Finite Group Databasebeta

← C2×#21461999 #30616459

Subgroups — C2×#21461999 #30616459

Order 15616.

1294 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D61⋊(C4⋊C8)C211yesmaximal, characteristic
2C2×C4×C4⋊#6923C211yesmaximal, characteristic
3#21461999C218yesmaximal, direct product factor
4C2×#16840635C212yesmaximal
5C2×#16840635C212yesmaximal
6C2×#125×D61C211yesmaximal, characteristic
7C22×D61⋊C8C2212yes—
8C22×C4⋊#6923C2212yes—
9C22×C4×#6923—21no—
10C2×#14519844—21no—
11D61⋊(C4⋊C8)—22no—
12D61⋊(C4⋊C8)C2214yes—
13C2×C4⋊#92420—21no—
14C4×C4⋊#6923—22no—
15C4×C4⋊#6923C2214yes—
16C2×(C2×#6923)⋊C4—21no—
17#16840635C2218yes—
18#16840635C2218yes—
19C2×#14519769C411yescharacteristic, complemented
20C2×#14519768C411yescharacteristic, complemented
21C2×#14519768C411yescharacteristic, complemented
22C2×#14519767—21no—
23C2×#14519767—21no—
24#125×D61—22no—
25#125×D61—22no—
26#125×D61C414yescomplemented
27#125×D61C2214yes—
28C2×C4⋊C4×D61—21no—
29C2×C4⋊C4×D61—21no—
30C2×C4⋊C4×D61C2211yescharacteristic
31C2×C4⋊C4×D61C2211yescharacteristic
32C2×C42×D61C2211yescharacteristic
33C2×#125×C61C411yescharacteristic, complemented
34C23×#6923—22no—
35C2×D61⋊C8—28no—
36C2×D61⋊C8—22no—
37C2×D61⋊C8C2314yes—
38C22×#92420—22no—
39C2×C4⋊#6923—22no—
40C2×C4⋊#6923—24no—
41C2×C4⋊#6923Q818yes—
42C2×C4⋊#6923C2314yes—
43C2×C4×#6923—24no—
44C2×C4×#6923—44no—
45#14519844—24no—
46C4⋊#92420—24no—
47(C2×#6923)⋊C4—24no—
48C22×C4×D61—21no—
49C22×C4×D61—21no—
50C22×C4×D61D411yescharacteristic
51C22×C4×D61D411yescharacteristic
52C22×C4×D61C2311yescharacteristic
53#14519769—22no—
54#14519769C2×C414yescomplemented
55#14519768—22no—
56#14519768—22no—
57#14519768C2×C414yescomplemented
58#14519768C2×C414yescomplemented
59#14519767—24no—
60#14519767—24no—
61#14519767—24no—
62#14519767—24no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×C4⋊Dic61C2×C411yescharacteristic
67C2×#11758196—21no—
68C2×#11758196—21no—
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C4⋊C4×D61—24no—
74C4⋊C4×D61—24no—
75C4⋊C4×D61—24no—
76C4⋊C4×D61—24no—
77C4⋊C4×D61—21no—
78C4⋊C4×D61—21no—
79C4⋊C4×D61—21no—
80C4⋊C4×D61—21no—
81C4⋊C4×D61C2×C412yes—
82C4⋊C4×D61C2×C412yes—
83C4⋊C4×D61C2312yes—
84C4⋊C4×D61C2312yes—
85C2×C4×Dic61C2×C411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C42×D61C2×C412yes—
89C42×D61D412yes—
90C42×D61D412yes—
91C42×D61C2312yes—
92#125×C61—22no—
93#125×C61C2×C414yescomplemented
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2×C411yescharacteristic
97C2×C4⋊C4×C61C2×C411yescharacteristic
98C2×C42×C61C2×C411yescharacteristic
99C23×D61C2×D411yescharacteristic
100C22×#6923—28no—
101C22×#6923—24no—
102C22×#6923—44no—
103C22×#6923—44no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61C22⋊C411yescharacteristic
107C22×C4×C61C22⋊C411yescharacteristic
108C22×C4×C61C22×C411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—21no—
