Finite Group Databasebeta

← C2×#21461724 #30615099

Subgroups — C2×#21461724 #30615099

Order 15616.

1574 conjugacy classes of subgroups.

The subgroup lattice has 578 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#21461724C218yesmaximal, direct product factor
4C2×#16840635C212yesmaximal, complemented
5C2×#16840635C212yesmaximal, complemented
6C2×#117×D61C211yesmaximal, characteristic
7C22×D61⋊C8C2212yes—
8C22×C4⋊#6923C2212yes—
9C22×C22⋊#6923—21no—
10C2×#14519834—21no—
11D61⋊(C22⋊C8)—22no—
12D61⋊(C22⋊C8)C2214yes—
13C2×C22⋊#92420—21no—
14#16840704—22no—
15#16840704C2214yes—
16C2×(C2×#6923)⋊C4—21no—
17#16840635C2218yescomplemented
18#16840635C2218yescomplemented
19C23×C4×D61C2211yescharacteristic
20C2×#14519736C411yescharacteristic, complemented
21C2×#14519735C411yescharacteristic, complemented
22C2×#14519735C411yescharacteristic, complemented
23C2×#14519734—21no—
24C2×#14519733—21no—
25#117×D61—22no—
26#117×D61—22no—
27#117×D61C414yescomplemented
28#117×D61C2214yes—
29C2×C4⋊C4×D61C2211yescharacteristic
30C2×C4⋊C4×D61C2211yescharacteristic
31C2×C22⋊C4×D61—21no—
32C2×C42×D61—21no—
33C2×#117×C61C411yescharacteristic, complemented
34C23×#6923—22no—
35C2×D61⋊C8—28no—
36C2×D61⋊C8—22no—
37C2×D61⋊C8C2314yes—
38C22×#92420—22no—
39C2×C4⋊#6923—28no—
40C2×C4⋊#6923—22no—
41C2×C4⋊#6923C2314yes—
42C2×C22⋊#6923—24no—
43C2×C22⋊#6923—44no—
44#14519834—24no—
45C22⋊#92420—24no—
46(C2×#6923)⋊C4—24no—
47C24×D61D411yescharacteristic
48C23×Dic61D411yescharacteristic
49C22×C4×D61—28no—
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C22×C4×D61C2×C412yes—
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61D412yes—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D412yes—
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2312yes—
59C22×C4×D61C2311yescharacteristic
60#14519736—22no—
61#14519736C2×C414yescomplemented
62#14519735—22no—
63#14519735—22no—
64#14519735C2×C414yescomplemented
65#14519735C2×C414yescomplemented
66#14519734—24no—
67#14519734—24no—
68#14519733—24no—
69#14519733—24no—
70C2×C4⋊Dic61C2×C411yescharacteristic
71C2×C4⋊Dic61C2×C411yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C4⋊C4×D61—21no—
75C4⋊C4×D61—21no—
76C4⋊C4×D61—21no—
77C4⋊C4×D61—21no—
78C4⋊C4×D61C2×C412yes—
79C4⋊C4×D61C2×C412yes—
80C4⋊C4×D61C2312yes—
81C4⋊C4×D61C2312yes—
82C2×C22⋊Dic61—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C22⋊C4×D61—24no—
86C22⋊C4×D61—24no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C42×D61—24no—
91C42×D61—24no—
92C23×C4×C61C2×C411yescharacteristic
93#117×C61—22no—
94#117×C61C2×C414yescomplemented
95C2×C4⋊C4×C61C2×C411yescharacteristic
96C2×C4⋊C4×C61C2×C411yescharacteristic
97C2×C22⋊C4×C61—21no—
98C2×C42×C61—21no—
99C24×C61C22⋊C411yescharacteristic
100C23×D61—24no—
101C23×D61—24no—
102C23×D61—22no—
103C23×D61—22no—
104C23×D61C22⋊C412yes—
105C23×D61C22⋊C411yescharacteristic
106C23×D61C2×D412yes—
107C23×D61C2×D411yescharacteristic
108C22×#6923—24no—
109C22×#6923—44no—
110C22×#6923—44no—
111C22×#6923—44no—
112C22×C4×C61—24no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61C22⋊C411yescharacteristic
116C22×C4×C61C22⋊C412yes—
117C22×C4×C61C22×C412yes—
