Finite Group Databasebeta

← #35226448

Subgroups — #35226448

Order 20864.

386 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#23782386C211yesmaximal, characteristic, complemented
2#23782317C211yesmaximal, characteristic
3#23782014C211yesmaximal, characteristic, complemented
4#23782008C211yesmaximal, characteristic, complemented
5#23782003C211yesmaximal, characteristic, complemented
6#23781944C211yesmaximal, characteristic, complemented
7#23781905C211yesmaximal, characteristic, complemented
8#23781873C211yesmaximal, characteristic, complemented
9#23781826C211yesmaximal, characteristic, complemented
10#23781825C211yesmaximal, characteristic, complemented
11#23781750C211yesmaximal, characteristic, complemented
12C8×C22⋊D163C211yesmaximal, characteristic, complemented
13#23781662C211yesmaximal, characteristic, complemented
14#23781660C211yesmaximal, characteristic, complemented
15#467×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579699C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104733C2211yescharacteristic
23#19104730—21no—
24#19104728—21no—
25#19104721—21no—
26#19104720—21no—
27#19104717—21no—
28#19104715C2211yescharacteristic
29#19104711C2211yescharacteristic
30#19104708—21no—
31Q8×Dic163C2211yescharacteristic
32#19104700—21no—
33#19104697C2211yescharacteristic
34C4×C22⋊D163C2211yescharacteristic
35#19104694C2211yescharacteristic
36#19104693C2211yescharacteristic
37C4⋊C4×D163—21no—
38#19104668C2211yescharacteristic
39#19104666C2211yescharacteristic
40Dic163⋊C8C2211yescharacteristic
41#19104660C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104659C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104655C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×Q16×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C22⋊Q8×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic
59#14579700—22no—
60#14579700—41no—
61#14579699—22no—
62#14579699—41no—
63C2×Dic326—21no—
64C2×Dic326C2311yescharacteristic
65C2×D652—21no—
66C2×D652C2311yescharacteristic
67C2×C22⋊D163—21no—
68C2×C22⋊D163D411yescharacteristic
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163C2311yescharacteristic
73#14579689—41no—
74#14579686—22no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C8×D163—41no—
78C4⋊Dic163—21no—
79C4⋊Dic163—21no—
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87C22⋊Dic163—21no—
88C22⋊Dic163D411yescharacteristic
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4D411yescharacteristic
94C4×Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163—21no—
97C4×Dic163C2311yescharacteristic
98C2×Q8×C163C2311yescharacteristic
99C2×Q8×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101Q16×C163—41no—
102C2×C8×C163—41no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C4⋊C4×C163—21no—
106C4⋊C4×C163—21no—
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C22⋊C4×C163—21no—
111C22×D163—21no—
112C22×D163C2×D411yescharacteristic
113Dic326—21no—
114Dic326—21no—
115Dic326—42no—
116D652—21no—
117D652—42no—
118D652#4112yes—
119C22⋊D163—22no—
120C22⋊D163—42no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C4×D163—21no—
128C4×D163—21no—
129C4×D163—42no—
130C4×D163—42no—
131C4×D163—42no—
132C163⋊C8—21no—
133C163⋊C8—21no—
134C23×C163C2×D411yescharacteristic
135Q8×C163—21no—
136Q8×C163—21no—
137Q8×C163—21no—
138Q8×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2411yescharacteristic
147C8×C163—21no—
148C8×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163C22×D411yescharacteristic
153C2×D163—21no—
154C2×D163—22no—
155C2×D163—21no—
156C2×D163—21no—
157C2×D163—41no—
158C2×D163—41no—
159C4×C163—21no—
160C4×C163—41no—
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#13512yes—
172C2×C163—41no—
173C2×C163#54611yescharacteristic
