Finite Group Databasebeta

← #35227357

Subgroups — #35227357

Order 20864.

346 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782375C211yesmaximal, characteristic, complemented
2#23782033C211yesmaximal, characteristic, complemented
3#23782024C211yesmaximal, characteristic, complemented
4#23782019C211yesmaximal, characteristic, complemented
5#23781972C211yesmaximal, characteristic, complemented
6#23781965C211yesmaximal, characteristic, complemented
7#23781950C211yesmaximal, characteristic, complemented
8#23781877C211yesmaximal, characteristic
9#23781877C211yesmaximal, characteristic
10#23781867C211yesmaximal, characteristic, complemented
11#23781796C211yesmaximal, characteristic, complemented
12#23781775C211yesmaximal, characteristic
13#23781763C211yesmaximal, characteristic, complemented
14C4×#14579686C211yesmaximal, characteristic, complemented
15#556×C163C211yesmaximal, characteristic, complemented
16C2×#14579686C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19#19104735C2211yescharacteristic
20#19104727C2211yescharacteristic
21#19104727C2211yescharacteristic
22#19104713—21no—
23#19104713—21no—
24#19104711—21no—
25C4×D652C2211yescharacteristic
26C4×D652C2211yescharacteristic
27C4×D652C2211yescharacteristic
28C4⋊C4×D163—21no—
29#19104668C2211yescharacteristic
30#19104666C2211yescharacteristic
31#19104666C2211yescharacteristic
32C4⋊(C163⋊C8)C2211yescharacteristic
33C4⋊(C163⋊C8)C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic
35#19104659C2211yescharacteristic
36#19104659C2211yescharacteristic
37#19104659C2211yescharacteristic
38#19104659C2211yescharacteristic
39#19104659C2211yescharacteristic
40#19104659C2211yescharacteristic
41#19104656C2211yescharacteristic
42#19104656C2211yescharacteristic
43#19104656C2211yescharacteristic
44#19104655C2211yescharacteristic
45#19104646C2211yescharacteristic
46#19104646C2211yescharacteristic
47C4×C163⋊C8C2211yescharacteristic
48C4⋊Q8×C163C2211yescharacteristic
49#126×C163—21no—
50#126×C163—21no—
51#124×C163C2211yescharacteristic
52#124×C163C2211yescharacteristic
53C22⋊Q8×C163—21no—
54C4×Q8×C163C2211yescharacteristic
55C4×Q8×C163C2211yescharacteristic
56C4×Q8×C163C2211yescharacteristic
57C4×D4×C163C2211yescharacteristic
58#117×C163—21no—
59C2×Dic326—21no—
60C2×D652C2311yescharacteristic
61C2×C4×D163—21no—
62C2×C4×D163—21no—
63C2×C4×D163—21no—
64#14579686—22no—
65#14579686—22no—
66#14579684—22no—
67C2×C163⋊C8C2311yescharacteristic
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C2×C163⋊C8C2311yescharacteristic
71C4⋊Dic163—21no—
72C4⋊Dic163C2311yescharacteristic
73C4⋊Dic163C2311yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79(C2×D163)⋊C4—21no—
80(C2×D163)⋊C4—21no—
81(C2×D163)⋊C4—21no—
82C2×Q8×C163D411yescharacteristic
83C2×Q8×C163C2311yescharacteristic
84C2×Q8×C163C2311yescharacteristic
85C2×D4×C163C2311yescharacteristic
86C22×C4×C163—21no—
87C4⋊C4×C163—21no—
88C4⋊C4×C163—21no—
89C4⋊C4×C163—21no—
90C4⋊C4×C163—21no—
91C4⋊C4×C163D411yescharacteristic
92C4⋊C4×C163D411yescharacteristic
93C4⋊C4×C163D411yescharacteristic
94C4⋊C4×C163C2311yescharacteristic
95C22⋊C4×C163—21no—
96C22⋊C4×C163—21no—
97C22⋊C4×C163—21no—
98C22⋊C4×C163—21no—
99C22⋊C4×C163—21no—
100C42×C163—21no—
101C42×C163—21no—
102C42×C163C2311yescharacteristic
103C42×C163C2311yescharacteristic
104C42×C163C2311yescharacteristic
105C22×D163—21no—
106Dic326—81no—
107D652—21no—
108D652#4112yes—
109C2×Dic163—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C4×D163—42no—
113C4×D163—81no—
114C4×D163—81no—
115C163⋊C8—21no—
116C163⋊C8—21no—
117C163⋊C8—21no—
118C163⋊C8—21no—
119C23×C163—21no—
120Q8×C163—21no—
121Q8×C163—21no—
122Q8×C163—21no—
123Q8×C163—21no—
124Q8×C163—41no—
125D4×C163—21no—
126D4×C163—21no—
127C2×C4×C163—21no—
128C2×C4×C163—21no—
129C2×C4×C163—21no—
130C2×C4×C163—42no—
131C2×C4×C163C2×D411yescharacteristic
132C2×C4×C163C2×D411yescharacteristic
133C2×C4×C163C2×D411yescharacteristic
134C2×C4×C163C2×D411yescharacteristic
135C2×C4×C163C2×D411yescharacteristic
136C2×C4×C163C2×D411yescharacteristic
137C2×C4×C163C2411yescharacteristic
138C22×C163—41no—
139C22×C163—41no—
140C22×C163—41no—
141C22×C163C22×D411yescharacteristic
142C2×D163—22no—
143C2×D163—41no—
144C2×D163—41no—
145C4×C163—21no—
146C4×C163—21no—
147C4×C163—21no—
148C4×C163—21no—