118C22×C4×C61C22×C411yescharacteristic
119C22×Dic61—24no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61C22⋊C411yescharacteristic
123C22×Dic61C22⋊C412yes—
124C22×Dic61C2×D412yes—
125C22×Dic61C2×D411yescharacteristic
126C2×C4×D61—24no—
127C2×C4×D61—24no—
128C2×C4×D61—24no—
129C2×C4×D61—28no—
130C2×C4×D61—28no—
131C2×C4×D61—28no—
132C2×C4×D61—28no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—44no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C4×D61C22⋊C412yes—
141C2×C4×D61C22⋊C412yes—
142C2×C4×D61C22⋊C412yes—
143C2×C4×D61C22×C412yes—
144C2×C4×D61C22×C411yescharacteristic
145C2×C4×D61C2×D412yes—
146C2×C4×D61C2×D412yes—
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61#4118yes—
150C2×C4×D61C2411yescharacteristic
151C22⋊#6923—48no—
152C2×#92420—24no—
153C2×#92420—44no—
154C4⋊#6923—28no—
155C4⋊#6923—44no—
156D61⋊C8—28no—
157D61⋊C8—44no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61C22×C412yes—
161C4⋊C4×C61C22×C412yes—
162C22⋊C4×C61—24no—
163C42×C61—24no—
164C22⋊Dic61—24no—
165#11758198—24no—
166#11758198—24no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61C22×C412yes—
170C4⋊Dic61C22×C412yes—
171#11758196—22no—
172#11758196—22no—
173#11758196—22no—
174#11758196—22no—
175C4×Dic61—24no—
176C4×Dic61—24no—
177C4×Dic61—24no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61C2×C22⋊C412yes—
182C23×C61C2×C22⋊C411yescharacteristic
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—24no—
187C22×D61—24no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—24no—
191C22×D61—24no—
192C22×D61—24no—
193C22×D61—48no—
194C22×D61—48no—
195C22×D61C2×C22⋊C412yes—
196C22×D61C2×C22⋊C411yescharacteristic
197C22×D61#12314yes—
198C22×D61#13514yes—
199C22×D61AGL(1,ℤ/8)12yes—
200C22×D61#13712yes—
201C22×D61C22×D411yescharacteristic
202C2×#6923—44no—
203C2×#6923—44no—
204C2×#6923—44no—
205C2×#6923—88no—
206C2×C4×C61—22no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61C2×C22⋊C412yes—
214C2×C4×C61C2×C22⋊C411yescharacteristic
215C2×C4×C61#11714yes—
216C2×C4×C61C23×C411yescharacteristic
217C2×Dic61—22no—
218C2×Dic61—24no—
219C2×Dic61—24no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61C2×C22⋊C412yes—
225C2×Dic61C2×C22⋊C411yescharacteristic
226C2×Dic61#12314yes—
227C2×Dic61C22×D411yescharacteristic
228C4×D61—24no—
229C4×D61—24no—
230C4×D61—24no—
231C4×D61—24no—
232C4×D61—24no—
233C4×D61—24no—
234C4×D61—44no—
235C4×D61—42no—
236C4×D61—42no—
237C4×D61—44no—
238C4×D61—44no—
239C4×D61#11714yes—
240C4×D61#12314yes—
241C4×D61#12314yes—
242C4×D61C2×#4114yes—
243#92420—44no—
244C2×#1825—611nomaximal, Sylow
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—44no—
249C22×C61#38112yes—
250C22×C61#41612yes—
251C22×C61#41712yes—
252C22×C61C22×C22⋊C411yescharacteristic
253C2×D61—22no—
254C2×D61—22no—
255C2×D61—22no—
256C2×D61—44no—
257C2×D61—44no—
258C2×D61—44no—
259C2×D61—48no—
260C2×D61#38112yes—
261C2×D61#41612yes—
262C2×D61#41712yes—
263C2×D61#48218yes—
264C2×D61#48518yes—
265C2×D61C2×#12312yes—
266C2×D61C2×#13512yes—
267C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
268C2×D61C2×#13711yescharacteristic
269#6923—84no—