174C2×C163C2×#13511yescharacteristic
175C2×C163#57811yescharacteristic
176D163—41no—
177D163—42no—
178C163#330211yescharacteristic, complemented, Sylow
179#3302—1631nomaximal, Sylow
180#556—1631no—
181#547—1631no—
182#487—1631no—
183#485—1631no—
184#485—1631no—
185#474—1631no—
186#467—1631no—
187#467D16311yescharacteristic, complemented
188#462—1631no—
189#452—1631no—
190#452—1631no—
191C4×SD16—1631no—
192C8×D4—1631no—
193#416—1631no—
194#416—1631no—
195C2×#41—1631no—
196C2×#41—1631no—
197C2×Q16—1631no—
198C2×Q16C2×D16311yescharacteristic
199C2×SD16—1631no—
200C22×C8—1631no—
201C22×C8C2×D16311yescharacteristic
202C4⋊Q8—1631no—
203C4⋊Q8—1631no—
204#126—3261no—
205#126—3261no—
206#124—3261no—
207#123—3261no—
208#123—3261no—
209C22⋊Q8—1631no—
210C22⋊Q8—1631no—
211C22⋊Q8C2×D16311yescharacteristic
212C22⋊Q8C2×D16311yescharacteristic
213#121—3261no—
214C4×Q8—3261no—
215C4×Q8—1631no—
216C4×D4—1631no—
217C4×D4—1631no—
218#117—1631no—
219#117—1631no—
220C2×C4⋊C4—3261no—
221#107—1631no—
222#107C2×D16311yescharacteristic
223#106—1631no—
224C4⋊C8—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103C2×D16311yescharacteristic
231#103C2×D16311yescharacteristic
232#102—1631no—
233#102—1631no—
234#102—1631no—
235C22⋊C8—1631no—
236C22⋊C8—1631no—
237C4×C8—1631no—
238#41—3262no—
239#41—3262no—
240#41—6521no—
241#41—6521no—
242C2×Q8—3261no—
243C2×Q8—1631no—
244C2×Q8C22×D16311yescharacteristic
245C2×Q8C22×D16311yescharacteristic
246C2×D4—3261no—
247C2×D4—3261no—
248C2×D4—1631no—
249C2×D4—1631no—
250C22×C4—3261no—
251C22×C4—3261no—
252C22×C4—3261no—
253C22×C4—1631no—
254C22×C4C22×D16311yescharacteristic
255Q16—41no—
256Q16—6521no—
257SD16—3262no—
258C2×C8—41no—
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—1631no—
273C4⋊C4—1631no—
274C4⋊C4—1631no—
275C4⋊C4C22×D16311yescharacteristic
276C4⋊C4C22×D16311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—1631no—
285C22⋊C4—1631no—
286C42—3261no—
287C42—3261no—
288C42—3261no—
289C42—1631no—
290C23—3261no—
291C23—1631no—
292C23D4×D16311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—3261no—
298Q8—3261no—
299Q8—6522no—
300D4—3262no—
301D4—3261no—
302D4—6522no—
303D4—6522no—
304D4—1632no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
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×C4—1631no—
322C2×C4D4×D16311yescharacteristic
323C2×C4C23×D16311yescharacteristic
324C8—21no—
325C8—21no—
326C8—3261no—
327C8—3261no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—3261no—
332C22—3261no—
333C22—3261no—
334C22—3262no—
335C22—6521no—
336C22—6521no—
337C22C2×D4×D16311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3261no—
344C4—3261no—
345C4—6521no—
346C4—6521no—
347C4—6521no—
348C4—1632no—
349C4#41×D16311yescharacteristic
350C4#1910478011yescharacteristic
351C2—41no—
352C2—6521no—
353C2—6522no—
354C2#2378230411yescharacteristic
355C2#135×D16311yescharacteristic
356C2#2378247811yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#467×C163C2
socleC22×C163C22×D4
prime radical#23782317C2
hypercenter#467D163
hypocenterC163#3302
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #23782386C2semidirect
3. #23782014C2semidirect
4. #23782008C2semidirect
5. #23782003C2semidirect
6. #23781944C2semidirect
7. #23781905C2semidirect
8. #23781873C2semidirect
9. #23781826C2semidirect
10. #23781825C2semidirect
11. #23781750C2semidirect
12. C8×C22⋊D163C2semidirect
13. #23781662C2semidirect
14. #23781660C2semidirect
15. #467×C163C2semidirect
178. C163#3302semidirect
187. #467D163semidirect

Maximal subgroups

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

subgroupindex
#237823862
#237823172
#237820142
#237820082
#237820032
#237819442
#237819052
#237818732
#237818262
#237818252
#237817502
C8×C22⋊D1632
#237816622
#237816602
#467×C1632
#3302163