Finite Group Databasebeta

← C2×#20261337 #27155093

Subgroups — C2×#20261337 #27155093

Order 12416.

1260 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
1#20261337C2116yesmaximal, direct product factor
2C2×#15690806C211yesmaximal, characteristic, complemented
3C2×#15690800C211yesmaximal, characteristic, complemented
4C2×#15690795C211yesmaximal, characteristic, complemented
5C2×#15690794C211yesmaximal, characteristic, complemented
6C2×#15690792C211yesmaximal, characteristic, complemented
7C2×#15690785C211yesmaximal, characteristic, complemented
8C2×#15690784C211yesmaximal, characteristic, complemented
9C2×#15690783C211yesmaximal, characteristic, complemented
10C2×#15690783C211yesmaximal, characteristic, complemented
11C2×#15690783C211yesmaximal, characteristic, complemented
12C2×#15690782C211yesmaximal, characteristic, complemented
13C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
14C2×#15690770C211yesmaximal, characteristic, complemented
15C2×#15690768C211yesmaximal, characteristic, complemented
16C2×#123×C97C211yesmaximal, characteristic, complemented
17C22×D388C2211yescharacteristic
18C22×D388C2211yescharacteristic
19C22×D388C2211yescharacteristic
20C22×C22⋊D97C2211yescharacteristic
21C22×C22⋊D97C2211yescharacteristic
22C22×C22⋊D97C2211yescharacteristic
23C22×C22⋊D97C2211yescharacteristic
24C22×C22⋊D97C2211yescharacteristic
25C22×C22⋊D97C2211yescharacteristic
26C23×Dic97C2211yescharacteristic
27C22×C4×D97C2211yescharacteristic
28C22×C4×D97C2211yescharacteristic
29C22×C4×D97C2211yescharacteristic
30#15690806C2218yescomplemented
31#15690800C2218yescomplemented
32#15690795C2218yescomplemented
33#15690794C2218yescomplemented
34#15690792C2218yescomplemented
35#15690785C2218yescomplemented
36#15690784C2218yescomplemented
37#15690783C2218yescomplemented
38#15690783C2218yescomplemented
39#15690783C2218yescomplemented
40#15690782C2218yescomplemented
41C4×C22⋊D97C2218yescomplemented
42#15690770C2218yescomplemented
43#15690768C2218yescomplemented
44C2×C4⋊Dic97C2211yescharacteristic
45C2×C4⋊Dic97C2211yescharacteristic
46C2×#13095490C2211yescharacteristic
47C2×#13095490C2211yescharacteristic
48C2×#13095490C2211yescharacteristic
49C2×#13095490C2211yescharacteristic
50C2×C22⋊Dic97C2211yescharacteristic
51C2×#13095488C2211yescharacteristic
52C2×#13095488C2211yescharacteristic
53C2×#13095488C2211yescharacteristic
54C2×#13095488C2211yescharacteristic
55C2×#13095488C2211yescharacteristic
56C2×#13095488C2211yescharacteristic
57C2×C4×Dic97C2211yescharacteristic
58C2×C4×Dic97C2211yescharacteristic
59C22×D4×C97C2211yescharacteristic
60C23×C4×C97C2211yescharacteristic
61#123×C97C2218yescomplemented
62C2×C4⋊C4×C97C2211yescharacteristic
63C2×C4⋊C4×C97C2211yescharacteristic
64C2×C22⋊C4×C97C2211yescharacteristic
65C2×C22⋊C4×C97C2211yescharacteristic
66C2×C22⋊C4×C97C2211yescharacteristic
67C23×D97C2311yescharacteristic
68C23×D97C2311yescharacteristic
69C23×D97C2311yescharacteristic
70C2×D388—42no—
71C2×D388—42no—
