Finite Group Databasebeta

← C2×#21461805 #30615598

Subgroups — C2×#21461805 #30615598

Order 15616.

1966 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519856C211yesmaximal, characteristic
2C22×#14519840C212yesmaximal, complemented
3C2×D61⋊(C22⋊C8)C211yesmaximal, characteristic
4#21461805C218yesmaximal, direct product factor
5C2×#16840635C212yesmaximal, complemented
6C2×C22⋊Q8×D61C211yesmaximal, characteristic
7C22×D61⋊C8C2212yes—
8C2×#14519856—28no—
9C2×#14519856—22no—
10C2×#14519856C2214yes—
11C22×C4⋊#6923C2212yes—
12C22×C22⋊#6923—21no—
13C22×C4×#6923—21no—
14C2×#14519840—28no—
15C2×#14519840C2218yescomplemented
16C2×#14519834—21no—
17D61⋊(C22⋊C8)—22no—
18D61⋊(C22⋊C8)C2214yes—
19C2×C22⋊#92420—21no—
20#16840635C2218yescomplemented
21C22×Q8×D61C2211yescharacteristic
22C23×C4×D61C2211yescharacteristic
23C2×#14519757C411yescharacteristic, complemented
24C2×#14519756C411yescharacteristic, complemented
25C2×#14519755—21no—
26C2×#14519754—21no—
27C2×#14519753C411yescharacteristic, complemented
28C22⋊Q8×D61—22no—
29C22⋊Q8×D61—22no—
30C22⋊Q8×D61C414yescomplemented
31C22⋊Q8×D61C2214yes—
32C2×C4⋊C4×D61—21no—
33C2×C4⋊C4×D61C2211yescharacteristic
34C2×C22⋊C4×D61—21no—
35C2×C22⋊Q8×C61C411yescharacteristic, complemented
36C23×#6923—22no—
37C2×D61⋊C8—24no—
38C2×D61⋊C8—24no—
39C2×D61⋊C8—22no—
40C2×D61⋊C8C2314yes—
41C22×#92420—22no—
42#14519856—216no—
43#14519856—216no—
44C2×C4⋊#6923—22no—
45C2×C4⋊#6923—24no—
46C2×C4⋊#6923D418yes—
47C2×C4⋊#6923C2314yes—
48C2×C22⋊#6923—28no—
49C2×C22⋊#6923—24no—
50C2×C4×#6923—28no—
51C2×C4×#6923—24no—
52#14519840—216no—
53#14519834—24no—
54C22⋊#92420—24no—
55C24×D61D411yescharacteristic
56C22×Dic122—21no—
57C22×Dic122C2×C411yescharacteristic
58C2×Q8×D61—21no—
59C2×Q8×D61—21no—
60C2×Q8×D61—24no—
61C2×Q8×D61—24no—
62C2×Q8×D61C2×C412yes—
63C2×Q8×D61C2312yes—
64C23×Dic61D411yescharacteristic
65C22×C4×D61—24no—
66C22×C4×D61—24no—
67C22×C4×D61—21no—
68C22×C4×D61—21no—
69C22×C4×D61C2×C412yes—
70C22×C4×D61C2×C411yescharacteristic
71C22×C4×D61D411yescharacteristic
72C22×C4×D61D412yes—
73C22×C4×D61D412yes—
74C22×C4×D61D411yescharacteristic
75C22×C4×D61C2312yes—
76C22×C4×D61C2311yescharacteristic
77#14519757—22no—
78#14519757C2×C414yescomplemented
79#14519756—22no—
80#14519756C2×C414yescomplemented
81#14519755—24no—
82#14519755—24no—
83#14519754—24no—
84#14519754—24no—
85#14519753—22no—
86#14519753C2×C414yescomplemented
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61C2×C411yescharacteristic
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C4⋊C4×D61—24no—
93C4⋊C4×D61—24no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61C2×C412yes—
97C4⋊C4×D61C2312yes—
98C2×C22⋊Dic61—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C22⋊C4×D61—24no—
102C22⋊C4×D61—24no—
103C22×Q8×C61C2×C411yescharacteristic
104C23×C4×C61C2×C411yescharacteristic
105C22⋊Q8×C61—22no—
106C22⋊Q8×C61C2×C414yescomplemented
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61C2×C411yescharacteristic
109C2×C22⋊C4×C61—21no—
110C24×C61C22⋊C411yescharacteristic
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—24no—
114C23×D61—24no—
115C23×D61C22⋊C412yes—
116C23×D61C22⋊C411yescharacteristic
117C23×D61C2×D412yes—
118C23×D61C2×D411yescharacteristic
119C22×#6923—28no—
120C22×#6923—24no—
121C22×#6923—44no—
122C22×#6923—44no—
123C2×Q8×C61—21no—
124C2×Q8×C61—24no—
125C2×Q8×C61C22×C412yes—
126C22×C4×C61—24no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C22⋊C411yescharacteristic
