Finite Group Databasebeta

← #35224706

Subgroups — #35224706

Order 20864.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782194C211yesmaximal, characteristic
2C4×Q8×D163C211yesmaximal, characteristic, complemented
3#23781999C211yesmaximal, characteristic, complemented
4#23781982C211yesmaximal, characteristic
5#23781973C211yesmaximal, characteristic, complemented
6#23781972C211yesmaximal, characteristic, complemented
7#23781884C211yesmaximal, characteristic, complemented
8#23781819C211yesmaximal, characteristic, complemented
9#23781802C211yesmaximal, characteristic
10#23781772C211yesmaximal, characteristic, complemented
11#23781769C211yesmaximal, characteristic, complemented
12#23781766C211yesmaximal, characteristic, complemented
13#23781762C211yesmaximal, characteristic, complemented
14#23781721C211yesmaximal, characteristic, complemented
15#464×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19#19104728—21no—
20#19104727C2211yescharacteristic
21#19104717—21no—
22#19104711C2211yescharacteristic
23#19104706C2211yescharacteristic
24Q8×Dic163C2211yescharacteristic
25Q8×Dic163C2211yescharacteristic
26C4×Dic326—21no—
27C4×Dic326C2211yescharacteristic
28#19104700—21no—
29#19104700C2211yescharacteristic
30D4×Dic163C2211yescharacteristic
31(C22⋊D163)⋊C4—21no—
32#19104691—21no—
33C4⋊C4×D163—21no—
34C4⋊C4×D163C2211yescharacteristic
35C42×D163—21no—
36C42×D163C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104667C2211yescharacteristic
40#19104666C2211yescharacteristic
41C4⋊(C163⋊C8)C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104656C2211yescharacteristic
48(C2×D163)⋊C8C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50#19104645C2211yescharacteristic
51#19104645C2211yescharacteristic
52C2×Q16×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54#124×C163C2211yescharacteristic
55C4×Q8×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61Q8×D163—22no—
62Q8×D163—22no—
63C2×C22⋊D163—21no—
64C22×Dic163—21no—
65C2×C4×D163—21no—
66C2×C4×D163—21no—
67C2×C4×D163D411yescharacteristic
68C2×C4×D163D411yescharacteristic
69C2×C4×D163C2311yescharacteristic
70#14579689—22no—
71#14579687—22no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C4⋊Dic163—21no—
75C4⋊Dic163—21no—
76C4⋊Dic163C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87(C2×D163)⋊C4—21no—
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163D411yescharacteristic
93C4×Dic163D411yescharacteristic
94C4×Dic163C2311yescharacteristic
95C2×Q8×C163C2311yescharacteristic
96C2×Q8×C163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98Q16×C163—22no—
99SD16×C163—22no—
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C4⋊C4×C163—21no—
103C4⋊C4×C163C2311yescharacteristic
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C42×C163—21no—
107C42×C163C2311yescharacteristic
108C22×D163C2×D411yescharacteristic
109Dic326—24no—
110Dic326—21no—
111Dic326#4112yes—
112C22⋊D163—42no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—42no—
118C2×Dic163C2×D411yescharacteristic
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C4×D163—24no—
122C4×D163—22no—
123C4×D163—21no—
124C4×D163—21no—
125C4×D163—41no—
126C4×D163—42no—
127C163⋊C8—21no—
128C163⋊C8—21no—
129C23×C163—21no—
130Q8×C163—21no—
131Q8×C163—21no—
132Q8×C163—21no—
133Q8×C163#4112yes—
134D4×C163—21no—
135D4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2411yescharacteristic
142C8×C163—21no—
143C8×C163—21no—
144C22×C163—41no—
145C22×C163—41no—
146C22×C163—41no—
147C22×C163C22×D411yescharacteristic
148C2×D163—21no—
149C2×D163—21no—
150C2×D163#13712yes—
151C4×C163—22no—
152C4×C163—21no—
153C4×C163—21no—
154C4×C163—41no—
155C4×C163—41no—
156C4×C163C2×#4111yescharacteristic
157C4×C163C2×#4111yescharacteristic
158Dic163—22no—
159Dic163—21no—
160Dic163—21no—
161Dic163—21no—
162Dic163—41no—
163Dic163—41no—
164Dic163—41no—
165C2×C163—81no—
166C2×C163#52511yescharacteristic
167C2×C163C2×#13711yescharacteristic
168C2×C163#57511yescharacteristic
169D163—22no—
