Finite Group Databasebeta

← #35224215

Subgroups — #35224215

Order 20864.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104759C211yesmaximal, characteristic, complemented
2C2×#19104741C211yesmaximal, characteristic, complemented
3#23782244C211yesmaximal, characteristic, complemented
4#23782231C211yesmaximal, characteristic, complemented
5#23781916C211yesmaximal, characteristic, complemented
6#23781915C211yesmaximal, characteristic, complemented
7#23781912C211yesmaximal, characteristic, complemented
8#23781909C211yesmaximal, characteristic, complemented
9#23781900C211yesmaximal, characteristic, complemented
10#23781894C211yesmaximal, characteristic, complemented
11#23781893C211yesmaximal, characteristic, complemented
12#23781706C211yesmaximal, characteristic, complemented
13#23781658C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#466×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C2×D4×D163C2211yescharacteristic
19#19104759—24no—
20C2×#14579688C2211yescharacteristic
21C2×#14579687C2211yescharacteristic
22C2×#14579684C2211yescharacteristic
23#19104741—24no—
24C2×#14579683C2211yescharacteristic, complemented
25C2×#14579682C2211yescharacteristic
26C2×#14579682C2211yescharacteristic, complemented
27C2×C8×D163C2211yescharacteristic
28C2×C8×D163C2211yescharacteristic, complemented
29#19104731—21no—
30#19104721—21no—
31#19104715C2211yescharacteristic, complemented
32#19104714C2211yescharacteristic, complemented
33#19104712—21no—
34#19104710C2211yescharacteristic
35#19104709C2211yescharacteristic
36#19104708—21no—
37#19104707—21no—
38#19104706—21no—
39#19104706C2211yescharacteristic
40#19104706C2211yescharacteristic, complemented
41#19104705—21no—
42#19104703—21no—
43#19104693C2211yescharacteristic
44C4⋊C4×D163C2211yescharacteristic
45#19104668C2211yescharacteristic, complemented
46#19104667C2211yescharacteristic, complemented
47#19104666C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104658C2211yescharacteristic, complemented
50#19104657C2211yescharacteristic
51#19104656C2211yescharacteristic
52#19104656C2211yescharacteristic, complemented
53#19104655C2211yescharacteristic
54C2×D8×C163C2211yescharacteristic
55C22×C8×C163C2211yescharacteristic, complemented
56#121×C163C2211yescharacteristic
57#121×C163C2211yescharacteristic, complemented
58#107×C163C2211yescharacteristic, complemented
59#102×C163C2211yescharacteristic
60#102×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62#14579699—22no—
63#14579699—22no—
64#14579698—22no—
65#14579698—22no—
66C2×Dic326D411yescharacteristic
67C2×Dic326C2311yescharacteristic
68C2×D652—21no—
69C2×D652D411yescharacteristic
70C2×D652C2311yescharacteristic
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163—21no—
75C2×C22⋊D163—21no—
76C2×C22⋊D163D411yescharacteristic
77C2×C22⋊D163D411yescharacteristic
78D4×D163—22no—
79D4×D163—22no—
80C22×Dic163—21no—
81C2×C4×D163—21no—
82C2×C4×D163C2311yescharacteristic
83C2×C4×D163C2311yescharacteristic
84#14579688—22no—
85#14579687—22no—
86#14579684—22no—
87#14579683—22no—
88#14579682—22no—
89#14579682D414yescomplemented
90C2×C163⋊C8C2311yescharacteristic
91C2×C163⋊C8C2311yescharacteristic
92C8×D163—22no—
93C8×D163—22no—
94C4⋊Dic163—21no—
95C4⋊Dic163C2311yescharacteristic
96C4⋊Dic163C2311yescharacteristic
97Dic163⋊C4—21no—
98Dic163⋊C4—21no—
99C22⋊Dic163—21no—
100C22⋊Dic163—21no—
101C22⋊Dic163—21no—
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106C4×Dic163—21no—
107C2×D4×C163—21no—
108C2×D4×C163—21no—
109C2×D4×C163C2311yescharacteristic
110C2×D4×C163C2311yescharacteristic
111C22×C4×C163C2311yescharacteristic
