Finite Group Databasebeta

← #35224255

Subgroups — #35224255

Order 20864.

452 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782465C211yesmaximal, characteristic, complemented
2#23782463C211yesmaximal, characteristic, complemented
3C2×#19104755C211yesmaximal, characteristic, complemented
4C2×#19104743C211yesmaximal, characteristic, complemented
5#23781939C212yesmaximal, complemented
6#23781934C212yesmaximal, complemented
7#23781931C212yesmaximal, complemented
8#23781927C212yesmaximal, complemented
9C2×#19104670C211yesmaximal, characteristic, complemented
10#23781647C211yesmaximal, characteristic, complemented
11#413×C163C211yesmaximal, characteristic, complemented
12C2×#14579699C2211yescharacteristic
13C2×#14579699C2211yescharacteristic
14#19104768—21no—
15#19104768C2212yes—
16#19104766C2212yes—
17#19104765—21no—
18#19104763—21no—
19#19104763C2212yes—
20#19104761C2212yes—
21#19104760—21no—
22#19104757—22no—
23#19104755—22no—
24#19104755—21no—
25#19104755—21no—
26#19104755—21no—
27#19104755—21no—
28C2×#14579689C2211yescharacteristic, complemented
29C2×#14579687C2211yescharacteristic, complemented
30C2×#14579686C2211yescharacteristic, complemented
31C2×#14579684C2211yescharacteristic, complemented
32#19104743—21no—
33#19104743—21no—
34#19104743C2212yes—
35#19104743C2212yes—
36C2×#14579683C2211yescharacteristic
37C2×#14579682—21no—
38C22×C163⋊C8C2211yescharacteristic
39C2×C8×D163—21no—
40#19104670C2212yescomplemented
41#19104670C2212yescomplemented
42#19104653C2212yescomplemented
43#19104651C2212yescomplemented
44C2×#41×C163C2211yescharacteristic
45C2×M16×C163C2211yescharacteristic
46C2×M16×C163C2211yescharacteristic
47#101×C163C2212yescomplemented
48#100×C163C2212yescomplemented
49#14579699—21no—
50#14579699—21no—
51#14579699—21no—
52#14579699—21no—
53#14579699—21no—
54#14579699—21no—
55#14579699D412yes—
56#14579699C2312yes—
57C2×Dic326D411yescharacteristic
58C2×Dic326C2311yescharacteristic
59C2×D652D411yescharacteristic
60C2×D652C2311yescharacteristic
61C2×C22⋊D163—21no—
62C2×C22⋊D163—21no—
63C2×C4×D163—21no—
64C2×C4×D163—21no—
65Dic652—22no—
66#14579689—22no—
67#14579689—21no—
68#14579689—21no—
69#14579688—22no—
70#14579688—22no—
71#14579687—22no—
72#14579687—21no—
73#14579687—21no—
74#14579686—22no—
75#14579686—21no—
76#14579686—21no—
77D1304—22no—
78#14579684—22no—
79#14579684—21no—
80#14579684—21no—
81#14579683—21no—
82#14579683C2312yes—
83#14579682—22no—
84#14579682—22no—
85C2×C163⋊C8—21no—
86C2×C163⋊C8D411yescharacteristic
87C2×C163⋊C8D411yescharacteristic
88C2×C163⋊C8D411yescharacteristic
89C2×C163⋊C8D411yescharacteristic
90C2×C163⋊C8C2311yescharacteristic
91C2×C163⋊C8C2311yescharacteristic
92C8×D163—22no—
93C8×D163—22no—
94#41×C163—21no—
95#41×C163—21no—
96#41×C163—21no—
97#41×C163—21no—
98C2×Q8×C163C2311yescharacteristic
99C2×D4×C163—21no—
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163—21no—
102C22×C4×C163C2311yescharacteristic
103M16×C163—21no—
104M16×C163—21no—
