Finite Group Databasebeta

← #35227157

Subgroups — #35227157

Order 20864.

354 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782379C211yesmaximal, characteristic
2#23782373C211yesmaximal, characteristic, complemented
3#23782069C211yesmaximal, characteristic, complemented
4#23782004C211yesmaximal, characteristic, complemented
5#23782003C211yesmaximal, characteristic, complemented
6#23781977C211yesmaximal, characteristic, complemented
7#23781976C211yesmaximal, characteristic, complemented
8#23781958C211yesmaximal, characteristic, complemented
9#23781953C211yesmaximal, characteristic, complemented
10#23781908C211yesmaximal, characteristic, complemented
11SD16×Dic163C211yesmaximal, characteristic, complemented
12#23781741C211yesmaximal, characteristic, complemented
13#23781707C211yesmaximal, characteristic, complemented
14#23781706C211yesmaximal, characteristic, complemented
15#468×C163C211yesmaximal, characteristic, complemented
16C2×#14579682C2211yescharacteristic
17#19104734C2211yescharacteristic
18#19104734C2211yescharacteristic
19#19104725—21no—
20#19104722—21no—
21#19104721—21no—
22#19104715—21no—
23#19104712—21no—
24#19104711C2211yescharacteristic
25#19104707—21no—
26#19104706C2211yescharacteristic
27Q8×Dic163C2211yescharacteristic
28D4×Dic163C2211yescharacteristic
29C4×C22⋊D163C2211yescharacteristic
30(C22⋊D163)⋊C4—21no—
31#19104694C2211yescharacteristic
32#19104693C2211yescharacteristic
33#19104691—21no—
34C2×C4⋊Dic163C2211yescharacteristic
35C4⋊C4×D163C2211yescharacteristic
36#19104669C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104667C2211yescharacteristic
42#19104667C2211yescharacteristic
43#19104666C2211yescharacteristic
44Dic163⋊C8C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×SD16×C163C2211yescharacteristic
53C2×M16×C163C2211yescharacteristic
54C22⋊Q8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×C22⋊D163—21no—
62C2×C22⋊D163D411yescharacteristic
63C22×Dic163—21no—
64C22×Dic163—21no—
65C2×C4×D163—21no—
66C2×C4×D163C2311yescharacteristic
67#14579682—22no—
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C4⋊Dic163—21no—
71C4⋊Dic163—21no—
72C4⋊Dic163C2311yescharacteristic
73C4⋊Dic163C2311yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75C4⋊Dic163C2311yescharacteristic
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4D411yescharacteristic
81C22⋊Dic163—21no—
82C22⋊Dic163—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163D411yescharacteristic
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4D411yescharacteristic
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163C2311yescharacteristic
92C2×Q8×C163C2311yescharacteristic
93C2×D4×C163—21no—
94C2×D4×C163C2311yescharacteristic
95C22×C4×C163C2311yescharacteristic
96SD16×C163—22no—
97M16×C163—22no—
98C2×C8×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C4⋊C4×C163—21no—
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C22⋊C4×C163—21no—
104C22⋊C4×C163—21no—
105C22×D163C2×D411yescharacteristic
106Dic326—81no—
107Dic326—81no—
108C22⋊D163—22no—
109C22⋊D163—81no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—42no—
115C2×Dic163—81no—
116C2×Dic163C2×D411yescharacteristic
117C2×Dic163C2×D411yescharacteristic
118C2×Dic163C2×D411yescharacteristic
119C4×D163—21no—
120C4×D163—21no—
121C4×D163—81no—
122C163⋊C8—21no—
123C163⋊C8—21no—
124C23×C163—21no—
125C23×C163C2×D411yescharacteristic
126Q8×C163—21no—
127Q8×C163—21no—
128D4×C163—21no—
129D4×C163—21no—
130D4×C163—81no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163C2×D411yescharacteristic
137C2×C4×C163C2411yescharacteristic
138C8×C163—21no—
139C8×C163#4112yes—
140C22×C163—21no—
141C22×C163—21no—
142C22×C163—21no—
143C22×C163—41no—
144C22×C163—41no—
145C22×C163—41no—
146C22×C163C22×D411yescharacteristic
147C2×D163—21no—
148C2×D163—21no—
149C2×D163—21no—
150C4×C163—21no—
151C4×C163—41no—
152C4×C163—41no—
153C4×C163—41no—
154C4×C163C2×#4111yescharacteristic
155C4×C1632−1+411yescharacteristic
156Dic163—21no—
157Dic163—21no—
158Dic163—41no—
159Dic163—41no—
160Dic163—41no—
161Dic163—41no—
162Dic163#13712yes—
163C2×C163—41no—
164C2×C163—81no—
165C2×C163#54711yescharacteristic
