Finite Group Databasebeta

← C2×#23782342 #35228034

Subgroups — C2×#23782342 #35228034

Order 20864.

1328 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782342C2116yesmaximal, direct product factor
2C2×#19104730C211yesmaximal, characteristic, complemented
3C2×#19104730C211yesmaximal, characteristic, complemented
4C2×#19104724C211yesmaximal, characteristic, complemented
5C2×#19104720C211yesmaximal, characteristic, complemented
6C2×#19104720C211yesmaximal, characteristic, complemented
7C2×#19104719C211yesmaximal, characteristic, complemented
8C2×#19104711C211yesmaximal, characteristic, complemented
9C2×#19104705C211yesmaximal, characteristic
10C2×#19104697C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊D163C211yesmaximal, characteristic, complemented
12C2×#19104693C211yesmaximal, characteristic, complemented
13C2×#19104691C211yesmaximal, characteristic, complemented
14C22×(C2×D163)⋊C4C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D163C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C163C211yesmaximal, characteristic, complemented
17C24×D163C2211yescharacteristic
18C22×D652C2211yescharacteristic
19C22×D652C2211yescharacteristic
20C22×C22⋊D163C2211yescharacteristic
21C23×Dic163C2211yescharacteristic
22C22×C4×D163C2211yescharacteristic
23C22×C4×D163C2211yescharacteristic
24C22×C4×D163C2211yescharacteristic, complemented
25#19104730C2218yescomplemented
26#19104730C2218yescomplemented
27#19104724C2218yescomplemented
28#19104720C2218yescomplemented
29#19104720C2218yescomplemented
30#19104719C2218yescomplemented
31#19104711C2218yescomplemented
32#19104705C2218yes—
33#19104697C2218yescomplemented
34C4×C22⋊D163C2218yescomplemented
35#19104693C2218yescomplemented
36#19104691C2218yescomplemented
37C2×C4⋊Dic163C2211yescharacteristic, complemented
38C2×C4⋊Dic163C2211yescharacteristic, complemented
39C2×Dic163⋊C4C2211yescharacteristic, complemented
40C2×Dic163⋊C4C2211yescharacteristic, complemented
41C2×Dic163⋊C4C2211yescharacteristic, complemented
42C2×C22⋊Dic163C2211yescharacteristic
43C2×(C2×D163)⋊C4C2211yescharacteristic
44C2×(C2×D163)⋊C4C2211yescharacteristic
45C2×(C2×D163)⋊C4C2211yescharacteristic
46C2×(C2×D163)⋊C4C2211yescharacteristic
47C2×(C2×D163)⋊C4C2211yescharacteristic
48C2×(C2×D163)⋊C4C2211yescharacteristic
49C2×(C2×D163)⋊C4C2211yescharacteristic
50C2×(C2×D163)⋊C4C2211yescharacteristic
51C2×(C2×D163)⋊C4C2218yescomplemented
52C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
53C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
54C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
55C22⋊C4×D163C2218yescomplemented
56C2×C4×Dic163C2211yescharacteristic, complemented
57C2×C4×Dic163C2211yescharacteristic, complemented
58C2×C4×Dic163C2211yescharacteristic, complemented
