Finite Group Databasebeta

← #35224830

Subgroups — #35224830

Order 20864.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782275C211yesmaximal, characteristic
2#23782143C211yesmaximal, characteristic
3#23782007C211yesmaximal, characteristic
4#23782003C211yesmaximal, characteristic
5#23781974C211yesmaximal, characteristic, complemented
6#23781970C211yesmaximal, characteristic, complemented
7#23781873C211yesmaximal, characteristic, complemented
8#23781873C211yesmaximal, characteristic, complemented
9#23781821C211yesmaximal, characteristic
10#23781772C211yesmaximal, characteristic, complemented
11#23781769C211yesmaximal, characteristic, complemented
12Q16×Dic163C211yesmaximal, characteristic, complemented
13#23781757C211yesmaximal, characteristic, complemented
14#23781723C211yesmaximal, characteristic
15#451×C163C211yesmaximal, characteristic
16C2×#14579689C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18#19104733C2211yescharacteristic
19#19104728—21no—
20#19104728—21no—
21#19104726—21no—
22#19104726C2211yescharacteristic
23#19104725—21no—
24#19104714C2211yescharacteristic
25Q8×Dic163—21no—
26Q8×Dic163—21no—
27Q8×Dic163C2211yescharacteristic
28Q8×Dic163C2211yescharacteristic
29Q8×Dic163C2211yescharacteristic
30Q8×Dic163C2211yescharacteristic
31Q8×Dic163C2211yescharacteristic
32#19104700C2211yescharacteristic
33#19104694—21no—
34#19104694—21no—
35#19104694C2211yescharacteristic
36#19104694C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104667C2211yescharacteristic
40#19104666C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104659C2211yescharacteristic
48#19104659C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C22×Q8×C163C2211yescharacteristic
53C2×Q16×C163C2211yescharacteristic
54C2×Q16×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59C2×Dic326C2311yescharacteristic
60#14579689—22no—
61#14579689—22no—
62C2×C163⋊C8C2311yescharacteristic
63C2×C163⋊C8C2311yescharacteristic
64C4⋊Dic163—21no—
65C4⋊Dic163—21no—
66C4⋊Dic163—21no—
67C4⋊Dic163—21no—
68C4⋊Dic163—21no—
69C4⋊Dic163—21no—
70C4⋊Dic163D411yescharacteristic
71C4⋊Dic163D411yescharacteristic
72C4⋊Dic163C2311yescharacteristic
73C4⋊Dic163C2311yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80C22⋊Dic163—21no—
81C22⋊Dic163—21no—
82C22⋊Dic163D411yescharacteristic
83C22⋊Dic163D411yescharacteristic
84C4×Dic163—21no—
85C4×Dic163—21no—
86C4×Dic163—21no—
87C4×Dic163—21no—
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163C2311yescharacteristic
92C4×Dic163C2311yescharacteristic
93C2×Q8×C163—22no—
94C2×Q8×C163—22no—
95C2×Q8×C163—21no—
96C2×Q8×C163C2311yescharacteristic
97C2×Q8×C163C2311yescharacteristic
98C2×Q8×C163C2311yescharacteristic
99C22×C4×C163—21no—
100C22×C4×C163C2311yescharacteristic
101Q16×C163—22no—
102Q16×C163—22no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C4⋊C4×C163—21no—
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108Dic326—21no—
109Dic326—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163C2×D411yescharacteristic
115C2×Dic163C2×D411yescharacteristic
116C2×Dic163C2×D411yescharacteristic
117C2×Dic163C2×D411yescharacteristic
118C163⋊C8—21no—
119C163⋊C8—21no—
120C23×C163C2×D411yescharacteristic
121Q8×C163—24no—
122Q8×C163—21no—
123Q8×C163—21no—
124Q8×C163—21no—
125Q8×C163—21no—
126Q8×C163#4112yes—
127Q8×C163#4112yes—
128C2×C4×C163—24no—
129C2×C4×C163—21no—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163C2×D411yescharacteristic
136C2×C4×C163C2411yescharacteristic
137C8×C163—21no—
138C8×C163—21no—
139C22×C163—21no—
140C22×C163—21no—
141C22×C163—21no—
142C22×C163C22×D411yescharacteristic
143C4×C163—22no—
144C4×C163—22no—
145C4×C163—21no—
146C4×C163—41no—
147C4×C163—41no—
148C4×C163C2×#4111yescharacteristic
149C4×C163C2×#4111yescharacteristic
150Dic163—21no—
151Dic163—21no—
152Dic163—21no—
153Dic163—21no—
154Dic163—41no—
155Dic163—41no—
156Dic163—41no—