170C163#311711yescharacteristic, complemented, Sylow
171#3117—1631nomaximal, Sylow
172#529—1631no—
173C2×C4×Q8—1631no—
174#484—1631no—
175#481—1631no—
176#479—1631no—
177#479—1631no—
178#464—1631no—
179#464D16311yescharacteristic, complemented
180#451—1631no—
181#448—1631no—
182#441—1631no—
183#441—1631no—
184#440—1631no—
185#440—1631no—
186#432—1631no—
187C22×Q8—1631no—
188C2×Q16—1631no—
189C2×Q16C2×D16311yescharacteristic
190C2×SD16—1631no—
191C2×SD16C2×D16311yescharacteristic
192#126—3261no—
193#125—1631no—
194#124C2×D16311yescharacteristic
195#123—3261no—
196C22⋊Q8—1631no—
197#121—1631no—
198C4×Q8—3261no—
199C4×Q8—3261no—
200C4×Q8—1631no—
201C4×Q8—1631no—
202C4×Q8—1631no—
203C4×Q8—1631no—
204C4×Q8C2×D16311yescharacteristic
205C4×D4—3261no—
206C4×D4—1631no—
207#117—3261no—
208C2×C4⋊C4—3261no—
209C2×C4⋊C4—1631no—
210C2×C42—3261no—
211C2×C42—1631no—
212#107—1631no—
213#107—1631no—
214#106—1631no—
215#106—1631no—
216C4⋊C8—1631no—
217C4⋊C8C2×D16311yescharacteristic
218#103—1631no—
219#103—1631no—
220#103—1631no—
221#103—1631no—
222#103—1631no—
223#103C2×D16311yescharacteristic
224#102—1631no—
225#102C2×D16311yescharacteristic
226C22⋊C8—1631no—
227C22⋊C8—1631no—
228#97—1631no—
229#97—1631no—
230C2×Q8—3261no—
231C2×Q8—3262no—
232C2×Q8—3262no—
233C2×Q8—1631no—
234C2×Q8C22×D16311yescharacteristic
235C2×Q8C22×D16311yescharacteristic
236C2×D4—3261no—
237C2×D4C22×D16311yescharacteristic
238C22×C4—3261no—
239C22×C4—3261no—
240C22×C4—3261no—
241C22×C4—1631no—
242C22×C4—1631no—
243C22×C4—1631no—
244Q16—22no—
245Q16—3262no—
246SD16—22no—
247SD16—3262no—
248C2×C8—1631no—
249C2×C8—1631no—
250C2×C8C22×D16311yescharacteristic
251C2×C8C22×D16311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4—3261no—
256C4⋊C4—3261no—
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—1631no—
262C4⋊C4—1631no—
263C4⋊C4—1631no—
264C4⋊C4C22×D16311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—3261no—
268C22⋊C4—3261no—
269C22⋊C4—3261no—
270C42—21no—
271C42—3261no—
272C42—3261no—
273C42—3261no—
274C42—3261no—
275C42—1631no—
276C42—1631no—
277C42—1631no—
278C42C22×D16311yescharacteristic
279C23—21no—
280C23—1631no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—3264no—
285Q8—3261no—
286Q8—1632no—
287Q8#1457969812yes—
288D4—21no—
289D4—21no—
290D4—6522no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—3264no—
295C2×C4—3262no—
296C2×C4—3261no—
297C2×C4—3261no—
298C2×C4—3261no—
299C2×C4—3261no—
300C2×C4—3261no—
301C2×C4—3261no—
302C2×C4—6521no—
303C2×C4—6522no—
304C2×C4—6522no—
305C2×C4—1631no—
306C2×C4—1631no—
307C2×C4—1631no—
308C2×C4D4×D16311yescharacteristic
309C2×C4D4×D16311yescharacteristic
310C2×C4C23×D16311yescharacteristic
311C8—21no—
312C8—21no—
313C8—3261no—
314C8—3261no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—3261no—
319C22—3261no—
320C22—1632no—
321C22C2×D4×D16311yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—3262no—
328C4—3261no—
329C4—3261no—
330C4—3261no—
331C4—6521no—
332C4—6521no—
333C4—6521no—
334C4#41×D16311yescharacteristic
335C4C2×#1457969811yescharacteristic
336C2—81no—
337C2—3262no—
338C2#2378217511yescharacteristic
339C2#137×D16311yescharacteristic
340C2#2378246211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#464×C163C2
socleC22×C163C22×D4
prime radical#19104706C22
hypercenter#464D163
hypocenterC163#3117
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D163C2semidirect
3. #23781999C2semidirect
5. #23781973C2semidirect
6. #23781972C2semidirect
7. #23781884C2semidirect
8. #23781819C2semidirect
10. #23781772C2semidirect
11. #23781769C2semidirect
12. #23781766C2semidirect
13. #23781762C2semidirect
14. #23781721C2semidirect
15. #464×C163C2semidirect
170. C163#3117semidirect
179. #464D163semidirect

Maximal subgroups

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

subgroupindex
#237821942
C4×Q8×D1632
#237819992
#237819822
#237819732
#237819722
#237818842
#237818192
#237818022
#237817722
#237817692
#237817662
#237817622
#237817212
#464×C1632
#3117163