59C22×Q8×C163C2211yescharacteristic, complemented
60C23×C4×C163C2211yescharacteristic
61C22⋊Q8×C163C2218yescomplemented
62C2×C4⋊C4×C163C2211yescharacteristic, complemented
63C2×C4⋊C4×C163C2211yescharacteristic, complemented
64C2×C4⋊C4×C163C2211yescharacteristic, complemented
65C2×C22⋊C4×C163C2211yescharacteristic
66C2×C22⋊C4×C163C2211yescharacteristic
67C23×D163—24no—
68C23×D163—24no—
69C23×D163—42no—
70C23×D163C2311yescharacteristic
71C23×D163C2311yescharacteristic
72C23×D163C2314yes—
73C23×D163C2311yescharacteristic
74C2×D652—24no—
75C2×D652—42no—
76C2×D652C2314yes—
77C2×D652C2314yes—
78C2×C22⋊D163—24no—
79C2×C22⋊D163C2314yes—
80C22×Dic163—24no—
81C22×Dic163C2311yescharacteristic
82C22×Dic163C2311yescharacteristic
83C22×Dic163C2311yescharacteristic
84C22×Dic163C2311yescharacteristic
85C22×Dic163C2314yes—
86C22×Dic163C2311yescharacteristic
87C2×C4×D163—24no—
88C2×C4×D163—42no—
89C2×C4×D163—42no—
90C2×C4×D163C2314yes—
91C2×C4×D163C2314yes—
92C2×C4×D163C2314yescomplemented
93C4⋊Dic163C2314yescomplemented
94C4⋊Dic163C2314yescomplemented
95Dic163⋊C4C2314yescomplemented
96Dic163⋊C4C2314yescomplemented
97Dic163⋊C4C2314yescomplemented
98C22⋊Dic163C2314yes—
99(C2×D163)⋊C4C2314yes—
100(C2×D163)⋊C4C2314yes—
101(C2×D163)⋊C4C2314yes—
102(C2×D163)⋊C4C2314yes—
103(C2×D163)⋊C4C2314yes—
104(C2×D163)⋊C4C2314yes—
105(C2×D163)⋊C4C2314yes—
106(C2×D163)⋊C4C2314yes—
107(C2×D163)⋊C4C2314yescomplemented
108(C2×D163)⋊C4C2314yescomplemented
109(C2×D163)⋊C4C2314yescomplemented
110C4×Dic163C2314yescomplemented
111C4×Dic163C2314yescomplemented
112C4×Dic163C2314yescomplemented
113C24×C163C2311yescharacteristic
114C2×Q8×C163—42no—
115C2×Q8×C163C2314yescomplemented
116C22×C4×C163—42no—
117C22×C4×C163C2311yescharacteristic
118C22×C4×C163C2311yescharacteristic
119C22×C4×C163C2311yescharacteristic
120C22×C4×C163C2311yescharacteristic
121C22×C4×C163C2311yescharacteristic
122C22×C4×C163C2311yescharacteristic
123C22×C4×C163C2314yes—
124C4⋊C4×C163C2314yescomplemented
125C4⋊C4×C163C2314yescomplemented
126C4⋊C4×C163C2314yescomplemented
127C22⋊C4×C163C2314yes—
128C22⋊C4×C163C2314yes—
129C22×D163—22no—
130C22×D163—22no—
131C22×D163—22no—
132C22×D163—22no—
133C22×D163—22no—
134C22×D163—28no—
135C22×D163—22no—
136C22×D163—22no—
137C22×D163—22no—
138C22×D163—28no—
139C22×D163—44no—
140C22×D163—416no—
141C22×D163#4114yes—
142C22×D163C2412yes—
143C22×D163C2412yes—
144C22×D163C2412yes—
145D652—28no—
146D652—44no—
147C22⋊D163—28no—
148C2×Dic163—22no—
149C2×Dic163—28no—
150C2×Dic163—22no—
151C2×Dic163—22no—
152C2×Dic163—22no—
153C2×Dic163—22no—
154C2×Dic163C2412yes—
155C2×Dic163C2412yes—
156C2×Dic163C2412yes—
157C2×Dic163C2412yes—
158C2×Dic163C2412yes—
159C4×D163—28no—
160C4×D163—44no—
161C4×D163—44no—
162C23×C163—22no—
