Finite Group Databasebeta

← #35225063

Subgroups — #35225063

Order 20864.

340 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782295C211yesmaximal, characteristic
2#23782292C211yesmaximal, characteristic
3#23782074C211yesmaximal, characteristic, complemented
4#23782073C211yesmaximal, characteristic, complemented
5#23782072C211yesmaximal, characteristic, complemented
6#23782066C211yesmaximal, characteristic, complemented
7#23782064C211yesmaximal, characteristic, complemented
8#23782062C211yesmaximal, characteristic, complemented
9#23782061C211yesmaximal, characteristic, complemented
10#23781706C211yesmaximal, characteristic
11#23781704C211yesmaximal, characteristic
12#23781702C211yesmaximal, characteristic
13#106×D163C211yesmaximal, characteristic
14#23781622C211yesmaximal, characteristic
15#501×C163C211yesmaximal, characteristic, complemented
16C2×#14579682C2211yescharacteristic
17C2×C8×D163C2211yescharacteristic
18#19104735C2211yescharacteristic
19#19104734—21no—
20#19104734C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104733—21no—
23#19104727C2211yescharacteristic
24#19104726C2211yescharacteristic
25#19104726C2211yescharacteristic
26#19104725—21no—
27#19104725—21no—
28#19104715—21no—
29#19104713—21no—
30#19104713—21no—
31#19104711—21no—
32#19104693C2211yescharacteristic
33#19104693C2211yescharacteristic
34#19104692C2211yescharacteristic
35C4⋊C4×D163C2211yescharacteristic
36C4⋊C4×D163C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104669C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104667C2211yescharacteristic
42#19104667C2211yescharacteristic
43#19104666C2211yescharacteristic
44#19104666C2211yescharacteristic
45#19104666C2211yescharacteristic
46#19104666C2211yescharacteristic
47#19104666C2211yescharacteristic
48#19104666C2211yescharacteristic
49#19104645C2211yescharacteristic
50C4×C163⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C4⋊Q8×C163C2211yescharacteristic
53#125×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic
56#106×C163C2211yescharacteristic
57#106×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×Dic326—21no—
62C2×C4×D163—21no—
63C2×C4×D163—21no—
64C2×C4×D163C2311yescharacteristic
65#14579682Q814yescomplemented
66C2×C163⋊C8C2311yescharacteristic
67C2×C163⋊C8C2311yescharacteristic
68C8×D163Q814yescomplemented
69C4⋊Dic163—21no—
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⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4D411yescharacteristic
85Dic163⋊C4D411yescharacteristic
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4D411yescharacteristic
89(C2×D163)⋊C4D411yescharacteristic
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163C2311yescharacteristic
93C2×Q8×C163—21no—
94C2×C8×C163C2311yescharacteristic
95C2×C8×C163C2311yescharacteristic
96C4⋊C4×C163—21no—
97C4⋊C4×C163—21no—
98C4⋊C4×C163—21no—
99C4⋊C4×C163C2311yescharacteristic
100C4⋊C4×C163C2311yescharacteristic
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C42×C163C2311yescharacteristic
104C22×D163C2×D411yescharacteristic
105Dic326—81no—
106Dic326—81no—
107Dic326—81no—
108C2×Dic163—21no—
109C2×Dic163—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163C2×D411yescharacteristic
113C2×Dic163C2×D411yescharacteristic
114C2×Dic163C2×D411yescharacteristic
115C4×D163—81no—
116C4×D163—81no—
117C4×D163C2×Q812yes—
118C4×D163C2×Q812yes—
119C163⋊C8C2×Q812yes—
120C163⋊C8C2×Q812yes—
121Q8×C163—81no—
122C2×C4×C163—21no—
123C2×C4×C163—21no—
124C2×C4×C163—21no—
125C2×C4×C163—21no—
126C2×C4×C163C2×D411yescharacteristic
127C2×C4×C163C2×D411yescharacteristic
128C2×C4×C163C2411yescharacteristic
129C8×C163C2×Q812yes—
130C8×C163C2×Q812yes—
131C22×C163C22×D411yescharacteristic
132C2×D163—21no—
133C2×D163—21no—
134C2×D163C4⋊Q812yes—
135C4×C163—21no—
136C4×C163—21no—
137C4×C163—41no—
138C4×C163—41no—
139C4×C163—41no—
140C4×C163—41no—
141C4×C163C22×Q811yescharacteristic
142C4×C163C22×Q811yescharacteristic
143Dic163—21no—
144Dic163—21no—
