Finite Group Databasebeta

← C2×#23782259 #35227884

Subgroups — C2×#23782259 #35227884

Order 20864.

1144 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782259C2116yesmaximal, direct product factor
2C2×#19104730C211yesmaximal, characteristic, complemented
3C2×#19104728C211yesmaximal, characteristic, complemented
4C2×#19104726C211yesmaximal, characteristic, complemented
5C2×#19104725C211yesmaximal, characteristic, complemented
6C2×#19104719C211yesmaximal, characteristic, complemented
7C2×#19104718C211yesmaximal, characteristic, complemented
8C2×#19104717C211yesmaximal, characteristic, complemented
9C2×#19104715C211yesmaximal, characteristic, complemented
10C2×#19104712C211yesmaximal, characteristic, complemented
11C2×#19104712C211yesmaximal, characteristic, complemented
12C2×#19104707C211yesmaximal, characteristic
13C2×D4×Dic163C211yesmaximal, characteristic, complemented
14C2×(C22⋊D163)⋊C4C211yesmaximal, characteristic, complemented
15C22×C4⋊Dic163C211yesmaximal, characteristic, complemented
16C2×#123×C163C211yesmaximal, characteristic, complemented
17C22×Dic326C2211yescharacteristic
18C22×C22⋊D163C2211yescharacteristic
19C22×C22⋊D163C2211yescharacteristic
20C23×Dic163C2211yescharacteristic
21C23×Dic163C2211yescharacteristic
22C23×Dic163C2211yescharacteristic
23C22×C4×D163C2211yescharacteristic
24#19104730C2218yescomplemented
25#19104728C2218yescomplemented
26#19104726C2218yescomplemented
27#19104725C2218yescomplemented
28#19104719C2218yescomplemented
29#19104718C2218yescomplemented
30#19104717C2218yescomplemented
31#19104715C2218yescomplemented
32#19104712C2218yescomplemented
33#19104712C2218yescomplemented
34#19104707C2218yes—
35D4×Dic163C2218yescomplemented
36(C22⋊D163)⋊C4C2218yescomplemented
37C2×C4⋊Dic163C2211yescharacteristic
38C2×C4⋊Dic163C2211yescharacteristic
39C2×C4⋊Dic163C2211yescharacteristic
40C2×C4⋊Dic163C2211yescharacteristic
41C2×C4⋊Dic163C2211yescharacteristic
42C2×C4⋊Dic163C2211yescharacteristic
43C2×C4⋊Dic163C2218yescomplemented
44C2×Dic163⋊C4C2211yescharacteristic
45C2×Dic163⋊C4C2211yescharacteristic
46C2×Dic163⋊C4C2211yescharacteristic
47C2×Dic163⋊C4C2211yescharacteristic
48C2×Dic163⋊C4C2211yescharacteristic
49C2×Dic163⋊C4C2211yescharacteristic
50C2×C22⋊Dic163C2211yescharacteristic
51C2×C22⋊Dic163C2211yescharacteristic
52C2×C22⋊Dic163C2211yescharacteristic
53C2×(C2×D163)⋊C4C2211yescharacteristic
54C2×(C2×D163)⋊C4C2211yescharacteristic
55C2×(C2×D163)⋊C4C2211yescharacteristic
56C2×(C2×D163)⋊C4C2211yescharacteristic
57C2×C4×Dic163C2211yescharacteristic
58C2×C4×Dic163C2211yescharacteristic
59C22×D4×C163C2211yescharacteristic
60C23×C4×C163C2211yescharacteristic
61#123×C163C2218yescomplemented
62C2×C4⋊C4×C163C2211yescharacteristic
63C2×C4⋊C4×C163C2211yescharacteristic
64C2×C22⋊C4×C163C2211yescharacteristic
65C2×C22⋊C4×C163C2211yescharacteristic
66C2×C22⋊C4×C163C2211yescharacteristic
67C23×D163C2311yescharacteristic