163C23×C163—22no—
164C23×C163#4114yes—
165C23×C163C2412yes—
166C23×C163C2411yescharacteristic
167Q8×C163—44no—
168C2×C4×C163—22no—
169C2×C4×C163—22no—
170C2×C4×C163—22no—
171C2×C4×C163—22no—
172C2×C4×C163—22no—
173C2×C4×C163—22no—
174C2×C4×C163—44no—
175C2×C4×C163C2412yes—
176C2×C4×C163C2412yes—
177C2×C4×C163C2412yes—
178C2×C4×C163C2412yes—
179C2×C4×C163C2412yes—
180C2×C4×C163C2412yes—
181C22×C163—28no—
182C22×C163—22no—
183C22×C163—22no—
184C22×C163C2×#4112yes—
185C22×C163C2×#4114yes—
186C22×C163C2×#4112yes—
187C22×C163H(5,𝔽2)12yes—
188C22×C163C2511yescharacteristic
189C2×D163—28no—
190C2×D163—22no—
191C2×D163—22no—
192C2×D163—22no—
193C2×D163—22no—
194C2×D163—22no—
195C2×D163—22no—
196C2×D163—22no—
197C2×D163—22no—
198C2×D163—416no—
199C2×D163—44no—
200C2×D163—44no—
201C2×D163C2×#4114yes—
202C4×C163—22no—
203C4×C163—22no—
204C4×C163—22no—
205C4×C163—22no—
206C4×C163—22no—
207C4×C163—22no—
208Dic163—22no—
209Dic163—22no—
210Dic163—22no—
211Dic163—22no—
212Dic163—22no—
213C2×C163—24no—
214C2×C163#55114yescomplemented
215C2×C163C22×#4111yescharacteristic
216C2×C163C22×#4111yescharacteristic
217C2×C163C2×H(5,𝔽2)11yescharacteristic
218D163—24no—
219D163—42no—
220D163—42no—
221C163C2×#55111yescharacteristic, complemented, Sylow
222C2×#551—1631nomaximal, Sylow
223#551—16316no—
224C2×#126—1631no—
225C2×#126—1631no—
226C2×#124—1631no—
227C2×#123—1631no—
228C2×#123—1631no—
229C2×#123—1631no—
230C2×C22⋊Q8—1631no—
231C2×C22⋊Q8D16311yescharacteristic, complemented
232C2×#120—1631no—
233C2×C4×D4—1631no—
234C2×C4×D4—1631no—
235C2×#117—1631no—
236C2×#117—1631no—
237C22×C22⋊C4—1631no—
238C22×C22⋊C4—1631no—
239C25—1631no—
240C22×Q8C2×D16311yescharacteristic, complemented
241C22×D4—1631no—
242C22×D4—1631no—
243C22×D4—1631no—
244C23×C4—1631no—
245C23×C4—1631no—
246C23×C4—1631no—
247C23×C4—1631no—
248C23×C4C2×D16311yescharacteristic
249#126—1638no—
250#126—1638no—
251#124—1638no—
252#123—1638no—
253#123—1638no—
254#123—1638no—
255C22⋊Q8—1638no—
256C22⋊Q8C2×D16318yescomplemented
257#120—1638no—
258C4×D4—1638no—
259C4×D4—1638no—
260#117—1638no—
261#117—1638no—
262C2×C4⋊C4—1631no—
263C2×C4⋊C4—1631no—
264C2×C4⋊C4—1631no—
265C2×C4⋊C4—1631no—
266C2×C4⋊C4—1631no—
267C2×C4⋊C4C2×D16311yescharacteristic, complemented
268C2×C4⋊C4C2×D16311yescharacteristic, complemented
269C2×C4⋊C4C2×D16311yescharacteristic, complemented
270C2×C22⋊C4—1631no—
271C2×C22⋊C4—1631no—
272C2×C22⋊C4—1631no—
273C2×C22⋊C4—1631no—
274C2×C22⋊C4—1631no—
275C2×C22⋊C4—1631no—
276C2×C22⋊C4—1631no—
277C2×C22⋊C4—1638no—
278C2×C22⋊C4—1631no—
279C2×C22⋊C4—1631no—
280C2×C22⋊C4—1631no—
281C2×C22⋊C4—1631no—
282C2×C22⋊C4—1631no—
283C2×C22⋊C4—1638no—
284C2×C22⋊C4C2×D16311yescharacteristic
285C2×C22⋊C4C2×D16311yescharacteristic
286C2×C42—1631no—
287C2×C42—1631no—
