Finite Group Databasebeta

← #35226236

Subgroups — #35226236

Order 20864.

504 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×#19104758C211yesmaximal, characteristic, complemented
2C22×#14579686C211yesmaximal, characteristic, complemented
3#23782326C211yesmaximal, characteristic, complemented
4#23782306C211yesmaximal, characteristic, complemented
5#23782050C211yesmaximal, characteristic, complemented
6#23781897C211yesmaximal, characteristic, complemented
7#23781896C211yesmaximal, characteristic, complemented
8#23781889C211yesmaximal, characteristic, complemented
9#23781883C211yesmaximal, characteristic, complemented
10#23781826C211yesmaximal, characteristic, complemented
11#23781813C211yesmaximal, characteristic, complemented
12#23781801C211yesmaximal, characteristic, complemented
13#23781750C211yesmaximal, characteristic, complemented
14(C22⋊D163)⋊C8C211yesmaximal, characteristic, complemented
15#451×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579700C2211yescharacteristic, complemented
18C22×D652C2211yescharacteristic
19#19104758—24no—
20C2×#14579686—24no—
21C2×#14579686C2211yescharacteristic
22C2×#14579686C2211yescharacteristic
23C2×#14579686C2211yescharacteristic
24C2×#14579686C2211yescharacteristic
25C2×#14579686C2211yescharacteristic, complemented
26C2×D1304C2211yescharacteristic
27C2×D1304C2211yescharacteristic
28C22×C163⋊C8C2211yescharacteristic
29C2×C8×D163C2211yescharacteristic, complemented
30#19104724—21no—
31#19104724C2211yescharacteristic
32#19104724C2211yescharacteristic, complemented
33#19104720—21no—
34#19104711—21no—
35#19104711C2211yescharacteristic, complemented
36#19104710—21no—
37#19104710C2211yescharacteristic
38#19104709—21no—
39#19104709C2211yescharacteristic
40#19104705—21no—
41Q8×Dic163C2211yescharacteristic, complemented
42#19104697C2211yescharacteristic
43C4×C22⋊D163—21no—
44C4×C22⋊D163C2211yescharacteristic
45C22⋊C4×D163—21no—
46#19104666C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104659C2211yescharacteristic, complemented
49#19104657C2211yescharacteristic
50#19104657C2211yescharacteristic
51#19104655C2211yescharacteristic, complemented
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53C8×Dic163C2211yescharacteristic, complemented
54C22×Q8×C163C2211yescharacteristic
55C2×Q16×C163C2211yescharacteristic, complemented
56C2×Q16×C163C2211yescharacteristic, complemented
57C22⋊Q8×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60C22⋊C8×C163C2211yescharacteristic
61C23×D163—21no—
62#14579700—22no—
63#14579700—22no—
64#14579700—41no—
65#14579700—41no—
66C2×D652—22no—
67C2×D652—21no—
68C2×D652—21no—
69C2×D652—21no—
70C2×D652—21no—
71C2×D652—41no—
72C2×D652C2311yescharacteristic
73C2×D652C2311yescharacteristic
74C2×D652C2311yescharacteristic
75C2×C22⋊D163—21no—
76C2×C22⋊D163D411yescharacteristic
77C2×C4×D163—21no—
78C2×C4×D163—21no—
79C2×C4×D163—21no—
80C2×C4×D163C2311yescharacteristic
81#14579686—22no—
82#14579686—22no—
83#14579686—22no—
84#14579686—22no—
85#14579686D414yescomplemented
86D1304—22no—
87D1304—41no—
88C2×C163⋊C8—22no—
89C2×C163⋊C8C2311yescharacteristic
90C2×C163⋊C8C2311yescharacteristic
91C8×D163—22no—
92C4⋊Dic163—21no—
93C4⋊Dic163C2311yescharacteristic
94Dic163⋊C4—21no—
95Dic163⋊C4D411yescharacteristic
96C22⋊Dic163D411yescharacteristic
97(C2×D163)⋊C4—21no—
98(C2×D163)⋊C4—21no—
99(C2×D163)⋊C4—21no—
100(C2×D163)⋊C4—21no—
101(C2×D163)⋊C4—21no—
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4D411yescharacteristic
106C4×Dic163—21no—
107C4×Dic163C2311yescharacteristic
108C2×Q8×C163—21no—
109C2×Q8×C163—22no—
110C2×Q8×C163—41no—
111C2×Q8×C163C2311yescharacteristic
112C2×Q8×C163C2311yescharacteristic