105M16×C163C2312yes—
106M16×C163C2312yes—
107C2×C8×C163—21no—
108C2×C8×C163—21no—
109C22×D163—21no—
110C22×D163—21no—
111Dic326—21no—
112Dic326—21no—
113Dic326—21no—
114Dic326C2×D412yes—
115D652—21no—
116D652—21no—
117D652—21no—
118D652C2×D412yes—
119C22⋊D163—22no—
120C22⋊D163—22no—
121C22⋊D163—41no—
122C22⋊D163—41no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C4×D163—22no—
126C4×D163—22no—
127C4×D163—41no—
128C4×D163—41no—
129C163⋊C8—22no—
130C163⋊C8C2×D411yescharacteristic
131C163⋊C8C2×D411yescharacteristic
132C163⋊C8C2×D411yescharacteristic
133C163⋊C8C2×D411yescharacteristic
134C23×C163—21no—
135C23×C163#4111yescharacteristic
136Q8×C163—21no—
137Q8×C163—21no—
138D4×C163—21no—
139D4×C163—21no—
140D4×C163—41no—
141D4×C163—41no—
142C2×C4×C163—21no—
143C2×C4×C163—41no—
144C2×C4×C163—41no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163#4111yescharacteristic
150C2×C4×C163C2411yescharacteristic
151C8×C163—22no—
152C8×C163—22no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163—81no—
156C22×C163#12111yescharacteristic
157C22×C163#12111yescharacteristic
158C22×C163C2×#4111yescharacteristic
159C2×D163—22no—
160C2×D163—42no—
161C2×D163—41no—
162C2×D163—81no—
163C4×C163—41no—
164C4×C163#12111yescharacteristic
165C4×C163#12111yescharacteristic
166C4×C163C22×D411yescharacteristic
167C4×C163C22×D411yescharacteristic
168Dic163—22no—
169Dic163—41no—
170C2×C163—21no—
171C2×C163—21no—
172C2×C163—21no—
173C2×C163—81no—
174C2×C163C2×#12111yescharacteristic
175D163—42no—
176D163—81no—
177C163#306711yescharacteristic, complemented, Sylow
178#3067—1631nomaximal, Sylow
179#575—1631no—
180#575—1631no—
181C2×#135—1631no—
182C2×#131—1631no—
183#473—1632no—
184#472—1632no—
185#471—1632no—
186#471—1632no—
187C2×#108—1631no—
188#413—1631no—
189#413D16311yescharacteristic, complemented
190C2×#41—1631no—
191C2×#41—1631no—
192C2×#41C2×D16311yescharacteristic
193#137—3261no—
194#137—3261no—
195#137—1632no—
196#137—1632no—
197AGL(1,ℤ/8)—3261no—
198AGL(1,ℤ/8)—3261no—
199AGL(1,ℤ/8)—1632no—
200AGL(1,ℤ/8)—1632no—
201#135—3262no—
202#135—3262no—
203#135—3261no—
204#135—3261no—
205#135—3261no—
206#135—3261no—
207C2×Q16—1631no—
208C2×SD16—1631no—
209C2×SD16—1631no—
210C2×D8—1631no—
211#131—3261no—
212#131—3261no—
213#131—1632no—
214#131—1632no—
215C2×M16—3261no—
216C2×M16—1631no—
217C2×M16C2×D16311yescharacteristic
218C2×M16C2×D16311yescharacteristic
219C22×C8—3261no—
220C22×C8—1631no—
221#108—1632no—
222#108—1632no—
223#101—1632no—
224#101C2×D16312yescomplemented
225#100—1632no—
226#100C2×D16312yescomplemented
227#41—21no—
228#41—21no—
229#41—21no—
230#41—21no—
231#41—3261no—
232#41—3261no—
233#41—3261no—
234#41—3261no—
235#41—3261no—
236#41—3261no—
237#41—1632no—
238#41—1632no—
239C2×Q8—1631no—
240C2×Q8—1631no—
241C2×Q8C22×D16311yescharacteristic
242C2×D4—21no—
243C2×D4—3261no—
244C2×D4—3261no—
245C2×D4—1631no—
246C2×D4—1631no—
247C2×D4C22×D16311yescharacteristic
248C22×C4—21no—
249C22×C4—3261no—
