Finite Group Databasebeta

← C2×#21446841 #29706905

Subgroups — C2×#21446841 #29706905

Order 15616.

2126 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519803C211yesmaximal, characteristic
2C2×#16840336C211yesmaximal, characteristic, complemented
3C22×#14519705C211yesmaximal, characteristic, complemented
4C2×#16839767C211yesmaximal, characteristic, complemented
5C22×#14519696C211yesmaximal, characteristic, complemented
6#21446841C218yesmaximal, direct product factor
7C2×#16839522C211yesmaximal, characteristic, complemented
8C22×#124×C61C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C2×#14519803—24no—
11C2×#14519803—24no—
12C2×#14519803—22no—
13C2×#14519803C2214yes—
14C22×#11758200—21no—
15C22×#11758199—21no—
16C22×#11758194C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18#16840336—22no—
19#16840336—22no—
20#16840336—22no—
21#16840336C2214yescomplemented
22C2×#14519775—21no—
23C2×#14519766—21no—
24C2×#14519760—22no—
25C2×#14519753—22no—
26C2×#14519736C2211yescharacteristic
27C2×#14519705—24no—
28C2×#14519705C2214yescomplemented
29#16839767—22no—
30#16839767C2214yescomplemented
31C2×#14519698—22no—
32C2×#14519696—24no—
33C2×#14519696C2214yescomplemented
34#16839522C2214yescomplemented
35C23×Q8×C61C2211yescharacteristic
36C23×D4×C61C2211yescharacteristic
37C2×#124×C61—22no—
38C2×#124×C61—24no—
39C2×#124×C61—24no—
40C2×#124×C61C2214yescomplemented
41C22×C22⋊C4×C61—22no—
42C22×C42×C61C2211yescharacteristic
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222D412yes—
50C2×#11758222C2312yes—
51C22×Dic122D411yescharacteristic
52C22×D244D411yescharacteristic
53C22×C22⋊D61—21no—
54C22×C4×D61—21no—
55#14519803—216no—
56#14519803—28no—
57#14519803—28no—
58C2×#11758200—28no—
59C2×#11758200—24no—
60C2×#11758199—28no—
61C2×#11758199—24no—
62C2×#11758194—22no—
63C2×#11758194—21no—
64C2×#11758194—21no—
65C2×#11758194—22no—
66C2×#11758194—22no—
67C2×#11758194D414yes—
68C2×#11758194C2312yes—
69C2×#11758194C2312yes—
70C22×#92418—21no—
71C22×#92418—21no—
72#14519775—24no—
73#14519775—24no—
74#14519766—24no—
75#14519766—24no—
76#14519760—28no—
77#14519760—28no—
78#14519753—28no—
79#14519753—28no—
80#14519736—21no—
81#14519736—21no—
82#14519736D412yes—
83#14519736C2312yes—
84C2×C4⋊Dic61D412yes—
85C2×#11758196—24no—
86C2×C22⋊Dic61—21no—
87C2×#11758198—24no—
88C2×C4×Dic61—21no—
89#14519705—28no—
90#14519698—28no—
91#14519696—28no—
92C25×C61—21no—
93C22×Q8×C61—22no—
94C22×Q8×C61—21no—
95C22×Q8×C61—21no—
96C22×Q8×C61—21no—
97C22×Q8×C61—21no—
98C22×Q8×C61—22no—
99C22×Q8×C61—22no—
