Finite Group Databasebeta

← C2×#21461723 #30615100

Subgroups — C2×#21461723 #30615100

Order 15616.

1878 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D61⋊(C22⋊C8)C211yesmaximal, characteristic
2C2×#16840704C211yesmaximal, characteristic
3#21461723C218yesmaximal, direct product factor
4C2×#16840635C214yesmaximal, complemented
5C22×C4⋊C4×D61C211yesmaximal, characteristic
6C22×D61⋊C8C2212yes—
7C22×C4⋊#6923C2212yes—
8C22×C22⋊#6923—21no—
9C2×#14519834—21no—
10D61⋊(C22⋊C8)—22no—
11D61⋊(C22⋊C8)C2214yes—
12C2×C22⋊#92420—21no—
13#16840704—22no—
14#16840704C2214yes—
15C2×(C2×#6923)⋊C4—21no—
16#16840635C22116yescomplemented
17C23×C4×D61—21no—
18C23×C4×D61C2211yescharacteristic
19C22×C4⋊Dic61C411yescharacteristic, complemented
20C22×#11758196—21no—
21C2×C4⋊C4×D61—22no—
22C2×C4⋊C4×D61—21no—
23C2×C4⋊C4×D61—22no—
24C2×C4⋊C4×D61—21no—
25C2×C4⋊C4×D61C412yescomplemented
26C2×C4⋊C4×D61C414yescomplemented
27C2×C4⋊C4×D61C2212yes—
28C2×C4⋊C4×D61C2214yes—
29C22×C4⋊C4×C61C411yescharacteristic, complemented
30C23×#6923—22no—
31C2×D61⋊C8—28no—
32C2×D61⋊C8—22no—
33C2×D61⋊C8C2314yes—
34C22×#92420—22no—
35C2×C4⋊#6923—28no—
36C2×C4⋊#6923—22no—
37C2×C4⋊#6923C2314yes—
38C2×C22⋊#6923—24no—
39C2×C22⋊#6923—44no—
40#14519834—24no—
41C22⋊#92420—24no—
42(C2×#6923)⋊C4—24no—
43C24×D61D411yescharacteristic
44C23×Dic61—21no—
45C23×Dic61D411yescharacteristic
46C22×C4×D61—28no—
47C22×C4×D61—24no—
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—24no—
51C22×C4×D61—48no—
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C412yes—
54C22×C4×D61D412yes—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D412yes—
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2312yes—
59C22×C4×D61C2311yescharacteristic
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2×C412yes—
63C2×C4⋊Dic61C2×C414yescomplemented
64C2×#11758196—24no—
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×#11758196—24no—
68C4⋊C4×D61—24no—
69C4⋊C4×D61—24no—
70C4⋊C4×D61—24no—
71C4⋊C4×D61—22no—
72C4⋊C4×D61—24no—
73C4⋊C4×D61—22no—
74C4⋊C4×D61—24no—
75C4⋊C4×D61C2×C414yes—
76C4⋊C4×D61C2×C418yescomplemented
77C4⋊C4×D61C2314yes—
78C23×C4×C61—21no—
79C23×C4×C61C2×C411yescharacteristic
80C2×C4⋊C4×C61—22no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61C2×C412yes—
83C2×C4⋊C4×C61C2×C414yescomplemented
84C24×C61C22⋊C411yescharacteristic
85C23×D61—24no—
86C23×D61—24no—
87C23×D61C22⋊C411yescharacteristic
88C23×D61C22⋊C412yes—
89C23×D61SD1614yes—
90C23×D61Q1614yes—
91C23×D61C2×D412yes—
92C23×D61C2×D411yescharacteristic
93C22×#6923—24no—
94C22×#6923—44no—
95C22×#6923—44no—
96C22×#6923—44no—
97C22×C4×C61—24no—
98C22×C4×C61—22no—
99C22×C4×C61—24no—
100C22×C4×C61—44no—
101C22×C4×C61C22⋊C411yescharacteristic
102C22×C4×C61C22⋊C412yes—
103C22×C4×C61C22×C412yes—
104C22×C4×C61C22×C411yescharacteristic
105C22×Dic61—24no—
106C22×Dic61—22no—
107C22×Dic61—24no—
108C22×Dic61—44no—
109C22×Dic61C22⋊C411yescharacteristic
110C22×Dic61C22⋊C412yes—
111C22×Dic61C2×D412yes—