130C22×C4×C61C22⋊C412yes—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C22×C411yescharacteristic
133C22×Dic61—24no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61C22⋊C411yescharacteristic
137C22×Dic61C22⋊C412yes—
138C22×Dic61C2×D412yes—
139C22×Dic61C2×D411yescharacteristic
140Q8×D61—28no—
141Q8×D61—24no—
142Q8×D61—24no—
143Q8×D61—42no—
144Q8×D61—44no—
145Q8×D61—42no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—28no—
151C2×C4×D61—24no—
152C2×C4×D61—28no—
153C2×C4×D61—24no—
154C2×C4×D61—24no—
155C2×C4×D61—24no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—44no—
163C2×C4×D61C22⋊C412yes—
164C2×C4×D61C22⋊C412yes—
165C2×C4×D61C22⋊C412yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C22×C411yescharacteristic
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×D412yes—
170C2×C4×D61C2×D414yes—
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×C4×D61#4114yes—
174C2×C4×D61C2411yescharacteristic
175C2×Dic122—24no—
176C2×Dic122—21no—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C2×Dic122—44no—
180C2×Dic122C22×C412yes—
181C22⋊#6923—216no—
182C2×#92420—24no—
183C2×#92420—44no—
184C4⋊#6923—28no—
185C4⋊#6923—24no—
186C4⋊#6923C2×D418yes—
187C4×#6923—216no—
188D61⋊C8—24no—
189D61⋊C8—24no—
190D61⋊C8—44no—
191C4⋊C4×C61—24no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61C22×C412yes—
194C22⋊C4×C61—24no—
195C22⋊Dic61—24no—
196#11758198—24no—
197#11758198—24no—
198C4⋊Dic61—21no—
199C4⋊Dic61—24no—
200C4⋊Dic61C22×C412yes—
201#11758196—24no—
202#11758196—24no—
203#11758196—22no—
204#11758196—22no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61C2×C22⋊C412yes—
209C23×C61C2×C22⋊C411yescharacteristic
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—24no—
216C22×D61—24no—
217C22×D61—24no—
218C22×D61—24no—
219C22×D61—24no—
220C22×D61—48no—
221C22×D61—48no—
222C22×D61C2×C22⋊C412yes—
223C22×D61C2×C22⋊C411yescharacteristic
224C22×D61#12012yes—
225C22×D61#12312yes—
226C22×D61#13514yes—
227C22×D61#13714yes—
228C22×D61C22×D411yescharacteristic
229C2×#6923—28no—
230C2×#6923—44no—
231C2×#6923—44no—
232C2×#6923—416no—
233Q8×C61—24no—
234Q8×C61—42no—
235C2×C4×C61—22no—
236C2×C4×C61—24no—
237C2×C4×C61—24no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C4×D414yes—
245C2×C4×C61C23×C411yescharacteristic
246C2×Dic61—22no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61#12114yes—
256C2×Dic61C22×D411yescharacteristic
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—24no—
260C4×D61—24no—
261C4×D61—24no—
262C4×D61—24no—
263C4×D61—24no—
264C4×D61—42no—
265C4×D61—42no—
266C4×D61—44no—
267C4×D61—44no—
268C4×D61—44no—
269C4×D61C4×D414yes—
270C4×D61#12012yes—
271C4×D61#12114yes—
272C4×D61#12312yes—
273C4×D61C22×D412yes—
274C4×D61C2×#4112yes—
275Dic122—24no—
276Dic122—42no—
277Dic122—44no—
278Dic122—44no—
279#92420—44no—
280C2×#1873—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—44no—
285C22×C61#38612yes—
286C22×C61#41612yes—
287C22×C61#41912yes—
288C22×C61C22×C22⋊C411yescharacteristic
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—44no—
293C2×D61—44no—
294C2×D61—48no—
295C2×D61—44no—
296C2×D61#38612yes—
297C2×D61#41612yes—
298C2×D61#41912yes—
299C2×D61#45218yes—
300C2×D61#48518yes—
301C2×D61C2×#12011yescharacteristic
302C2×D61C2×#12311yescharacteristic
303C2×D61C2×#13512yes—
304C2×D61C2×#13712yes—
305#6923—48no—
306C4×C61—22no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61#38612yes—
