Finite Group Databasebeta

← C2×#23782412 #35228662

Subgroups — C2×#23782412 #35228662

Order 20864.

1152 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782412C2116yesmaximal, direct product factor
2C2×#19104726C211yesmaximal, characteristic, complemented
3C2×#19104721C211yesmaximal, characteristic, complemented
4C2×#19104721C211yesmaximal, characteristic, complemented
5C2×#19104720C211yesmaximal, characteristic, complemented
6C2×#19104719C211yesmaximal, characteristic, complemented
7C2×#19104717C211yesmaximal, characteristic, complemented
8C2×#19104715C211yesmaximal, characteristic, complemented
9C2×#19104714C211yesmaximal, characteristic, complemented
10C2×#19104713C211yesmaximal, characteristic, complemented
11C2×#19104712C211yesmaximal, characteristic, complemented
12C2×#19104709C211yesmaximal, characteristic, complemented
13C2×#19104708C211yesmaximal, characteristic, complemented
14C2×#19104707C211yesmaximal, characteristic, complemented
15C2×#19104706C211yesmaximal, characteristic, complemented
16C2×#123×C163C211yesmaximal, characteristic, complemented
17C22×Dic326C2211yescharacteristic
18C22×Dic326C2211yescharacteristic
19C22×D652C2211yescharacteristic
20C22×C22⋊D163C2211yescharacteristic
21C22×C22⋊D163C2211yescharacteristic
22C22×C22⋊D163C2211yescharacteristic
23C22×C22⋊D163C2211yescharacteristic
24C23×Dic163C2211yescharacteristic
25C22×C4×D163C2211yescharacteristic
26C22×C4×D163C2211yescharacteristic
27#19104726C2218yescomplemented
28#19104721C2218yescomplemented
29#19104721C2218yescomplemented
30#19104720C2218yescomplemented
31#19104719C2218yescomplemented
32#19104717C2218yescomplemented
33#19104715C2218yescomplemented
34#19104714C2218yescomplemented
35#19104713C2218yescomplemented
36#19104712C2218yescomplemented
37#19104709C2218yescomplemented
38#19104708C2218yescomplemented
39#19104707C2218yescomplemented
40#19104706C2218yescomplemented
41C2×C4⋊Dic163C2211yescharacteristic
42C2×C4⋊Dic163C2211yescharacteristic
43C2×C4⋊Dic163C2211yescharacteristic
44C2×C4⋊Dic163C2211yescharacteristic
45C2×Dic163⋊C4C2211yescharacteristic
46C2×Dic163⋊C4C2211yescharacteristic
47C2×Dic163⋊C4C2211yescharacteristic
48C2×Dic163⋊C4C2211yescharacteristic
49C2×C22⋊Dic163C2211yescharacteristic
50C2×C22⋊Dic163C2211yescharacteristic
51C2×C22⋊Dic163C2211yescharacteristic
52C2×(C2×D163)⋊C4C2211yescharacteristic
53C2×(C2×D163)⋊C4C2211yescharacteristic
54C2×(C2×D163)⋊C4C2211yescharacteristic
55C2×(C2×D163)⋊C4C2211yescharacteristic
56C2×(C2×D163)⋊C4C2211yescharacteristic
57C2×(C2×D163)⋊C4C2211yescharacteristic
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
68C23×D163C2311yescharacteristic
69C2×Dic326—42no—
70C2×Dic326—42no—
71C2×Dic326C2314yes—
72C2×Dic326C2314yes—
73C2×D652—42no—
74C2×D652C2314yes—