112C22×Dic61C2×D411yescharacteristic
113C2×C4×D61—24no—
114C2×C4×D61—24no—
115C2×C4×D61—24no—
116C2×C4×D61—28no—
117C2×C4×D61—28no—
118C2×C4×D61—28no—
119C2×C4×D61—28no—
120C2×C4×D61—24no—
121C2×C4×D61—28no—
122C2×C4×D61—24no—
123C2×C4×D61—48no—
124C2×C4×D61—416no—
125C2×C4×D61—416no—
126C2×C4×D61—48no—
127C2×C4×D61C22⋊C412yes—
128C2×C4×D61C22⋊C412yes—
129C2×C4×D61C22⋊C412yes—
130C2×C4×D61C22×C412yes—
131C2×C4×D61C22×C411yescharacteristic
132C2×C4×D61C2×D412yes—
133C2×C4×D61C2×D412yes—
134C2×C4×D61C2×D411yescharacteristic
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61#4118yes—
137C2×C4×D61C2411yescharacteristic
138C22⋊#6923—48no—
139C2×#92420—24no—
140C2×#92420—44no—
141C4⋊#6923—28no—
142C4⋊#6923—44no—
143D61⋊C8—28no—
144D61⋊C8—44no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61C22×C414yes—
148C4⋊Dic61—22no—
149C4⋊Dic61—24no—
150C4⋊Dic61C22×C414yes—
151#11758196—24no—
152#11758196—28no—
153#11758196—24no—
154C23×C61—22no—
155C23×C61—22no—
156C23×C61#10314yes—
157C23×C61C2×C22⋊C412yes—
158C23×C61C2×C22⋊C411yescharacteristic
159C22×D61—22no—
160C22×D61—22no—
161C22×D61—24no—
162C22×D61—24no—
163C22×D61—24no—
164C22×D61—24no—
165C22×D61—216no—
166C22×D61—216no—
167C22×D61#10314yes—
168C22×D61#10318yes—
169C22×D61C2×C22⋊C412yes—
170C22×D61C2×C22⋊C411yescharacteristic
171C22×D61#12314yes—
172C22×D61C2×SD1614yes—
173C22×D61C2×SD1612yes—
174C22×D61C2×Q1614yes—
175C22×D61C2×Q1612yes—
176C22×D61AGL(1,ℤ/8)14yes—
177C22×D61C22×D411yescharacteristic
178C2×#6923—44no—
179C2×#6923—44no—
180C2×#6923—44no—
181C2×#6923—88no—
182C2×C4×C61—22no—
183C2×C4×C61—24no—
184C2×C4×C61—24no—
185C2×C4×C61—24no—
186C2×C4×C61—48no—
187C2×C4×C61—44no—
188C2×C4×C61C2×C22⋊C412yes—
189C2×C4×C61C2×C22⋊C411yescharacteristic
190C2×C4×C61#11714yes—
191C2×C4×C61C23×C411yescharacteristic
192C2×Dic61—22no—
193C2×Dic61—24no—
194C2×Dic61—24no—
195C2×Dic61—24no—
196C2×Dic61—48no—
197C2×Dic61—44no—
198C2×Dic61C2×C22⋊C412yes—
199C2×Dic61C2×C22⋊C411yescharacteristic
200C2×Dic61#12314yes—
201C2×Dic61C22×D411yescharacteristic
202C4×D61—24no—
203C4×D61—24no—
204C4×D61—24no—
205C4×D61—24no—
206C4×D61—24no—
207C4×D61—24no—
208C4×D61—416no—
209C4×D61—48no—
210C4×D61—48no—
211C4×D61#11714yes—
212C4×D61#12314yes—
213C4×D61#12314yes—
214C4×D61C2×#4114yes—
215#92420—44no—
216C2×#1824—611nomaximal, Sylow
217C22×C61—28no—
218C22×C61—22no—
219C22×C61—22no—
220C22×C61#38112yes—
221C22×C61C2×#10314yes—
222C22×C61C2×#10312yes—
223C22×C61#41812yes—
224C22×C61C22×C22⋊C411yescharacteristic
225C2×D61—28no—
226C2×D61—22no—
227C2×D61—22no—
228C2×D61—28no—
229C2×D61—28no—
230C2×D61—216no—
231C2×D61#38112yes—
232C2×D61C2×#10314yes—
233C2×D61C2×#10312yes—
234C2×D61#41812yes—
235C2×D61#48118yes—
236C2×D61#48418yes—
237C2×D61C2×#12312yes—
238C2×D61C22×SD1611yescharacteristic
239C2×D61C22×Q1611yescharacteristic
240C2×D61C2×AGL(1,ℤ/8)12yes—
241#6923—84no—
242C4×C61—22no—
243C4×C61—44no—
244C4×C61#38112yes—
245C4×C61C2×#11712yes—