166C2×C163C2×#13711yescharacteristic
167C2×C163#57611yescharacteristic
168D163—41no—
169C163#335711yescharacteristic, complemented, Sylow
170#3357—1631nomaximal, Sylow
171#555—1631no—
172#554—1631no—
173#500—1631no—
174#485—1631no—
175#485—1631no—
176#480—1631no—
177#480—1631no—
178#477—1631no—
179#476—1631no—
180#468—1631no—
181#468D16311yescharacteristic, complemented
182C4×SD16—1631no—
183#436—1631no—
184#428—1631no—
185#428—1631no—
186C2×SD16C2×D16311yescharacteristic
187C2×M16—1631no—
188C2×M16C2×D16311yescharacteristic
189C4⋊Q8—1631no—
190C4⋊Q8—1631no—
191#125—3261no—
192#124—3261no—
193#124—3261no—
194C22⋊Q8—3261no—
195C22⋊Q8—3261no—
196C22⋊Q8—1631no—
197C22⋊Q8C2×D16311yescharacteristic
198#121—3261no—
199#121—1631no—
200#121C2×D16311yescharacteristic
201C4×Q8—1631no—
202C4×D4—3261no—
203C4×D4—1631no—
204C4×D4—1631no—
205#117—3261no—
206#117—1631no—
207#117—1631no—
208C2×C4⋊C4—1631no—
209C2×C4⋊C4—1631no—
210#107—1631no—
211#107—1631no—
212#107—1631no—
213#107—1631no—
214#107—1631no—
215#107C2×D16311yescharacteristic
216#106—1631no—
217#106—1631no—
218#106—1631no—
219C4⋊C8—1631no—
220#103—1631no—
221#103—1631no—
222#103C2×D16311yescharacteristic
223#102—1631no—
224#102—1631no—
225#102C2×D16311yescharacteristic
226C22⋊C8—1631no—
227C22⋊C8—1631no—
228C4×C8—1631no—
229C2×Q8—3261no—
230C2×Q8—3261no—
231C2×Q8C22×D16311yescharacteristic
232C2×D4—21no—
233C2×D4—3261no—
234C2×D4—1631no—
235C2×D4C22×D16311yescharacteristic
236C22×C4—3261no—
237C22×C4—3261no—
238C22×C4—3261no—
239C22×C4—1631no—
240C22×C4C22×D16311yescharacteristic
241SD16—22no—
242M16—22no—
243M16—3262no—
244C2×C8—1631no—
245C2×C8—1631no—
246C2×C8C22×D16311yescharacteristic
247C2×C8C22×D16311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—3261no—
250C4⋊C4—3261no—
251C4⋊C4—3261no—
252C4⋊C4—3261no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4—1631no—
256C4⋊C4—1631no—
257C4⋊C4—1631no—
258C4⋊C4—1631no—
259C4⋊C4—1631no—
260C4⋊C4C22×D16311yescharacteristic
261C4⋊C4C22×D16311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—3261no—
265C22⋊C4—3261no—
266C22⋊C4—3261no—
267C22⋊C4—3261no—
268C22⋊C4—3261no—
269C22⋊C4—1631no—
270C22⋊C4—1631no—
271C42—3261no—
272C42—3261no—
273C42—3261no—
274C42—1631no—
275C23—21no—
276C23—1631no—
277C23D4×D16311yescharacteristic
278Q8—21no—
279Q8—21no—
280Q8—13041no—
281Q8—13041no—
282D4—21no—
283D4—21no—
284D4—81no—
285D4—13041no—
286D4—3262no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—13041no—
293C2×C4—13041no—
294C2×C4—3261no—
295C2×C4—3261no—
296C2×C4—3261no—
297C2×C4—3261no—
298C2×C4—3261no—
299C2×C4—3261no—
300C2×C4—6522no—
301C2×C4—1631no—
302C2×C4—1631no—
303C2×C4—1631no—
304C2×C4D4×D16311yescharacteristic
305C2×C4C23×D16311yescharacteristic
306C8—21no—
307C8—3261no—
308C8—3261no—
309C8#1457969812yes—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—3261no—
317C22—3261no—
318C22—3261no—
319C22C2×D4×D16311yescharacteristic
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—3261no—
325C4—3261no—
326C4—6521no—
327C4—6521no—
328C4—6521no—
329C4—6521no—
330C4—1632no—
331C4C2×#1457969811yescharacteristic
332C4#1910478211yescharacteristic
333C2—41no—
334C2—81no—
335C2—6521no—
336C2#2378231511yescharacteristic
337C2#137×D16311yescharacteristic
338C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#468×C163C2
socleC22×C163C22×D4
prime radical#23782379C2
hypercenter#468D163
hypocenterC163#3357
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #23782373C2semidirect
3. #23782069C2semidirect
4. #23782004C2semidirect
5. #23782003C2semidirect
6. #23781977C2semidirect
7. #23781976C2semidirect
8. #23781958C2semidirect
9. #23781953C2semidirect
10. #23781908C2semidirect
11. SD16×Dic163C2semidirect
12. #23781741C2semidirect
13. #23781707C2semidirect
14. #23781706C2semidirect
15. #468×C163C2semidirect
169. C163#3357semidirect
181. #468D163semidirect

Maximal subgroups

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

subgroupindex
#237823792
#237823732
#237820692
#237820042
#237820032
#237819772
#237819762
#237819582
#237819532
#237819082
SD16×Dic1632
#237817412
#237817072
#237817062
#468×C1632
#3357163