Finite Group Databasebeta

← C2×#21461004 #30608778

Subgroups — C2×#21461004 #30608778

Order 15616.

3246 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊#6923C211yesmaximal, characteristic
2C22×(C2×#6923)⋊C4C212yesmaximal
3C22×(C2×#6923)⋊C4C211yesmaximal, characteristic
4C22×(C2×#6923)⋊C4C212yesmaximal
5#21461004C218yesmaximal, direct product factor
6C22×C4⋊C4×D61C211yesmaximal, characteristic
7C24×#6923C2212yes—
8C24×#6923C2212yes—
9C22×C4⋊#6923—28no—
10C22×C4⋊#6923—24no—
11C22×C4⋊#6923C2214yes—
12C2×(C2×#6923)⋊C4—28no—
13C2×(C2×#6923)⋊C4—24no—
14C2×(C2×#6923)⋊C4—28no—
15C2×(C2×#6923)⋊C4C2218yes—
16C2×(C2×#6923)⋊C4C2214yes—
17C2×(C2×#6923)⋊C4C2218yes—
18C23×C4×D61C2211yescharacteristic
19C23×C4×D61C2211yescharacteristic
20C23×C4×D61C2211yescharacteristic
21C22×C4⋊Dic61C411yescharacteristic, complemented
22C22×#11758196C411yescharacteristic, complemented
23C22×#11758196C411yescharacteristic, complemented
24C2×C4⋊C4×D61—24no—
25C2×C4⋊C4×D61—24no—
26C2×C4⋊C4×D61C414yescomplemented
27C2×C4⋊C4×D61C2214yes—
28C22×C4⋊C4×C61C411yescharacteristic, complemented
29C23×#6923—24no—
30C23×#6923—24no—
31C23×#6923—24no—
32C23×#6923—24no—
33C23×#6923—24no—
34C23×#6923—24no—
35C23×#6923—24no—
36C23×#6923—24no—
37C23×#6923—24no—
38C23×#6923—24no—
39C23×#6923—24no—
40C23×#6923—24no—
41C23×#6923D418yes—
42C23×#6923Q818yes—
43C23×#6923C2314yes—
44C23×#6923C2314yes—
45C2×C4⋊#6923—232no—
46C2×C4⋊#6923—216no—
47C2×C4⋊#6923—28no—
48(C2×#6923)⋊C4—216no—
49(C2×#6923)⋊C4—28no—
50(C2×#6923)⋊C4—216no—
51C24×D61C2311yescharacteristic
52C23×Dic61C2×C411yescharacteristic
53C23×Dic61C2×C411yescharacteristic
54C23×Dic61C2×C411yescharacteristic
55C22×C4×D61—22no—
56C22×C4×D61—22no—
57C22×C4×D61—24no—
58C22×C4×D61—22no—
59C22×C4×D61—22no—
60C22×C4×D61—24no—
61C22×C4×D61—22no—
62C22×C4×D61—22no—
63C22×C4×D61—24no—
64C22×C4×D61—24no—
65C22×C4×D61C2×C412yes—
66C22×C4×D61C2×C412yes—
67C22×C4×D61C2×C412yes—
68C22×C4×D61D414yes—
69C22×C4×D61D414yes—
70C22×C4×D61D414yes—
71C22×C4×D61Q814yes—
72C22×C4×D61C2312yes—
73C22×C4×D61C2312yes—
74C22×C4×D61C2312yes—
75C2×C4⋊Dic61—24no—
76C2×C4⋊Dic61C2×C414yescomplemented
77C2×#11758196—24no—
78C2×#11758196—24no—
79C2×#11758196C2×C414yescomplemented
80C2×#11758196C2×C414yescomplemented
81C4⋊C4×D61—216no—
82C4⋊C4×D61—28no—
83C4⋊C4×D61—28no—
84C23×C4×C61C2×C411yescharacteristic
85C23×C4×C61C2×C411yescharacteristic
86C23×C4×C61C2×C411yescharacteristic
87C2×C4⋊C4×C61—24no—
88C2×C4⋊C4×C61C2×C414yescomplemented
89C24×C61C22×C411yescharacteristic
90C23×D61C22×C411yescharacteristic
91C23×D61C2×D412yes—
92C23×D61C2×D412yes—
93C23×D61C2×D412yes—
94C23×D61C2×D414yes—
95C23×D61C2×Q812yes—
96C23×D61C2×Q814yes—
97C23×D61#4112yes—
98C23×D61#4112yes—
99C23×D61#4114yes—
100C23×D61#4114yes—
101C23×D61C2411yescharacteristic
102C22×#6923—216no—
103C22×#6923—216no—
104C22×#6923—216no—
105C22×#6923—216no—
106C22×#6923—216no—
107C22×#6923—216no—
108C22×#6923—24no—
109C22×#6923—24no—
110C22×#6923—24no—
111C22×#6923—24no—
112C22×#6923—24no—
113C22×#6923—24no—