100C22×Q8×C61—22no—
101C22×Q8×C61D412yes—
102C22×Q8×C61C2312yes—
103C22×D4×C61—21no—
104C22×D4×C61—21no—
105C22×D4×C61—21no—
106C22×D4×C61—21no—
107C22×D4×C61—22no—
108C22×D4×C61—21no—
109C22×D4×C61—22no—
110C22×D4×C61—22no—
111C22×D4×C61D414yes—
112C22×D4×C61C2312yes—
113C23×C4×C61—21no—
114C23×C4×C61D412yes—
115C23×C4×C61C2311yescharacteristic
116#124×C61—216no—
117#124×C61—28no—
118#124×C61—28no—
119C2×C22⋊C4×C61—216no—
120C2×C22⋊C4×C61—28no—
121C2×C42×C61—21no—
122C2×C42×C61—22no—
123C2×C42×C61D414yes—
124C2×C42×C61C2312yes—
125C24×C61—24no—
126C24×C61—22no—
127C24×C61—48no—
128C24×C61—41no—
129C24×C61—41no—
130C24×C61—42no—
131C24×C61C2×D411yescharacteristic
132C23×D61—21no—
133C2×Q8×C61—24no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C2×Q8×C61—24no—
137C2×Q8×C61—22no—
138C2×Q8×C61—22no—
139C2×Q8×C61—22no—
140C2×Q8×C61—22no—
141C2×Q8×C61—22no—
142C2×Q8×C61—22no—
143C2×Q8×C61—42no—
144C2×Q8×C61—42no—
145C2×Q8×C61—42no—
146C2×Q8×C61—42no—
147C2×Q8×C61—42no—
148C2×Q8×C61—42no—
149C2×Q8×C61—42no—
150C2×Q8×C61—42no—
151C2×D4×C61—24no—
152C2×D4×C61—24no—
153C2×D4×C61—24no—
154C2×D4×C61—24no—
155C2×D4×C61—24no—
156C2×D4×C61—22no—
157C2×D4×C61—22no—
158C2×D4×C61—22no—
159C2×D4×C61—22no—
160C2×D4×C61—22no—
161C2×D4×C61—41no—
162C2×D4×C61—41no—
163C2×D4×C61—42no—
164C2×D4×C61—42no—
165C2×D4×C61—42no—
166C2×D4×C61—42no—
167C2×D4×C61—42no—
168C2×D4×C61C2×D414yes—
169C22×C4×C61—24no—
170C22×C4×C61—24no—
171C22×C4×C61—22no—
172C22×C4×C61—22no—
173C22×C4×C61—24no—
174C22×C4×C61—22no—
175C22×C4×C61—44no—
176C22×C4×C61C2×D414yes—
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2×D412yes—
181C22×C4×C61C2×D412yes—
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2×D411yescharacteristic
184C22×C4×C61C2×D412yes—
185C22×C4×C61#4112yes—
186C22×C4×C61C2411yescharacteristic
187C2×C22⋊D61—22no—
188C2×C22⋊D61—22no—
189C2×C22⋊D61—42no—
190C2×C22⋊D61—42no—
191C22×Dic61—21no—
192C22×Dic61—22no—
193#11758222—24no—
194#11758222—24no—
195#11758222—22no—
196#11758222—22no—
197#11758222—22no—
198#11758222—22no—
199#11758222—42no—
200#11758222—42no—
201#11758222—42no—
202#11758222—42no—
203C2×D244—22no—
204C2×D244—22no—
205C2×D244—21no—
206C2×D244C2×D412yes—
207C2×C4×D61—22no—
208C2×C4×D61—22no—
209C2×C4×D61—42no—
210C2×C4×D61—42no—
211C2×Dic122—22no—
212C2×Dic122—22no—
213C2×Dic122—21no—
214C2×Dic122C2×D412yes—
215C22⋊C4×C61—232no—
216C42×C61—24no—
217C42×C61—22no—
218C42×C61C2×D414yes—
219C22⋊Dic61—22no—
220C22⋊Dic61—22no—
221#11758200—216no—
222#11758199—216no—
223#11758198—216no—
224C4⋊Dic61—22no—