72C2×D388—42no—
73C2×D388C2314yes—
74C2×D388C2314yes—
75C2×D388C2314yes—
76C2×C22⋊D97—24no—
77C2×C22⋊D97—24no—
78C2×C22⋊D97—42no—
79C2×C22⋊D97—42no—
80C2×C22⋊D97—42no—
81C2×C22⋊D97—42no—
82C2×C22⋊D97C2314yes—
83C2×C22⋊D97C2314yes—
84C2×C22⋊D97C2314yes—
85C2×C22⋊D97C2314yes—
86C2×C22⋊D97C2314yes—
87C2×C22⋊D97C2314yes—
88C22×Dic97—42no—
89C22×Dic97C2311yescharacteristic
90C22×Dic97C2311yescharacteristic
91C22×Dic97C2311yescharacteristic
92C22×Dic97C2311yescharacteristic
93C22×Dic97C2314yes—
94C22×Dic97C2311yescharacteristic
95C2×C4×D97—24no—
96C2×C4×D97—42no—
97C2×C4×D97—42no—
98C2×C4×D97C2314yes—
99C2×C4×D97C2314yes—
100C2×C4×D97C2314yes—
101C4⋊Dic97C2314yes—
102C4⋊Dic97C2314yes—
103#13095490C2314yes—
104#13095490C2314yes—
105#13095490C2314yes—
106#13095490C2314yes—
107C22⋊Dic97C2314yes—
108#13095488C2314yes—
109#13095488C2314yes—
110#13095488C2314yes—
111#13095488C2314yes—
112#13095488C2314yes—
113#13095488C2314yes—
114C4×Dic97C2314yes—
115C4×Dic97C2314yes—
116C24×C97C2311yescharacteristic
117C24×C97C2311yescharacteristic
118C2×D4×C97—42no—
119C2×D4×C97C2314yes—
120C22×C4×C97—42no—
121C22×C4×C97C2311yescharacteristic
122C22×C4×C97C2311yescharacteristic
123C22×C4×C97C2311yescharacteristic
124C22×C4×C97C2311yescharacteristic
125C22×C4×C97C2311yescharacteristic
126C22×C4×C97C2314yes—
127C4⋊C4×C97C2314yes—
128C4⋊C4×C97C2314yes—
129C22⋊C4×C97C2314yes—
130C22⋊C4×C97C2314yes—
131C22⋊C4×C97C2314yes—
132C23×C97—22no—
133C23×C97—22no—
134C23×C97—22no—
135C23×C97—22no—
136C23×C97—22no—
137C23×C97—22no—
138C23×C97C2412yes—
139C23×C97C2412yes—
140C23×C97C2411yescharacteristic
141C22×D97—22no—
142C22×D97—22no—
143C22×D97—22no—
144C22×D97—22no—
145C22×D97—22no—
146C22×D97—22no—
147C22×D97—22no—
148C22×D97—22no—
149C22×D97—22no—
150C22×D97C2412yes—
151C22×D97C2412yes—
152C22×D97C2412yes—
153D4×C97—44no—
154C2×C4×C97—22no—
155C2×C4×C97—22no—
156C2×C4×C97—22no—
157C2×C4×C97—22no—
158C2×C4×C97—22no—
159C2×C4×C97—44no—
160C2×C4×C97C2412yes—
161C2×C4×C97C2412yes—
162C2×C4×C97C2412yes—
163C2×C4×C97C2412yes—
164C2×C4×C97C2412yes—
165C22⋊D97—28no—
166C22⋊D97—28no—
167C22⋊D97—44no—
168C22⋊D97—44no—
169C22⋊D97—44no—
170C22⋊D97—44no—
171C2×Dic97—22no—
172C2×Dic97—22no—
173C2×Dic97—22no—
174C2×Dic97—22no—
175C2×Dic97—44no—
176C2×Dic97#4114yes—
177C2×Dic97C2412yes—
178C2×Dic97C2412yes—
179C2×Dic97C2412yes—
180C2×Dic97C2412yes—
181C2×Dic97C2412yes—
182D388—44no—
183D388—44no—
184D388—44no—
185C4×D97—28no—
186C4×D97—44no—
187C4×D97—44no—
188C22×C97—22no—
189C22×C97—22no—
190C22×C97—22no—
191C22×C97—22no—
192C22×C97—22no—
193C22×C97—22no—
194C22×C97—44no—
195C22×C97—44no—
196C22×C97C2×#4112yes—
197C22×C97H(5,𝔽2)12yes—
198C22×C97H(5,𝔽2)12yes—
199C22×C97C2511yescharacteristic
200C2×D97—22no—
201C2×D97—22no—
202C2×D97—22no—
203C2×D97—22no—
204C2×D97—22no—
205C2×D97—22no—
206C2×D97—22no—
207C2×D97—22no—
208C2×D97—22no—
209C2×D97—44no—
210C2×D97—44no—
211C2×D97—44no—
212C4×C97—22no—
213C4×C97—22no—
214C4×C97—22no—
215C4×C97—22no—