112D8×C163—22no—
113C2×C8×C163—22no—
114C2×C8×C163C2311yescharacteristic
115C2×C8×C163C2311yescharacteristic
116C4⋊C4×C163C2311yescharacteristic
117C4⋊C4×C163C2311yescharacteristic
118C22⋊C4×C163—21no—
119C22⋊C4×C163—21no—
120C22×D163—21no—
121C22×D163—42no—
122C22×D163—81no—
123C22×D163—81no—
124C22×D163C2×D411yescharacteristic
125C22×D163C2×D411yescharacteristic
126Dic326—22no—
127Dic326—21no—
128Dic326—21no—
129D652—22no—
130D652—21no—
131D652—21no—
132D652—81no—
133C22⋊D163—22no—
134C22⋊D163—22no—
135C22⋊D163—42no—
136C22⋊D163—42no—
137C22⋊D163—81no—
138C22⋊D163—81no—
139C22⋊D163—81no—
140C2×Dic163—21no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—42no—
144C2×Dic163C2×D411yescharacteristic
145C2×Dic163C2×D411yescharacteristic
146C4×D163—21no—
147C4×D163—21no—
148C4×D163—81no—
149C4×D163C2×D412yes—
150C4×D163#4112yes—
151C163⋊C8—21no—
152C163⋊C8C2×D412yes—
153C23×C163—21no—
154C23×C163—21no—
155C23×C163C2×D411yescharacteristic
156D4×C163—21no—
157D4×C163—21no—
158D4×C163—21no—
159D4×C163—21no—
160D4×C163—81no—
161D4×C163—81no—
162C2×C4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163C2×D411yescharacteristic
167C2×C4×C163C2411yescharacteristic
168C8×C163—21no—
169C8×C163C2×D412yes—
170C22×C163—21no—
171C22×C163—21no—
172C22×C163—21no—
173C22×C163—41no—
174C22×C163—41no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163—41no—
179C22×C163C22×D411yescharacteristic
180C2×D163—21no—
181C2×D163—21no—
182C2×D163—21no—
183C2×D163—21no—
184C2×D163—21no—
185C2×D163—41no—
186C2×D163—41no—
187C2×D163—41no—
188C2×D163—84no—
189C2×D163#12112yes—
190C4×C163—21no—
191C4×C163—41no—
192C4×C163—41no—
193C4×C163C22×D411yescharacteristic
194C4×C163C2×#4111yescharacteristic
195Dic163—21no—
196Dic163—41no—
197Dic163—41no—
198Dic163—41no—
199Dic163#12112yes—
200C2×C163—41no—
201C2×C163—81no—
202C2×C163—81no—
203C2×C163C2×#12111yescharacteristic
204C2×C163#57611yescharacteristic
205C2×C163#57711yescharacteristic
206D163—22no—
207D163—41no—
208D163—81no—
209C163#306211yescharacteristic, complemented, Sylow
210#3062—1631nomaximal, Sylow
211C2×AGL(1,ℤ/8)—1631no—
212C2×#131—1631no—
213#537—1631no—
214#535—1631no—
215#469—1631no—
216#469—1631no—
217#468—1631no—
218#468—1631no—
219#466—1631no—
220#466D16311yescharacteristic, complemented
221#465—1631no—
222#465—1631no—
223#428—1631no—
224#416—1631no—
225C2×#102—1631no—
226C2×#41—1631no—
227C2×#41—1631no—
228C22×D4—1631no—
229AGL(1,ℤ/8)—3264no—
230C2×SD16—1631no—
231C2×SD16—1631no—
232C2×D8—1631no—
233C2×D8C2×D16311yescharacteristic
234#131—3264no—
235C2×M16—1631no—
236C2×M16—1631no—
237C2×M16—1631no—
238C22×C8—1631no—
239C22×C8—1631no—
240C22×C8C2×D16311yescharacteristic, complemented
241#127—3261no—
242#124—3261no—
243C22⋊Q8—3261no—
244C22⋊Q8—1631no—
245C22⋊Q8—1631no—
246#121—3261no—
247#121—3261no—
248#121—3261no—
249#121—1631no—
250#121—1631no—
251#121—1631no—
252#121—1631no—
253#121C2×D16311yescharacteristic
254#121C2×D16311yescharacteristic, complemented
255#120—3261no—
256#120—3261no—
257#117—1631no—
258C2×C4⋊C4—1631no—
259#107—1631no—
260#107C2×D16311yescharacteristic, complemented
261#106—1631no—
262#106—1631no—
263#103—1631no—
264#103—1631no—
265#102—1631no—
266#102—1631no—
267#102—1631no—
268#102—1631no—
269#102C2×D16311yescharacteristic
270#102C2×D16311yescharacteristic, complemented
271C24—3261no—
272#41—3262no—
273#41—3262no—
274#41—3262no—
275#41—3262no—
276C2×Q8—1631no—
277C2×Q8—1631no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—3261no—
281C2×D4—3261no—
282C2×D4—3261no—
283C2×D4—3261no—
284C2×D4—3261no—
285C2×D4—3262no—