246Dic61—22no—
247Dic61—44no—
248Dic61#38112yes—
249Dic61C2×#12312yes—
250C23×C4⋊C4—611no—
251C22×C22⋊C8—611no—
252C2×#380—611no—
253#1824—618no—
254C2×#340—614no—
255C2×C61—24no—
256C2×C61#182414yescomplemented
257C2×C61C2×#38111yescharacteristic
258C2×C61C22×#10311yescharacteristic
259C2×C61C2×#41811yescharacteristic
260D61—24no—
261D61#182414yescomplemented
262D61C2×#48112yes—
263D61C2×#48412yes—
264C24×C4—611no—
265C24×C4—1221no—
266C23×C8—612no—
267C22×C4⋊C4—611no—
268C22×C4⋊C4—612no—
269C22×C4⋊C4—612no—
270C22×C4⋊C4—614no—
271C22×C4⋊C4—614no—
272C22×C4⋊C4—612no—
273C22×C4⋊C4—1221no—
274C22×C4⋊C4—1222no—
275C22×C4⋊C4—1222no—
276C22×C4⋊C4—1221no—
277C22×C4⋊C4—1221no—
278C22×C4⋊C4#692311yescharacteristic, complemented
279C22×C22⋊C4—1221no—
280C2×C22⋊C8—614no—
281C2×C22⋊C8—1221no—
282C2×C22⋊C8—1222no—
283C2×C22⋊C8—1221no—
284#380—614no—
285#380—1222no—
286C2×#95—1221no—
287#340—6116no—
288C61C2×#182411yescharacteristic, complemented, Sylow
289C25—611no—
290C23×C4—21no—
291C23×C4—611no—
292C23×C4—611no—
293C23×C4—611no—
294C23×C4—612no—
295C23×C4—611no—
296C23×C4—612no—
297C23×C4—612no—
298C23×C4—611no—
299C23×C4—612no—
300C23×C4—1228no—
301C23×C4—1224no—
302C23×C4—1224no—
303C23×C4—1222no—
304C23×C4—1222no—
305C23×C4—1221no—
306C23×C4—1222no—
307C23×C4—2448no—
308C23×C4C2×#692311yescharacteristic
309C22×C8—614no—
310C22×C8—1222no—
311C22×C8—1228no—
312C22×C8—1222no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—612no—
316C2×C4⋊C4—614no—
317C2×C4⋊C4—614no—
318C2×C4⋊C4—614no—
319C2×C4⋊C4—618no—
320C2×C4⋊C4—614no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4—1222no—
323C2×C4⋊C4—1224no—
324C2×C4⋊C4—1224no—
325C2×C4⋊C4—1224no—
326C2×C4⋊C4—1224no—
327C2×C4⋊C4—1224no—
328C2×C4⋊C4—1224no—
329C2×C4⋊C4—1224no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1222no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1228no—
335C2×C4⋊C4—1222no—
336C2×C4⋊C4C2×#692312yes—
337C2×C4⋊C4C2×#692314yescomplemented
338C2×C22⋊C4—1224no—
339C2×C22⋊C4—2444no—
340C22⋊C8—1224no—
341C22⋊C8—1224no—
342#95—1224no—
343C24—611no—
344C24—614no—
345C24—614no—
346C24—612no—
347C24—612no—
348C24—611no—
349C24—1224no—
350C24—1224no—
351C24C22⋊#692311yescharacteristic
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—44no—
356C22×C4—611no—
357C22×C4—611no—
358C22×C4—611no—
359C22×C4—618no—
360C22×C4—611no—
361C22×C4—611no—
362C22×C4—612no—
363C22×C4—612no—
364C22×C4—611no—
365C22×C4—612no—
366C22×C4—612no—
367C22×C4—612no—
368C22×C4—612no—
369C22×C4—612no—
370C22×C4—612no—
371C22×C4—1224no—
372C22×C4—1228no—
373C22×C4—1228no—
374C22×C4—1228no—
375C22×C4—1228no—
376C22×C4—1224no—
377C22×C4—1224no—
378C22×C4—1224no—
379C22×C4—1228no—
380C22×C4—1224no—
381C22×C4—1224no—
382C22×C4—1224no—
383C22×C4—1222no—
384C22×C4—1224no—
385C22×C4—2448no—
386C22×C4—24416no—
387C22×C4—24416no—
388C22×C4—2448no—
389C22×C4—2444no—
390C22×C4—2444no—
391C22×C4—2444no—
392C22×C4—2444no—
393C22×C4C22⋊#692312yes—
