Finite Group Databasebeta

← #35224382

Subgroups — #35224382

Order 20864.

376 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782292C211yesmaximal, characteristic, complemented
2#23782144C211yesmaximal, characteristic
3#23781973C211yesmaximal, characteristic, complemented
4#23781971C211yesmaximal, characteristic, complemented
5#23781967C211yesmaximal, characteristic, complemented
6#23781965C211yesmaximal, characteristic, complemented
7#23781959C211yesmaximal, characteristic, complemented
8#23781957C211yesmaximal, characteristic, complemented
9#23781943C211yesmaximal, characteristic, complemented
10#23781706C211yesmaximal, characteristic, complemented
11#107×D163C211yesmaximal, characteristic, complemented
12#23781690C211yesmaximal, characteristic, complemented
13#23781663C211yesmaximal, characteristic
14#23781658C211yesmaximal, characteristic
15#474×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×#14579682C2211yescharacteristic
18C2×C8×D163C2211yescharacteristic
19#19104734C2211yescharacteristic
20#19104733—21no—
21#19104727C2211yescharacteristic
22#19104726C2211yescharacteristic
23#19104725—21no—
24#19104713—21no—
25#19104711—21no—
26C4×Dic326C2211yescharacteristic
27#19104700C2211yescharacteristic
28D4×Dic163C2211yescharacteristic
29C4×C22⋊D163—21no—
30(C22⋊D163)⋊C4—21no—
31#19104693C2211yescharacteristic
32#19104692C2211yescharacteristic
33#19104691—21no—
34C2×Dic163⋊C4—21no—
35C4⋊C4×D163C2211yescharacteristic
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#19104658C2211yescharacteristic
49#19104658C2211yescharacteristic
50#19104656C2211yescharacteristic
51#19104656C2211yescharacteristic
52C4⋊Q8×C163C2211yescharacteristic
53C4×D4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#14579698—22no—
60#14579698Q814yescomplemented
61C2×Dic326—21no—
62C2×Dic326C2311yescharacteristic
63C2×C22⋊D163—21no—
64C22×Dic163—21no—
65C22×Dic163—21no—
66C2×C4×D163—21no—
67C2×C4×D163—21no—
68C2×C4×D163C2311yescharacteristic
69#14579682—22no—
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C8×D163—22no—
73C4⋊Dic163—21no—
74C4⋊Dic163C2311yescharacteristic
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4D411yescharacteristic
86Dic163⋊C4D411yescharacteristic
87C22⋊Dic163—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4D411yescharacteristic
90(C2×D163)⋊C4D411yescharacteristic
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163—21no—
94C4×Dic163C2311yescharacteristic
95C2×Q8×C163—21no—
96C2×D4×C163C2311yescharacteristic
97C22×C4×C163—21no—
98C2×C8×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C4⋊C4×C163—21no—
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C22⋊C4×C163—21no—
105C42×C163C2311yescharacteristic
106C22×D163C2×D411yescharacteristic
107Dic326—21no—
108Dic326—81no—
109Dic326C2×Q812yes—
110C22⋊D163—24no—
111C2×Dic163—21no—
112C2×Dic163—24no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—81no—
117C2×Dic163C2×D411yescharacteristic
118C2×Dic163C2×D411yescharacteristic
119C2×Dic163C2×D411yescharacteristic
120C4×D163—81no—
121C4×D163—81no—
122C4×D163C2×Q812yes—
123C4×D163#4112yes—
124C163⋊C8—21no—
125C163⋊C8—21no—
126C23×C163—21no—
127Q8×C163—81no—
128D4×C163—21no—
129D4×C163C2×Q812yes—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—81no—
134C2×C4×C163C2×D411yescharacteristic
135C2×C4×C163C2×D411yescharacteristic
136C2×C4×C163C2411yescharacteristic
137C8×C163—21no—
138C8×C163—21no—
139C22×C163—22no—
140C22×C163—41no—
141C22×C163—41no—
142C22×C163C22×D411yescharacteristic
143C2×D163—21no—
144C2×D163—21no—
145C2×D163C22⋊Q812yes—
146C4×C163—21no—
147C4×C163—21no—
148C4×C163—41no—
149C4×C163—41no—
150C4×C163—41no—
151C4×C163C22×Q811yescharacteristic
152C4×C163C2×#4111yescharacteristic
153Dic163—22no—
154Dic163—21no—
155Dic163—21no—
156Dic163—41no—
157Dic163—41no—
158Dic163—41no—
159Dic163C22⋊Q812yes—
160C2×C163—42no—
161C2×C163C2×C22⋊Q811yescharacteristic
162C2×C163#57611yescharacteristic
163C2×C163#57811yescharacteristic
164D163—22no—
