Finite Group Databasebeta

← #35226063

Subgroups — #35226063

Order 20864.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104759C211yesmaximal, characteristic, complemented
2C2×#19104754C211yesmaximal, characteristic, complemented
3#23782319C211yesmaximal, characteristic, complemented
4#23782310C211yesmaximal, characteristic, complemented
5#23782059C211yesmaximal, characteristic, complemented
6#23781924C211yesmaximal, characteristic, complemented
7#23781895C211yesmaximal, characteristic, complemented
8#23781880C211yesmaximal, characteristic, complemented
9#23781870C211yesmaximal, characteristic, complemented
10#23781828C211yesmaximal, characteristic, complemented
11#23781828C211yesmaximal, characteristic, complemented
12#23781817C211yesmaximal, characteristic, complemented
13#23781811C211yesmaximal, characteristic, complemented
14#23781738C211yesmaximal, characteristic, complemented
15#442×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C2×#14579698C2211yescharacteristic, complemented
19C2×D4×D163C2211yescharacteristic
20#19104759—24no—
21#19104754—24no—
22C2×Dic652C2211yescharacteristic, complemented
23C2×Dic652C2211yescharacteristic, complemented
24C2×#14579688C2211yescharacteristic
25C2×#14579688C2211yescharacteristic
26C2×#14579687C2211yescharacteristic
27C2×#14579687C2211yescharacteristic
28C2×#14579687C2211yescharacteristic, complemented
29C2×#14579684C2211yescharacteristic
30C2×#14579682C2211yescharacteristic
31C2×C8×D163C2211yescharacteristic, complemented
32#19104735C2211yescharacteristic, complemented
33#19104723C2211yescharacteristic
34#19104719—21no—
35#19104719—21no—
36#19104719—21no—
37#19104715C2211yescharacteristic
38#19104715C2211yescharacteristic
39#19104715C2211yescharacteristic, complemented
40#19104710C2211yescharacteristic
41#19104709—21no—
42#19104709—21no—
43#19104705—21no—
44C4×D652C2211yescharacteristic
45C2×C4⋊Dic163—21no—
46C2×(C2×D163)⋊C4—21no—
47C4⋊(C163⋊C8)C2211yescharacteristic, complemented
48#19104658C2211yescharacteristic, complemented
49#19104658C2211yescharacteristic, complemented
50#19104658C2211yescharacteristic, complemented
51#19104655C2211yescharacteristic
52(C2×D163)⋊C8C2211yescharacteristic
53(C2×D163)⋊C8C2211yescharacteristic, complemented
54C2×D8×C163C2211yescharacteristic
55C4×D4×C163C2211yescharacteristic
56C4×D4×C163C2211yescharacteristic, complemented
57#106×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic
59#102×C163C2211yescharacteristic, complemented
60#97×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62#14579698—22no—
63#14579698—22no—
64#14579698—22no—
65#14579698—41no—
66#14579698—41no—
67#14579698—41no—
68C2×Dic326C2311yescharacteristic
69C2×Dic326C2311yescharacteristic
70C2×Dic326C2311yescharacteristic
71C2×D652—21no—
72C2×D652D411yescharacteristic
73C2×D652C2311yescharacteristic
74C2×C22⋊D163—21no—
75C2×C22⋊D163—21no—
76C2×C22⋊D163—21no—
77C2×C22⋊D163—21no—
78D4×D163—22no—
79D4×D163—41no—
80C22×Dic163—21no—
81C22×Dic163—21no—
82C22×Dic163—21no—
83C2×C4×D163C2311yescharacteristic
84C2×C4×D163C2311yescharacteristic
85Dic652—22no—
86Dic652—41no—
87#14579688—22no—
88#14579688—41no—
89#14579687—22no—
90#14579687—22no—
91#14579687—22no—
92#14579684—22no—
93#14579682—22no—
94C2×C163⋊C8C2311yescharacteristic
95C2×C163⋊C8C2311yescharacteristic
96C8×D163—22no—
97C4⋊Dic163—21no—
98C4⋊Dic163—21no—
99C4⋊Dic163—21no—
100C4⋊Dic163—21no—
101C4⋊Dic163—21no—
102C4⋊Dic163D411yescharacteristic
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4—21no—
108(C2×D163)⋊C4—21no—
109(C2×D163)⋊C4D411yescharacteristic
110(C2×D163)⋊C4D411yescharacteristic