68C2×Dic326—42no—
69C2×Dic326C2314yes—
70C2×C22⋊D163—24no—
71C2×C22⋊D163—42no—
72C2×C22⋊D163C2314yes—
73C2×C22⋊D163C2314yes—
74C22×Dic163—42no—
75C22×Dic163—42no—
76C22×Dic163—42no—
77C22×Dic163C2311yescharacteristic
78C22×Dic163C2311yescharacteristic
79C22×Dic163C2311yescharacteristic
80C22×Dic163C2311yescharacteristic
81C22×Dic163C2311yescharacteristic
82C22×Dic163C2311yescharacteristic
83C22×Dic163C2311yescharacteristic
84C22×Dic163C2314yes—
85C22×Dic163C2314yes—
86C22×Dic163C2314yes—
87C2×C4×D163—42no—
88C2×C4×D163C2314yes—
89C4⋊Dic163C2314yes—
90C4⋊Dic163C2314yes—
91C4⋊Dic163C2314yes—
92C4⋊Dic163C2314yes—
93C4⋊Dic163C2314yes—
94C4⋊Dic163C2314yes—
95Dic163⋊C4C2314yes—
96Dic163⋊C4C2314yes—
97Dic163⋊C4C2314yes—
98Dic163⋊C4C2314yes—
99Dic163⋊C4C2314yes—
100Dic163⋊C4C2314yes—
101C22⋊Dic163C2314yes—
102C22⋊Dic163C2314yes—
103C22⋊Dic163C2314yes—
104(C2×D163)⋊C4C2314yes—
105(C2×D163)⋊C4C2314yes—
106(C2×D163)⋊C4C2314yes—
107(C2×D163)⋊C4C2314yes—
108C4×Dic163C2314yes—
109C4×Dic163C2314yes—
110C24×C163C2311yescharacteristic
111C24×C163C2311yescharacteristic
112C2×D4×C163—24no—
113C2×D4×C163C2314yes—
114C22×C4×C163—24no—
115C22×C4×C163C2311yescharacteristic
116C22×C4×C163C2311yescharacteristic
117C22×C4×C163C2311yescharacteristic
118C22×C4×C163C2311yescharacteristic
119C22×C4×C163C2311yescharacteristic
120C22×C4×C163C2314yes—
121C4⋊C4×C163C2314yes—
122C4⋊C4×C163C2314yes—
123C22⋊C4×C163C2314yes—
124C22⋊C4×C163C2314yes—
125C22⋊C4×C163C2314yes—
126C22×D163—22no—
127C22×D163—22no—
128C22×D163—22no—
129C22×D163C2412yes—
130Dic326—44no—
131C22⋊D163—28no—
132C22⋊D163—44no—
133C2×Dic163—22no—
134C2×Dic163—22no—
135C2×Dic163—22no—
136C2×Dic163—22no—
137C2×Dic163—22no—
138C2×Dic163—22no—
139C2×Dic163—22no—
140C2×Dic163—44no—
141C2×Dic163—44no—
142C2×Dic163—44no—
143C2×Dic163C2412yes—
144C2×Dic163C2412yes—
145C2×Dic163C2412yes—
146C2×Dic163C2412yes—
147C2×Dic163C2412yes—
148C2×Dic163C2412yes—
149C2×Dic163C2412yes—
150C4×D163—44no—
151C23×C163—22no—
152C23×C163—22no—
153C23×C163—22no—
154C23×C163—22no—
155C23×C163—22no—
156C23×C163#4114yes—
157C23×C163C2412yes—
158C23×C163C2412yes—
159C23×C163C2411yescharacteristic
160D4×C163—28no—
161C2×C4×C163—28no—
162C2×C4×C163—22no—
163C2×C4×C163—22no—
164C2×C4×C163—22no—
165C2×C4×C163—22no—
166C2×C4×C163—22no—
167C2×C4×C163C2412yes—
168C2×C4×C163C2412yes—
169C2×C4×C163C2412yes—
170C2×C4×C163C2412yes—
171C2×C4×C163C2412yes—
172C22×C163—28no—
173C22×C163—22no—
174C22×C163—22no—
175C22×C163—22no—
176C22×C163—22no—
177C22×C163—22no—
178C22×C163—44no—
179C22×C163C2×#4114yes—
180C22×C163C2×#4112yes—
181C22×C163H(5,𝔽2)12yes—
182C22×C1632−1+412yes—
183C22×C163C2511yescharacteristic
184C2×D163—22no—
185C2×D163—22no—
186C2×D163—22no—
187C2×D163—44no—
188C4×C163—22no—
189C4×C163—22no—
190C4×C163—22no—
191C4×C163—22no—
192C4×C163—22no—
193Dic163—22no—
194Dic163—22no—
195Dic163—22no—
196Dic163—22no—