114C22×#6923—24no—
115C22×#6923—24no—
116C22×#6923—24no—
117C22×#6923—24no—
118C22×#6923—24no—
119C22×#6923—24no—
120C22×#6923—48no—
121C22×#6923—48no—
122C22×#6923—48no—
123C22×#6923—48no—
124C22×#6923—48no—
125C22×#6923—48no—
126C22×#6923—48no—
127C22×#6923—48no—
128C22×#6923C2×D418yes—
129C22×#6923C2×Q818yes—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—24no—
133C22×C4×C61—22no—
134C22×C4×C61—24no—
135C22×C4×C61C22⋊C414yes—
136C22×C4×C61C4⋊C414yescomplemented
137C22×C4×C61C22×C412yes—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C22×C412yes—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—24no—
143C22×Dic61—22no—
144C22×Dic61—24no—
145C22×Dic61C22⋊C414yes—
146C22×Dic61C4⋊C414yescomplemented
147C22×Dic61C22×C412yes—
148C22×Dic61C22×C412yes—
149C22×Dic61C22×C412yes—
150C2×C4×D61—24no—
151C2×C4×D61—24no—
152C2×C4×D61—28no—
153C2×C4×D61—28no—
154C2×C4×D61—28no—
155C2×C4×D61—28no—
156C2×C4×D61—216no—
157C2×C4×D61—28no—
158C2×C4×D61—28no—
159C2×C4×D61—28no—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—24no—
166C2×C4×D61—24no—
167C2×C4×D61—28no—
168C2×C4×D61—24no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—24no—
172C2×C4×D61—44no—
173C2×C4×D61—48no—
174C2×C4×D61—44no—
175C2×C4×D61C22⋊C418yes—
176C2×C4×D61C4⋊C418yescomplemented
177C2×C4×D61C2×D414yes—
178C2×C4×D61C2×D414yes—
179C2×C4×D61C2×D414yes—
180C2×C4×D61C2×Q814yes—
181C4⋊#6923—232no—
182C4⋊C4×C61—28no—
183C4⋊Dic61—28no—
184#11758196—28no—
185#11758196—28no—
186C23×C61C2×C22⋊C412yes—
187C23×C61C2×C4⋊C412yes—
188C23×C61#11712yes—
189C23×C61C4×D414yes—
190C23×C61C4×Q814yes—
191C23×C61C23×C411yescharacteristic
192C22×D61C2×C22⋊C412yes—
193C22×D61C2×C4⋊C412yes—
194C22×D61#11712yes—
195C22×D61C4×D414yes—
196C22×D61C4×Q814yes—
197C22×D61#12014yes—
198C22×D61#12114yes—
199C22×D61#12114yes—
200C22×D61#12118yes—
201C22×D61#12118yes—
202C22×D61C22⋊Q814yes—
203C22×D61C22⋊Q814yes—
204C22×D61C22⋊Q818yes—
205C22×D61C22⋊Q818yes—
206C22×D61C22⋊Q818yes—
207C22×D61#12314yes—
208C22×D61#12318yes—
209C22×D61#12414yes—
210C22×D61#12518yes—
211C22×D61#12518yes—
212C22×D61#12618yes—
213C22×D61C4⋊Q818yes—
214C22×D61C4⋊Q814yes—
215C22×D61C22×D412yes—
216C22×D61C22×D411yescharacteristic
217C22×D61C22×D411yescharacteristic
218C22×D61C22×D411yescharacteristic
219C22×D61C22×Q812yes—
220C22×D61C22×Q811yescharacteristic
221C22×D61C2×#4111yescharacteristic
222C22×D61C2×#4111yescharacteristic
223C22×D61C2×#4112yes—
224C22×D61C2×#4112yes—
225C2×#6923—28no—
226C2×#6923—28no—
227C2×#6923—28no—
228C2×#6923—28no—
229C2×#6923—28no—
230C2×#6923—28no—
231C2×#6923—44no—
232C2×#6923—44no—
233C2×#6923—44no—
234C2×#6923—44no—
235C2×#6923—44no—
236C2×#6923—44no—
237C2×#6923—44no—
238C2×#6923—44no—
239C2×#6923—432no—
240C2×#6923—432no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—24no—
244C2×C4×C61—28no—
245C2×C4×C61—28no—
246C2×C4×C61—22no—
247C2×C4×C61—24no—
248C2×C4×C61—44no—
249C2×C4×C61C2×C22⋊C414yes—
250C2×C4×C61C2×C4⋊C414yescomplemented
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—24no—