225C4⋊Dic61C2×D414yes—
226#11758196—216no—
227C4×Dic61—22no—
228C4×Dic61—22no—
229#11758194—24no—
230#11758194—22no—
231#11758194—22no—
232#11758194—22no—
233#11758194—42no—
234#11758194C2×D414yes—
235C2×#92418—24no—
236C2×#92418—22no—
237C2×#92418—22no—
238C2×#92418—42no—
239C23×C61—24no—
240C23×C61—24no—
241C23×C61—48no—
242C23×C61—416no—
243C23×C61—42no—
244C23×C61—42no—
245C23×C61—44no—
246C23×C61—44no—
247C23×C61—84no—
248C23×C61—88no—
249C23×C61—84no—
250C23×C61#12011yescharacteristic
251C23×C61#12012yes—
252C23×C61#12011yescharacteristic
253C23×C61#12112yes—
254C23×C61C22×D411yescharacteristic
255C22×D61—22no—
256C22×D61—42no—
257C22×D61—44no—
258Q8×C61—42no—
259Q8×C61—42no—
260Q8×C61—42no—
261Q8×C61—42no—
262Q8×C61—44no—
263Q8×C61—44no—
264D4×C61—24no—
265D4×C61—24no—
266D4×C61—42no—
267D4×C61—42no—
268D4×C61—44no—
269D4×C61—44no—
270C2×C4×C61—24no—
271C2×C4×C61—24no—
272C2×C4×C61—24no—
273C2×C4×C61—24no—
274C2×C4×C61—44no—
275C2×C4×C61—48no—
276C2×C4×C61—44no—
277C2×C4×C61—44no—
278C2×C4×C61#12012yes—
279C2×C4×C61#12012yes—
280C2×C4×C61#12011yescharacteristic
281C2×C4×C61#12011yescharacteristic
282C2×C4×C61#12012yes—
283C2×C4×C61#12112yes—
284C2×C4×C61#12112yes—
285C2×C4×C61#12112yes—
286C2×C4×C61#12112yes—
287C2×C4×C61#12112yes—
288C2×C4×C61#12712yes—
289C2×C4×C61#12712yes—
290C2×C4×C61C22×D411yescharacteristic
291C2×C4×C61C22×D411yescharacteristic
292C2×C4×C61C22×D411yescharacteristic
293C2×C4×C61C22×D411yescharacteristic
294C2×C4×C61C22×D411yescharacteristic
295C2×C4×C61C2×#4111yescharacteristic
296C22⋊D61—42no—
297C22⋊D61—42no—
298C22⋊D61—44no—
299C2×Dic61—24no—
300C2×Dic61—22no—
301C2×Dic61—42no—
302C2×Dic61—44no—
303D244—22no—
304D244—22no—
305D244—42no—
306C4×D61—42no—
307C4×D61—42no—
308C4×D61—44no—
309Dic122—22no—
310Dic122—22no—
311Dic122—42no—
312#92418—24no—
313#92418—42no—
314C2×#2003—611nomaximal, Sylow
315C22×C61—22no—
316C22×C61—22no—
317C22×C61—24no—
318C22×C61—24no—
319C22×C61—48no—
320C22×C61—44no—
321C22×C61—84no—
322C22×C61—88no—
323C22×C61—84no—
324C22×C61—88no—
325C22×C61#39212yes—
326C22×C61#39212yes—
327C22×C61#45414yes—
328C22×C61C2×#12011yescharacteristic
329C22×C61C2×#12011yescharacteristic
330C22×C61C2×#12111yescharacteristic
331C2×D61—42no—
332C2×D61—44no—
333C2×D61—84no—
334C4×C61—44no—
335C4×C61—44no—
336C4×C61#39212yes—
337C4×C61#39212yes—
338C4×C61C2×#12011yescharacteristic
339C4×C61C2×#12111yescharacteristic
340C4×C61C2×#12111yescharacteristic
341C4×C61C2×#12711yescharacteristic
342Dic61—42no—
343Dic61—44no—
344C22×AGL(1,ℤ/8)—611no—
345C2×#536—611no—