270C4×C61—22no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61#38112yes—
274C4×C61C2×#11712yes—
275Dic61—22no—
276Dic61—42no—
277Dic61—42no—
278Dic61#38112yes—
279Dic61C2×#12312yes—
280C22×#117—611no—
281C22×C22⋊C8—611no—
282C2×#380—611no—
283#1825—618no—
284C2×#340—612no—
285C2×#340—612no—
286C2×C61—42no—
287C2×C61#182514yescomplemented
288C2×C61C2×#38111yescharacteristic
289C2×C61C2×#41611yescharacteristic
290C2×C61C2×#41711yescharacteristic
291D61—42no—
292D61#182514yescomplemented
293D61C2×#48212yes—
294D61C2×#48512yes—
295C24×C4—611no—
296C23×C8—612no—
297C2×#117—611no—
298C2×#117—614no—
299C2×#117—614no—
300C2×#117—611no—
301C2×#117—611no—
302C2×#117—1221no—
303C2×#117—1221no—
304C2×#117—1222no—
305C2×#117—1222no—
306C2×#117#692311yescharacteristic, complemented
307C22×C4⋊C4—611no—
308C22×C4⋊C4—611no—
309C22×C4⋊C4—612no—
310C22×C22⋊C4—1221no—
311C22×C22⋊C4—1221no—
312C22×C42—1221no—
313C2×C22⋊C8—614no—
314C2×C22⋊C8—1221no—
315C2×C22⋊C8—1222no—
316C2×C22⋊C8—1221no—
317#380—614no—
318#380—1222no—
319C2×#95—1221no—
320#340—618no—
321#340—618no—
322C61C2×#182511yescharacteristic, complemented, Sylow
323C25—611no—
324C23×C4—611no—
325C23×C4—611no—
326C23×C4—612no—
327C23×C4—611no—
328C23×C4—612no—
329C23×C4—611no—
330C23×C4—612no—
331C23×C4—611no—
332C23×C4—612no—
333C23×C4—1228no—
334C23×C4—1221no—
335C23×C4—1221no—
336C23×C4—1222no—
337C23×C4C2×#692311yescharacteristic
338C22×C8—614no—
339C22×C8—1222no—
340C22×C8—1228no—
341C22×C8—1222no—
342#117—22no—
343#117—614no—
344#117—614no—
345#117—614no—
346#117—1224no—
347#117—1224no—
348#117—1224no—
349#117—1224no—
350#117—1222no—
351#117—1222no—
352#117—1222no—
353#117C2×#692314yescomplemented
354C2×C4⋊C4—611no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—612no—
360C2×C4⋊C4—614no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1228no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4C2×#692311yescharacteristic
370C2×C4⋊C4C2×#692311yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1224no—
373C2×C22⋊C4—1224no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—2444no—
379C2×C42—21no—
380C2×C42—1224no—
381C2×C42—1224no—
382C2×C42—1221no—
383C2×C42—1221no—
384C2×C42—1221no—
385C22⋊C8—1224no—
386C22⋊C8—1224no—
387#95—1224no—
388C24—611no—
389C24—612no—
390C24—612no—
391C24—611no—
392C24—1222no—
393C24—1224no—
394C24—1224no—
395C24—1222no—
396C24C22⋊#692311yescharacteristic
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—611no—
403C22×C4—618no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—611no—
409C22×C4—612no—
410C22×C4—612no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—612no—
414C22×C4—612no—
415C22×C4—1224no—
416C22×C4—1228no—
417C22×C4—1228no—
418C22×C4—1228no—
419C22×C4—1228no—
420C22×C4—1224no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1224no—
426C22×C4—1224no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1224no—
430C22×C4—2444no—
431C22×C4—2442no—
432C22×C4—2442no—
433C22×C4—2444no—
434C22×C4—2444no—
435C22×C4—2444no—
436C22×C4C22⋊#692312yes—
437C22×C4C22⋊#692311yescharacteristic
438C22×C4C22×#692312yes—
439C22×C4C22×#692311yescharacteristic
440C2×C8—1224no—
441C2×C8—1228no—
442C2×C8—2444no—