254C2×Dic61—28no—
255C2×Dic61—28no—
256C2×Dic61—22no—
257C2×Dic61—24no—
258C2×Dic61—44no—
259C2×Dic61C2×C22⋊C414yes—
260C2×Dic61C2×C4⋊C414yescomplemented
261C4×D61—24no—
262C4×D61—24no—
263C4×D61—24no—
264C4×D61—28no—
265C4×D61—24no—
266C4×D61—24no—
267C4×D61—24no—
268C4×D61—216no—
269C4×D61—216no—
270C4×D61—42no—
271C4×D61—44no—
272C4×D61—42no—
273C4×D61—48no—
274C2×#1441—611nomaximal, Sylow
275C22×C61#38014yes—
276C22×C61#38614yes—
277C22×C61#38718yes—
278C22×C61#38918yes—
279C22×C61#39114yes—
280C22×C61C22×C22⋊C411yescharacteristic
281C22×C61C22×C4⋊C411yescharacteristic
282C22×C61C2×#11711yescharacteristic
283C22×C61C2×C4×D412yes—
284C22×C61C2×C4×Q812yes—
285C2×D61#38014yes—
286C2×D61#38614yes—
287C2×D61#38718yes—
288C2×D61#38918yes—
289C2×D61#39114yes—
290C2×D61#39418yes—
291C2×D61#39518yes—
292C2×D61#396116yes—
293C2×D61#39818yes—
294C2×D61#398116yes—
295C2×D61#40018yes—
296C2×D61C2×#12011yescharacteristic
297C2×D61C2×#12112yes—
298C2×D61C2×#12111yescharacteristic
299C2×D61C2×#12111yescharacteristic
300C2×D61C2×#12112yes—
301C2×D61C2×C22⋊Q812yes—
302C2×D61C2×C22⋊Q811yescharacteristic
303C2×D61C2×C22⋊Q812yes—
304C2×D61C2×C22⋊Q812yes—
305C2×D61C2×C22⋊Q811yescharacteristic
306C2×D61C2×#12312yes—
307C2×D61C2×#12311yescharacteristic
308C2×D61C2×#12411yescharacteristic
309C2×D61C2×#12512yes—
310C2×D61C2×#12512yes—
311C2×D61C2×#12612yes—
312C2×D61C2×C4⋊Q812yes—
313C2×D61C2×C4⋊Q811yescharacteristic
314#6923—48no—
315#6923—48no—
316C4×C61—24no—
317C4×C61—24no—
318C4×C61—42no—
319Dic61—24no—
320Dic61—24no—
321Dic61—42no—
322C23×C4⋊C4—611no—
323C23×C4⋊C4—611no—
324C22×#95—611no—
325C22×#95—612no—
326C22×#95—612no—
327#1441—618no—
328C2×C61#144118yescomplemented
329C2×C61C2×#38011yescharacteristic
330C2×C61C2×#38611yescharacteristic
331C2×C61C2×#38712yes—
332C2×C61C2×#38912yes—
333C2×C61C2×#39111yescharacteristic
334D61#144118yescomplemented
335D61C2×#39411yescharacteristic
336D61C2×#39511yescharacteristic
337D61C2×#39612yes—
338D61C2×#39811yescharacteristic
339D61C2×#39812yes—
340D61C2×#40011yescharacteristic
341C24×C4—611no—
342C24×C4—611no—
343C24×C4—612no—
344C24×C4—612no—
345C24×C4—611no—
346C22×C4⋊C4—611no—
347C22×C4⋊C4—611no—
348C22×C4⋊C4—611no—
349C22×C4⋊C4—614no—
350C22×C4⋊C4—614no—
351C22×C4⋊C4—614no—
352C22×C4⋊C4—1224no—
353C22×C4⋊C4—1224no—
354C22×C4⋊C4—1224no—
355C22×C4⋊C4—1228no—
356C22×C4⋊C4#692311yescharacteristic, complemented
357C2×#95—614no—
358C2×#95—618no—
359C2×#95—618no—
360C2×#95—1228no—
361C2×#95—1224no—
362C2×#95—1228no—
363C61C2×#144111yescharacteristic, complemented, Sylow
364C25—611no—
365C23×C4—611no—
366C23×C4—611no—
367C23×C4—614no—
368C23×C4—614no—
369C23×C4—614no—
370C23×C4—614no—
371C23×C4—612no—
372C23×C4—612no—
373C23×C4—612no—
374C23×C4—612no—
375C23×C4—614no—
376C23×C4—618no—
377C23×C4—618no—
378C23×C4—614no—
379C23×C4—611no—
380C23×C4—612no—
381C23×C4—612no—
382C23×C4—1222no—
383C23×C4—1222no—
384C23×C4—1224no—
385C23×C4—1222no—
386C23×C4—1222no—
387C23×C4—1224no—
388C23×C4—1222no—
