Finite Group Databasebeta

← #35224371

Subgroups — #35224371

Order 20864.

362 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782292C211yesmaximal, characteristic
2#23782141C211yesmaximal, characteristic
3#23781975C211yesmaximal, characteristic, complemented
4#23781973C211yesmaximal, characteristic, complemented
5#23781972C211yesmaximal, characteristic, complemented
6#23781970C211yesmaximal, characteristic, complemented
7#23781958C211yesmaximal, characteristic, complemented
8#23781957C211yesmaximal, characteristic, complemented
9#23781949C211yesmaximal, characteristic, complemented
10#23781704C211yesmaximal, characteristic
11#23781700C211yesmaximal, characteristic
12#23781690C211yesmaximal, characteristic
13#23781672C211yesmaximal, characteristic
14#103×D163C211yesmaximal, characteristic
15#475×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×#14579682C2211yescharacteristic
18C2×C8×D163C2211yescharacteristic
19#19104734C2211yescharacteristic
20#19104733—21no—
21#19104727C2211yescharacteristic
22#19104726C2211yescharacteristic
23#19104725—21no—
24#19104713—21no—
25#19104711—21no—
26Q8×Dic163C2211yescharacteristic
27C4×Dic326—21no—
28C4×Dic326C2211yescharacteristic
29#19104700—21no—
30#19104700C2211yescharacteristic
31#19104693—21no—
32#19104693C2211yescharacteristic
33#19104693C2211yescharacteristic
34#19104692—21no—
35#19104692C2211yescharacteristic
36C4⋊C4×D163C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104667C2211yescharacteristic
42#19104666C2211yescharacteristic
43C4⋊(C163⋊C8)C2211yescharacteristic
44Dic163⋊C8C2211yescharacteristic
45Dic163⋊C8C2211yescharacteristic
46#19104660C2211yescharacteristic
47#19104660C2211yescharacteristic
48#19104659C2211yescharacteristic
49#19104659C2211yescharacteristic
50#19104658C2211yescharacteristic
51#19104658C2211yescharacteristic
52C4⋊Q8×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54#106×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×Dic326C2311yescharacteristic
62Q8×D163—22no—
63Q8×D163Q814yescomplemented
64C2×C4×D163—21no—
65C2×C4×D163—21no—
66C2×C4×D163C2311yescharacteristic
67#14579682—22no—
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C8×D163—22no—
71C4⋊Dic163—21no—
72C4⋊Dic163—21no—
73C4⋊Dic163C2311yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75C4⋊Dic163C2311yescharacteristic
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4D411yescharacteristic
84Dic163⋊C4D411yescharacteristic
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4D411yescharacteristic
88(C2×D163)⋊C4D411yescharacteristic
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163—21no—
94C4×Dic163C2311yescharacteristic
95C2×Q8×C163—21no—
96C2×Q8×C163C2311yescharacteristic
97C2×C8×C163C2311yescharacteristic
98C2×C8×C163C2311yescharacteristic
99C4⋊C4×C163—21no—
100C4⋊C4×C163—21no—
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C42×C163—21no—
105C42×C163C2311yescharacteristic
106C22×D163C2×D411yescharacteristic
107Dic326—21no—
108Dic326—24no—
109Dic326—81no—
110Dic326C2×Q812yes—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163C2×D411yescharacteristic
116C2×Dic163C2×D411yescharacteristic
117C2×Dic163C2×D411yescharacteristic
118C4×D163—24no—
119C4×D163—81no—
120C4×D163C2×Q812yes—
121C4×D163#4112yes—
122C163⋊C8—21no—
123C163⋊C8—21no—
124Q8×C163—21no—
125Q8×C163—81no—
126Q8×C163C2×Q812yes—
127C2×C4×C163—21no—
128C2×C4×C163—21no—
129C2×C4×C163—21no—
130C2×C4×C163—21no—
131C2×C4×C163C2×D411yescharacteristic
132C2×C4×C163C2×D411yescharacteristic
133C2×C4×C163C2411yescharacteristic
134C8×C163—21no—
135C8×C163—21no—
136C22×C163C22×D411yescharacteristic
137C2×D163—21no—
138C2×D163—21no—
139C2×D163C22⋊Q812yes—
140C4×C163—22no—
141C4×C163—21no—
142C4×C163—21no—
143C4×C163—41no—
144C4×C163—41no—
145C4×C163—41no—
146C4×C163C22×Q811yescharacteristic
147C4×C163C2×#4111yescharacteristic
148Dic163—22no—
149Dic163—21no—
150Dic163—21no—
151Dic163—41no—
152Dic163—41no—