346C22×#124D6111yescharacteristic, complemented
347C22×#104—611no—
348C2×#418—611no—
349C22×#101—611no—
350#2003—618no—
351C2×#339—611no—
352C2×C61—21no—
353C2×C61—21no—
354C2×C61—22no—
355C2×C61—84no—
356C2×C61#200314yescomplemented
357C2×C61C2×#39211yescharacteristic
358C2×C61C2×#45412yes—
359D61—82no—
360C22×#41—611no—
361C23×Q8C2×D6111yescharacteristic
362C23×D4C2×D6111yescharacteristic
363C2×AGL(1,ℤ/8)—614no—
364C2×AGL(1,ℤ/8)—1224no—
365C2×AGL(1,ℤ/8)—1222no—
366C2×AGL(1,ℤ/8)—1224no—
367C22×SD16—1221no—
368C22×D8—1221no—
369C22×M16—611no—
370C22×M16—611no—
371#536—614no—
372#536—1222no—
373#536—1222no—
374#536—1222no—
375C2×C4⋊Q8—1221no—
376C2×#124—22no—
377C2×#124—24no—
378C2×#124—24no—
379C2×#124—1221no—
380C2×#124C2×D6114yescomplemented
381C2×#123—1222no—
382C2×C22⋊Q8—1222no—
383C2×#117—611no—
384C22×C22⋊C4—22no—
385C22×C42C2×D6111yescharacteristic
386C2×#104—614no—
387C2×#104—1224no—
388#418—614no—
389#418—1222no—
390C2×#102—1222no—
391C2×#101—614no—
392C2×#101—1224no—
393#339—614no—
394C61C2×#200311yescharacteristic, complemented, Sylow
395C25—21no—
396C2×#41—612no—
397C2×#41—612no—
398C2×#41—1221no—
399C2×#41—1221no—
400C2×#41—1222no—
401C2×#41—1222no—
402C2×#41—1222no—
403C2×#41—1222no—
404C22×Q8—21no—
405C22×Q8—21no—
406C22×Q8—21no—
407C22×Q8—21no—
408C22×Q8—22no—
409C22×Q8—22no—
410C22×Q8—22no—
411C22×Q8—22no—
412C22×Q8—611no—
413C22×Q8C22⋊D6112yes—
414C22×Q8C22×D6112yes—
415C22×D4—21no—
416C22×D4—21no—
417C22×D4—22no—
418C22×D4—21no—
419C22×D4—21no—
420C22×D4—22no—
421C22×D4—22no—
422C22×D4—21no—
423C22×D4—611no—
424C22×D4—1221no—
425C22×D4C22⋊D6114yes—
426C22×D4C22×D6112yes—
427C23×C4—21no—
428C23×C4—1221no—
429C23×C4C22⋊D6112yes—
430C23×C4C22×D6111yescharacteristic
431AGL(1,ℤ/8)—1228no—
432AGL(1,ℤ/8)—12216no—
433AGL(1,ℤ/8)—1228no—
434C2×SD16—1224no—
435C2×SD16—1228no—
436C2×D8—1224no—
437C2×D8—1228no—
438C2×M16—614no—
439C2×M16—612no—
440C2×M16—612no—
441C2×M16—1221no—
442C2×M16—1222no—
443C2×M16—1221no—
444C2×M16—1222no—
445C2×M16—1222no—
446C22×C8—1221no—
447C22×C8—1221no—
448C4⋊Q8—1224no—
449C4⋊Q8—1224no—
450#124—216no—
451#124—28no—
452#124—28no—
453#124—1224no—
454#124—1224no—
455#123—1228no—
456#123—1228no—
457C22⋊Q8—1228no—
458C22⋊Q8—1228no—
459#117—612no—
460#117—612no—
461#117—1221no—
462#117—1221no—
463C2×C4⋊C4—612no—
464C2×C4⋊C4—1224no—
465C2×C22⋊C4—216no—
466C2×C22⋊C4—28no—
467C2×C22⋊C4—1224no—
468C2×C22⋊C4—1221no—
469C2×C42—22no—
470C2×C42—21no—
471C2×C42—1221no—
472C2×C42C22⋊D6114yes—
473C2×C42C22×D6112yes—
474#104—1228no—
475#102—1228no—
476#101—1228no—
