Finite Group Databasebeta

← #35224509

Subgroups — #35224509

Order 20864.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782244C211yesmaximal, characteristic
2#23782146C211yesmaximal, characteristic, complemented
3#23782008C211yesmaximal, characteristic, complemented
4#23782007C211yesmaximal, characteristic, complemented
5#23782002C211yesmaximal, characteristic, complemented
6#23782000C211yesmaximal, characteristic, complemented
7#23781988C211yesmaximal, characteristic, complemented
8#23781986C211yesmaximal, characteristic, complemented
9#23781977C211yesmaximal, characteristic, complemented
10#23781706C211yesmaximal, characteristic, complemented
11#107×D163C211yesmaximal, characteristic, complemented
12#23781663C211yesmaximal, characteristic, complemented
13#23781658C211yesmaximal, characteristic, complemented
14#23781630C211yesmaximal, characteristic, complemented
15#480×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C2×#14579682C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20#19104715C2211yescharacteristic
21#19104714C2211yescharacteristic
22#19104712—21no—
23#19104708—21no—
24#19104707—21no—
25#19104706—21no—
26#19104706C2211yescharacteristic
27#19104700—21no—
28#19104697—21no—
29C4×C22⋊D163—21no—
30C4×C22⋊D163—21no—
31#19104694C2211yescharacteristic
32#19104693C2211yescharacteristic
33#19104693C2211yescharacteristic
34C4⋊C4×D163C2211yescharacteristic
35C4⋊C4×D163C2211yescharacteristic
36C4⋊C4×D163C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104667C2211yescharacteristic
42#19104666C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic
52#121×C163C2211yescharacteristic
53C2×C4⋊C4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59#14579699—22no—
60#14579699—22no—
61#14579698—22no—
62#14579698—22no—
63C2×Dic326D411yescharacteristic
64C2×Dic326C2311yescharacteristic
65C2×D652D411yescharacteristic
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C22⋊D163D411yescharacteristic
69C2×C22⋊D163D411yescharacteristic
70C22×Dic163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163C2311yescharacteristic
75C2×C4×D163C2311yescharacteristic
76#14579682—22no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C8×D163—22no—
80C4⋊Dic163—21no—
81C4⋊Dic163C2311yescharacteristic
82C4⋊Dic163C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94C4×Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163—21no—
97C2×D4×C163—21no—
98C2×D4×C163C2311yescharacteristic
99C22×C4×C163—21no—
100C22×C4×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C2×C8×C163C2311yescharacteristic
103C4⋊C4×C163—21no—
104C4⋊C4×C163C2311yescharacteristic
105C4⋊C4×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108C22×D163C2×D411yescharacteristic
109C22×D163C2×D411yescharacteristic
110Dic326—22no—
111Dic326—21no—
112Dic326—21no—
113D652—22no—
114C22⋊D163—22no—
115C22⋊D163—22no—
116C22⋊D163—42no—
117C22⋊D163—81no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—42no—
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C4×D163—21no—
126C4×D163—21no—
127C4×D163—81no—
128C4×D163—81no—
129C4×D163—81no—
130C4×D163#4112yes—
131C4×D163#4112yes—
132C163⋊C8—21no—
133C163⋊C8—21no—
134C23×C163—21no—
135C23×C163C2×D411yescharacteristic
136D4×C163—21no—
137D4×C163—21no—
138D4×C163—81no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—81no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2411yescharacteristic
147C8×C163—21no—
148C8×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163C22×D411yescharacteristic
156C2×D163—21no—
157C2×D163—21no—
158C2×D163—21no—
159C2×D163—21no—
160C2×D163—21no—
161C2×D163#12312yes—
162C4×C163—21no—
163C4×C163—41no—
164C4×C163—41no—
165C4×C163—41no—
166C4×C163C2×#4111yescharacteristic
167C4×C163C2×#4111yescharacteristic
168Dic163—21no—
169Dic163—41no—
170Dic163—41no—
171Dic163—41no—
172Dic163—41no—
173Dic163#12312yes—
174C2×C163—41no—
175C2×C163—81no—
176C2×C163C2×#12311yescharacteristic
177C2×C163#57611yescharacteristic
178C2×C163#57811yescharacteristic
179D163—22no—
180D163—41no—