443C2×C8—2444no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—612no—
447C4⋊C4—612no—
448C4⋊C4—1222no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4—1228no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—2444no—
456C4⋊C4C22×#692312yes—
457C4⋊C4C22×#692312yes—
458C22⋊C4—24no—
459C22⋊C4—1224no—
460C22⋊C4—1224no—
461C22⋊C4—1224no—
462C22⋊C4—2448no—
463C42—24no—
464C42—1224no—
465C42—1224no—
466C42—1224no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—614no—
471C23—614no—
472C23—612no—
473C23—612no—
474C23—611no—
475C23—611no—
476C23—612no—
477C23—1222no—
478C23—1224no—
479C23—1224no—
480C23—1224no—
481C23—1224no—
482C23—1222no—
483C23—1224no—
484C23—1222no—
485C23—1222no—
486C23—1222no—
487C23—2448no—
488C23—2448no—
489C23C2×C22⋊#692312yes—
490C23C2×C22⋊#692311yescharacteristic
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—614no—
499C2×C4—611no—
500C2×C4—611no—
501C2×C4—614no—
502C2×C4—614no—
503C2×C4—614no—
504C2×C4—614no—
505C2×C4—612no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—1224no—
511C2×C4—1224no—
512C2×C4—1222no—
513C2×C4—1224no—
514C2×C4—1224no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—4888no—
518C2×C4—2444no—
519C2×C4—2444no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2444no—
523C2×C4—2444no—
524C2×C4—2444no—
525C2×C4—2444no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4C2×C22⋊#692312yes—
529C2×C4C2×C22⋊#692311yescharacteristic
530C2×C4#1451985614yes—
531C2×C4C23×#692311yescharacteristic
532C8—2444no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—44no—
537C22—612no—
538C22—612no—
539C22—611no—
540C22—611no—
541C22—612no—
542C22—618no—
543C22—618no—
544C22—612no—
545C22—612no—
546C22—1222no—
547C22—1222no—
548C22—1222no—
549C22—2444no—
550C22—2444no—
551C22—2448no—
552C22—2444no—
553C22#1684070512yes—
554C22#1684075212yes—
555C22#1684075312yes—
556C22C22×C22⋊#692311yescharacteristic
557C4—22no—
558C4—42no—
559C4—42no—
560C4—612no—
561C4—612no—
562C4—1222no—
563C4—4884no—
564C4—2442no—
565C4—2442no—
566C4#1684070512yes—
567C4C2×#1451985612yes—
568C2—42no—
569C2—612no—
570C2—612no—
571C2—614no—
572C2—2442no—
573C2#2146172414yesdirect product factor
574C2C2×#1684070511yescharacteristic
575C2C2×#1684075211yescharacteristic
576C2C2×#1684075311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#117×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC2×C4×D61C24
hypercenterC2×#117#6923
hypocenterC61C2×#1825
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #21461724C2direct (N × H)
4. C2×#16840635C2semidirect
5. C2×#16840635C2semidirect
17. #16840635C22semidirect
18. #16840635C22semidirect
20. C2×#14519736C4semidirect
21. C2×#14519735C4semidirect
22. C2×#14519735C4semidirect
27. #117×D61C4semidirect
33. C2×#117×C61C4semidirect
61. #14519736C2×C4semidirect
64. #14519735C2×C4semidirect
65. #14519735C2×C4semidirect
94. #117×C61C2×C4semidirect
287. C2×C61#1825semidirect
292. D61#1825semidirect
306. C2×#117#6923semidirect
322. C61C2×#1825semidirect
353. #117C2×#6923semidirect
573. C2#21461724direct (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
#214617242
C2×#168406352
C2×#117×D612
C2×#182561