477C24—22no—
478C24—24no—
479C24—48no—
480C24—41no—
481C24—41no—
482C24—42no—
483C24—1221no—
484C24C2×C22⋊D6111yescharacteristic
485#41—1222no—
486#41—1222no—
487#41—1222no—
488#41—1222no—
489#41—1224no—
490#41—1224no—
491#41—2442no—
492#41—2442no—
493#41—2442no—
494#41—2442no—
495C2×Q8—22no—
496C2×Q8—22no—
497C2×Q8—22no—
498C2×Q8—22no—
499C2×Q8—22no—
500C2×Q8—22no—
501C2×Q8—22no—
502C2×Q8—22no—
503C2×Q8—24no—
504C2×Q8—24no—
505C2×Q8—42no—
506C2×Q8—42no—
507C2×Q8—42no—
508C2×Q8—42no—
509C2×Q8—42no—
510C2×Q8—42no—
511C2×Q8—42no—
512C2×Q8—42no—
513C2×Q8—612no—
514C2×Q8—1221no—
515C2×Q8—1222no—
516C2×Q8—1222no—
517C2×D4—22no—
518C2×D4—22no—
519C2×D4—22no—
520C2×D4—22no—
521C2×D4—22no—
522C2×D4—24no—
523C2×D4—24no—
524C2×D4—24no—
525C2×D4—24no—
526C2×D4—24no—
527C2×D4—41no—
528C2×D4—41no—
529C2×D4—42no—
530C2×D4—42no—
531C2×D4—42no—
532C2×D4—42no—
533C2×D4—42no—
534C2×D4—612no—
535C2×D4—1222no—
536C2×D4—1222no—
537C2×D4—1222no—
538C2×D4—1221no—
539C2×D4—1222no—
540C2×D4—2442no—
541C2×D4—2442no—
542C2×D4C2×C22⋊D6114yes—
543C22×C4—22no—
544C22×C4—22no—
545C22×C4—22no—
546C22×C4—24no—
547C22×C4—24no—
548C22×C4—24no—
549C22×C4—44no—
550C22×C4—1222no—
551C22×C4—1221no—
552C22×C4—1222no—
553C22×C4—1222no—
554C22×C4—2442no—
555C22×C4—2442no—
556C22×C4D4×D6111yescharacteristic
557C22×C4D4×D6111yescharacteristic
558C22×C4D4×D6112yes—
559C22×C4D4×D6111yescharacteristic
560C22×C4D4×D6111yescharacteristic
561C22×C4#1175822412yes—
562C22×C4C2×C22⋊D6114yes—
563C22×C4C2×C22⋊D6112yes—
564C22×C4C2×C22⋊D6112yes—
565C22×C4C2×C22⋊D6111yescharacteristic
566C22×C4C23×D6111yescharacteristic
567SD16—12216no—
568D8—12216no—
569M16—614no—
570M16—1222no—
571M16—1224no—
572M16—1222no—
573M16—1222no—
574M16—2442no—
575C2×C8—1222no—
576C2×C8—1224no—
577C2×C8—1222no—
578C2×C8—2442no—
579C4⋊C4—614no—
580C4⋊C4—12216no—
581C4⋊C4—1222no—
582C22⋊C4—232no—
583C22⋊C4—12216no—
584C22⋊C4—1222no—
585C22⋊C4—1222no—
586C42—22no—
587C42—24no—
588C42—1222no—
589C42—1222no—
590C42C2×C22⋊D6114yes—
591C23—24no—
592C23—24no—
593C23—48no—
594C23—416no—
595C23—42no—
596C23—42no—
597C23—44no—
598C23—44no—
599C23—84no—
600C23—88no—
601C23—84no—
602C23—1222no—
603C23—2442no—
604C23—2444no—
605C23#1451974511yescharacteristic
606C23#1451974511yescharacteristic
607C23#1451974612yes—
608C23#1451974912yes—
609C23C22×C22⋊D6111yescharacteristic
610Q8—42no—
611Q8—42no—
612Q8—42no—
613Q8—42no—
614Q8—44no—
615Q8—44no—
616Q8—1222no—
617Q8—1222no—
618Q8—2442no—
619D4—24no—
620D4—24no—
621D4—42no—
622D4—42no—
623D4—44no—
624D4—44no—
625D4—1222no—