157Dic163—41no—
158C2×C163—41no—
159C2×C163#52511yescharacteristic
160C2×C163#57811yescharacteristic
161C2×C163#57811yescharacteristic
162C163#312711yescharacteristic, complemented, Sylow
163#3127—1631nomaximal, Sylow
164#541—1631no—
165#519—1631no—
166#485—1631no—
167#485—1631no—
168#479—1631no—
169#479—1631no—
170#462—1631no—
171#462—1631no—
172#451—1631no—
173#451D16311yescharacteristic
174#441—1631no—
175#441—1631no—
176C4×Q16—1631no—
177C4×Q16—1631no—
178#433—1631no—
179C22×Q8C2×D16311yescharacteristic
180C2×Q16—1631no—
181C2×Q16—1631no—
182C2×Q16C2×D16311yescharacteristic
183C2×Q16C2×D16311yescharacteristic
184C4⋊Q8—1631no—
185#126—3261no—
186#126—3261no—
187#125—3261no—
188#125—3261no—
189#125—1631no—
190C22⋊Q8—1631no—
191C22⋊Q8C2×D16311yescharacteristic
192C4×Q8—3261no—
193C4×Q8—3261no—
194C4×Q8—1631no—
195C4×Q8—1631no—
196C4×Q8—1631no—
197C4×Q8—1631no—
198C4×Q8—1631no—
199C4×Q8—1631no—
200#117—3261no—
201#117—3261no—
202#117—1631no—
203#117—1631no—
204#107—1631no—
205#107—1631no—
206#106—1631no—
207#106—1631no—
208C4⋊C8—1631no—
209C4⋊C8—1631no—
210#103—1631no—
211#103—1631no—
212#103—1631no—
213#103—1631no—
214#103—1631no—
215#103—1631no—
216#103C2×D16311yescharacteristic
217#103C2×D16311yescharacteristic
218C22⋊C8—1631no—
219C22⋊C8C2×D16311yescharacteristic
220#97—1631no—
221C4×C8—1631no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—1631no—
226C2×Q8C22×D16311yescharacteristic
227C2×Q8C22×D16311yescharacteristic
228C2×Q8C22×D16311yescharacteristic
229C22×C4—21no—
230C22×C4C22×D16311yescharacteristic
231Q16—22no—
232Q16—22no—
233Q16—3262no—
234Q16—3262no—
235C2×C8—1631no—
236C2×C8—1631no—
237C2×C8C22×D16311yescharacteristic
238C2×C8C22×D16311yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—3261no—
241C4⋊C4—3261no—
242C4⋊C4—3261no—
243C4⋊C4—3261no—
244C4⋊C4—3261no—
245C4⋊C4—3261no—
246C4⋊C4—3261no—
247C4⋊C4—3261no—
248C4⋊C4—3261no—
249C4⋊C4—3261no—
250C4⋊C4—3261no—
251C4⋊C4—1631no—
252C4⋊C4—1631no—
253C4⋊C4—1631no—
254C4⋊C4—1631no—
255C4⋊C4—1631no—
256C4⋊C4C22×D16311yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—3261no—
259C22⋊C4—3261no—
260C22⋊C4—1631no—
261C22⋊C4—1631no—
262C42—3261no—
263C42—3261no—
264C42—3261no—
265C42—3261no—
266C42—3261no—
267C42—3261no—
268C42—3261no—
269C42—1631no—
270C42—1631no—
271C23D4×D16311yescharacteristic
272Q8—24no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—3261no—
278Q8—3261no—
279Q8#1457969812yes—
280Q8#1457969812yes—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—3261no—
289C2×C4—3261no—
290C2×C4—3261no—
291C2×C4—3261no—
292C2×C4—1631no—
293C2×C4—1631no—
294C2×C4—1631no—
295C2×C4—1631no—
296C2×C4D4×D16311yescharacteristic
297C2×C4C23×D16311yescharacteristic
298C8—21no—
299C8—21no—
300C8—3261no—
301C8—3261no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22C2×D4×D16311yescharacteristic
306C4—22no—
307C4—22no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—3261no—
312C4—3261no—
313C4—3261no—
314C4—3261no—
315C4—6521no—
316C4—6521no—
317C4—6521no—
318C4—6521no—
319C4C2×#1457969811yescharacteristic
320C4C2×#1457969811yescharacteristic
321C2—41no—
322C2#2378217111yescharacteristic
323C2#2378247811yescharacteristic
324C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#451×C163C2
socleC22×C163C22×D4
prime radicalC23×C163C2×D4
hypercenter#451D163
hypocenterC163#3127
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
5. #23781974C2semidirect
6. #23781970C2semidirect
7. #23781873C2semidirect
8. #23781873C2semidirect
10. #23781772C2semidirect
11. #23781769C2semidirect
12. Q16×Dic163C2semidirect
13. #23781757C2semidirect
162. C163#3127semidirect

Maximal subgroups

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

subgroupindex
#237822752
#237821432
#237820072
#237820032
#237819742
#237819702
#237818732
#237818212
#237817722
#237817692
Q16×Dic1632
#237817572
#237817232
#451×C1632
#3127163