197Dic163—22no—
198Dic163—22no—
199Dic163—22no—
200C2×C163—24no—
201C2×C163—42no—
202C2×C163#53914yescomplemented
203C2×C163C22×#4111yescharacteristic
204C2×C163C2×H(5,𝔽2)11yescharacteristic
205C2×C163C2×2−1+411yescharacteristic
206D163—42no—
207C163C2×#53911yescharacteristic, complemented, Sylow
208C2×#539—1631nomaximal, Sylow
209#539—16316no—
210C2×#126—1631no—
211C2×#126—1631no—
212C2×#125—1631no—
213C2×#125—1631no—
214C2×#123—1631no—
215C2×#123—1631no—
216C2×#123—1631no—
217C2×#123D16311yescharacteristic, complemented
218C2×C22⋊Q8—1631no—
219C2×C22⋊Q8—1631no—
220C2×C22⋊Q8—1631no—
221C2×#121—1631no—
222C2×C4×D4—1631no—
223C2×C4×D4—1631no—
224C22×C4⋊C4—1631no—
225C22×Q8—1631no—
226C22×D4—1631no—
227C22×D4—1631no—
228C22×D4C2×D16311yescharacteristic
229C23×C4—1631no—
230C23×C4—1631no—
231C23×C4—1631no—
232C23×C4—1631no—
233C23×C4C2×D16311yescharacteristic
234#126—1638no—
235#126—1638no—
236#125—1638no—
237#125—1638no—
238#123—1638no—
239#123—1638no—
240#123—1638no—
241#123C2×D16318yescomplemented
242C22⋊Q8—1638no—
243C22⋊Q8—1638no—
244C22⋊Q8—1638no—
245#121—1638no—
246C4×D4—1638no—
247C4×D4—1638no—
248C2×C4⋊C4—1631no—
249C2×C4⋊C4—1631no—
250C2×C4⋊C4—1631no—
251C2×C4⋊C4—1631no—
252C2×C4⋊C4—1638no—
253C2×C4⋊C4—1631no—
254C2×C4⋊C4—1631no—
255C2×C4⋊C4—1631no—
256C2×C4⋊C4—1631no—
257C2×C4⋊C4—1631no—
258C2×C4⋊C4—1631no—
259C2×C4⋊C4—1631no—
260C2×C4⋊C4—1631no—
261C2×C4⋊C4C2×D16311yescharacteristic
262C2×C4⋊C4C2×D16311yescharacteristic
263C2×C22⋊C4—1631no—
264C2×C22⋊C4—1631no—
265C2×C22⋊C4—1631no—
266C2×C22⋊C4—1631no—
267C2×C22⋊C4—1631no—
268C2×C22⋊C4—1631no—
269C2×C22⋊C4—1631no—
270C2×C22⋊C4C2×D16311yescharacteristic
271C2×C22⋊C4C2×D16311yescharacteristic
272C2×C22⋊C4C2×D16311yescharacteristic
273C2×C42—1631no—
274C2×C42—1631no—
275C24—1631no—
276C24C22×D16311yescharacteristic
277C24C22×D16311yescharacteristic
278C2×Q8—6522no—
279C2×Q8—1634no—
280C2×D4—24no—
281C2×D4—3264no—
282C2×D4—6522no—
283C2×D4—1634no—
284C2×D4—1634no—
285C2×D4C22×D16314yes—
286C22×C4—24no—
287C22×C4—6522no—
288C22×C4—6522no—
289C22×C4—6522no—
290C22×C4—6522no—
291C22×C4—1631no—
292C22×C4—1631no—
293C22×C4—1631no—
294C22×C4—1631no—
295C22×C4—1631no—
296C22×C4—1631no—
297C22×C4—1631no—
298C22×C4—1634no—
299C22×C4—1634no—
300C22×C4—1634no—
301C22×C4—1634no—
302C22×C4C22×D16314yes—
303C22×C4C22×D16311yescharacteristic
304C22×C4C22×D16311yescharacteristic
305C22×C4C22×D16311yescharacteristic
306C22×C4C22×D16311yescharacteristic
307C22×C4C22×D16311yescharacteristic
308C4⋊C4—1634no—
309C4⋊C4—1634no—
310C4⋊C4—1634no—
311C4⋊C4—1634no—
312C4⋊C4—1634no—
313C4⋊C4—1634no—
314C4⋊C4—1634no—
315C4⋊C4—1634no—
316C4⋊C4—1634no—
317C4⋊C4—1634no—
318C4⋊C4—1634no—
319C4⋊C4—1634no—
320C4⋊C4C22×D16314yes—
321C4⋊C4C22×D16314yes—
322C22⋊C4—1634no—
323C22⋊C4—1634no—
324C22⋊C4—1634no—
325C22⋊C4—1634no—
326C22⋊C4—1634no—
327C22⋊C4—1634no—
328C22⋊C4—1634no—
329C22⋊C4C22×D16314yes—