75C2×C22⋊D163—42no—
76C2×C22⋊D163—42no—
77C2×C22⋊D163—42no—
78C2×C22⋊D163—42no—
79C2×C22⋊D163C2314yes—
80C2×C22⋊D163C2314yes—
81C2×C22⋊D163C2314yes—
82C2×C22⋊D163C2314yes—
83C22×Dic163—42no—
84C22×Dic163C2311yescharacteristic
85C22×Dic163C2311yescharacteristic
86C22×Dic163C2311yescharacteristic
87C22×Dic163C2311yescharacteristic
88C22×Dic163C2311yescharacteristic
89C22×Dic163C2311yescharacteristic
90C22×Dic163C2314yes—
91C2×C4×D163—42no—
92C2×C4×D163—42no—
93C2×C4×D163C2314yes—
94C2×C4×D163C2314yes—
95C4⋊Dic163C2314yes—
96C4⋊Dic163C2314yes—
97C4⋊Dic163C2314yes—
98C4⋊Dic163C2314yes—
99Dic163⋊C4C2314yes—
100Dic163⋊C4C2314yes—
101Dic163⋊C4C2314yes—
102Dic163⋊C4C2314yes—
103C22⋊Dic163C2314yes—
104C22⋊Dic163C2314yes—
105C22⋊Dic163C2314yes—
106(C2×D163)⋊C4C2314yes—
107(C2×D163)⋊C4C2314yes—
108(C2×D163)⋊C4C2314yes—
109(C2×D163)⋊C4C2314yes—
110(C2×D163)⋊C4C2314yes—
111(C2×D163)⋊C4C2314yes—
112C4×Dic163C2314yes—
113C24×C163C2311yescharacteristic
114C24×C163C2311yescharacteristic
115C2×D4×C163—42no—
116C2×D4×C163C2314yes—
117C22×C4×C163—42no—
118C22×C4×C163C2311yescharacteristic
119C22×C4×C163C2311yescharacteristic
120C22×C4×C163C2311yescharacteristic
121C22×C4×C163C2311yescharacteristic
122C22×C4×C163C2311yescharacteristic
123C22×C4×C163C2314yes—
124C4⋊C4×C163C2314yes—
125C4⋊C4×C163C2314yes—
126C22⋊C4×C163C2314yes—
127C22⋊C4×C163C2314yes—
128C22⋊C4×C163C2314yes—
129C22×D163—22no—
130C22×D163—22no—
131C22×D163—22no—
132C22×D163—22no—
133C22×D163—22no—
134C22×D163—22no—
135C22×D163C2412yes—
136C22×D163C2412yes—
137Dic326—44no—
138Dic326—44no—
139D652—44no—
140C22⋊D163—44no—
141C22⋊D163—44no—
142C22⋊D163—44no—
143C22⋊D163—44no—
144C2×Dic163—22no—
145C2×Dic163—22no—
146C2×Dic163—22no—
147C2×Dic163—22no—
148C2×Dic163—22no—
149C2×Dic163—22no—
150C2×Dic163—44no—
151C2×Dic163C2412yes—
152C2×Dic163C2412yes—
153C2×Dic163C2412yes—
154C2×Dic163C2412yes—
155C2×Dic163C2412yes—
156C2×Dic163C2412yes—
157C4×D163—44no—
158C4×D163—44no—
159C23×C163—22no—
160C23×C163—22no—
161C23×C163—22no—
162C23×C163—22no—
163C23×C163—22no—
164C23×C163—22no—
165C23×C163C2412yes—
166C23×C163C2412yes—
167C23×C163C2411yescharacteristic
168D4×C163—44no—
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—
180C22×C163—22no—
181C22×C163—22no—
182C22×C163—22no—
183C22×C163—22no—
184C22×C163—22no—
185C22×C163—22no—
186C22×C163—44no—
187C22×C163—44no—
188C22×C163H(5,𝔽2)12yes—
189C22×C163H(5,𝔽2)12yes—
190C22×C1632−1+412yes—
191C22×C163C2511yescharacteristic
192C2×D163—22no—
193C2×D163—22no—
194C2×D163—22no—
195C2×D163—22no—
196C2×D163—22no—
197C2×D163—22no—
198C2×D163—44no—
199C2×D163—44no—
200C4×C163—22no—
201C4×C163—22no—
202C4×C163—22no—
203C4×C163—22no—
204C4×C163—22no—
205Dic163—22no—
206Dic163—22no—
207Dic163—22no—
208Dic163—22no—