626D4—1222no—
627D4—2442no—
628D4—2442no—
629D4—2444no—
630D4—2442no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—44no—
636C2×C4—48no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—1224no—
640C2×C4—1222no—
641C2×C4—2442no—
642C2×C4—2444no—
643C2×C4—2442no—
644C2×C4—2442no—
645C2×C4—2444no—
646C2×C4#1451974512yes—
647C2×C4#1451974512yes—
648C2×C4#1451974511yescharacteristic
649C2×C4#1451974511yescharacteristic
650C2×C4#1451974612yes—
651C2×C4#1451974812yes—
652C2×C4#1451974812yes—
653C2×C4#1451974912yes—
654C2×C4#1451974912yes—
655C2×C4#1451974912yes—
656C2×C4#1451977312yes—
657C2×C4#1451977312yes—
658C2×C4C2×D4×D6111yescharacteristic
659C2×C4C2×D4×D6111yescharacteristic
660C2×C4C2×D4×D6111yescharacteristic
661C2×C4C22×C22⋊D6111yescharacteristic
662C2×C4C22×C22⋊D6111yescharacteristic
663C2×C4C2×#1175822411yescharacteristic
664C8—1224no—
665C8—2442no—
666C22—22no—
667C22—22no—
668C22—24no—
669C22—24no—
670C22—48no—
671C22—44no—
672C22—84no—
673C22—88no—
674C22—84no—
675C22—88no—
676C22—4884no—
677C22—2444no—
678C22—2442no—
679C22#1683967012yes—
680C22#1683967012yes—
681C22#1683993414yes—
682C22C2×#1451974511yescharacteristic
683C22C2×#1451974611yescharacteristic
684C22C2×#1451974911yescharacteristic
685C4—44no—
686C4—44no—
687C4—2444no—
688C4—2442no—
689C4#1683967012yes—
690C4#1683967012yes—
691C4C2×#1451974511yescharacteristic
692C4C2×#1451974811yescharacteristic
693C4C2×#1451974911yescharacteristic
694C4C2×#1451977311yescharacteristic
695C2—21no—
696C2—21no—
697C2—22no—
698C2—84no—
699C2—4882no—
700C2#2144684114yesdirect product factor
701C2C2×#1683967011yescharacteristic
702C2C2×#1683993412yes—
subgroupG/H
centerC23#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#124×C61C2
socleC23×C61#120
prime radicalC22×#14519803C2
hypercenterC22×#124D61
hypocenterC61C2×#2003
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
2. C2×#16840336C2semidirect
3. C22×#14519705C2semidirect
4. C2×#16839767C2semidirect
5. C22×#14519696C2semidirect
6. #21446841C2direct (N × H)
7. C2×#16839522C2semidirect
8. C22×#124×C61C2semidirect
21. #16840336C22semidirect
28. C2×#14519705C22semidirect
30. #16839767C22semidirect
33. C2×#14519696C22semidirect
34. #16839522C22semidirect
40. C2×#124×C61C22semidirect
346. C22×#124D61semidirect
356. C2×C61#2003semidirect
380. C2×#124C2×D61semidirect
394. C61C2×#2003semidirect
700. C2#21446841direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198032
C2×#168403362
C22×#145197052
C2×#168397672
C22×#145196962
#214468412
C2×#168395222
C22×#124×C612
C2×#200361