330C22⋊C4C22×D16314yes—
331C22⋊C4C22×D16314yes—
332C42—1634no—
333C42—1634no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—3262no—
340C23—3262no—
341C23—3262no—
342C23—1632no—
343C23#1457969814yes—
344C23C23×D16312yes—
345C23C23×D16312yes—
346C23C23×D16311yescharacteristic
347Q8—6524no—
348D4—28no—
349D4—3268no—
350D4—6524no—
351C2×C4—28no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—3262no—
358C2×C4—3262no—
359C2×C4—3262no—
360C2×C4—3262no—
361C2×C4—3262no—
362C2×C4—3262no—
363C2×C4—3262no—
364C2×C4—6524no—
365C2×C4—6524no—
366C2×C4—6524no—
367C2×C4—6524no—
368C2×C4—1632no—
369C2×C4—1632no—
370C2×C4—1632no—
371C2×C4—1632no—
372C2×C4—1632no—
373C2×C4—1632no—
374C2×C4—1632no—
375C2×C4C23×D16312yes—
376C2×C4C23×D16312yes—
377C2×C4C23×D16312yes—
378C2×C4C23×D16312yes—
379C2×C4C23×D16312yes—
380C22—22no—
381C22—22no—
382C22—28no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—44no—
387C22—3262no—
388C22—3262no—
389C22—3262no—
390C22—6524no—
391C22C2×#1457969814yes—
392C22C2×#1457969812yes—
393C22#1910478112yes—
394C22#1910478212yes—
395C22C24×D16311yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—3262no—
402C4—3262no—
403C4—3262no—
404C4—3262no—
405C4—3262no—
406C4—3262no—
407C4—3262no—
408C2—24no—
409C2—42no—
410C2—6522no—
411C2#2378225914yesdirect product factor
412C2C22×#1457969811yescharacteristic
413C2C2×#1910478111yescharacteristic
414C2C2×#1910478211yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×#123×C163C2
socleC23×C163C24
prime radicalC2×#19104707C2
hypercenterC2×#123D163
hypocenterC163C2×#539
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #23782259C2direct (N × H)
2. C2×#19104730C2semidirect
3. C2×#19104728C2semidirect
4. C2×#19104726C2semidirect
5. C2×#19104725C2semidirect
6. C2×#19104719C2semidirect
7. C2×#19104718C2semidirect
8. C2×#19104717C2semidirect
9. C2×#19104715C2semidirect
10. C2×#19104712C2semidirect
11. C2×#19104712C2semidirect
13. C2×D4×Dic163C2semidirect
14. C2×(C22⋊D163)⋊C4C2semidirect
15. C22×C4⋊Dic163C2semidirect
16. C2×#123×C163C2semidirect
24. #19104730C22semidirect
25. #19104728C22semidirect
26. #19104726C22semidirect
27. #19104725C22semidirect
28. #19104719C22semidirect
29. #19104718C22semidirect
30. #19104717C22semidirect
31. #19104715C22semidirect
32. #19104712C22semidirect
33. #19104712C22semidirect
35. D4×Dic163C22semidirect
36. (C22⋊D163)⋊C4C22semidirect
43. C2×C4⋊Dic163C22semidirect
61. #123×C163C22semidirect
202. C2×C163#539semidirect
207. C163C2×#539semidirect
217. C2×#123D163semidirect
241. #123C2×D163semidirect
411. C2#23782259direct (N × H)

Maximal subgroups

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

subgroupindex
#237822592
C2×#191047302
C2×#191047282
C2×#191047262
C2×#191047252
C2×#191047192
C2×#191047182
C2×#191047172
C2×#191047152
C2×#191047122
C2×#191047072
C2×D4×Dic1632
C2×(C22⋊D163)⋊C42
C22×C4⋊Dic1632
C2×#123×C1632
C2×#539163