Finite Group Databasebeta

← #35226629

Subgroups — #35226629

Order 20864.

382 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782353C211yesmaximal, characteristic, complemented
2#23782319C211yesmaximal, characteristic
3#23782035C211yesmaximal, characteristic, complemented
4#23782009C211yesmaximal, characteristic, complemented
5#23782002C211yesmaximal, characteristic, complemented
6#23781924C211yesmaximal, characteristic, complemented
7#23781907C211yesmaximal, characteristic, complemented
8#23781876C211yesmaximal, characteristic, complemented
9#23781828C211yesmaximal, characteristic, complemented
10#23781828C211yesmaximal, characteristic, complemented
11#23781765C211yesmaximal, characteristic, complemented
12#23781735C211yesmaximal, characteristic, complemented
13#23781663C211yesmaximal, characteristic, complemented
14#23781658C211yesmaximal, characteristic, complemented
15#478×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C2×Dic652C2211yescharacteristic
19C2×Dic652C2211yescharacteristic
20C2×#14579687C2211yescharacteristic
21C2×#14579682C2211yescharacteristic
22C2×C8×D163C2211yescharacteristic
23#19104735C2211yescharacteristic
24#19104730—21no—
25#19104726—21no—
26#19104720—21no—
27#19104719—21no—
28#19104719—21no—
29#19104715C2211yescharacteristic
30#19104715C2211yescharacteristic
31#19104715C2211yescharacteristic
32#19104715C2211yescharacteristic
33#19104709—21no—
34C4×Dic326C2211yescharacteristic
35C4×D652C2211yescharacteristic
36#19104693—21no—
37#19104693C2211yescharacteristic
38C2×C4⋊Dic163—21no—
39C4⋊C4×D163C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104660C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104658C2211yescharacteristic
44#19104658C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104656C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104646C2211yescharacteristic
52#125×C163C2211yescharacteristic
53C4×D4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#14579698—22no—
60#14579698—22no—
61#14579698—41no—
62#14579698—41no—
63C2×Dic326C2311yescharacteristic
64C2×Dic326C2311yescharacteristic
65C2×Dic326C2311yescharacteristic
66C2×D652D411yescharacteristic
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C22×Dic163—21no—
70C22×Dic163—21no—
71C2×C4×D163—21no—
72C2×C4×D163C2311yescharacteristic
73C2×C4×D163C2311yescharacteristic
74Dic652—41no—
75Dic652—41no—
76#14579687—22no—
77#14579682—41no—
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C8×D163—41no—
81C4⋊Dic163—21no—
82C4⋊Dic163—21no—
83C4⋊Dic163—21no—
84C4⋊Dic163—21no—
85C4⋊Dic163—21no—
86C4⋊Dic163—21no—
87C4⋊Dic163D411yescharacteristic
88C4⋊Dic163C2311yescharacteristic
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4D411yescharacteristic
97(C2×D163)⋊C4D411yescharacteristic
98C4×Dic163—21no—
99C4×Dic163—21no—
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163—21no—
102C2×C8×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C4⋊C4×C163—21no—
105C4⋊C4×C163—21no—
106C4⋊C4×C163C2311yescharacteristic
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163C2×D411yescharacteristic
112C22×D163C2×D411yescharacteristic
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116Dic326—21no—
117Dic326—21no—
118Dic326—21no—
119D652—41no—
120C22⋊D163—42no—
121C22⋊D163—42no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—42no—
127C2×Dic163—42no—
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C4×D163—21no—
131C4×D163—21no—
132C4×D163—21no—
133C4×D163—21no—
134C4×D163—42no—
135C163⋊C8—21no—
136C163⋊C8—21no—
137C23×C163—21no—
138D4×C163—21no—
139D4×C163#4112yes—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—42no—
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2411yescharacteristic
147C8×C163—21no—
148C8×C163—21no—
149C22×C163—22no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163C22×D411yescharacteristic
153C2×D163—21no—
154C2×D163—21no—
155C2×D163—21no—
156C2×D163—21no—
157C2×D163—21no—
158C2×D163—21no—
159C4×C163—21no—
160C4×C163—21no—
161C4×C163—41no—
162C4×C163—41no—
163C4×C163—41no—
164C4×C163C2×#4111yescharacteristic
165C4×C163H(5,𝔽2)11yescharacteristic
166Dic163—21no—
167Dic163—21no—
