—
229#102C2×D16311yescharacteristic
230C22⋊C8—1631no—
231C22⋊C8C2×D16311yescharacteristic
232#97—1631no—
233C4×C8—1631no—
234#41—22no—
235#41—22no—
236C2×Q8—3261no—
237C2×Q8—1631no—
238C2×Q8C22×D16311yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4C22×D16311yescharacteristic
242C2×D4C22×D16311yescharacteristic
243C22×C4—21no—
244C22×C4—3261no—
245C22×C4—3261no—
246C22×C4—3261no—
247C22×C4—3261no—
248C22×C4C22×D16311yescharacteristic
249Q16—3262no—
250SD16—22no—
251SD16—3262no—
252D8—22no—
253C2×C8—1631no—
254C2×C8—1631no—
255C2×C8C22×D16311yescharacteristic
256C2×C8C22×D16311yescharacteristic
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—1631no—
264C4⋊C4—1631no—
265C4⋊C4—1631no—
266C4⋊C4—1631no—
267C4⋊C4—1631no—
268C4⋊C4C22×D16311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—3261no—
271C22⋊C4—3261no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C22⋊C4—3261no—
276C22⋊C4—1631no—
277C22⋊C4—1631no—
278C42—3261no—
279C42—3261no—
280C42—3261no—
281C42—1631no—
282C42—1631no—
283C23—21no—
284C23—21no—
285C23D4×D16311yescharacteristic
286Q8—21no—
287Q8—13041no—
288Q8—3261no—
289Q8—3261no—
290Q8#1457969812yes—
291D4—24no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4#1457969812yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—13041no—
303C2×C4—13041no—
304C2×C4—13041no—
305C2×C4—13041no—
306C2×C4—3261no—
307C2×C4—3261no—
308C2×C4—3261no—
309C2×C4—3261no—
310C2×C4—1631no—
311C2×C4—1631no—
312C2×C4—1631no—
313C2×C4—1631no—
314C2×C4D4×D16311yescharacteristic
315C2×C4C23×D16311yescharacteristic
316C8—21no—
317C8—21no—
318C8—3261no—
319C8—3261no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22C2×D4×D16311yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—3261no—
334C4—3261no—
335C4—3261no—
336C4—3261no—
337C4—6521no—
338C4—6521no—
339C4—6521no—
340C4—6521no—
341C4C2×#1457969811yescharacteristic
342C4C2×#1457969811yescharacteristic
343C2—41no—
344C2—42no—
345C2—81no—
346C2#2378217111yescharacteristic
347C2#2378247211yescharacteristic
348C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting, prime radical#449×C163C2
socleC22×C163C22×D4
hypercenter#449D163
hypocenterC163#3125
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. #23782282C2semidirect
2. #23782148C2semidirect
3. #23781996C2semidirect
4. #23781981C2semidirect
5. #23781953C2semidirect
6. #23781948C2semidirect
7. #23781885C2semidirect
8. #23781878C2semidirect
9. #23781827C2semidirect
10. #23781777C2semidirect
11. #23781769C2semidirect
12. #23781756C2semidirect
13. SD16×Dic163C2semidirect
14. #23781723C2semidirect
174. C163#3125semidirect

Maximal subgroups

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

subgroupindex
#237822822
#237821482
#237819962
#237819812
#237819532
#237819482
#237818852
#237818782
#237818272
#237817772
#237817692
#237817562
SD16×Dic1632
#237817232
#449×C1632
#3125163