Finite Group Databasebeta

← C2×#21445908 #29679798

Subgroups — C2×#21445908 #29679798

Order 15616.

1878 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D61C211yesmaximal, characteristic
2C22×#14519689C211yesmaximal, characteristic
3C2×#16839619C211yesmaximal, characteristic
4#21445908C218yesmaximal, direct product factor
5C2×#16839525C211yesmaximal, characteristic, complemented
6C2×#16839524C211yesmaximal, characteristic, complemented
7C2×#16839524C211yesmaximal, characteristic, complemented
8C2×C61×#340C211yesmaximal, characteristic, complemented
9C23×C4×D61—21no—
10C23×C4×D61C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196—21no—
15C2×C4⋊C4×D61—24no—
16C2×C4⋊C4×D61—22no—
17C2×C4⋊C4×D61C418yescomplemented
18C2×C4⋊C4×D61C2214yes—
19C22×#11758198—21no—
20C2×#14519689—24no—
21C2×#14519689C2214yes—
22#16839619—22no—
23#16839619C2214yes—
24C2×#14519684—21no—
25#16839525C2214yescomplemented
26#16839524C2214yescomplemented
27#16839524C2214yescomplemented
28C23×C8×C61C2211yescharacteristic
29C22×C4⋊C4×C61C2211yescharacteristic
30C22×C4⋊C4×C61C2211yescharacteristic
31C61×#340C2214yescomplemented
32C24×D61D411yescharacteristic
33C23×Dic61—21no—
34C23×Dic61—21no—
35C23×Dic61D411yescharacteristic
36C22×C4×D61—22no—
37C22×C4×D61—22no—
38C22×C4×D61—22no—
39C22×C4×D61—22no—
40C22×C4×D61—28no—
41C22×C4×D61—24no—
42C22×C4×D61—44no—
43C22×C4×D61—44no—
44C22×C4×D61C2×C414yes—
45C22×C4×D61D412yes—
46C22×C4×D61D414yes—
47C22×C4×D61C2312yes—
48C22×#92418—22no—
49C22×#92418—22no—
50C22×#92418—22no—
51C22×#92418C2312yes—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2×C414yes—
58C2×C4⋊Dic61C2312yes—
59C2×C4⋊Dic61C2312yes—
60C2×#11758196—28no—
61C2×#11758196—24no—
62C4⋊C4×D61—216no—
63C4⋊C4×D61—28no—
64C4⋊C4×D61C2×C4116yescomplemented
65C2×#11758198—24no—
66C2×#11758198—44no—
67#14519689—28no—
68#14519684—24no—
69C23×C4×C61—21no—
70C23×C4×C61—21no—
71C23×C4×C61C2311yescharacteristic
72C22×C8×C61—22no—
73C22×C8×C61—22no—
74C22×C8×C61—22no—
75C22×C8×C61C2312yes—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61C2×C414yes—
82C2×C4⋊C4×C61C2312yes—
83C2×C4⋊C4×C61C2312yes—
84C24×C61C2×D411yescharacteristic
85C23×D61—22no—
86C23×D61—22no—
87C23×D61—22no—
88C23×D61—22no—
89C23×D61C22⋊C414yes—
90C23×D61SD1614yes—
91C23×D61Q1614yes—
92C23×D61C2×D412yes—
93C22×C4×C61—24no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—42no—
97C22×C4×C61—42no—
98C22×C4×C61—42no—
99C22×C4×C61—42no—
100C22×C4×C61—42no—
101C22×C4×C61C22×C412yes—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D412yes—
104C22×C4×C61#4112yes—
105C22×C4×C61#4112yes—
106C22×C4×C61#4112yes—
107C22×C4×C61#4112yes—
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—22no—
110C22×Dic61—22no—
111C22×Dic61—24no—
112C22×Dic61—22no—
113C22×Dic61—22no—
114C22×Dic61—42no—
115C22×Dic61—42no—
116C22×Dic61—42no—
117C22×Dic61—42no—
118C22×Dic61—42no—
119C22×Dic61C22⋊C414yes—
120C22×Dic61C2×D412yes—
121C2×C4×D61—28no—
122C2×C4×D61—28no—
123C2×C4×D61—28no—