288C2×C42—1631no—
289C24—3264no—
290C24—3264no—
291C24—6522no—
292C24—1631no—
293C24—1631no—
294C24—1634no—
295C24—1631no—
296C24C22×D16311yescharacteristic
297C2×Q8—42no—
298C2×Q8C22×D16314yescomplemented
299C2×D4—3264no—
300C2×D4—3264no—
301C2×D4—6522no—
302C2×D4—1634no—
303C2×D4—1634no—
304C2×D4—1634no—
305C22×C4—42no—
306C22×C4—3264no—
307C22×C4—3264no—
308C22×C4—6522no—
309C22×C4—6522no—
310C22×C4—1634no—
311C22×C4—1631no—
312C22×C4—1631no—
313C22×C4—1631no—
314C22×C4—1631no—
315C22×C4—1631no—
316C22×C4—1634no—
317C22×C4—1634no—
318C22×C4—1634no—
319C22×C4C22×D16311yescharacteristic
320C22×C4C22×D16311yescharacteristic
321C22×C4C22×D16311yescharacteristic
322C22×C4C22×D16311yescharacteristic
323C22×C4C22×D16311yescharacteristic
324C22×C4C22×D16311yescharacteristic
325C22×C4C22×D16314yes—
326C4⋊C4—1634no—
327C4⋊C4—1634no—
328C4⋊C4—1634no—
329C4⋊C4—1634no—
330C4⋊C4—1634no—
331C4⋊C4C22×D16314yescomplemented
332C4⋊C4C22×D16314yescomplemented
333C4⋊C4C22×D16314yescomplemented
334C22⋊C4—1634no—
335C22⋊C4—1634no—
336C22⋊C4—1634no—
337C22⋊C4—1634no—
338C22⋊C4—1634no—
339C22⋊C4—1634no—
340C22⋊C4—1634no—
341C22⋊C4—1634no—
342C22⋊C4—1634no—
343C22⋊C4—1634no—
344C22⋊C4—1634no—
345C22⋊C4—1634no—
346C22⋊C4C22×D16314yes—
347C22⋊C4C22×D16314yes—
348C42—1634no—
349C42—1634no—
350C42—1634no—
351C23—22no—
352C23—22no—
353C23—3268no—
354C23—3262no—
355C23—3262no—
356C23—3262no—
357C23—3262no—
358C23—3262no—
359C23—3268no—
360C23—3262no—
361C23—3262no—
362C23—3262no—
363C23—6524no—
364C23—65216no—
365C23—1632no—
366C23—1632no—
367C23—1634no—
368C23—1632no—
369C23#1457970014yes—
370C23C23×D16312yes—
371C23C23×D16311yescharacteristic
372Q8—44no—
373D4—3268no—
374D4—3268no—
375D4—6524no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—44no—
383C2×C4—3268no—
384C2×C4—3262no—
385C2×C4—3268no—
386C2×C4—3262no—
387C2×C4—3262no—
388C2×C4—3262no—
389C2×C4—3262no—
390C2×C4—6524no—
391C2×C4—6524no—
392C2×C4—1632no—
393C2×C4—1632no—
394C2×C4—1632no—
395C2×C4—1632no—
396C2×C4—1632no—
397C2×C4C23×D16312yes—
398C2×C4C23×D16312yes—
399C2×C4C23×D16312yes—
400C2×C4C23×D16312yes—
401C2×C4C23×D16312yes—
402C2×C4C23×D16312yes—
403C22—22no—
404C22—22no—
405C22—28no—
406C22—3262no—
407C22—3262no—
408C22—3268no—
409C22—3262no—
410C22—3262no—
411C22—3262no—
412C22—3262no—
413C22—3262no—
414C22—3262no—
415C22—65216no—
416C22—6524no—
417C22—6524no—
418C22—1634no—
419C22#41×D16312yes—
420C22C2×#1457970014yes—
421C22C2×#1457970012yes—
422C22#1910478012yes—
423C22C24×D16311yescharacteristic
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—3262no—
431C4—3262no—
432C4—3262no—
433C4—3262no—
434C4—3262no—
435C2—24no—
436C2—3264no—
437C2—6522no—
438C2—6522no—
439C2#2378234214yesdirect product factor
440C2C2×#41×D16311yescharacteristic
