Finite Group Databasebeta

← C2×#20265825 #27184586

Subgroups — C2×#20265825 #27184586

Order 12928.

1632 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13108074C211yesmaximal, characteristic, complemented
2C22×#13108073C211yesmaximal, characteristic, complemented
3#20265825C2116yesmaximal, direct product factor
4C2×#15692254C211yesmaximal, characteristic, complemented
5C2×#15692240C211yesmaximal, characteristic, complemented
6C2×#15692239C211yesmaximal, characteristic, complemented
7C2×#15692237C211yesmaximal, characteristic, complemented
8C2×#15692233C211yesmaximal, characteristic, complemented
9C2×#15692232C211yesmaximal, characteristic, complemented
10C2×#15692228C211yesmaximal, characteristic, complemented
11C2×#15692227C211yesmaximal, characteristic, complemented
12C2×#15692215C211yesmaximal, characteristic, complemented
13C2×#15692215C211yesmaximal, characteristic, complemented
14C22×#13108053C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D101C211yesmaximal, characteristic, complemented
16C2×#123×C101C211yesmaximal, characteristic, complemented
17C2×#13108074—28no—
18C2×#13108074C2218yescomplemented
19C2×#13108073—28no—
20C2×#13108073C2218yescomplemented
21C22×Dic202C2211yescharacteristic
22C22×Dic202C2211yescharacteristic
23C22×D404C2211yescharacteristic
24C22×C22⋊D101C2211yescharacteristic
25C22×C22⋊D101C2211yescharacteristic
26C22×C22⋊D101C2211yescharacteristic
27C22×C22⋊D101C2211yescharacteristic
28C23×Dic101C2211yescharacteristic
29C23×Dic101C2211yescharacteristic
30C23×Dic101C2211yescharacteristic
31C22×C4×D101C2211yescharacteristic
32C22×C4×D101C2211yescharacteristic
33C22×C4×D101C2211yescharacteristic
34C22×C4×D101C2211yescharacteristic
35#15692254C2218yescomplemented
36#15692240C2218yescomplemented
37#15692239C2218yescomplemented
38#15692237C2218yescomplemented
39#15692233C2218yescomplemented
40#15692232C2218yescomplemented
41#15692228C2218yescomplemented
42#15692227C2218yescomplemented
43#15692215C2218yescomplemented
44#15692215C2218yescomplemented
45C2×C4⋊Dic101C2211yescharacteristic
46C2×C4⋊Dic101C2211yescharacteristic
47C2×#13108053C2211yescharacteristic
48C2×#13108053C2211yescharacteristic
49C2×#13108053C2211yescharacteristic
50C2×#13108053C2211yescharacteristic
51C2×#13108053C2211yescharacteristic
52C2×#13108053C2211yescharacteristic
53C2×#13108053C2218yescomplemented
54C4⋊C4×D101C2218yescomplemented
55C2×C22⋊Dic101C2211yescharacteristic
56C2×#13108051C2211yescharacteristic
57C2×#13108051C2211yescharacteristic
58C2×#13108051C2211yescharacteristic
59C2×#13108051C2211yescharacteristic
60C2×C4×Dic101C2211yescharacteristic
61C22×D4×C101C2211yescharacteristic
62C23×C4×C101C2211yescharacteristic
63#123×C101C2218yescomplemented
64C2×C4⋊C4×C101C2211yescharacteristic
65C2×C4⋊C4×C101C2211yescharacteristic
66C2×C22⋊C4×C101C2211yescharacteristic
67C2×C22⋊C4×C101C2211yescharacteristic
68C2×C22⋊C4×C101C2211yescharacteristic
69C23×D101C2311yescharacteristic
70C23×D101C2311yescharacteristic
71#13108074—232no—
72#13108073—232no—
73C2×Dic202—24no—
74C2×Dic202—24no—
75C2×Dic202C2314yes—
76C2×Dic202C2314yes—
77C2×D404—24no—
78C2×D404C2314yes—
79C2×C22⋊D101—24no—