124C2×C4×D61—28no—
125C2×C4×D61—24no—
126C2×C4×D61—24no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—216no—
132C2×C4×D61—48no—
133C2×C4×D61—48no—
134C2×C4×D61—432no—
135C2×C4×D61C22⋊C418yes—
136C2×C4×D61C22×C414yes—
137C2×C4×D61C2×D414yes—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—44no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—42no—
147C4⋊C4×C61C22×C414yes—
148#11758198—48no—
149C4⋊Dic61—24no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C4⋊Dic61—42no—
154C4⋊Dic61C22×C414yes—
155#11758196—216no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—44no—
160C23×C61C2×C22⋊C412yes—
161C23×C61#12312yes—
162C23×C61#12312yes—
163C23×C61C2×SD1612yes—
164C23×C61C2×Q1612yes—
165C23×C61AGL(1,ℤ/8)12yes—
166C23×C61AGL(1,ℤ/8)12yes—
167C23×C61C22×D411yescharacteristic
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—22no—
171C22×D61—22no—
172C22×D61—28no—
173C22×D61—28no—
174C22×D61—28no—
175C22×D61—28no—
176C22×D61—28no—
177C22×D61—28no—
178C22×D61#103116yes—
179C22×D61C2×C22⋊C414yes—
180C22×D61C2×SD1614yes—
181C22×D61C2×Q1614yes—
182C2×C4×C61—24no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—48no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—84no—
190C2×C4×C61C2×C22⋊C412yes—
191C2×C4×C61#11714yes—
192C2×C4×C61#11714yes—
193C2×C4×C61#12314yes—
194C2×C4×C61#12314yes—
195C2×C4×C61#12312yes—
196C2×C4×C61#12312yes—
197C2×C4×C61C23×C411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—28no—
206C2×Dic61—24no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—48no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—84no—
214C2×Dic61C2×C22⋊C414yes—
215C4×D61—24no—
216C4×D61—24no—
217C4×D61—24no—
218C4×D61—24no—
219C4×D61—216no—
220C4×D61—432no—
221C8×C61—42no—
222#92418—42no—
223C2×#1824—611nomaximal, Sylow
224C22×C61#38114yes—
225C22×C61C2×#10314yes—
226C22×C61#41814yes—
227C22×C61#48114yes—
228C22×C61#48114yes—
229C22×C61#48414yes—
230C22×C61#48414yes—
231C22×C61C22×C22⋊C411yescharacteristic
232C22×C61C2×#12311yescharacteristic
233C22×C61C2×#12311yescharacteristic
234C22×C61C22×SD1611yescharacteristic
235C22×C61C22×Q1611yescharacteristic
236C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C2×D61—24no—
239C2×D61—24no—
240C2×D61—24no—
241C2×D61—24no—
242C2×D61—232no—
243C2×D61—24no—
244C2×D61—24no—
245C2×D61C2×#10318yes—
246C4×C61—44no—
247C4×C61—82no—
248C4×C61#38114yes—
249C4×C61C2×#11711yescharacteristic
250C4×C61C2×#11711yescharacteristic
251C4×C61C2×#12311yescharacteristic
252C4×C61C2×#12311yescharacteristic
253Dic61—24no—
254Dic61—44no—
255Dic61—82no—
256C23×C4⋊C4—611no—
257C22×C22⋊C8—611no—
258C2×#380—611no—
259#1824—618no—
260C2×#340—611no—
261C2×#340—611no—
262C2×#340—611no—
263C2×#340D6111yescharacteristic, complemented
264C2×C61#182418yescomplemented
265C2×C61C2×#38111yescharacteristic
266C2×C61C22×#10311yescharacteristic
267C2×C61C2×#41811yescharacteristic
268C2×C61C2×#48111yescharacteristic
269C2×C61C2×#48111yescharacteristic
270C2×C61C2×#48411yescharacteristic