153Dic163—41no—
154Dic163C22⋊Q812yes—
155C2×C163C2×C22⋊Q811yescharacteristic
156C2×C163#57711yescharacteristic
157C2×C163#57811yescharacteristic
158D163—22no—
159C163#308211yescharacteristic, complemented, Sylow
160#3082—1631nomaximal, Sylow
161#544—1631no—
162#519—1631no—
163#479—1631no—
164#479—1631no—
165#479—1631no—
166#479—1631no—
167#477—1631no—
168#477—1631no—
169#475—1631no—
170#475D16311yescharacteristic, complemented
171#428—1631no—
172#427—1631no—
173#424—1631no—
174#419—1631no—
175C2×#103—1631no—
176C22×Q8—1631no—
177C2×M16—1631no—
178C22×C8—1631no—
179C4⋊Q8—3261no—
180C4⋊Q8—1631no—
181C4⋊Q8C2×D16311yescharacteristic
182#125—3261no—
183#125—1631no—
184#125—1631no—
185C22⋊Q8—3261no—
186C22⋊Q8—3261no—
187C4×Q8—3261no—
188C4×Q8—3261no—
189C4×Q8—1631no—
190C4×Q8—1631no—
191C4×Q8—1631no—
192C4×Q8C2×D16311yescharacteristic
193#117—3261no—
194#117—3261no—
195#117—1631no—
196#117—1631no—
197#117—1631no—
198C2×C4⋊C4—1631no—
199#107—1631no—
200#107—1631no—
201#107—1631no—
202#107—1631no—
203#106—1631no—
204#106—1631no—
205#106C2×D16311yescharacteristic
206#106C2×D16311yescharacteristic
207C4⋊C8—1631no—
208C4⋊C8—1631no—
209C4⋊C8—1631no—
210C4⋊C8C2×D16311yescharacteristic
211#103—1631no—
212#103—1631no—
213#103—1631no—
214#103—1631no—
215#103—1631no—
216#103—1631no—
217#103C2×D16311yescharacteristic
218#103C2×D16311yescharacteristic
219C2×Q8—21no—
220C2×Q8—3261no—
221C2×Q8—3261no—
222C2×Q8—3262no—
223C2×Q8—1634no—
224C2×Q8—1631no—
225C2×Q8C22×D16311yescharacteristic
226C22×C4—3261no—
227C22×C4—3261no—
228C22×C4—1631no—
229M16—3262no—
230C2×C8—3262no—
231C2×C8—1631no—
232C2×C8—1631no—
233C2×C8C22×D16311yescharacteristic
234C2×C8C22×D16311yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—3261no—
238C4⋊C4—3261no—
239C4⋊C4—3261no—
240C4⋊C4—3261no—
241C4⋊C4—3261no—
242C4⋊C4—3261no—
243C4⋊C4—3261no—
244C4⋊C4—3261no—
245C4⋊C4—3261no—
246C4⋊C4—1631no—
247C4⋊C4—1631no—
248C4⋊C4—1631no—
249C4⋊C4—1631no—
250C4⋊C4—1631no—
251C4⋊C4C22×D16311yescharacteristic
252C4⋊C4C22×D16311yescharacteristic
253C4⋊C4C22×D16311yescharacteristic
254C22⋊C4—3261no—
255C22⋊C4—3261no—
256C22⋊C4—1631no—
257C22⋊C4—1631no—
258C42—21no—
259C42—3261no—
260C42—3261no—
261C42—3261no—
262C42—3261no—
263C42—3261no—
264C42—1631no—
265C42C22×D16311yescharacteristic
266C23—1631no—
267Q8—21no—
268Q8—81no—
269Q8—13041no—
270Q8—3264no—
271Q8—3261no—
272Q8—1632no—
273Q8Q8×D16312yes—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—13041no—
279C2×C4—3264no—
280C2×C4—3261no—
281C2×C4—3261no—
282C2×C4—3261no—
283C2×C4—3261no—
284C2×C4—1631no—
285C2×C4—1631no—
286C2×C4—1631no—
287C2×C4—1632no—
288C2×C4—1632no—
289C2×C4D4×D16311yescharacteristic
290C2×C4D4×D16311yescharacteristic
291C2×C4C23×D16311yescharacteristic
292C8—21no—
293C8—21no—
294C8—3261no—
295C8—3261no—
296C22—3261no—
297C22—3261no—
298C22—1632no—
299C22C2×D4×D16311yescharacteristic
300C4—22no—
301C4—21no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—3262no—
307C4—3261no—
308C4—3261no—
309C4—6521no—
310C4—6521no—
311C4—6521no—
312C4—1632no—
313C4C2×Q8×D16311yescharacteristic
314C4#41×D16311yescharacteristic
315C2—3262no—
316C2C22⋊Q8×D16311yescharacteristic
317C2#2378247111yescharacteristic
318C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#475×C163C2
socleC22×C163C22×D4
prime radicalC22×D163C2×D4
hypercenter#475D163
hypocenterC163#3082
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #23781975C2semidirect
4. #23781973C2semidirect
5. #23781972C2semidirect
6. #23781970C2semidirect
7. #23781958C2semidirect
8. #23781957C2semidirect
9. #23781949C2semidirect
15. #475×C163C2semidirect
63. Q8×D163Q8semidirect
159. C163#3082semidirect
170. #475D163semidirect

Maximal subgroups

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

subgroupindex
#237822922
#237821412
#237819752
#237819732
#237819722
#237819702
#237819582
#237819572
#237819492
#237817042
#237817002
#237816902
#237816722
#103×D1632
#475×C1632
#3082163