113C2×Q8×C163C2311yescharacteristic
114C22×C4×C163—21no—
115C22×C4×C163C2311yescharacteristic
116Q16×C163—22no—
117Q16×C163—41no—
118C2×C8×C163C2311yescharacteristic
119C2×C8×C163C2311yescharacteristic
120C4⋊C4×C163—21no—
121C4⋊C4×C163C2311yescharacteristic
122C22⋊C4×C163—21no—
123C22×D163—21no—
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—42no—
127C22×D163—42no—
128C22×D163—81no—
129C22×D163C2×D411yescharacteristic
130D652—21no—
131D652—21no—
132D652—21no—
133D652—21no—
134D652—21no—
135D652—42no—
136D652—42no—
137D652—42no—
138D652—81no—
139D652C2×D412yes—
140C22⋊D163—22no—
141C22⋊D163—42no—
142C2×Dic163—21no—
143C2×Dic163C2×D411yescharacteristic
144C2×Dic163C2×D411yescharacteristic
145C2×Dic163C2×D411yescharacteristic
146C4×D163—21no—
147C4×D163—21no—
148C4×D163—42no—
149C4×D163—42no—
150C4×D163—81no—
151C163⋊C8—21no—
152C163⋊C8C2×D412yes—
153C23×C163C2×D411yescharacteristic
154Q8×C163—21no—
155Q8×C163—21no—
156Q8×C163—21no—
157Q8×C163—21no—
158Q8×C163—21no—
159Q8×C163—42no—
160Q8×C163C2×D412yes—
161C2×C4×C163—21no—
162C2×C4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—42no—
168C2×C4×C163C2×D411yescharacteristic
169C2×C4×C163C2411yescharacteristic
170C8×C163—21no—
171C8×C163—21no—
172C22×C163—21no—
173C22×C163—21no—
174C22×C163#13512yescomplemented
175C22×C163C22×D411yescharacteristic
176C2×D163—21no—
177C2×D163—22no—
178C2×D163—21no—
179C2×D163—21no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—41no—
183C2×D163—41no—
184C2×D163—41no—
185C2×D163—41no—
186C2×D163—41no—
187C2×D163—41no—
188C2×D163—84no—
189C4×C163—21no—
190C4×C163—22no—
191C4×C163—41no—
192C4×C163—41no—
193C4×C163—41no—
194C4×C163C22×D411yescharacteristic
195C4×C163H(5,𝔽2)11yescharacteristic
196Dic163—21no—
197Dic163—21no—
198Dic163—21no—
199Dic163—41no—
200C2×C163—22no—
201C2×C163D4211yescharacteristic
202C2×C163C2×#13511yescharacteristic
203C2×C163#57711yescharacteristic
204D163—41no—
205D163—42no—
206D163—81no—
207D163—81no—
208C163#328211yescharacteristic, complemented, Sylow
209#3282—1631nomaximal, Sylow
210C2×#135—1631no—
211C22×SD16—1631no—
212#548—1631no—
213#546—1631no—
214#495—1631no—
215#465—1631no—
216#465—1631no—
217#464—1631no—
218#463—1631no—
219#452—1631no—
220#451D16311yescharacteristic, complemented
221#449—1631no—
222#447—1631no—
223C4×SD16—1631no—
224C8×D4—1631no—
225C2×#41—1631no—
226C2×#41—1631no—
227C22×Q8C2×D16311yescharacteristic
228C22×D4—1631no—
229#135—3264no—
230C2×Q16C2×D16311yescharacteristic, complemented
231C2×Q16C2×D16311yescharacteristic, complemented
232C2×SD16—3264no—
233C2×SD16—1631no—
234C2×SD16—1631no—
235C2×SD16—1631no—
236C2×SD16—1631no—
237C2×SD16—1631no—
238C2×D8—1631no—
239C2×D8—1631no—
240C22×C8—1631no—
241C22×C8—1631no—
242#124—3261no—
243#124—1631no—
244#124—1631no—
245#123—3261no—
246C22⋊Q8—3261no—
247C22⋊Q8—1631no—
248C22⋊Q8C2×D16311yescharacteristic
249#121—3261no—
250#121—3261no—
251#121—1631no—
252#121—1631no—
253#120—3261no—
254C4×Q8—1631no—
255C4×D4—3261no—
256C4×D4—1631no—
257C4×D4—1631no—
258C2×C22⋊C4—3261no—
259#106—1631no—
260C4⋊C8—1631no—
261#103—1631no—
262#103C2×D16311yescharacteristic, complemented
263#103C2×D16311yescharacteristic, complemented
264#102—1631no—
265#102—1631no—
266#102—1631no—
267C22⋊C8—1631no—
268C22⋊C8C2×D16311yescharacteristic
269C4×C8—1631no—
270C24—3261no—
271#41—3262no—
272#41—3262no—
273#41—6521no—
274#41—6521no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8C22×D16311yescharacteristic
279C2×Q8C22×D16311yescharacteristic
280C2×Q8C22×D16311yescharacteristic
281C2×D4—3261no—
282C2×D4—3262no—
283C2×D4—3261no—
284C2×D4—3261no—
285C2×D4—3261no—
286C2×D4—3261no—
287C2×D4—6521no—
288C2×D4—1631no—