441C2C22×#1457970011yescharacteristic
442C2C2×#1910478011yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×C22⋊Q8×C163C2
socleC23×C163C24
prime radicalC2×#19104705C2
hypercenterC2×C22⋊Q8D163
hypocenterC163C2×#551
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #23782342C2direct (N × H)
2. C2×#19104730C2semidirect
3. C2×#19104730C2semidirect
4. C2×#19104724C2semidirect
5. C2×#19104720C2semidirect
6. C2×#19104720C2semidirect
7. C2×#19104719C2semidirect
8. C2×#19104711C2semidirect
10. C2×#19104697C2semidirect
11. C2×C4×C22⋊D163C2semidirect
12. C2×#19104693C2semidirect
13. C2×#19104691C2semidirect
14. C22×(C2×D163)⋊C4C2semidirect
15. C2×C22⋊C4×D163C2semidirect
16. C2×C22⋊Q8×C163C2semidirect
24. C22×C4×D163C22semidirect
25. #19104730C22semidirect
26. #19104730C22semidirect
27. #19104724C22semidirect
28. #19104720C22semidirect
29. #19104720C22semidirect
30. #19104719C22semidirect
31. #19104711C22semidirect
33. #19104697C22semidirect
34. C4×C22⋊D163C22semidirect
35. #19104693C22semidirect
36. #19104691C22semidirect
37. C2×C4⋊Dic163C22semidirect
38. C2×C4⋊Dic163C22semidirect
39. C2×Dic163⋊C4C22semidirect
40. C2×Dic163⋊C4C22semidirect
41. C2×Dic163⋊C4C22semidirect
51. C2×(C2×D163)⋊C4C22semidirect
52. C2×(C2×D163)⋊C4C22semidirect
53. C2×(C2×D163)⋊C4C22semidirect
54. C2×(C2×D163)⋊C4C22semidirect
55. C22⋊C4×D163C22semidirect
56. C2×C4×Dic163C22semidirect
57. C2×C4×Dic163C22semidirect
58. C2×C4×Dic163C22semidirect
59. C22×Q8×C163C22semidirect
61. C22⋊Q8×C163C22semidirect
62. C2×C4⋊C4×C163C22semidirect
63. C2×C4⋊C4×C163C22semidirect
64. C2×C4⋊C4×C163C22semidirect
92. C2×C4×D163C23semidirect
93. C4⋊Dic163C23semidirect
94. C4⋊Dic163C23semidirect
95. Dic163⋊C4C23semidirect
96. Dic163⋊C4C23semidirect
97. Dic163⋊C4C23semidirect
107. (C2×D163)⋊C4C23semidirect
108. (C2×D163)⋊C4C23semidirect
109. (C2×D163)⋊C4C23semidirect
110. C4×Dic163C23semidirect
111. C4×Dic163C23semidirect
112. C4×Dic163C23semidirect
115. C2×Q8×C163C23semidirect
124. C4⋊C4×C163C23semidirect
125. C4⋊C4×C163C23semidirect
126. C4⋊C4×C163C23semidirect
214. C2×C163#551semidirect
221. C163C2×#551semidirect
231. C2×C22⋊Q8D163semidirect
240. C22×Q8C2×D163semidirect
256. C22⋊Q8C2×D163semidirect
267. C2×C4⋊C4C2×D163semidirect
268. C2×C4⋊C4C2×D163semidirect
269. C2×C4⋊C4C2×D163semidirect
298. C2×Q8C22×D163semidirect
331. C4⋊C4C22×D163semidirect
332. C4⋊C4C22×D163semidirect
333. C4⋊C4C22×D163semidirect
439. C2#23782342direct (N × H)

Maximal subgroups

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

subgroupindex
#237823422
C2×#191047302
C2×#191047242
C2×#191047202
C2×#191047192
C2×#191047112
C2×#191047052
C2×#191046972
C2×C4×C22⋊D1632
C2×#191046932
C2×#191046912
C22×(C2×D163)⋊C42
C2×C22⋊C4×D1632
C2×C22⋊Q8×C1632
C2×#551163