80C2×C22⋊D101—24no—
81C2×C22⋊D101—24no—
82C2×C22⋊D101D418yescomplemented
83C2×C22⋊D101C2314yes—
84C2×C22⋊D101C2314yes—
85C2×C22⋊D101C2314yes—
86C2×C22⋊D101C2314yes—
87C22×Dic101—24no—
88C22×Dic101—24no—
89C22×Dic101—24no—
90C22×Dic101C2311yescharacteristic
91C22×Dic101C2311yescharacteristic
92C22×Dic101C2311yescharacteristic
93C22×Dic101C2314yes—
94C22×Dic101C2314yes—
95C22×Dic101C2311yescharacteristic
96C22×Dic101C2314yes—
97C22×Dic101C2311yescharacteristic
98C22×Dic101C2311yescharacteristic
99C2×C4×D101—24no—
100C2×C4×D101—24no—
101C2×C4×D101—24no—
102C2×C4×D101—24no—
103C2×C4×D101C2314yes—
104C2×C4×D101C2314yes—
105C2×C4×D101C2314yes—
106C2×C4×D101C2314yes—
107C4⋊Dic101C2314yes—
108C4⋊Dic101C2314yes—
109#13108053C2314yes—
110#13108053C2314yes—
111#13108053C2314yes—
112#13108053C2314yes—
113#13108053C2314yes—
114#13108053C2314yes—
115C22⋊Dic101C2314yes—
116#13108051C2314yes—
117#13108051C2314yes—
118#13108051C2314yes—
119#13108051C2314yes—
120C4×Dic101C2314yes—
121C24×C101C2311yescharacteristic
122C24×C101C2311yescharacteristic
123C2×D4×C101—24no—
124C2×D4×C101C2314yes—
125C22×C4×C101—24no—
126C22×C4×C101C2311yescharacteristic
127C22×C4×C101C2311yescharacteristic
128C22×C4×C101C2311yescharacteristic
129C22×C4×C101C2311yescharacteristic
130C22×C4×C101C2314yes—
131C22×C4×C101C2311yescharacteristic
132C4⋊C4×C101C2314yes—
133C4⋊C4×C101C2314yes—
134C22⋊C4×C101C2314yes—
135C22⋊C4×C101C2314yes—
136C22⋊C4×C101C2314yes—
137C23×C101—22no—
138C23×C101—22no—
139C23×C101—22no—
140C23×C101—22no—
141C23×C101—22no—
142C23×C101C2×D414yes—
143C23×C101C2412yes—
144C23×C101C2412yes—
145C23×C101C2411yescharacteristic
146C22×D101—22no—
147C22×D101—22no—
148C22×D101—22no—
149C22×D101—22no—
150C22×D101—22no—
151C22×D101C2×D414yes—
152C22×D101C2412yes—
153C22×D101C2412yes—
154D4×C101—28no—
155C2×C4×C101—22no—
156C2×C4×C101—22no—
157C2×C4×C101—28no—
158C2×C4×C101—22no—
159C2×C4×C101—22no—
160C2×C4×C101—22no—
161C2×C4×C101C2412yes—
162C2×C4×C101C2412yes—
163C2×C4×C101C2412yes—
164C2×C4×C101C2412yes—
165C2×C4×C101C2412yes—
166C22⋊D101—28no—
167C22⋊D101—28no—
168C22⋊D101—28no—
169C22⋊D101C2×D4116yescomplemented
170C2×Dic101—22no—
171C2×Dic101—22no—
172C2×Dic101—22no—
173C2×Dic101—28no—
174C2×Dic101—22no—
175C2×Dic101—28no—
176C2×Dic101—28no—
177C2×Dic101C2×D414yes—
178C2×Dic101#4114yescomplemented
179C2×Dic101C2412yes—
180C2×Dic101C2412yes—
181C2×Dic101C2412yes—
182C2×Dic101C2412yes—
183C2×Dic101C2412yes—
184C2×Dic101C2412yes—
185D404—28no—
186C4×D101—28no—
187C4×D101—28no—
188C4×D101—28no—
189C4×D101—28no—
190Dic202—28no—
191Dic202—28no—
192C22×C101—28no—
193C22×C101—22no—
194C22×C101—22no—
195C22×C101—22no—
196C22×C101—22no—
197C22×C101—22no—
198C22×C101—44no—
199C22×C101C22×D414yes—
200C22×C101C22×D412yes—
201C22×C101C2×#4112yes—
202C22×C1012−1+412yes—
203C22×C101C2511yescharacteristic
204C2×D101—28no—
205C2×D101—22no—
206C2×D101—22no—
207C2×D101—22no—
208C2×D101—22no—
209C2×D101—22no—
210C2×D101—44no—
211C2×D101C22×D414yes—
212C4×C101—22no—
213C4×C101—22no—
214C4×C101—22no—
215C4×C101—22no—
216C4×C101—22no—
217Dic101—22no—
218Dic101—22no—
219Dic101—22no—
220Dic101—22no—
221Dic101C22×D414yes—