168Dic163—41no—
169Dic163—41no—
170Dic163—41no—
171Dic163—41no—
172C2×C163—42no—
173C2×C163#54611yescharacteristic
174C2×C163#57511yescharacteristic
175C2×C163#57811yescharacteristic
176D163—41no—
177D163—41no—
178C163#331511yescharacteristic, complemented, Sylow
179#3315—1631nomaximal, Sylow
180#552—1631no—
181#547—1631no—
182#491—1631no—
183#485—1631no—
184#484—1631no—
185#478D16311yescharacteristic, complemented
186#470—1631no—
187#467—1631no—
188#462—1631no—
189#452—1631no—
190#452—1631no—
191#440—1631no—
192#435—1631no—
193#417—1631no—
194#416—1631no—
195C2×#41—1631no—
196C2×#41—1631no—
197C2×Q16—1631no—
198C2×Q16—1631no—
199C2×SD16—1631no—
200C2×M16—1631no—
201C22×C8—1631no—
202C4⋊Q8—1631no—
203#126—3261no—
204#125—3261no—
205#125C2×D16311yescharacteristic
206#123—3261no—
207#123—3261no—
208#123—3261no—
209C22⋊Q8—1631no—
210C22⋊Q8—1631no—
211C22⋊Q8—1631no—
212C22⋊Q8—1631no—
213#121—3261no—
214C4×Q8—1631no—
215C4×D4—1631no—
216C4×D4C2×D16311yescharacteristic
217#117—3261no—
218#117—1631no—
219C2×C4⋊C4—3261no—
220C2×C4⋊C4—1631no—
221#107—1631no—
222#107C2×D16311yescharacteristic
223#106C2×D16311yescharacteristic
224C4⋊C8C2×D16311yescharacteristic
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103—1631no—
231#103—1631no—
232#102—1631no—
233#102C2×D16311yescharacteristic
234#102C2×D16311yescharacteristic
235C22⋊C8—1631no—
236C22⋊C8—1631no—
237#97—1631no—
238#41—3262no—
239#41—3262no—
240#41—6521no—
241#41—6521no—
242C2×Q8—1631no—
243C2×Q8—1631no—
244C2×Q8—1631no—
245C2×D4—3261no—
246C2×D4—3261no—
247C2×D4—1631no—
248C2×D4C22×D16311yescharacteristic
249C22×C4—21no—
250C22×C4—3261no—
251C22×C4—3261no—
252C22×C4—3261no—
253C22×C4—1631no—
254C22×C4—1631no—
255Q16—6521no—
256Q16—6521no—
257SD16—3262no—
258M16—6521no—
259C2×C8—6521no—
260C2×C8—1631no—
261C2×C8—1631no—
262C2×C8C22×D16311yescharacteristic
263C2×C8C22×D16311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—3261no—
272C4⋊C4—3261no—
273C4⋊C4—3261no—
274C4⋊C4—1631no—
275C4⋊C4—1631no—
276C4⋊C4C22×D16311yescharacteristic
277C4⋊C4C22×D16311yescharacteristic
278C4⋊C4C22×D16311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4—1631no—
286C22⋊C4—1631no—
287C42—3261no—
288C42—3261no—
289C42C22×D16311yescharacteristic
290C23—21no—
291C23—1631no—
292C23—1631no—
293Q8—3261no—
294Q8—3261no—
295Q8—3261no—
296Q8—3261no—
297Q8—3261no—
298Q8—3261no—
299D4—21no—
300D4—6521no—
301D4—6522no—
302D4—6522no—
303D4#1457970012yes—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—3261no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—6522no—
317C2×C4—6522no—
318C2×C4—6522no—
319C2×C4—1631no—
320C2×C4—1631no—
321C2×C4D4×D16311yescharacteristic
322C2×C4D4×D16311yescharacteristic
323C2×C4C23×D16311yescharacteristic
324C8—21no—
325C8—21no—
326C8—3261no—
327C8—3261no—
328C22—22no—
329C22—41no—
330C22—41no—
331C22—3261no—
332C22—3261no—
333C22—3261no—
334C22—3261no—
335C22—3261no—
336C22—3261no—
337C22C2×D4×D16311yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3261no—
344C4—3261no—
345C4—6521no—
346C4—6521no—
347C4—6521no—
348C4—6521no—
349C4C2×#1457970011yescharacteristic
350C4#1910478111yescharacteristic
351C2—42no—
352C2—6521no—
353C2—6521no—
354C2#2378230611yescharacteristic
355C2#2378246111yescharacteristic
356C2#2378247911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#478×C163C2
socleC22×C163C22×D4
prime radical#23782319C2
hypercenter#478D163
hypocenterC163#3315
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #23782353C2semidirect
3. #23782035C2semidirect
4. #23782009C2semidirect
5. #23782002C2semidirect
6. #23781924C2semidirect
7. #23781907C2semidirect
8. #23781876C2semidirect
9. #23781828C2semidirect
10. #23781828C2semidirect
11. #23781765C2semidirect
12. #23781735C2semidirect
13. #23781663C2semidirect
14. #23781658C2semidirect
15. #478×C163C2semidirect
178. C163#3315semidirect
185. #478D163semidirect

Maximal subgroups

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

subgroupindex
#237823532
#237823192
#237820352
#237820092
#237820022
#237819242
#237819072
#237818762
#237818282
#237817652
#237817352
#237816632
#237816582
#478×C1632
#3315163