216C4×C97—22no—
217Dic97—22no—
218Dic97—22no—
219Dic97—22no—
220Dic97—22no—
221Dic97C2×#4114yes—
222C2×C97—42no—
223C2×C97—42no—
224C2×C97#54014yescomplemented
225C2×C97C22×#4111yescharacteristic
226C2×C97C2×H(5,𝔽2)11yescharacteristic
227C2×C97C2×H(5,𝔽2)11yescharacteristic
228D97—42no—
229D97—42no—
230D97—42no—
231C2×#540—971nomaximal, Sylow
232C97C2×#54011yescharacteristic, complemented, Sylow
233#540—9716no—
234C2×#127—971no—
235C2×#125—971no—
236C2×#123—971no—
237C2×#123—971no—
238C2×#123—971no—
239C2×#123D9711yescharacteristic, complemented
240C2×#121—971no—
241C2×#121—971no—
242C2×#121—971no—
243C2×#121—971no—
244C2×#121—971no—
245C2×#121—971no—
246C2×C4×D4—971no—
247C2×C4×D4—971no—
248C2×#117—971no—
249C22×D4—971no—
250C22×D4—971no—
251C22×D4—971no—
252C22×D4—971no—
253C22×D4—971no—
254C22×D4—971no—
255C22×D4—971no—
256C22×D4—971no—
257C22×D4—971no—
258C22×D4C2×D9711yescharacteristic
259C23×C4—971no—
260C23×C4—971no—
261C23×C4—971no—
262C23×C4—971no—
263C23×C4C2×D9711yescharacteristic
264#127—978no—
265#125—978no—
266#123—978no—
267#123—978no—
268#123—978no—
269#123C2×D9718yescomplemented
270#121—978no—
271#121—978no—
272#121—978no—
273#121—978no—
274#121—978no—
275#121—978no—
276C4×D4—978no—
277C4×D4—978no—
278#117—978no—
279C2×C4⋊C4—971no—
280C2×C4⋊C4—971no—
281C2×C4⋊C4—971no—
282C2×C4⋊C4—971no—
283C2×C4⋊C4—971no—
284C2×C4⋊C4—971no—
285C2×C4⋊C4C2×D9711yescharacteristic
286C2×C4⋊C4C2×D9711yescharacteristic
287C2×C22⋊C4—971no—
288C2×C22⋊C4—971no—
289C2×C22⋊C4—971no—
290C2×C22⋊C4—971no—
291C2×C22⋊C4—971no—
292C2×C22⋊C4—971no—
293C2×C22⋊C4—971no—
294C2×C22⋊C4C2×D9711yescharacteristic
295C2×C22⋊C4C2×D9711yescharacteristic
296C2×C22⋊C4C2×D9711yescharacteristic
297C2×C42—971no—
298C2×C42—971no—
299C24—971no—
300C24—971no—
301C24—971no—
302C24C22×D9711yescharacteristic
303C24C22×D9711yescharacteristic
304C2×D4—42no—
305C2×D4—974no—
306C2×D4—974no—
307C2×D4—974no—
308C2×D4—974no—
309C2×D4—974no—
310C2×D4—974no—
311C2×D4—974no—
312C2×D4—974no—
313C2×D4—974no—
314C2×D4—3882no—
315C2×D4—3882no—
316C2×D4—3882no—
317C2×D4—3882no—
318C2×D4—3882no—
319C2×D4—3882no—
320C2×D4—3882no—
321C2×D4—1944no—
322C2×D4—1944no—
323C2×D4C22×D9714yes—
324C22×C4—42no—
325C22×C4—971no—
326C22×C4—974no—
327C22×C4—971no—
328C22×C4—971no—
329C22×C4—971no—
330C22×C4—971no—
331C22×C4—974no—
332C22×C4—974no—
333C22×C4—974no—
334C22×C4—3882no—
335C22×C4—3882no—
336C22×C4—3882no—
337C22×C4—1944no—
338C22×C4C22×D9711yescharacteristic
339C22×C4C22×D9711yescharacteristic
340C22×C4C22×D9711yescharacteristic
341C22×C4C22×D9711yescharacteristic
342C22×C4C22×D9711yescharacteristic
343C22×C4C22×D9714yes—
344C4⋊C4—974no—
345C4⋊C4—974no—
346C4⋊C4—974no—
347C4⋊C4—974no—
348C4⋊C4—974no—
349C4⋊C4—974no—
350C4⋊C4C22×D9714yes—
351C4⋊C4C22×D9714yes—
352C22⋊C4—974no—
353C22⋊C4—974no—
354C22⋊C4—974no—
355C22⋊C4—974no—
356C22⋊C4—974no—
357C22⋊C4—974no—
358C22⋊C4—974no—
359C22⋊C4C22×D9714yes—
360C22⋊C4C22×D9714yes—
361C22⋊C4C22×D9714yes—
362C42—974no—