250C22×C4—3261no—
251C22×C4C22×D16311yescharacteristic
252Q16—3262no—
253Q16—3262no—
254Q16—3261no—
255Q16—3261no—
256SD16—3262no—
257SD16—3262no—
258SD16—3262no—
259SD16—3262no—
260SD16—3261no—
261SD16—3261no—
262SD16—3261no—
263SD16—3261no—
264D8—3262no—
265D8—3262no—
266D8—3261no—
267D8—3261no—
268M16—21no—
269M16—21no—
270M16—3262no—
271M16—3262no—
272M16—3261no—
273M16—1632no—
274M16C22×D16312yes—
275M16C22×D16312yes—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—3262no—
279C2×C8—3262no—
280C2×C8—3261no—
281C2×C8—1631no—
282C2×C8—1631no—
283C2×C8—1631no—
284C2×C8—1631no—
285C2×C8—1631no—
286C2×C8—1631no—
287C23—21no—
288C23—3261no—
289C23—3261no—
290C23#1457969911yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—3261no—
294Q8—3261no—
295Q8—3261no—
296Q8—1632no—
297D4—21no—
298D4—21no—
299D4—41no—
300D4—41no—
301D4—3262no—
302D4—3262no—
303D4—3261no—
304D4—3261no—
305D4—3261no—
306D4—6521no—
307D4—6521no—
308D4—1632no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—3261no—
313C2×C4—3262no—
314C2×C4—3262no—
315C2×C4—3261no—
316C2×C4—6521no—
317C2×C4—6521no—
318C2×C4D4×D16311yescharacteristic
319C2×C4D4×D16311yescharacteristic
320C2×C4D4×D16311yescharacteristic
321C2×C4D4×D16311yescharacteristic
322C2×C4#1457969911yescharacteristic
323C2×C4C23×D16311yescharacteristic
324C8—22no—
325C8—22no—
326C8—3262no—
327C8—1631no—
328C8—1631no—
329C8—1631no—
330C8—1631no—
331C22—41no—
332C22—41no—
333C22—81no—
334C22—13041no—
335C22—3262no—
336C22—6522no—
337C22—6521no—
338C22#1910470811yescharacteristic
339C22#1910470811yescharacteristic
340C22C2×#1457969911yescharacteristic
341C4—41no—
342C4—3262no—
343C4—6521no—
344C4#1910470811yescharacteristic
345C4#1910470811yescharacteristic
346C4C2×D4×D16311yescharacteristic
347C4C2×D4×D16311yescharacteristic
348C2—21no—
349C2—21no—
350C2—21no—
351C2—81no—
352C2—13041no—
353C2—6522no—
354C2C2×#1910470811yescharacteristic
subgroupG/H
centerC4#19104708
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#413×C163C2
socleC2×C163C2×#121
hypercenter#413D163
hypocenterC163#3067
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #23782465C2semidirect
2. #23782463C2semidirect
3. C2×#19104755C2semidirect
4. C2×#19104743C2semidirect
5. #23781939C2semidirect
6. #23781934C2semidirect
7. #23781931C2semidirect
8. #23781927C2semidirect
9. C2×#19104670C2semidirect
10. #23781647C2semidirect
11. #413×C163C2semidirect
28. C2×#14579689C22semidirect
29. C2×#14579687C22semidirect
30. C2×#14579686C22semidirect
31. C2×#14579684C22semidirect
40. #19104670C22semidirect
41. #19104670C22semidirect
42. #19104653C22semidirect
43. #19104651C22semidirect
47. #101×C163C22semidirect
48. #100×C163C22semidirect
177. C163#3067semidirect
189. #413D163semidirect
224. #101C2×D163semidirect
226. #100C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824652
#237824632
C2×#191047552
C2×#191047432
#237819392
#237819342
#237819312
#237819272
C2×#191046702
#237816472
#413×C1632
#3067163