389C23×C4—1222no—
390C23×C4—1224no—
391C23×C4—1224no—
392C23×C4—1224no—
393C23×C4—1224no—
394C23×C4—1224no—
395C23×C4—1224no—
396C23×C4—1224no—
397C23×C4—1224no—
398C23×C4—1224no—
399C23×C4—1224no—
400C23×C4—1224no—
401C23×C4—1224no—
402C23×C4—1224no—
403C23×C4—1224no—
404C23×C4C2×#692311yescharacteristic
405C23×C4C2×#692311yescharacteristic
406C23×C4C2×#692311yescharacteristic
407C2×C4⋊C4—24no—
408C2×C4⋊C4—614no—
409C2×C4⋊C4—614no—
410C2×C4⋊C4—614no—
411C2×C4⋊C4—1228no—
412C2×C4⋊C4—1228no—
413C2×C4⋊C4—12216no—
414C2×C4⋊C4—1224no—
415C2×C4⋊C4—1224no—
416C2×C4⋊C4—1228no—
417C2×C4⋊C4—12216no—
418C2×C4⋊C4—12232no—
419C2×C4⋊C4—1224no—
420C2×C4⋊C4C2×#692314yescomplemented
421#95—1228no—
422#95—12216no—
423#95—12216no—
424C24—612no—
425C24—612no—
426C24—612no—
427C24—612no—
428C24—612no—
429C24—612no—
430C24—614no—
431C24—614no—
432C24—614no—
433C24—614no—
434C24—611no—
435C24—611no—
436C24C22×#692311yescharacteristic
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—614no—
443C22×C4—614no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—614no—
447C22×C4—614no—
448C22×C4—618no—
449C22×C4—614no—
450C22×C4—618no—
451C22×C4—614no—
452C22×C4—618no—
453C22×C4—618no—
454C22×C4—612no—
455C22×C4—1228no—
456C22×C4—1224no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1224no—
460C22×C4—1228no—
461C22×C4—1228no—
462C22×C4—1228no—
463C22×C4—1228no—
464C22×C4—12216no—
465C22×C4—1228no—
466C22×C4—1224no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1228no—
470C22×C4—1224no—
471C22×C4—1224no—
472C22×C4—1224no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1228no—
476C22×C4—1224no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4—1224no—
482C22×C4—1224no—
483C22×C4—1224no—
484C22×C4—12216no—
485C22×C4—1224no—
486C22×C4—1224no—
487C22×C4—12216no—
488C22×C4—12216no—
489C22×C4—12216no—
490C22×C4—12216no—
491C22×C4—12216no—
492C22×C4—1224no—
493C22×C4—1224no—
494C22×C4—1224no—
495C22×C4—1224no—
496C22×C4—1224no—
497C22×C4—1224no—
498C22×C4—1222no—
499C22×C4—1224no—
500C22×C4—2444no—
501C22×C4—2448no—
502C22×C4—2444no—
503C22×C4—2448no—
504C22×C4—2448no—
505C22×C4—2448no—
506C22×C4—2448no—
507C22×C4—2448no—
508C22×C4—2448no—
509C22×C4—2448no—
510C22×C4—2448no—
511C22×C4C4⋊#692314yescomplemented
512C22×C4C22⋊#692314yes—
513C22×C4C22×#692312yes—
514C22×C4C22×#692312yes—
515C22×C4C22×#692312yes—
516C4⋊C4—28no—
517C4⋊C4—1228no—
518C4⋊C4—1228no—
519C4⋊C4—12232no—
520C4⋊C4—1228no—
521C23—614no—
522C23—614no—
523C23—614no—
524C23—614no—
525C23—614no—
526C23—618no—
527C23—614no—
528C23—618no—
529C23—618no—
530C23—618no—
531C23—618no—
532C23—618no—
533C23—618no—
534C23—618no—
535C23—618no—
536C23—618no—
537C23—614no—
538C23—614no—
539C23—612no—
540C23—611no—
541C23—611no—
542C23—612no—
543C23—611no—
544C23—611no—
545C23—612no—
546C23—611no—
547C23—611no—
548C23—614no—
549C23—612no—
550C23—612no—
551C23—614no—
552C23—612no—
553C23—612no—
554C23C2×C22⋊#692312yes—
555C23C2×C4⋊#692312yes—
556C23#1451985612yes—
557C23D4×#692314yes—