222Dic101C2×#4114yescomplemented
223C2×C101—24no—
224C2×C101—42no—
225C2×C101#54714yescomplemented
226C2×C101C23×D411yescharacteristic
227C2×C101C22×#4111yescharacteristic
228C2×C101C2×2−1+411yescharacteristic
229D101—24no—
230D101—42no—
231C2×#547—1011nomaximal, Sylow
232C101C2×#54711yescharacteristic, complemented, Sylow
233C22×#41—1011no—
234C22×#41—1011no—
235#547—10116no—
236C2×C4⋊Q8—1011no—
237C2×#123—1011no—
238C2×#123—1011no—
239C2×#123—1011no—
240C2×#123D10111yescharacteristic, complemented
241C2×C22⋊Q8—1011no—
242C2×C22⋊Q8—1011no—
243C2×#121—1011no—
244C2×#121—1011no—
245C2×C4×D4—1011no—
246C2×C4×D4—1011no—
247C22×C4⋊C4—1011no—
248C22×C4⋊C4—1011no—
249C2×#41—1018no—
250C2×#41—1018no—
251C2×#41—2028no—
252C2×#41—2028no—
253C22×Q8—1011no—
254C22×Q8—1011no—
255C22×D4—1011no—
256C22×D4—1011no—
257C22×D4—1011no—
258C22×D4—1011no—
259C22×D4—1011no—
260C22×D4C2×D10111yescharacteristic
261C23×C4—1011no—
262C23×C4—1011no—
263C23×C4—1011no—
264C23×C4—1011no—
265C23×C4—1011no—
266C23×C4—1011no—
267C23×C4—1011no—
268C23×C4C2×D10111yescharacteristic
269C4⋊Q8—1018no—
270#123—1018no—
271#123—1018no—
272#123—1018no—
273#123C2×D10118yescomplemented
274C22⋊Q8—1018no—
275C22⋊Q8—1018no—
276#121—1018no—
277#121—1018no—
278C4×D4—1018no—
279C4×D4—1018no—
280C2×C4⋊C4—1011no—
281C2×C4⋊C4—1011no—
282C2×C4⋊C4—1011no—
283C2×C4⋊C4—1011no—
284C2×C4⋊C4—1011no—
285C2×C4⋊C4—1011no—
286C2×C4⋊C4—1018no—
287C2×C4⋊C4—1011no—
288C2×C4⋊C4—1011no—
289C2×C4⋊C4—1018no—
290C2×C4⋊C4C2×D10111yescharacteristic
291C2×C4⋊C4C2×D10111yescharacteristic
292C2×C22⋊C4—1011no—
293C2×C22⋊C4—1011no—
294C2×C22⋊C4—1011no—
295C2×C22⋊C4—1011no—
296C2×C22⋊C4—1011no—
297C2×C22⋊C4C2×D10111yescharacteristic
298C2×C22⋊C4C2×D10111yescharacteristic
299C2×C22⋊C4C2×D10111yescharacteristic
300C2×C42—1011no—
301C24—1011no—
302C24—1011no—
303C24C22×D10111yescharacteristic
304C24C22×D10111yescharacteristic
305#41—20232no—
306#41—20232no—
307C2×Q8—1014no—
308C2×Q8—1014no—
309C2×Q8—2024no—
310C2×Q8—2024no—
311C2×D4—24no—
312C2×D4—1018no—
313C2×D4—1014no—
314C2×D4—1014no—
315C2×D4—1014no—
316C2×D4—1014no—
317C2×D4—1014no—
318C2×D4—2024no—
319C2×D4—2024no—
320C2×D4—2024no—
321C2×D4—2024no—
322C2×D4C22×D10114yes—
323C22×C4—24no—
324C22×C4—1011no—
325C22×C4—1011no—
326C22×C4—1014no—
327C22×C4—1014no—
328C22×C4—1014no—
329C22×C4—1011no—
330C22×C4—1011no—
331C22×C4—1011no—
332C22×C4—1011no—
333C22×C4—1014no—
334C22×C4—1014no—
335C22×C4—1014no—
336C22×C4—1014no—
337C22×C4—2024no—
338C22×C4—2024no—
339C22×C4—2024no—
340C22×C4—2024no—
341C22×C4—2024no—
342C22×C4—2024no—
343C22×C4—2024no—
344C22×C4C22×D10114yes—
345C22×C4C22×D10111yescharacteristic
346C22×C4C22×D10111yescharacteristic
347C22×C4C22×D10111yescharacteristic
348C22×C4C22×D10111yescharacteristic
349C22×C4C22×D10111yescharacteristic
350C4⋊C4—1014no—
351C4⋊C4—1014no—
352C4⋊C4—1014no—
353C4⋊C4—1014no—
354C4⋊C4—1014no—
355C4⋊C4—1014no—
356C4⋊C4—1014no—
357C4⋊C4—1014no—
358C4⋊C4C22×D10114yes—
359C4⋊C4C22×D10114yes—
360C22⋊C4—1014no—
361C22⋊C4—1014no—
362C22⋊C4—1014no—
363C22⋊C4—1014no—
364C22⋊C4—1014no—
365C22⋊C4C22×D10114yes—
366C22⋊C4C22×D10114yes—
367C22⋊C4C22×D10114yes—