111C22×Dic61C22⋊C411yescharacteristic
112C22×Dic61C22⋊C411yescharacteristic
113C22×Dic61C2×D411yescharacteristic
114C2×C4×D61—24no—
115C2×C4×D61—22no—
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—44no—
122C2×C4×D61—42no—
123C2×C4×D61—42no—
124C2×C4×D61C22⋊C412yes—
125C2×C4×D61C22⋊C412yes—
126C2×C4×D61C22×C411yescharacteristic
127C2×C4×D61C2×D412yes—
128C2×C4×D61C2×D412yes—
129C2×C4×D61C2×D411yescharacteristic
130C2×C4×D61C2×D411yescharacteristic
131C2×C4×D61C2×Q814yes—
132C2×C4×D61#4114yes—
133C2×C4×D61C2411yescharacteristic
134C2×#92420—24no—
135C2×#92420—44no—
136C4⋊#6923—28no—
137C4⋊#6923—24no—
138C4⋊#6923C2×Q818yes—
139C4×#6923—48no—
140D61⋊C8—28no—
141D61⋊C8—44no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61C22×C412yes—
147C4⋊C4×C61C22×C412yes—
148C42×C61C22⋊C412yes—
149C42×C61C22×C412yes—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—24no—
153C4⋊Dic61—24no—
154C4⋊Dic61C22×C412yes—
155C4⋊Dic61C22×C412yes—
156#11758196—24no—
157#11758196—24no—
158#11758196—24no—
159#11758196—24no—
160#11758196—22no—
161#11758196—22no—
162#11758196—22no—
163#11758196—22no—
164C4×Dic61C22⋊C412yes—
165C4×Dic61C22⋊C412yes—
166C4×Dic61C22⋊C412yes—
167C4×Dic61C22×C412yes—
168C4×Dic61C2×D412yes—
169C4×Dic61C2×D412yes—
170C23×C61C2×C22⋊C411yescharacteristic
171C22×D61C2×C22⋊C411yescharacteristic
172C22×D61C22⋊Q814yes—
173C22×D61#13514yes—
174C22×D61AGL(1,ℤ/8)12yes—
175C22×D61#13712yes—
176C22×D61C22×D411yescharacteristic
177C2×#6923—28no—
178C2×#6923—44no—
179C2×#6923—44no—
180C2×#6923—416no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61C2×C22⋊C412yes—
188C2×C4×C61C2×C22⋊C412yes—
189C2×C4×C61C2×C22⋊C411yescharacteristic
190C2×C4×C61C4×Q814yes—
191C2×C4×C61C23×C411yescharacteristic
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61C2×C22⋊C412yes—
199C2×Dic61C2×C22⋊C412yes—
200C2×Dic61C2×C22⋊C411yescharacteristic
201C2×Dic61#12412yes—
202C2×Dic61C4⋊Q812yes—
203C2×Dic61C22×D411yescharacteristic
204C4×D61—24no—
205C4×D61—22no—
206C4×D61—22no—
207C4×D61—24no—
208C4×D61—24no—
209C4×D61—44no—
210C4×D61—42no—
211C4×D61—42no—
212C4×D61—44no—
213C4×D61—44no—
214C4×D61C4×Q814yes—
215C4×D61C22⋊Q814yes—
216C4×D61#12412yes—
217C4×D61C4⋊Q812yes—
218C4×D61C22×Q812yes—
219C4×D61C2×#4112yes—
220#92420—44no—
221C2×#1989—611nomaximal, Sylow
222C22×C61#39112yes—
223C22×C61#41612yes—
224C22×C61#41712yes—
225C22×C61C22×C22⋊C411yescharacteristic
226C2×D61#39112yes—
227C2×D61#41612yes—
228C2×D61#41712yes—
229C2×D61#47818yes—
230C2×D61#47918yes—
231C2×D61C2×C22⋊Q812yes—
232C2×D61C2×#13512yes—
233C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
234C2×D61C2×#13711yescharacteristic
235#6923—48no—
236C4×C61—22no—
237C4×C61—22no—
238C4×C61—42no—
239C4×C61—42no—
240C4×C61#39112yes—
241C4×C61C2×C4×Q812yes—
242Dic61—22no—
243Dic61—22no—
244Dic61—42no—
245Dic61—42no—
246Dic61#39112yes—
247Dic61C2×#12411yescharacteristic