209Dic163—22no—
210Dic163—22no—
211C2×C163—42no—
212C2×C163—42no—
213C2×C163#56214yescomplemented
214C2×C163C2×H(5,𝔽2)11yescharacteristic
215C2×C163C2×H(5,𝔽2)11yescharacteristic
216C2×C163C2×2−1+411yescharacteristic
217D163—42no—
218D163—42no—
219C163C2×#56211yescharacteristic, complemented, Sylow
220C2×#562—1631nomaximal, Sylow
221#562—16316no—
222C2×#125—1631no—
223C2×#124—1631no—
224C2×#124—1631no—
225C2×#123—1631no—
226C2×#123—1631no—
227C2×#123—1631no—
228C2×#123D16311yescharacteristic, complemented
229C2×C22⋊Q8—1631no—
230C2×C22⋊Q8—1631no—
231C2×C22⋊Q8—1631no—
232C2×C22⋊Q8—1631no—
233C2×#121—1631no—
234C2×#121—1631no—
235C2×#121—1631no—
236C2×#121—1631no—
237C22×Q8—1631no—
238C22×Q8—1631no—
239C22×D4—1631no—
240C22×D4—1631no—
241C22×D4—1631no—
242C22×D4—1631no—
243C22×D4—1631no—
244C22×D4C2×D16311yescharacteristic
245C23×C4—1631no—
246C23×C4—1631no—
247C23×C4—1631no—
248C23×C4C2×D16311yescharacteristic
249#125—1638no—
250#124—1638no—
251#124—1638no—
252#123—1638no—
253#123—1638no—
254#123—1638no—
255#123C2×D16318yescomplemented
256C22⋊Q8—1638no—
257C22⋊Q8—1638no—
258C22⋊Q8—1638no—
259C22⋊Q8—1638no—
260#121—1638no—
261#121—1638no—
262#121—1638no—
263#121—1638no—
264C2×C4⋊C4—1631no—
265C2×C4⋊C4—1631no—
266C2×C4⋊C4—1631no—
267C2×C4⋊C4—1631no—
268C2×C4⋊C4—1631no—
269C2×C4⋊C4—1631no—
270C2×C4⋊C4—1631no—
271C2×C4⋊C4—1631no—
272C2×C4⋊C4C2×D16311yescharacteristic
273C2×C4⋊C4C2×D16311yescharacteristic
274C2×C22⋊C4—1631no—
275C2×C22⋊C4—1631no—
276C2×C22⋊C4—1631no—
277C2×C22⋊C4—1631no—
278C2×C22⋊C4—1631no—
279C2×C22⋊C4—1631no—
280C2×C22⋊C4—1631no—
281C2×C22⋊C4—1631no—
282C2×C22⋊C4—1631no—
283C2×C22⋊C4C2×D16311yescharacteristic
284C2×C22⋊C4C2×D16311yescharacteristic
285C2×C22⋊C4C2×D16311yescharacteristic
286C2×C42—1631no—
287C24—1631no—
288C24—1631no—
289C24C22×D16311yescharacteristic
290C24C22×D16311yescharacteristic
291C2×Q8—6522no—
292C2×Q8—6522no—
293C2×Q8—1634no—
294C2×Q8—1634no—
295C2×D4—42no—
296C2×D4—6522no—
297C2×D4—6522no—
298C2×D4—6522no—
299C2×D4—6522no—
300C2×D4—6522no—
301C2×D4—1634no—
302C2×D4—1634no—
303C2×D4—1634no—
304C2×D4—1634no—
305C2×D4—1634no—
306C2×D4C22×D16314yes—
307C22×C4—42no—
308C22×C4—6522no—
309C22×C4—6522no—
310C22×C4—6522no—
311C22×C4—1631no—
312C22×C4—1631no—
313C22×C4—1631no—
314C22×C4—1631no—
315C22×C4—1631no—
316C22×C4—1631no—
317C22×C4—1634no—
318C22×C4—1634no—
319C22×C4—1634no—
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⋊C4—1634no—
332C4⋊C4—1634no—
333C4⋊C4—1634no—
334C4⋊C4C22×D16314yes—
335C4⋊C4C22×D16314yes—
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⋊C4C22×D16314yes—
346C22⋊C4C22×D16314yes—
347C22⋊C4C22×D16314yes—
348C42—1634no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—3262no—
356C23—3262no—
357C23—3262no—
358C23—3262no—
359C23—3262no—