368C42—1014no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—1014no—
375C23—1012no—
376C23—1012no—
377C23—2022no—
378C23—2022no—
379C23—2022no—
380C23—2022no—
381C23—2022no—
382C23D4×D10114yes—
383C23C23×D10112yes—
384C23C23×D10112yes—
385C23C23×D10111yescharacteristic
386Q8—2028no—
387Q8—2028no—
388D4—28no—
389D4—10116no—
390D4—2028no—
391D4—2028no—
392D4—2028no—
393D4—2028no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—28no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—1014no—
401C2×C4—1014no—
402C2×C4—1012no—
403C2×C4—1012no—
404C2×C4—1012no—
405C2×C4—1012no—
406C2×C4—1012no—
407C2×C4—1012no—
408C2×C4—2028no—
409C2×C4—2028no—
410C2×C4—2028no—
411C2×C4—2028no—
412C2×C4—2028no—
413C2×C4—2028no—
414C2×C4—2022no—
415C2×C4—2022no—
416C2×C4—2022no—
417C2×C4—2028no—
418C2×C4—2022no—
419C2×C4C23×D10112yes—
420C2×C4C23×D10112yes—
421C2×C4C23×D10112yes—
422C2×C4C23×D10112yes—
423C2×C4C23×D10112yes—
424C22—22no—
425C22—22no—
426C22—28no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—44no—
431C22—1014no—
432C22—4044no—
433C22—2022no—
434C22—2022no—
435C22—2022no—
436C22—2022no—
437C22—2022no—
438C22—2028no—
439C22C2×D4×D10114yes—
440C22C2×D4×D10112yes—
441C22#41×D10112yes—
442C22#1569230212yes—
443C22C24×D10111yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—1014no—
450C4—1014no—
451C4—2022no—
452C4—2022no—
453C4—2022no—
454C4—2022no—
455C2—24no—
456C2—42no—
457C2—4042no—
458C2—2024no—
459C2#2026582514yesdirect product factor
460C2C22×D4×D10111yescharacteristic
461C2C2×#41×D10111yescharacteristic
462C2C2×#1569230211yescharacteristic
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#123×C101C2
socleC23×C101C24
hypercenterC2×#123D101
hypocenterC101C2×#547
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×#13108074C2semidirect
2. C22×#13108073C2semidirect
3. #20265825C2direct (N × H)
4. C2×#15692254C2semidirect
5. C2×#15692240C2semidirect
6. C2×#15692239C2semidirect
7. C2×#15692237C2semidirect
8. C2×#15692233C2semidirect
9. C2×#15692232C2semidirect
10. C2×#15692228C2semidirect
11. C2×#15692227C2semidirect
12. C2×#15692215C2semidirect
13. C2×#15692215C2semidirect
14. C22×#13108053C2semidirect
15. C2×C4⋊C4×D101C2semidirect
16. C2×#123×C101C2semidirect
18. C2×#13108074C22semidirect
20. C2×#13108073C22semidirect
35. #15692254C22semidirect
36. #15692240C22semidirect
37. #15692239C22semidirect
38. #15692237C22semidirect
39. #15692233C22semidirect
40. #15692232C22semidirect
41. #15692228C22semidirect
42. #15692227C22semidirect
43. #15692215C22semidirect
44. #15692215C22semidirect
53. C2×#13108053C22semidirect
54. C4⋊C4×D101C22semidirect
63. #123×C101C22semidirect
82. C2×C22⋊D101D4semidirect
169. C22⋊D101C2×D4semidirect
178. C2×Dic101#41semidirect
222. Dic101C2×#41semidirect
225. C2×C101#547semidirect
232. C101C2×#547semidirect
240. C2×#123D101semidirect
273. #123C2×D101semidirect
459. C2#20265825direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080742
C22×#131080732
#202658252
C2×#156922542
C2×#156922402
C2×#156922392
C2×#156922372
C2×#156922332
C2×#156922322
C2×#156922282
C2×#156922272
C2×#156922152
C22×#131080532
C2×C4⋊C4×D1012
C2×#123×C1012
C2×#547101