248Dic61C2×C4⋊Q811yescharacteristic
249C22×#125—611no—
250C22×C4⋊C8—611no—
251C2×C4×C4⋊C4—611no—
252#1989—618no—
253C2×#340—612no—
254C2×#340—612no—
255C2×C61#198914yescomplemented
256C2×C61C2×#39111yescharacteristic
257C2×C61C2×#41611yescharacteristic
258C2×C61C2×#41711yescharacteristic
259D61#198914yescomplemented
260D61C2×#47812yes—
261D61C2×#47912yes—
262C23×C8—612no—
263C2×#125—614no—
264C2×#125—614no—
265C2×#125—611no—
266C2×#125—611no—
267C2×#125—611no—
268C2×#125—1221no—
269C2×#125—1221no—
270C2×#125—1222no—
271C2×#125—1222no—
272C2×#125#692311yescharacteristic, complemented
273C22×C4⋊C4—612no—
274C22×C4⋊C4—611no—
275C22×C4⋊C4—611no—
276C22×C4⋊C4—1221no—
277C22×C4⋊C4—1221no—
278C22×C42—611no—
279C22×C42—1221no—
280C2×C4⋊C8—614no—
281C2×C4⋊C8—1221no—
282C2×C4⋊C8—1222no—
283C2×C4⋊C8—1221no—
284C4×C4⋊C4—614no—
285C4×C4⋊C4—1222no—
286C2×#95—1221no—
287#340—618no—
288#340—618no—
289C61C2×#198911yescharacteristic, complemented, Sylow
290C23×C4—611no—
291C23×C4—611no—
292C23×C4—611no—
293C23×C4—1221no—
294C23×C4—1221no—
295C23×C4—1222no—
296C22×C8—614no—
297C22×C8—1222no—
298C22×C8—1228no—
299C22×C8—1222no—
300#125—22no—
301#125—614no—
302#125—614no—
303#125—614no—
304#125—1224no—
305#125—1224no—
306#125—1224no—
307#125—1224no—
308#125—1222no—
309#125—1222no—
310#125—1222no—
311#125C2×#692314yescomplemented
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—618no—
315C2×C4⋊C4—614no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—612no—
319C2×C4⋊C4—612no—
320C2×C4⋊C4—612no—
321C2×C4⋊C4—612no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4—1221no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4—1224no—
327C2×C4⋊C4—1224no—
328C2×C4⋊C4—1224no—
329C2×C4⋊C4—1224no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1224no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4C2×#692311yescharacteristic
341C2×C4⋊C4C2×#692311yescharacteristic
342C2×C42—611no—
343C2×C42—611no—
344C2×C42—611no—
345C2×C42—612no—
346C2×C42—612no—
347C2×C42—612no—
348C2×C42—612no—
349C2×C42—1224no—
350C2×C42—2444no—
351C2×C42C2×#692311yescharacteristic
352C4⋊C8—1224no—
353C4⋊C8—1224no—
354#95—1224no—
355C24—611no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—611no—
359C22×C4—611no—
360C22×C4—611no—
361C22×C4—614no—
362C22×C4—614no—
363C22×C4—611no—
364C22×C4—611no—
365C22×C4—611no—
366C22×C4—611no—
367C22×C4—612no—
368C22×C4—612no—
369C22×C4—612no—
370C22×C4—612no—
371C22×C4—1222no—
372C22×C4—1224no—
373C22×C4—1222no—
374C22×C4—1222no—
375C22×C4—1222no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1224no—
379C22×C4—1228no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—2444no—
383C22×C4—2442no—
384C22×C4—2442no—
385C22×C4—2444no—
386C22×C4—2444no—
387C22×C4C22⋊#692311yescharacteristic
388C22×C4C22⋊#692311yescharacteristic
389C22×C4C22×#692311yescharacteristic
390C2×C8—1224no—
391C2×C8—1228no—