558C23Q8×#692314yes—
559C23C23×#692311yescharacteristic
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—44no—
568C2×C4—614no—
569C2×C4—614no—
570C2×C4—1224no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1224no—
575C2×C4—12216no—
576C2×C4—1228no—
577C2×C4—1224no—
578C2×C4—12216no—
579C2×C4—1228no—
580C2×C4—1228no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1224no—
584C2×C4—1228no—
585C2×C4—1228no—
586C2×C4—1228no—
587C2×C4—1228no—
588C2×C4—1228no—
589C2×C4—1228no—
590C2×C4—1222no—
591C2×C4—1224no—
592C2×C4—2442no—
593C2×C4—2444no—
594C2×C4—2442no—
595C2×C4—2448no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—2444no—
599C2×C4—24432no—
600C2×C4—24432no—
601C2×C4—2444no—
602C2×C4—2444no—
603C2×C4—2444no—
604C2×C4—2444no—
605C2×C4—2444no—
606C2×C4—2444no—
607C2×C4C2×C22⋊#692314yes—
608C2×C4C2×C4⋊#692314yescomplemented
609C22—612no—
610C22—612no—
611C22—612no—
612C22—612no—
613C22—618no—
614C22—612no—
615C22—611no—
616C22—611no—
617C22—614no—
618C22—618no—
619C22—6116no—
620C22—618no—
621C22—618no—
622C22—6116no—
623C22—618no—
624C22—611no—
625C22—611no—
626C22—611no—
627C22—611no—
628C22—614no—
629C22—611no—
630C22—612no—
631C22—611no—
632C22—614no—
633C22—612no—
634C22—612no—
635C22—612no—
636C22—612no—
637C22—618no—
638C22#1684070414yes—
639C22#1684071014yes—
640C22#1684071118yes—
641C22#1684071318yes—
642C22(C2×#6923)⋊Q814yes—
643C22C22×C22⋊#692311yescharacteristic
644C22C22×C4⋊#692311yescharacteristic
645C22C2×#1451985611yescharacteristic
646C22C2×D4×#692312yes—
647C22C2×Q8×#692312yes—
648C4—24no—
649C4—24no—
650C4—42no—
651C4—1224no—
652C4—1224no—
653C4—2448no—
654C4—2448no—
655C4—2442no—
656C2—611no—
657C2—612no—
658C2—611no—
659C2—611no—
660C2—612no—
661C2—611no—
662C2—618no—
663C2#2146100418yesdirect product factor
664C2C2×#1684070411yescharacteristic
665C2C2×#1684071011yescharacteristic
666C2C2×#1684071112yes—
667C2C2×#1684071312yes—
668C2C2×(C2×#6923)⋊Q811yescharacteristic
subgroupG/H
centerC24C22×#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC22×C4⋊C4×C61C4
socleC24×C61C22×C4
prime radicalC24×D61C23
cosocleC23×D61C24
hypercenterC22×C4⋊C4#6923
hypocenterC61C2×#1441
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #21461004C2direct (N × H)
21. C22×C4⋊Dic61C4semidirect
22. C22×#11758196C4semidirect
23. C22×#11758196C4semidirect
26. C2×C4⋊C4×D61C4semidirect
28. C22×C4⋊C4×C61C4semidirect
76. C2×C4⋊Dic61C2×C4semidirect
79. C2×#11758196C2×C4semidirect
80. C2×#11758196C2×C4semidirect
88. C2×C4⋊C4×C61C2×C4semidirect
136. C22×C4×C61C4⋊C4semidirect
146. C22×Dic61C4⋊C4semidirect
176. C2×C4×D61C4⋊C4semidirect
250. C2×C4×C61C2×C4⋊C4semidirect
260. C2×Dic61C2×C4⋊C4semidirect
328. C2×C61#1441semidirect
334. D61#1441semidirect
356. C22×C4⋊C4#6923semidirect
363. C61C2×#1441semidirect
420. C2×C4⋊C4C2×#6923semidirect
511. C22×C4C4⋊#6923semidirect
608. C2×C4C2×C4⋊#6923semidirect
663. C2#21461004direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊#69232
C22×(C2×#6923)⋊C42
#214610042
C22×C4⋊C4×D612
C2×#144161