145Dic163—41no—
146Dic163—41no—
147Dic163—41no—
148Dic163—41no—
149Dic163C4⋊Q812yes—
150C2×C163C2×C4⋊Q811yescharacteristic
151C2×C163#57711yescharacteristic
152C2×C163#57711yescharacteristic
153D163—22no—
154C163#316311yescharacteristic, complemented, Sylow
155#3163—1631nomaximal, Sylow
156#544—1631no—
157#544—1631no—
158#501—1631no—
159#501—1631no—
160#501—1631no—
161#501D16311yescharacteristic, complemented
162#499—1631no—
163#499—1631no—
164#498—1631no—
165#498—1631no—
166#428—1631no—
167#428—1631no—
168#427—1631no—
169C2×#106—1631no—
170#405—1631no—
171C2×M16—1631no—
172C22×C8—1631no—
173C4⋊Q8—3261no—
174C4⋊Q8—3261no—
175C4⋊Q8—1631no—
176C4⋊Q8—1631no—
177C4⋊Q8—1631no—
178C4⋊Q8C2×D16311yescharacteristic
179#125—3261no—
180#125—3261no—
181#125—1631no—
182#125—1631no—
183#125—1631no—
184#125C2×D16311yescharacteristic
185C22⋊Q8—3261no—
186C22⋊Q8—3261no—
187C22⋊Q8—3261no—
188C22⋊Q8—3261no—
189#117—1631no—
190#117—1631no—
191#117—1631no—
192C2×C4⋊C4—1631no—
193C2×C4⋊C4—1631no—
194#107—1631no—
195#107—1631no—
196#107—1631no—
197#107—1631no—
198#107C2×D16311yescharacteristic
199#107C2×D16311yescharacteristic
200#106—1631no—
201#106—1631no—
202#106—1631no—
203#106—1631no—
204#106—1631no—
205#106—1631no—
206#106—1631no—
207#106—1631no—
208#106C2×D16311yescharacteristic
209#106C2×D16311yescharacteristic
210#97—1631no—
211#97C2×D16311yescharacteristic
212C4×C8—1631no—
213C4×C8—1631no—
214C2×Q8—21no—
215C2×Q8—3261no—
216C2×Q8—3261no—
217C2×Q8—3261no—
218C22×C4—3261no—
219C22×C4—3261no—
220C22×C4—1631no—
221M16—1634no—
222C2×C8—1631no—
223C2×C8—1631no—
224C2×C8—1634no—
225C2×C8C22×D16311yescharacteristic
226C2×C8C22×D16311yescharacteristic
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—3261no—
231C4⋊C4—3261no—
232C4⋊C4—3261no—
233C4⋊C4—3261no—
234C4⋊C4—3261no—
235C4⋊C4—3261no—
236C4⋊C4—3261no—
237C4⋊C4—3261no—
238C4⋊C4—3261no—
239C4⋊C4—3261no—
240C4⋊C4—3261no—
241C4⋊C4—1631no—
242C4⋊C4—1631no—
243C4⋊C4—1631no—
244C4⋊C4—1631no—
245C4⋊C4—1631no—
246C4⋊C4—1631no—
247C4⋊C4C22×D16311yescharacteristic
248C4⋊C4C22×D16311yescharacteristic
249C4⋊C4C22×D16311yescharacteristic
250C4⋊C4C22×D16311yescharacteristic
251C22⋊C4—3261no—
252C22⋊C4—3261no—
253C22⋊C4—1631no—
254C22⋊C4—1631no—
255C42—3261no—
256C42—3261no—
257C42—1631no—
258C42C22×D16311yescharacteristic
259C23—1631no—
260Q8—81no—
261Q8—13041no—
262Q8—13041no—
263Q8—13041no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—13041no—
269C2×C4—13041no—
270C2×C4—3261no—
271C2×C4—3261no—
272C2×C4—3261no—
273C2×C4—3261no—
274C2×C4—1631no—
275C2×C4—1631no—
276C2×C4—1631no—
277C2×C4—1632no—
278C2×C4—1632no—
279C2×C4D4×D16311yescharacteristic
280C2×C4D4×D16311yescharacteristic
281C2×C4C23×D16311yescharacteristic
282C8—1632no—
283C8—1632no—
284C8Q8×D16312yes—
285C8Q8×D16312yes—
286C22—3261no—
287C22—3261no—
288C22—1632no—
289C22C2×D4×D16311yescharacteristic
290C4—21no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—3261no—
297C4—3261no—
298C4—6521no—
299C4—6521no—
300C4—6521no—
301C4—6521no—
302C4—1632no—
303C4C2×Q8×D16311yescharacteristic
304C4C2×Q8×D16311yescharacteristic
305C2—3262no—
306C2C4⋊Q8×D16311yescharacteristic
307C2#2378247411yescharacteristic
308C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#501×C163C2
socleC22×C163C22×D4
prime radicalC22×D163C2×D4
hypercenter#501D163
hypocenterC163#3163
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #23782074C2semidirect
4. #23782073C2semidirect
5. #23782072C2semidirect
6. #23782066C2semidirect
7. #23782064C2semidirect
8. #23782062C2semidirect
9. #23782061C2semidirect
15. #501×C163C2semidirect
65. #14579682Q8semidirect
68. C8×D163Q8semidirect
154. C163#3163semidirect
161. #501D163semidirect

Maximal subgroups

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

subgroupindex
#237822952
#237822922
#237820742
#237820732
#237820722
#237820662
#237820642
#237820622
#237820612
#237817062
#237817042
#237817022
#106×D1632
#237816222
#501×C1632
#3163163