392C2×C8—2444no—
393C2×C8—2444no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—618no—
399C4⋊C4—612no—
400C4⋊C4—612no—
401C4⋊C4—1222no—
402C4⋊C4—1222no—
403C4⋊C4—1222no—
404C4⋊C4—1222no—
405C4⋊C4—1224no—
406C4⋊C4—1224no—
407C4⋊C4—1224no—
408C4⋊C4—1224no—
409C4⋊C4—1228no—
410C4⋊C4—1224no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4—1224no—
414C4⋊C4—1224no—
415C4⋊C4C22×#692312yes—
416C4⋊C4C22×#692312yes—
417C42—612no—
418C42—612no—
419C42—612no—
420C42—612no—
421C42—612no—
422C42—612no—
423C42—2448no—
424C42C22⋊#692312yes—
425C42C22×#692312yes—
426C23—614no—
427C23—614no—
428C23—612no—
429C23—612no—
430C23—611no—
431C23—611no—
432C23C2×C22⋊#692311yescharacteristic
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—612no—
440C2×C4—612no—
441C2×C4—611no—
442C2×C4—611no—
443C2×C4—614no—
444C2×C4—612no—
445C2×C4—612no—
446C2×C4—614no—
447C2×C4—612no—
448C2×C4—612no—
449C2×C4—612no—
450C2×C4—612no—
451C2×C4—1222no—
452C2×C4—1224no—
453C2×C4—1224no—
454C2×C4—1222no—
455C2×C4—1224no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1228no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—2444no—
462C2×C4—2444no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2444no—
466C2×C4—2444no—
467C2×C4—2444no—
468C2×C4—24416no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4C2×C22⋊#692312yes—
472C2×C4C2×C22⋊#692312yes—
473C2×C4C2×C22⋊#692311yescharacteristic
474C2×C4Q8×#692314yes—
475C2×C4C23×#692311yescharacteristic
476C8—2444no—
477C22—612no—
478C22—612no—
479C22—612no—
480C22—612no—
481C22—611no—
482C22—611no—
483C22—612no—
484C22—618no—
485C22—618no—
486C22(C2×#6923)⋊Q812yes—
487C22#1684075212yes—
488C22#1684075312yes—
489C22C22×C22⋊#692311yescharacteristic
490C4—22no—
491C4—22no—
492C4—42no—
493C4—42no—
494C4—611no—
495C4—611no—
496C4—612no—
497C4—1222no—
498C4—1222no—
499C4—2448no—
500C4—2442no—
501C4—2442no—
502C4(C2×#6923)⋊Q812yes—
503C4C2×Q8×#692312yes—
504C2—612no—
505C2—612no—
506C2—614no—
507C2#2146199914yesdirect product factor
508C2C2×(C2×#6923)⋊Q811yescharacteristic
509C2C2×#1684075211yescharacteristic
510C2C2×#1684075311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#125×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenterC2×#125#6923
hypocenterC61C2×#1989
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
3. #21461999C2direct (N × H)
19. C2×#14519769C4semidirect
20. C2×#14519768C4semidirect
21. C2×#14519768C4semidirect
26. #125×D61C4semidirect
33. C2×#125×C61C4semidirect
54. #14519769C2×C4semidirect
57. #14519768C2×C4semidirect
58. #14519768C2×C4semidirect
93. #125×C61C2×C4semidirect
255. C2×C61#1989semidirect
259. D61#1989semidirect
272. C2×#125#6923semidirect
289. C61C2×#1989semidirect
311. #125C2×#6923semidirect
507. C2#21461999direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D61⋊(C4⋊C8)2
C2×C4×C4⋊#69232
#214619992
C2×#168406352
C2×#125×D612
C2×#198961