310C4×C61C2×C4×D412yes—
311Dic61—22no—
312Dic61—42no—
313Dic61—42no—
314Dic61#38612yes—
315Dic61C2×#12112yes—
316C22×C22⋊Q8—611no—
317C22×#117—611no—
318C22×#103—612no—
319C22×C22⋊C8—611no—
320#1873—618no—
321C2×#340—612no—
322C2×C61—42no—
323C2×C61#187314yescomplemented
324C2×C61C2×#38611yescharacteristic
325C2×C61C2×#41611yescharacteristic
326C2×C61C2×#41911yescharacteristic
327D61—42no—
328D61#187314yescomplemented
329D61C2×#45212yes—
330D61C2×#48512yes—
331C23×Q8—611no—
332C24×C4—611no—
333C23×C8—612no—
334C2×C22⋊Q8—611no—
335C2×C22⋊Q8—611no—
336C2×C22⋊Q8—614no—
337C2×C22⋊Q8—614no—
338C2×C22⋊Q8—611no—
339C2×C22⋊Q8—1221no—
340C2×C22⋊Q8—1221no—
341C2×C22⋊Q8—1222no—
342C2×C22⋊Q8—1222no—
343C2×C22⋊Q8#692311yescharacteristic, complemented
344C2×#117—614no—
345C2×#117—1228no—
346C2×#117—1222no—
347C22×C4⋊C4—612no—
348C22×C4⋊C4—611no—
349C22×C4⋊C4—1221no—
350C22×C22⋊C4—1221no—
351C22×C22⋊C4—1221no—
352C22×C42—1221no—
353C2×#103—618no—
354C2×#103—1228no—
355C2×C22⋊C8—614no—
356C2×C22⋊C8—1221no—
357C2×C22⋊C8—1222no—
358C2×C22⋊C8—1221no—
359#340—618no—
360C61C2×#187311yescharacteristic, complemented, Sylow
361C25—611no—
362C22×Q8—612no—
363C22×Q8—612no—
364C22×Q8—611no—
365C22×Q8—1221no—
366C22×Q8—1221no—
367C22×Q8—1221no—
368C22×Q8—1224no—
369C22×Q8—1224no—
370C22×Q8C2×#692311yescharacteristic
371C23×C4—611no—
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—612no—
375C23×C4—611no—
376C23×C4—612no—
377C23×C4—612no—
378C23×C4—612no—
379C23×C4—611no—
380C23×C4—1224no—
381C23×C4—1224no—
382C23×C4—1221no—
383C23×C4—1221no—
384C23×C4—1222no—
385C23×C4C2×#692311yescharacteristic
386C22×C8—614no—
387C22×C8—1222no—
388C22×C8—1224no—
389C22×C8—1222no—
390C22×C8—1224no—
391C22⋊Q8—22no—
392C22⋊Q8—614no—
393C22⋊Q8—614no—
394C22⋊Q8—614no—
395C22⋊Q8—1224no—
396C22⋊Q8—1224no—
397C22⋊Q8—1224no—
398C22⋊Q8—1224no—
399C22⋊Q8—1222no—
400C22⋊Q8—1222no—
401C22⋊Q8—1222no—
402C22⋊Q8C2×#692314yescomplemented
403#117—12216no—
404#117—12216no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—614no—
407C2×C4⋊C4—618no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1224no—
414C2×C4⋊C4—1224no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1224no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4C2×#692311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1224no—
424C2×C22⋊C4—1224no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1224no—
428C2×C22⋊C4—1228no—
429C2×C22⋊C4—1221no—
430C2×C42—1224no—
431C2×C42—1228no—
432#103—12216no—
433C22⋊C8—1224no—
434C22⋊C8—1224no—
435C24—611no—
436C24—612no—
437C24—612no—
438C24—611no—
439C24—1222no—
440C24—1222no—
441C24—1224no—
442C24—1224no—
443C24C22⋊#692311yescharacteristic
444C2×Q8—21no—
445C2×Q8—24no—
446C2×Q8—612no—
447C2×Q8—1222no—
448C2×Q8—1222no—
449C2×Q8—1224no—
450C2×Q8—1228no—
451C2×Q8—1224no—
452C2×Q8—1221no—
453C2×Q8—1224no—
454C2×Q8—2444no—
455C2×Q8—2442no—
456C2×Q8—2444no—
457C2×Q8—2442no—
458C2×Q8C22×#692312yes—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—614no—
466C22×C4—614no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—612no—
470C22×C4—612no—
471C22×C4—611no—
472C22×C4—612no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—612no—
478C22×C4—1224no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4—1228no—
482C22×C4—1224no—
483C22×C4—1228no—
484C22×C4—1224no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1224no—
492C22×C4—1224no—
493C22×C4—1228no—