111C2×D4×C163C2311yescharacteristic
112C2×D4×C163C2311yescharacteristic
113C22×C4×C163—21no—
114C22×C4×C163—21no—
115D8×C163D414yescomplemented
116C2×C8×C163C2311yescharacteristic
117C2×C8×C163C2311yescharacteristic
118C4⋊C4×C163C2311yescharacteristic
119C4⋊C4×C163C2311yescharacteristic
120C22⋊C4×C163—21no—
121C22⋊C4×C163—21no—
122C42×C163C2311yescharacteristic
123C22×D163—21no—
124C22×D163—42no—
125C22×D163—42no—
126C22×D163—81no—
127C22×D163C2×D411yescharacteristic
128C22×D163C2×D411yescharacteristic
129Dic326—21no—
130Dic326—21no—
131Dic326—21no—
132Dic326—21no—
133Dic326—21no—
134Dic326—21no—
135D652—21no—
136D652—21no—
137D652—41no—
138D652—81no—
139C22⋊D163—42no—
140C22⋊D163—42no—
141C22⋊D163—42no—
142C22⋊D163—42no—
143C2×Dic163—21no—
144C2×Dic163—21no—
145C2×Dic163—21no—
146C2×Dic163—42no—
147C2×Dic163—42no—
148C2×Dic163—42no—
149C2×Dic163C2×D411yescharacteristic
150C2×Dic163C2×D411yescharacteristic
151C4×D163—21no—
152C4×D163—21no—
153C4×D163—21no—
154C4×D163—21no—
155C163⋊C8—21no—
156C163⋊C8—21no—
157C23×C163—21no—
158C23×C163—21no—
159D4×C163—21no—
160D4×C163—21no—
161D4×C163C2×D412yes—
162D4×C163C2×D412yes—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—42no—
166C2×C4×C163—81no—
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2×D411yescharacteristic
169C2×C4×C163C2411yescharacteristic
170C8×C163—21no—
171C8×C163C2×D412yes—
172C22×C163—22no—
173C22×C163—22no—
174C22×C163—41no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163C22×D411yescharacteristic
179C2×D163—21no—
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—
189C4×C163—21no—
190C4×C163—21no—
191C4×C163—41no—
192C4×C163—41no—
193C4×C163C22×D411yescharacteristic
194C4×C163H(5,𝔽2)11yescharacteristic
195Dic163—21no—
196Dic163—21no—
197Dic163—41no—
198Dic163—41no—
199Dic163—41no—
200C2×C163—42no—
201C2×C163—42no—
202C2×C163D4211yescharacteristic
203C2×C163#57511yescharacteristic
204C2×C163#57711yescharacteristic
205D163—41no—
206D163—41no—
207D163—81no—
208C163#326711yescharacteristic, complemented, Sylow
209#3267—1631nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1631no—
211C2×#135—1631no—
212#547—1631no—
213#546—1631no—
214#497—1631no—
215#470—1631no—
216#465—1631no—
217#463—1631no—
218#461—1631no—
219#452—1631no—
220#452—1631no—
221#450—1631no—
222#449—1631no—
223#442D16311yescharacteristic, complemented
224#435—1631no—
225C2×#41—1631no—
226C2×#41—1631no—
227C2×#41—1631no—
228C22×D4—1631no—
229AGL(1,ℤ/8)—3264no—
230#135—3264no—
231C2×Q16—1631no—
232C2×Q16—1631no—
233C2×SD16—1631no—
234C2×SD16—1631no—
235C2×SD16—1631no—
236C2×SD16—1631no—
237C2×SD16—1631no—
238C2×D8—1631no—
239C2×D8C2×D16311yescharacteristic
240C2×M16—1631no—
241C22×C8—1631no—
242C4⋊Q8—1631no—
243#124—1631no—
244#123—3261no—
245#123—3261no—
246#123—3261no—
247C22⋊Q8—1631no—
248C22⋊Q8—1631no—
249C22⋊Q8—1631no—
250#121—3261no—
251#121—3261no—
252#121—1631no—
253#120—3261no—
254C4×D4—1631no—
255C4×D4C2×D16311yescharacteristic
256C4×D4C2×D16311yescharacteristic, complemented
257C2×C4⋊C4—3261no—
258C2×C22⋊C4—3261no—
259#106C2×D16311yescharacteristic, complemented
260C4⋊C8—1631no—
261#103—1631no—
262#103—1631no—
263#103—1631no—
264#102—1631no—
265#102C2×D16311yescharacteristic
266#102C2×D16311yescharacteristic, complemented
267C22⋊C8—1631no—
268C22⋊C8—1631no—
269#97C2×D16311yescharacteristic, complemented
270C24—3261no—
271#41—3262no—
272#41—3262no—
273#41—3262no—
274#41—6521no—
275#41—6521no—
276#41—6521no—
277C2×Q8—1631no—
278C2×Q8—1631no—
279C2×Q8—1631no—
280C2×D4—3261no—
281C2×D4—3262no—
282C2×D4—3261no—
283C2×D4—3261no—
284C2×D4—3261no—