363C42—974no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—972no—
371C23—972no—
372C23—972no—
373C23—1942no—
374C23—1942no—
375C23—1942no—
376C23—1942no—
377C23—1942no—
378C23—1942no—
379C23—1942no—
380C23—1942no—
381C23—1942no—
382C23C23×D9712yes—
383C23C23×D9712yes—
384C23C23×D9711yescharacteristic
385D4—44no—
386D4—3884no—
387D4—3884no—
388D4—3884no—
389D4—3884no—
390D4—3884no—
391D4—3884no—
392D4—3884no—
393D4—1948no—
394D4—1948no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—44no—
401C2×C4—974no—
402C2×C4—972no—
403C2×C4—972no—
404C2×C4—972no—
405C2×C4—972no—
406C2×C4—972no—
407C2×C4—3884no—
408C2×C4—3884no—
409C2×C4—3884no—
410C2×C4—1948no—
411C2×C4—1942no—
412C2×C4—1942no—
413C2×C4—1942no—
414C2×C4—1942no—
415C2×C4C23×D9712yes—
416C2×C4C23×D9712yes—
417C2×C4C23×D9712yes—
418C2×C4C23×D9712yes—
419C2×C4C23×D9712yes—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—44no—
427C22—44no—
428C22—3884no—
429C22—3884no—
430C22—3884no—
431C22—1942no—
432C22—1942no—
433C22—1942no—
434C22—1942no—
435C22—1942no—
436C22—1942no—
437C22—1942no—
438C22—1942no—
439C22—1942no—
440C22#41×D9712yes—
441C22#1569085512yes—
442C22#1569085612yes—
443C22C24×D9711yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—974no—
450C4—1942no—
451C4—1942no—
452C4—1942no—
453C4—1942no—
454C2—42no—
455C2—42no—
456C2—3882no—
457C2—3882no—
458C2—3882no—
459C2#2026133714yesdirect product factor
460C2C2×#41×D9711yescharacteristic
461C2C2×#1569085511yescharacteristic
462C2C2×#1569085611yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#123×C97C2
socleC23×C97C24
hypercenterC2×#123D97
hypocenterC97C2×#540
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #20261337C2direct (N × H)
2. C2×#15690806C2semidirect
3. C2×#15690800C2semidirect
4. C2×#15690795C2semidirect
5. C2×#15690794C2semidirect
6. C2×#15690792C2semidirect
7. C2×#15690785C2semidirect
8. C2×#15690784C2semidirect
9. C2×#15690783C2semidirect
10. C2×#15690783C2semidirect
11. C2×#15690783C2semidirect
12. C2×#15690782C2semidirect
13. C2×C4×C22⋊D97C2semidirect
14. C2×#15690770C2semidirect
15. C2×#15690768C2semidirect
16. C2×#123×C97C2semidirect
30. #15690806C22semidirect
31. #15690800C22semidirect
32. #15690795C22semidirect
33. #15690794C22semidirect
34. #15690792C22semidirect
35. #15690785C22semidirect
36. #15690784C22semidirect
37. #15690783C22semidirect
38. #15690783C22semidirect
39. #15690783C22semidirect
40. #15690782C22semidirect
41. C4×C22⋊D97C22semidirect
42. #15690770C22semidirect
43. #15690768C22semidirect
61. #123×C97C22semidirect
224. C2×C97#540semidirect
232. C97C2×#540semidirect
239. C2×#123D97semidirect
269. #123C2×D97semidirect
459. C2#20261337direct (N × H)

Maximal subgroups

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

subgroupindex
#202613372
C2×#156908062
C2×#156908002
C2×#156907952
C2×#156907942
C2×#156907922
C2×#156907852
C2×#156907842
C2×#156907832
C2×#156907822
C2×C4×C22⋊D972
C2×#156907702
C2×#156907682
C2×#123×C972
C2×#54097