271C2×C61C2×#48411yescharacteristic
272D61—28no—
273C24×C4—611no—
274C24×C4—1221no—
275C23×C8—611no—
276C23×C8C2×D6111yescharacteristic
277C22×C4⋊C4—611no—
278C22×C4⋊C4—618no—
279C22×C4⋊C4—614no—
280C22×C4⋊C4—611no—
281C22×C4⋊C4—1222no—
282C22×C4⋊C4—1224no—
283C22×C4⋊C4—1221no—
284C22×C4⋊C4C2×D6111yescharacteristic
285C22×C4⋊C4C2×D6111yescharacteristic
286C22×C22⋊C4—1221no—
287C2×C22⋊C8—614no—
288C2×C22⋊C8—1224no—
289#380—614no—
290#380—1222no—
291C2×#95—1221no—
292#340—614no—
293#340—614no—
294#340—614no—
295#340C2×D6114yescomplemented
296C61C2×#182411yescharacteristic, complemented, Sylow
297C25—611no—
298C23×C4—21no—
299C23×C4—21no—
300C23×C4—611no—
301C23×C4—614no—
302C23×C4—612no—
303C23×C4—614no—
304C23×C4—612no—
305C23×C4—1224no—
306C23×C4—1228no—
307C23×C4—1222no—
308C23×C4—1222no—
309C23×C4—1222no—
310C23×C4—1222no—
311C23×C4—1221no—
312C23×C4—1221no—
313C23×C4—2444no—
314C23×C4—2444no—
315C23×C4C22×D6111yescharacteristic
316C22×C8—22no—
317C22×C8—22no—
318C22×C8—22no—
319C22×C8—612no—
320C22×C8—1222no—
321C22×C8—1222no—
322C22×C8—1222no—
323C22×C8C22×D6112yes—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—614no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4—6116no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—1228no—
334C2×C4⋊C4—12216no—
335C2×C4⋊C4—1224no—
336C2×C4⋊C4—1228no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4C4×D6114yes—
343C2×C4⋊C4C22×D6112yes—
344C2×C4⋊C4C22×D6112yes—
345C2×C22⋊C4—1224no—
346C2×C22⋊C4—2444no—
347C22⋊C8—1228no—
348#95—1224no—
349C24—612no—
350C24—614no—
351C24—614no—
352C24—614no—
353C24—1222no—
354C24—1222no—
355C24—1222no—
356C24—1222no—
357C24D4×D6111yescharacteristic
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—612no—
367C22×C4—614no—
368C22×C4—614no—
369C22×C4—614no—
370C22×C4—618no—
371C22×C4—12216no—
372C22×C4—1222no—
373C22×C4—1222no—
374C22×C4—1222no—
375C22×C4—1222no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1228no—
379C22×C4—1228no—
380C22×C4—1224no—
381C22×C4—1228no—
382C22×C4—1224no—
383C22×C4—1228no—
384C22×C4—1222no—
385C22×C4—1224no—
386C22×C4—1222no—
387C22×C4—24432no—
388C22×C4—2448no—
389C22×C4—2448no—
390C22×C4—2442no—
391C22×C4—2442no—
392C22×C4—2442no—
393C22×C4—2442no—
394C22×C4—2442no—
395C22×C4C2×C4×D6112yes—
396C22×C4#1175822212yes—
397C22×C4#1175822212yes—
398C22×C4D4×D6112yes—
399C22×C4D4×D6111yescharacteristic
400C22×C4#1175822412yes—
401C22×C4#1175822612yes—
402C22×C4C23×D6111yescharacteristic
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—44no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—2444no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—614no—
417C4⋊C4—12216no—
418C4⋊C4—1222no—
419C4⋊C4—1224no—
420C4⋊C4—1222no—
421C4⋊C4—1222no—
422C4⋊C4—2442no—
423C4⋊C4C2×C4×D6114yes—
424C22⋊C4—2448no—
425C23—614no—
426C23—6116no—
427C23—614no—
428C23—614no—
429C23—1222no—
430C23—1222no—
431C23—1228no—
432C23—1222no—
433C23—1222no—
434C23—1228no—
435C23—1228no—
436C23—1228no—