286C2×D4—3262no—
287C2×D4—3261no—
288C2×D4—1631no—
289C2×D4—1631no—
290C2×D4—1631no—
291C2×D4—1631no—
292C2×D4C22×D16311yescharacteristic
293C2×D4C22×D16311yescharacteristic
294C22×C4—3261no—
295C22×C4—3261no—
296C22×C4—1631no—
297C22×C4—1631no—
298C22×C4C22×D16311yescharacteristic
299SD16—3262no—
300SD16—3262no—
301D8—22no—
302D8—3262no—
303M16—3262no—
304M16—3262no—
305M16—1634no—
306C2×C8—22no—
307C2×C8—3262no—
308C2×C8—3262no—
309C2×C8—1631no—
310C2×C8—1631no—
311C2×C8C22×D16311yescharacteristic
312C2×C8C22×D16311yescharacteristic
313C4⋊C4—3261no—
314C4⋊C4—3261no—
315C4⋊C4—3261no—
316C4⋊C4—1631no—
317C4⋊C4—1631no—
318C4⋊C4C22×D16311yescharacteristic
319C4⋊C4C22×D16311yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4—3261no—
325C22⋊C4—3261no—
326C22⋊C4—3261no—
327C22⋊C4—3261no—
328C22⋊C4—3261no—
329C42—3261no—
330C23—21no—
331C23—21no—
332C23—13041no—
333C23—13041no—
334C23—3261no—
335C23—6522no—
336C23—1631no—
337C23—1631no—
338C23D4×D16311yescharacteristic
339Q8—3262no—
340Q8—3261no—
341Q8—3261no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—81no—
347D4—81no—
348D4—13041no—
349D4—13041no—
350D4—13041no—
351D4—13041no—
352D4—3262no—
353D4—3262no—
354D4—3262no—
355D4—3261no—
356D4—3261no—
357D4—6522no—
358D4—6522no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—13041no—
364C2×C4—3261no—
365C2×C4—3261no—
366C2×C4—3261no—
367C2×C4—3261no—
368C2×C4—3261no—
369C2×C4—6522no—
370C2×C4—1631no—
371C2×C4—1631no—
372C2×C4—1632no—
373C2×C4—1632no—
374C2×C4D4×D16311yescharacteristic
375C2×C4C23×D16311yescharacteristic
376C8—21no—
377C8—3261no—
378C8—1632no—
379C8D4×D16312yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—13044no—
390C22—3261no—
391C22—3261no—
392C22—3261no—
393C22—3261no—
394C22—3261no—
395C22—6521no—
396C22—6521no—
397C22—6521no—
398C22—1632no—
399C22C2×D4×D16311yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—3261no—
404C4—6521no—
405C4—6521no—
406C4—6521no—
407C4—1632no—
408C4C2×D4×D16311yescharacteristic
409C4#41×D16311yescharacteristic
410C2—41no—
411C2—81no—
412C2—81no—
413C2—13041no—
414C2—3262no—
415C2—6521no—
416C2#121×D16311yescharacteristic
417C2#2378246611yescharacteristic
418C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#466×C163C2
socleC22×C163C22×D4
hypercenter#466D163
hypocenterC163#3062
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#19104759C2semidirect
2. C2×#19104741C2semidirect
3. #23782244C2semidirect
4. #23782231C2semidirect
5. #23781916C2semidirect
6. #23781915C2semidirect
7. #23781912C2semidirect
8. #23781909C2semidirect
9. #23781900C2semidirect
10. #23781894C2semidirect
11. #23781893C2semidirect
12. #23781706C2semidirect
13. #23781658C2semidirect
14. #102×D163C2semidirect
15. #466×C163C2semidirect
24. C2×#14579683C22semidirect
26. C2×#14579682C22semidirect
28. C2×C8×D163C22semidirect
31. #19104715C22semidirect
32. #19104714C22semidirect
40. #19104706C22semidirect
45. #19104668C22semidirect
46. #19104667C22semidirect
47. #19104666C22semidirect
48. #19104660C22semidirect
49. #19104658C22semidirect
52. #19104656C22semidirect
55. C22×C8×C163C22semidirect
57. #121×C163C22semidirect
58. #107×C163C22semidirect
60. #102×C163C22semidirect
89. #14579682D4semidirect
209. C163#3062semidirect
220. #466D163semidirect
240. C22×C8C2×D163semidirect
254. #121C2×D163semidirect
260. #107C2×D163semidirect
270. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047592
C2×#191047412
#237822442
#237822312
#237819162
#237819152
#237819122
#237819092
#237819002
#237818942
#237818932
#237817062
#237816582
#102×D1632
#466×C1632
#3062163