289C2×D4—1631no—
290C2×D4—1631no—
291C2×D4—1631no—
292C22×C4—21no—
293C22×C4—3261no—
294C22×C4—3261no—
295C22×C4—3261no—
296C22×C4—1631no—
297C22×C4C22×D16311yescharacteristic
298Q16—22no—
299Q16—41no—
300SD16—3262no—
301SD16—3262no—
302SD16—3262no—
303SD16—3262no—
304SD16—1634no—
305D8—3262no—
306D8—6521no—
307C2×C8—3262no—
308C2×C8—3262no—
309C2×C8—1631no—
310C2×C8—1631no—
311C2×C8C22×D16311yescharacteristic
312C2×C8C22×D16311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3261no—
315C4⋊C4—3261no—
316C4⋊C4—1631no—
317C4⋊C4—1631no—
318C4⋊C4C22×D16311yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—3261no—
321C22⋊C4—3261no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4—3261no—
325C22⋊C4—3261no—
326C22⋊C4—3261no—
327C22⋊C4—3261no—
328C22⋊C4—1631no—
329C22⋊C4—1631no—
330C42—3261no—
331C42—1631no—
332C23—13041no—
333C23—3261no—
334C23—3261no—
335C23—3261no—
336C23—6522no—
337C23—6522no—
338C23—1631no—
339C23D4×D16311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—42no—
346Q8D4×D16312yes—
347D4—13041no—
348D4—3262no—
349D4—3261no—
350D4—3261no—
351D4—3261no—
352D4—3261no—
353D4—3261no—
354D4—6522no—
355D4—6522no—
356D4—6522no—
357D4—6522no—
358D4—1632no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—13041no—
367C2×C4—3261no—
368C2×C4—3261no—
369C2×C4—3261no—
370C2×C4—6522no—
371C2×C4—6522no—
372C2×C4—1631no—
373C2×C4—1631no—
374C2×C4—1631no—
375C2×C4D4×D16311yescharacteristic
376C2×C4C23×D16311yescharacteristic
377C8—21no—
378C8—21no—
379C8—3261no—
380C8—1632no—
381C22—21no—
382C22—21no—
383C22—13044no—
384C22—3261no—
385C22—3261no—
386C22—3261no—
387C22—3262no—
388C22—6521no—
389C22—6521no—
390C22—6521no—
391C22—6521no—
392C22—6521no—
393C22—6521no—
394C22—6521no—
395C22—6521no—
396C22#1910475812yescomplemented
397C22C2×D4×D16311yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—3261no—
404C4—3261no—
405C4—3261no—
406C4—6521no—
407C4C2×D4×D16311yescharacteristic
408C4#1910478011yescharacteristic
409C2—22no—
410C2—13041no—
411C2—13041no—
412C2—6522no—
413C2—6521no—
414C2#2378229711yescharacteristic
415C2C2×#1910475811yescharacteristic
416C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#451×C163C2
socleC22×C163C22×D4
hypercenter#451D163
hypocenterC163#3282
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#19104758C2semidirect
2. C22×#14579686C2semidirect
3. #23782326C2semidirect
4. #23782306C2semidirect
5. #23782050C2semidirect
6. #23781897C2semidirect
7. #23781896C2semidirect
8. #23781889C2semidirect
9. #23781883C2semidirect
10. #23781826C2semidirect
11. #23781813C2semidirect
12. #23781801C2semidirect
13. #23781750C2semidirect
14. (C22⋊D163)⋊C8C2semidirect
15. #451×C163C2semidirect
17. C2×#14579700C22semidirect
25. C2×#14579686C22semidirect
29. C2×C8×D163C22semidirect
32. #19104724C22semidirect
35. #19104711C22semidirect
41. Q8×Dic163C22semidirect
46. #19104666C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104659C22semidirect
51. #19104655C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. C8×Dic163C22semidirect
55. C2×Q16×C163C22semidirect
56. C2×Q16×C163C22semidirect
58. #103×C163C22semidirect
59. #103×C163C22semidirect
85. #14579686D4semidirect
174. C22×C163#135semidirect
208. C163#3282semidirect
220. #451D163semidirect
230. C2×Q16C2×D163semidirect
231. C2×Q16C2×D163semidirect
262. #103C2×D163semidirect
263. #103C2×D163semidirect
396. C22#19104758semidirect

Maximal subgroups

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

subgroupindex
C2×#191047582
C22×#145796862
#237823262
#237823062
#237820502
#237818972
#237818962
#237818892
#237818832
#237818262
#237818132
#237818012
#237817502
(C22⋊D163)⋊C82
#451×C1632
#3282163