149C4×C163—21no—
150C4×C163—21no—
151C4×C163—41no—
152C4×C163—41no—
153C4×C163—41no—
154C4×C163C2×#4111yescharacteristic
155C4×C1632−1+411yescharacteristic
156Dic163—41no—
157Dic163—41no—
158Dic163—41no—
159C2×C163—81no—
160C2×C163#54811yescharacteristic
161C2×C163#57511yescharacteristic
162C2×C163#57711yescharacteristic
163D163—42no—
164C163#337611yescharacteristic, complemented, Sylow
165#3376—1631nomaximal, Sylow
166#556D16311yescharacteristic, complemented
167#554—1631no—
168#491—1631no—
169#489—1631no—
170#488—1631no—
171#479—1631no—
172#478—1631no—
173#475—1631no—
174#463—1631no—
175#463—1631no—
176#461—1631no—
177#446—1631no—
178#442—1631no—
179#440—1631no—
180C4×SD16—1631no—
181C2×SD16—1631no—
182C2×SD16—1631no—
183C2×D8—1631no—
184C4⋊Q8—1631no—
185C4⋊Q8C2×D16311yescharacteristic
186#126—21no—
187#126—21no—
188#125—1631no—
189#125—1631no—
190#124C2×D16311yescharacteristic
191#124C2×D16311yescharacteristic
192C22⋊Q8—21no—
193C22⋊Q8—3261no—
194C22⋊Q8—3261no—
195C22⋊Q8—3261no—
196C4×Q8C2×D16311yescharacteristic
197C4×Q8C2×D16311yescharacteristic
198C4×Q8C2×D16311yescharacteristic
199C4×D4—1631no—
200C4×D4—1631no—
201C4×D4—1631no—
202C4×D4C2×D16311yescharacteristic
203#117—21no—
204C2×C4⋊C4—3261no—
205#107—1631no—
206#106—1631no—
207#106—1631no—
208C4⋊C8—1631no—
209C4⋊C8—1631no—
210C4⋊C8—1631no—
211#103—1631no—
212#103—1631no—
213#103—1631no—
214#103—1631no—
215#103—1631no—
216#103—1631no—
217#102—1631no—
218#102—1631no—
219#102—1631no—
220#102—1631no—
221#97—1631no—
222#97—1631no—
223C4×C8—1631no—
224C2×Q8—3261no—
225C2×Q8C22⋊D16311yescharacteristic
226C2×Q8C22×D16311yescharacteristic
227C2×Q8C22×D16311yescharacteristic
228C2×D4—1631no—
229C2×D4C22×D16311yescharacteristic
230C22×C4—21no—
231C22×C4—3261no—
232C22×C4—3261no—
233C22×C4—3261no—
234SD16—3262no—
235SD16—3262no—
236D8—3262no—
237C2×C8—1631no—
238C2×C8—1631no—
239C2×C8—1631no—
240C2×C8—1631no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—3261no—
246C4⋊C4—3261no—
247C4⋊C4—3261no—
248C4⋊C4—3261no—
249C4⋊C4—3261no—
250C4⋊C4—1631no—
251C4⋊C4—1631no—
252C4⋊C4—1631no—
253C4⋊C4C22⋊D16311yescharacteristic
254C4⋊C4C22⋊D16311yescharacteristic
255C4⋊C4C22⋊D16311yescharacteristic
256C4⋊C4C22×D16311yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—3261no—
263C22⋊C4—3261no—
264C22⋊C4—3261no—
265C42—21no—
266C42—21no—
267C42C22×D16311yescharacteristic
268C42C22×D16311yescharacteristic
269C42C22×D16311yescharacteristic
270C23—21no—
271C23—3261no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—41no—
277Q8—13041no—
278D4—21no—
279D4—21no—
280D4—3261no—
281D4—1632no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—13041no—
287C2×C4—13041no—
288C2×C4—3261no—
289C2×C4—3261no—
290C2×C4—3261no—
291C2×C4—6522no—
292C2×C4C2×C22⋊D16311yescharacteristic
293C2×C4C2×C22⋊D16311yescharacteristic
294C2×C4C2×C22⋊D16311yescharacteristic
295C2×C4C2×C22⋊D16311yescharacteristic
296C2×C4C2×C22⋊D16311yescharacteristic
297C2×C4C2×C22⋊D16311yescharacteristic
298C2×C4C23×D16311yescharacteristic
299C8—3261no—
300C8—3261no—
301C8—3261no—
302C8—3261no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—3262no—
307C22—6521no—
308C22—6521no—
309C22C22×C22⋊D16311yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—6521no—
320C4—6521no—
321C4—6521no—
322C4#41×D16311yescharacteristic
323C4#1910478211yescharacteristic
324C2—81no—
325C2—6522no—
326C2#2378232211yescharacteristic
327C2#2378246311yescharacteristic
328C2#2378247311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#556×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579684C22
hypercenter#556D163
hypocenterC163#3376
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #23782375C2semidirect
2. #23782033C2semidirect
3. #23782024C2semidirect
4. #23782019C2semidirect
5. #23781972C2semidirect
6. #23781965C2semidirect
7. #23781950C2semidirect
10. #23781867C2semidirect
11. #23781796C2semidirect
13. #23781763C2semidirect
14. C4×#14579686C2semidirect
15. #556×C163C2semidirect
164. C163#3376semidirect
166. #556D163semidirect

Maximal subgroups

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

subgroupindex
#237823752
#237820332
#237820242
#237820192
#237819722
#237819652
#237819502
#237818772
#237818672
#237817962
#237817752
#237817632
C4×#145796862
#556×C1632
#3376163