394C22×C4C22⋊#692311yescharacteristic
395C22×C4C22×#692312yes—
396C22×C4C22×#692311yescharacteristic
397C2×C8—1224no—
398C2×C8—1228no—
399C2×C8—2444no—
400C2×C8—2444no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—614no—
404C4⋊C4—1224no—
405C4⋊C4—1224no—
406C4⋊C4—1228no—
407C4⋊C4—1228no—
408C4⋊C4—1224no—
409C4⋊C4—1222no—
410C4⋊C4—2444no—
411C4⋊C4C22×#692314yes—
412C22⋊C4—2448no—
413C23—22no—
414C23—22no—
415C23—614no—
416C23—612no—
417C23—612no—
418C23—614no—
419C23—611no—
420C23—611no—
421C23—618no—
422C23—614no—
423C23—614no—
424C23—614no—
425C23—612no—
426C23—1224no—
427C23—1224no—
428C23—1224no—
429C23—1224no—
430C23—12216no—
431C23—12216no—
432C23—1222no—
433C23—1222no—
434C23#1451984014yes—
435C23C2×C22⋊#692312yes—
436C23C2×C22⋊#692311yescharacteristic
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—48no—
442C2×C4—44no—
443C2×C4—614no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—614no—
447C2×C4—614no—
448C2×C4—614no—
449C2×C4—614no—
450C2×C4—612no—
451C2×C4—1224no—
452C2×C4—1224no—
453C2×C4—1224no—
454C2×C4—1224no—
455C2×C4—1224no—
456C2×C4—1224no—
457C2×C4—1222no—
458C2×C4—1224no—
459C2×C4—1224no—
460C2×C4—1224no—
461C2×C4—4888no—
462C2×C4—2448no—
463C2×C4—2448no—
464C2×C4—24416no—
465C2×C4—2444no—
466C2×C4—2444no—
467C2×C4—2444no—
468C2×C4—2448no—
469C2×C4—2444no—
470C2×C4C2×C22⋊#692312yes—
471C2×C4C2×C22⋊#692311yescharacteristic
472C2×C4#1451985614yes—
473C2×C4C23×#692311yescharacteristic
474C8—2444no—
475C22—22no—
476C22—28no—
477C22—22no—
478C22—611no—
479C22—611no—
480C22—612no—
481C22—612no—
482C22—612no—
483C22—618no—
484C22—618no—
485C22—612no—
486C22—612no—
487C22—614no—
488C22—1228no—
489C22—1228no—
490C22—12216no—
491C22—1222no—
492C22—1228no—
493C22—1222no—
494C22#1684070512yes—
495C22C2×#1451984014yes—
496C22C2×#1451984012yes—
497C22#1684075412yes—
498C22C22×C22⋊#692311yescharacteristic
499C4—22no—
500C4—44no—
501C4—612no—
502C4—612no—
503C4—1222no—
504C4—4884no—
505C4—2444no—
506C4#1684070512yes—
507C4C2×#1451985612yes—
508C2—24no—
509C2—612no—
510C2—612no—
511C2—614no—
512C2—1224no—
513C2#2146172314yesdirect product factor
514C2C2×#1684070511yescharacteristic
515C2C22×#1451984011yescharacteristic
516C2C2×#1684075411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC2×C4×D61C24
hypercenterC22×C4⋊C4#6923
hypocenterC61C2×#1824
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21461723C2direct (N × H)
4. C2×#16840635C2semidirect
16. #16840635C22semidirect
19. C22×C4⋊Dic61C4semidirect
25. C2×C4⋊C4×D61C4semidirect
26. C2×C4⋊C4×D61C4semidirect
29. C22×C4⋊C4×C61C4semidirect
63. C2×C4⋊Dic61C2×C4semidirect
76. C4⋊C4×D61C2×C4semidirect
83. C2×C4⋊C4×C61C2×C4semidirect
256. C2×C61#1824semidirect
261. D61#1824semidirect
278. C22×C4⋊C4#6923semidirect
288. C61C2×#1824semidirect
337. C2×C4⋊C4C2×#6923semidirect
513. C2#21461723direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D61⋊(C22⋊C8)2
C2×#168407042
#214617232
C2×#168406352
C22×C4⋊C4×D612
C2×#182461