285C2×D4—3261no—
286C2×D4—6521no—
287C2×D4—1631no—
288C2×D4—1631no—
289C2×D4C22×D16311yescharacteristic
290C2×D4C22×D16311yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—3261no—
294C22×C4—3261no—
295C22×C4—3261no—
296C22×C4—1631no—
297C22×C4—1631no—
298Q16—3262no—
299Q16—6521no—
300SD16—3262no—
301SD16—3262no—
302SD16—3262no—
303SD16—3262no—
304SD16—6521no—
305D8—3262no—
306D8D65214yescomplemented
307M16—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—3261no—
317C4⋊C4—3261no—
318C4⋊C4—1631no—
319C4⋊C4C22×D16311yescharacteristic
320C4⋊C4C22×D16311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—3261no—
324C22⋊C4—3261no—
325C22⋊C4—3261no—
326C22⋊C4—3261no—
327C22⋊C4—3261no—
328C22⋊C4—3261no—
329C22⋊C4—1631no—
330C22⋊C4—1631no—
331C42C22×D16311yescharacteristic
332C23—21no—
333C23—21no—
334C23—13041no—
335C23—3261no—
336C23—6522no—
337C23—6522no—
338C23—1631no—
339C23—1631no—
340Q8—3261no—
341Q8—3261no—
342Q8—3261no—
343Q8—3261no—
344Q8—3261no—
345Q8—3261no—
346D4—21no—
347D4—21no—
348D4—13041no—
349D4—3261no—
350D4—3261no—
351D4—6521no—
352D4—6522no—
353D4—6522no—
354D4—6522no—
355D4—6522no—
356D4C2×D65212yes—
357D4C2×D65212yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—3261no—
363C2×C4—3261no—
364C2×C4—3261no—
365C2×C4—3261no—
366C2×C4—3261no—
367C2×C4—3261no—
368C2×C4—3261no—
369C2×C4—6522no—
370C2×C4—6522no—
371C2×C4—6522no—
372C2×C4—1631no—
373C2×C4—1631no—
374C2×C4D4×D16311yescharacteristic
375C2×C4D4×D16311yescharacteristic
376C2×C4C23×D16311yescharacteristic
377C8—21no—
378C8—3261no—
379C8—3261no—
380C8C2×D65212yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—13044no—
388C22—3261no—
389C22—3261no—
390C22—3261no—
391C22—3261no—
392C22—3261no—
393C22—3261no—
394C22—6521no—
395C22—6521no—
396C22—6521no—
397C22C2×D4×D16311yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3261no—
403C4—3261no—
404C4—6521no—
405C4—6521no—
406C4—6521no—
407C4C22×D65211yescharacteristic
408C4#1910478111yescharacteristic
409C2—42no—
410C2—42no—
411C2—13041no—
412C2—6521no—
413C2—6521no—
414C2D4×D65211yescharacteristic
415C2#2378246111yescharacteristic
416C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#442×C163C2
socleC22×C163C22×D4
hypercenter#442D163
hypocenterC163#3267
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#19104759C2semidirect
2. C2×#19104754C2semidirect
3. #23782319C2semidirect
4. #23782310C2semidirect
5. #23782059C2semidirect
6. #23781924C2semidirect
7. #23781895C2semidirect
8. #23781880C2semidirect
9. #23781870C2semidirect
10. #23781828C2semidirect
11. #23781828C2semidirect
12. #23781817C2semidirect
13. #23781811C2semidirect
14. #23781738C2semidirect
15. #442×C163C2semidirect
18. C2×#14579698C22semidirect
22. C2×Dic652C22semidirect
23. C2×Dic652C22semidirect
28. C2×#14579687C22semidirect
31. C2×C8×D163C22semidirect
32. #19104735C22semidirect
39. #19104715C22semidirect
47. C4⋊(C163⋊C8)C22semidirect
48. #19104658C22semidirect
49. #19104658C22semidirect
50. #19104658C22semidirect
53. (C2×D163)⋊C8C22semidirect
56. C4×D4×C163C22semidirect
57. #106×C163C22semidirect
59. #102×C163C22semidirect
60. #97×C163C22semidirect
115. D8×C163D4semidirect
208. C163#3267semidirect
223. #442D163semidirect
256. C4×D4C2×D163semidirect
259. #106C2×D163semidirect
266. #102C2×D163semidirect
269. #97C2×D163semidirect
306. D8D652semidirect

Maximal subgroups

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

subgroupindex
C2×#191047592
C2×#191047542
#237823192
#237823102
#237820592
#237819242
#237818952
#237818802
#237818702
#237818282
#237818172
#237818112
#237817382
#442×C1632
#3267163