494C22×C4—1224no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4—2444no—
500C22×C4—2444no—
501C22×C4—2444no—
502C22×C4C22⋊#692312yes—
503C22×C4C22⋊#692311yescharacteristic
504C22×C4C22×#692312yes—
505C22×C4C22×#692311yescharacteristic
506C2×C8—1224no—
507C2×C8—1224no—
508C2×C8—1224no—
509C2×C8—2444no—
510C2×C8—2444no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—618no—
514C4⋊C4—612no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1224no—
518C4⋊C4—1224no—
519C4⋊C4—1228no—
520C4⋊C4—1224no—
521C4⋊C4—1221no—
522C4⋊C4—1224no—
523C4⋊C4C22×#692312yes—
524C22⋊C4—24no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C22⋊C4—12216no—
528C22⋊C4—1224no—
529C42—12216no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—612no—
534C23—612no—
535C23—614no—
536C23—614no—
537C23—611no—
538C23—611no—
539C23—612no—
540C23—1222no—
541C23—1222no—
542C23—1224no—
543C23—1224no—
544C23—1224no—
545C23—1224no—
546C23—1224no—
547C23—1222no—
548C23—1222no—
549C23—1222no—
550C23—2448no—
551C23—2448no—
552C23C2×C22⋊#692312yes—
553C23C2×C22⋊#692311yescharacteristic
554Q8—24no—
555Q8—42no—
556Q8—1224no—
557Q8—2444no—
558Q8—2444no—
559Q8—2442no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—614no—
568C2×C4—611no—
569C2×C4—611no—
570C2×C4—612no—
571C2×C4—614no—
572C2×C4—612no—
573C2×C4—614no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1224no—
582C2×C4—1224no—
583C2×C4—1224no—
584C2×C4—1222no—
585C2×C4—1224no—
586C2×C4—1224no—
587C2×C4—1228no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—2444no—
591C2×C4—2444no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2444no—
595C2×C4—24416no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4C2×C22⋊#692312yes—
601C2×C4C2×C22⋊#692311yescharacteristic
602C2×C4D4×#692314yes—
603C2×C4C23×#692311yescharacteristic
604C8—2444no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—44no—
609C22—618no—
610C22—618no—
611C22—612no—
612C22—612no—
613C22—612no—
614C22—612no—
615C22—611no—
616C22—611no—
617C22—612no—
618C22—1222no—
619C22—1222no—
620C22—1222no—
621C22—2444no—
622C22—2444no—
623C22—2448no—
624C22—2444no—
625C22#1684071012yes—
626C22#1684075212yes—
627C22#1684075512yes—
628C22C22×C22⋊#692311yescharacteristic
629C4—22no—
630C4—42no—
631C4—42no—
632C4—612no—
633C4—612no—
634C4—1222no—
635C4—2448no—
636C4—2442no—
637C4—2442no—
638C4#1684071012yes—
639C4C2×D4×#692312yes—
640C2—42no—
641C2—612no—
642C2—612no—
643C2—614no—
644C2—2442no—
645C2#2146180514yesdirect product factor
646C2C2×#1684071011yescharacteristic
647C2C2×#1684075211yescharacteristic
648C2C2×#1684075511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC2×C4×D61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61C2×#1873
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×#14519840C2semidirect
4. #21461805C2direct (N × H)
5. C2×#16840635C2semidirect
15. C2×#14519840C22semidirect
20. #16840635C22semidirect
23. C2×#14519757C4semidirect
24. C2×#14519756C4semidirect
27. C2×#14519753C4semidirect
30. C22⋊Q8×D61C4semidirect
35. C2×C22⋊Q8×C61C4semidirect
78. #14519757C2×C4semidirect
80. #14519756C2×C4semidirect
86. #14519753C2×C4semidirect
106. C22⋊Q8×C61C2×C4semidirect
323. C2×C61#1873semidirect
328. D61#1873semidirect
343. C2×C22⋊Q8#6923semidirect
360. C61C2×#1873semidirect
402. C22⋊Q8C2×#6923semidirect
645. C2#21461805direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198562
C22×#145198402
C2×D61⋊(C22⋊C8)2
#214618052
C2×#168406352
C2×C22⋊Q8×D612
C2×#187361