181C163#309811yescharacteristic, complemented, Sylow
182#3098—1631nomaximal, Sylow
183#537—1631no—
184#520—1631no—
185#485—1631no—
186#485—1631no—
187#484—1631no—
188#484—1631no—
189#482—1631no—
190#482—1631no—
191#480—1631no—
192#480D16311yescharacteristic, complemented
193#428—1631no—
194C2×#107—1631no—
195#417—1631no—
196#416—1631no—
197#408—1631no—
198C2×#41—1631no—
199C2×#41—1631no—
200C2×M16—1631no—
201C22×C8—1631no—
202C22⋊Q8—3261no—
203C22⋊Q8—1631no—
204C22⋊Q8—1631no—
205#121—3261no—
206#121—3261no—
207#121—3261no—
208#121—1631no—
209#121C2×D16311yescharacteristic
210C4×Q8—3261no—
211C4×D4—3261no—
212C4×D4—3261no—
213C4×D4—3261no—
214#117—1631no—
215#117—1631no—
216#117—1631no—
217C2×C4⋊C4—1631no—
218C2×C4⋊C4—1631no—
219C2×C4⋊C4—1631no—
220C2×C4⋊C4C2×D16311yescharacteristic
221#107—1631no—
222#107—1631no—
223#107—1631no—
224#107—1631no—
225#107C2×D16311yescharacteristic
226#107C2×D16311yescharacteristic
227#106—1631no—
228#106—1631no—
229#103—1631no—
230#103—1631no—
231#103—1631no—
232#103—1631no—
233#102—1631no—
234#102—1631no—
235#102C2×D16311yescharacteristic
236#102C2×D16311yescharacteristic
237C22⋊C8—1631no—
238C22⋊C8—1631no—
239C22⋊C8—1631no—
240C22⋊C8C2×D16311yescharacteristic
241#41—3262no—
242#41—3262no—
243#41—3262no—
244#41—3262no—
245C2×Q8—1631no—
246C2×Q8—1631no—
247C2×D4—21no—
248C2×D4—3261no—
249C2×D4—3261no—
250C2×D4—1631no—
251C2×D4—1631no—
252C2×D4—1631no—
253C2×D4C22×D16311yescharacteristic
254C22×C4—21no—
255C22×C4—3261no—
256C22×C4—3261no—
257C22×C4—3261no—
258C22×C4—3261no—
259C22×C4—1631no—
260C22×C4—1631no—
261C22×C4C22×D16311yescharacteristic
262M16—3262no—
263C2×C8—3262no—
264C2×C8—1631no—
265C2×C8—1631no—
266C2×C8C22×D16311yescharacteristic
267C2×C8C22×D16311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—3261no—
272C4⋊C4—3261no—
273C4⋊C4—3261no—
274C4⋊C4—1631no—
275C4⋊C4—1631no—
276C4⋊C4—1631no—
277C4⋊C4C22×D16311yescharacteristic
278C4⋊C4C22×D16311yescharacteristic
279C4⋊C4C22×D16311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C22⋊C4—3261no—
287C42—3261no—
288C42—3261no—
289C42—3261no—
290C23—21no—
291C23—1631no—
292C23—1631no—
293C23D4×D16311yescharacteristic
294Q8—3262no—
295Q8—3261no—
296Q8—3261no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—13041no—
301D4—3262no—
302D4—3262no—
303D4—3262no—
304D4—6522no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—13041no—
312C2×C4—13041no—
313C2×C4—13041no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—6522no—
321C2×C4—1631no—
322C2×C4—1631no—
323C2×C4—1632no—
324C2×C4—1632no—
325C2×C4D4×D16311yescharacteristic
326C2×C4C23×D16311yescharacteristic
327C8—21no—
328C8—21no—
329C8—3261no—
330C8—3261no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—3261no—
338C22—3261no—
339C22—3261no—
340C22—3261no—
341C22—3261no—
342C22—1632no—
343C22C2×D4×D16311yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—3261no—
349C4—6521no—
350C4—6521no—
351C4—6521no—
352C4—6521no—
353C4—1632no—
354C4#41×D16311yescharacteristic
355C4#41×D16311yescharacteristic
356C2—41no—
357C2—81no—
358C2—3262no—
359C2—6521no—
360C2#123×D16311yescharacteristic
361C2#2378246611yescharacteristic
362C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#480×C163C2
socleC22×C163C22×D4
prime radical#23782244C2
hypercenter#480D163
hypocenterC163#3098
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #23782146C2semidirect
3. #23782008C2semidirect
4. #23782007C2semidirect
5. #23782002C2semidirect
6. #23782000C2semidirect
7. #23781988C2semidirect
8. #23781986C2semidirect
9. #23781977C2semidirect
10. #23781706C2semidirect
11. #107×D163C2semidirect
12. #23781663C2semidirect
13. #23781658C2semidirect
14. #23781630C2semidirect
15. #480×C163C2semidirect
181. C163#3098semidirect
192. #480D163semidirect

Maximal subgroups

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

subgroupindex
#237822442
#237821462
#237820082
#237820072
#237820022
#237820002
#237819882
#237819862
#237819772
#237817062
#107×D1632
#237816632
#237816582
#237816302
#480×C1632
#3098163