165C163#308311yescharacteristic, complemented, Sylow
166#3083—1631nomaximal, Sylow
167#544—1631no—
168#520—1631no—
169#479—1631no—
170#479—1631no—
171#478—1631no—
172#478—1631no—
173#477—1631no—
174#477—1631no—
175#474—1631no—
176#474D16311yescharacteristic, complemented
177#428—1631no—
178C2×#107—1631no—
179#424—1631no—
180#417—1631no—
181#416—1631no—
182C2×#41—1631no—
183C2×M16—1631no—
184C22×C8—1631no—
185C4⋊Q8—3261no—
186C4⋊Q8—1631no—
187C4⋊Q8C2×D16311yescharacteristic
188#125—3261no—
189#125—1631no—
190#125—1631no—
191C22⋊Q8—3261no—
192C22⋊Q8—3261no—
193C4×Q8—1631no—
194C4×Q8—1631no—
195C4×D4—3261no—
196C4×D4—3261no—
197C4×D4—1631no—
198C4×D4C2×D16311yescharacteristic
199#117—3261no—
200#117—1631no—
201#117—1631no—
202C2×C4⋊C4—3261no—
203C2×C4⋊C4—1631no—
204C2×C4⋊C4—1631no—
205#107—1631no—
206#107—1631no—
207#107—1631no—
208#107—1631no—
209#107C2×D16311yescharacteristic
210#107C2×D16311yescharacteristic
211#106—1631no—
212#106—1631no—
213C4⋊C8—1631no—
214C4⋊C8—1631no—
215C4⋊C8—1631no—
216C4⋊C8C2×D16311yescharacteristic
217#103—1631no—
218#103—1631no—
219#103—1631no—
220#103—1631no—
221#102—1631no—
222#102—1631no—
223#102C2×D16311yescharacteristic
224#102C2×D16311yescharacteristic
225#41—3262no—
226#41—1634no—
227C2×Q8—21no—
228C2×Q8—3261no—
229C2×Q8—1631no—
230C2×D4—3261no—
231C2×D4C22×D16311yescharacteristic
232C22×C4—21no—
233C22×C4—3261no—
234C22×C4—3261no—
235C22×C4—3261no—
236C22×C4—3261no—
237C22×C4—1631no—
238M16—3262no—
239C2×C8—3262no—
240C2×C8—1631no—
241C2×C8—1631no—
242C2×C8C22×D16311yescharacteristic
243C2×C8C22×D16311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—3261no—
246C4⋊C4—3261no—
247C4⋊C4—3261no—
248C4⋊C4—3261no—
249C4⋊C4—3261no—
250C4⋊C4—3261no—
251C4⋊C4—3261no—
252C4⋊C4—3261no—
253C4⋊C4—3261no—
254C4⋊C4—1631no—
255C4⋊C4—1631no—
256C4⋊C4—1631no—
257C4⋊C4—1631no—
258C4⋊C4—1631no—
259C4⋊C4C22×D16311yescharacteristic
260C4⋊C4C22×D16311yescharacteristic
261C4⋊C4C22×D16311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—3261no—
264C22⋊C4—3261no—
265C22⋊C4—1631no—
266C22⋊C4—1631no—
267C42—3261no—
268C42—3261no—
269C42—3261no—
270C42—1631no—
271C42C22×D16311yescharacteristic
272C23—21no—
273C23—1631no—
274Q8—81no—
275Q8—13041no—
276Q8—3261no—
277Q8—1632no—
278D4—21no—
279D4—3264no—
280D4Q8×D16312yes—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—13041no—
286C2×C4—13041no—
287C2×C4—13041no—
288C2×C4—3261no—
289C2×C4—3264no—
290C2×C4—3261no—
291C2×C4—3261no—
292C2×C4—3261no—
293C2×C4—1631no—
294C2×C4—1631no—
295C2×C4—1631no—
296C2×C4—1632no—
297C2×C4—1632no—
298C2×C4D4×D16311yescharacteristic
299C2×C4D4×D16311yescharacteristic
300C2×C4C23×D16311yescharacteristic
301C8—21no—
302C8—21no—
303C8—3261no—
304C8—3261no—
305C22—22no—
306C22—41no—
307C22—41no—
308C22—3261no—
309C22—3261no—
310C22—1632no—
311C22C2×D4×D16311yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—3262no—
318C4—3261no—
319C4—3261no—
320C4—6521no—
321C4—6521no—
322C4—6521no—
323C4—1632no—
324C4C2×Q8×D16311yescharacteristic
325C4#41×D16311yescharacteristic
326C2—42no—
327C2—3262no—
328C2C22⋊Q8×D16311yescharacteristic
329C2#2378246611yescharacteristic
330C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#474×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579698C22
hypercenter#474D163
hypocenterC163#3083
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #23782292C2semidirect
3. #23781973C2semidirect
4. #23781971C2semidirect
5. #23781967C2semidirect
6. #23781965C2semidirect
7. #23781959C2semidirect
8. #23781957C2semidirect
9. #23781943C2semidirect
10. #23781706C2semidirect
11. #107×D163C2semidirect
12. #23781690C2semidirect
15. #474×C163C2semidirect
60. #14579698Q8semidirect
165. C163#3083semidirect
176. #474D163semidirect

Maximal subgroups

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

subgroupindex
#237822922
#237821442
#237819732
#237819712
#237819672
#237819652
#237819592
#237819572
#237819432
#237817062
#107×D1632
#237816902
#237816632
#237816582
#474×C1632
#3083163