360C23—3262no—
361C23—1632no—
362C23—1632no—
363C23C23×D16312yes—
364C23C23×D16312yes—
365C23C23×D16311yescharacteristic
366Q8—6524no—
367Q8—6524no—
368D4—44no—
369D4—6524no—
370D4—6524no—
371D4—6524no—
372D4—6524no—
373D4—6524no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—44no—
380C2×C4—3262no—
381C2×C4—3262no—
382C2×C4—3262no—
383C2×C4—3262no—
384C2×C4—3262no—
385C2×C4—3262no—
386C2×C4—6524no—
387C2×C4—6524no—
388C2×C4—6524no—
389C2×C4—1632no—
390C2×C4—1632no—
391C2×C4—1632no—
392C2×C4—1632no—
393C2×C4—1632no—
394C2×C4—1632no—
395C2×C4C23×D16312yes—
396C2×C4C23×D16312yes—
397C2×C4C23×D16312yes—
398C2×C4C23×D16312yes—
399C2×C4C23×D16312yes—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—44no—
407C22—44no—
408C22—3262no—
409C22—3262no—
410C22—3262no—
411C22—3262no—
412C22—3262no—
413C22—3262no—
414C22—6524no—
415C22—6524no—
416C22#1910478012yes—
417C22#1910478112yes—
418C22#1910478212yes—
419C22C24×D16311yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—3262no—
426C4—3262no—
427C4—3262no—
428C4—3262no—
429C4—3262no—
430C4—3262no—
431C2—42no—
432C2—42no—
433C2—6522no—
434C2—6522no—
435C2#2378241214yesdirect product factor
436C2C2×#1910478011yescharacteristic
437C2C2×#1910478111yescharacteristic
438C2C2×#1910478211yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×#123×C163C2
socleC23×C163C24
hypercenterC2×#123D163
hypocenterC163C2×#562
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #23782412C2direct (N × H)
2. C2×#19104726C2semidirect
3. C2×#19104721C2semidirect
4. C2×#19104721C2semidirect
5. C2×#19104720C2semidirect
6. C2×#19104719C2semidirect
7. C2×#19104717C2semidirect
8. C2×#19104715C2semidirect
9. C2×#19104714C2semidirect
10. C2×#19104713C2semidirect
11. C2×#19104712C2semidirect
12. C2×#19104709C2semidirect
13. C2×#19104708C2semidirect
14. C2×#19104707C2semidirect
15. C2×#19104706C2semidirect
16. C2×#123×C163C2semidirect
27. #19104726C22semidirect
28. #19104721C22semidirect
29. #19104721C22semidirect
30. #19104720C22semidirect
31. #19104719C22semidirect
32. #19104717C22semidirect
33. #19104715C22semidirect
34. #19104714C22semidirect
35. #19104713C22semidirect
36. #19104712C22semidirect
37. #19104709C22semidirect
38. #19104708C22semidirect
39. #19104707C22semidirect
40. #19104706C22semidirect
61. #123×C163C22semidirect
213. C2×C163#562semidirect
219. C163C2×#562semidirect
228. C2×#123D163semidirect
255. #123C2×D163semidirect
435. C2#23782412direct (N × H)

Maximal subgroups

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

subgroupindex
#237824122
C2×#191047262
C2×#191047212
C2×#191047202
C2×#191047192
C2×#191047172
C2×#191047152
C2×#191047142
C2×#191047132
C2×#191047122
C2×#191047092
C2×#191047082
C2×#191047072
C2×#191047062
C2×#123×C1632
C2×#562163