437C23—1228no—
438C23—1228no—
439C23C22⋊C4×D6112yes—
440C23#1451975912yes—
441C23#1451976212yes—
442C23SD16×D6112yes—
443C23Q16×D6112yes—
444C23#1451980112yes—
445C23#1451980412yes—
446C23C2×D4×D6111yescharacteristic
447C2×C4—24no—
448C2×C4—48no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—84no—
455C2×C4—614no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1228no—
459C2×C4—1224no—
460C2×C4—1224no—
461C2×C4—1224no—
462C2×C4—1224no—
463C2×C4—12216no—
464C2×C4—1224no—
465C2×C4—4884no—
466C2×C4—24432no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2448no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4C22⋊C4×D6112yes—
474C2×C4#1451973414yes—
475C2×C4#1451973514yes—
476C2×C4#1451975914yes—
477C2×C4#1451975912yes—
478C2×C4#1451976214yes—
479C2×C4#1451976212yes—
480C2×C4C22×C4×D6111yescharacteristic
481C2×C4C2×D4×D6111yescharacteristic
482C2×C4C2×#1175822411yescharacteristic
483C2×C4C2×#1175822211yescharacteristic
484C2×C4C2×#1175822211yescharacteristic
485C2×C4C2×#1175822611yescharacteristic
486C8—42no—
487C8—2442no—
488C22—618no—
489C22—1224no—
490C22—12232no—
491C22—1224no—
492C22—1224no—
493C22—1224no—
494C22—1224no—
495C22—1224no—
496C22#1683962414yes—
497C22#103×D6114yes—
498C22#1683976514yes—
499C22#1684007914yes—
500C22#1684008214yes—
501C22#1684009614yes—
502C22#1684009914yes—
503C22C2×C22⋊C4×D6111yescharacteristic
504C22C2×#1451975911yescharacteristic
505C22C2×#1451976211yescharacteristic
506C22C2×SD16×D6111yescharacteristic
507C22C2×Q16×D6111yescharacteristic
508C22C2×#1451980111yescharacteristic
509C22C2×#1451980411yescharacteristic
510C4—44no—
511C4—82no—
512C4—1224no—
513C4—4882no—
514C4—2444no—
515C4#1683962414yes—
516C4C2×#1451973411yescharacteristic
517C4C2×#1451973511yescharacteristic
518C4C2×#1451975911yescharacteristic
519C4C2×#1451976211yescharacteristic
520C2—1228no—
521C2#2144590818yesdirect product factor
522C2C2×#1683962411yescharacteristic
523C2C2×#103×D6111yescharacteristic
524C2C2×#1683976511yescharacteristic
525C2C2×#1684007911yescharacteristic
526C2C2×#1684008211yescharacteristic
527C2C2×#1684009611yescharacteristic
528C2C2×#1684009911yescharacteristic
subgroupG/H
centerC24D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#340C2
socleC24×C61C2×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenterC2×#340D61
hypocenterC61C2×#1824
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
4. #21445908C2direct (N × H)
5. C2×#16839525C2semidirect
6. C2×#16839524C2semidirect
7. C2×#16839524C2semidirect
8. C2×C61×#340C2semidirect
17. C2×C4⋊C4×D61C4semidirect
25. #16839525C22semidirect
26. #16839524C22semidirect
27. #16839524C22semidirect
31. C61×#340C22semidirect
64. C4⋊C4×D61C2×C4semidirect
263. C2×#340D61semidirect
264. C2×C61#1824semidirect
295. #340C2×D61semidirect
296. C61C2×#1824semidirect
521. C2#21445908direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D612
C22×#145196892
C2×#168396192
#214459082
C2×#168395252
C2×#168395242
C2×C61×#3402
C2×#182461