Finite Group Databasebeta

← C2×#21446857 #29706915

Subgroups — C2×#21446857 #29706915

Order 15616.

1998 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519810C211yesmaximal, characteristic
2C2×#16840336C211yesmaximal, characteristic
3C22×#14519775C211yesmaximal, characteristic, complemented
4C22×#14519703C211yesmaximal, characteristic
5C2×#16839772C211yesmaximal, characteristic, complemented
6#21446857C218yesmaximal, direct product factor
7C2×#16839523C211yesmaximal, characteristic, complemented
8C22×#101×C61C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519810—24no—
12C2×#14519810—22no—
13C2×#14519810—24no—
14C2×#14519810C2214yes—
15C22×Dic244—21no—
16C22×#11758190—21no—
17#16840336—22no—
18#16840336—22no—
19#16840336—22no—
20#16840336C2214yes—
21C2×#14519775—24no—
22C2×#14519775—22no—
23C2×#14519775—24no—
24C2×#14519775—21no—
25C2×#14519775C2214yescomplemented
26C2×#14519766—21no—
27C2×#14519760—22no—
28C2×#14519753—22no—
29C2×#14519736C2211yescharacteristic
30C22×#11758196—22no—
31C22×C4×Dic61C2211yescharacteristic
32C2×#14519703—24no—
33C2×#14519703C2214yes—
34#16839772—22no—
35#16839772C2214yescomplemented
36C2×#14519702—22no—
37#16839523C2214yescomplemented
38C23×Q8×C61C2211yescharacteristic
39C22×M16×C61C2211yescharacteristic
40C22×M16×C61C2211yescharacteristic
41C2×#101×C61—24no—
42C2×#101×C61C2214yescomplemented
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222D412yes—
50C2×#11758222C2312yes—
51C22×Dic122—21no—
52C22×Dic122—21no—
53C22×Dic122—21no—
54C22×Dic122—21no—
55C22×Dic122—22no—
56C22×Dic122—22no—
57C22×Dic122—21no—
58C22×Dic122—22no—
59C22×Dic122D411yescharacteristic
60C22×Dic122D414yes—
61C22×Dic122C2312yes—
62C22×D244D411yescharacteristic
63C22×C22⋊D61—21no—
64C23×Dic61—21no—
65C23×Dic61—21no—
66C22×C4×D61—21no—
67#14519810—216no—
68#14519810—28no—
69#14519810—28no—
70C2×Dic244—28no—
71C2×Dic244—24no—
72C2×#11758190—28no—
73C2×#11758190—24no—
74#14519775—216no—
75#14519775—24no—
76#14519775—28no—
77#14519775—24no—
78#14519775—28no—
79#14519766—24no—
80#14519766—24no—
81#14519760—28no—
82#14519760—28no—
83#14519753—28no—
84#14519753—28no—
85#14519736—21no—
86#14519736—21no—
87#14519736D412yes—
88#14519736C2312yes—
89C2×C4⋊Dic61D412yes—
90C2×#11758196—216no—
91C2×#11758196—24no—
92C2×#11758196—28no—
93C2×C22⋊Dic61—21no—
94C2×#11758198—24no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—22no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61D414yes—
99C2×C4×Dic61C2312yes—
100#14519703—28no—
101#14519702—28no—
102C22×Q8×C61—21no—
103C22×Q8×C61—21no—
104C22×Q8×C61—22no—
105C22×Q8×C61—22no—
106C22×Q8×C61—22no—
107C22×Q8×C61—21no—
108C22×Q8×C61—21no—
109C22×Q8×C61—22no—
110C22×Q8×C61D412yes—
111C22×Q8×C61C2312yes—
112C23×C4×C61D412yes—
113C23×C4×C61C2311yescharacteristic
114C2×M16×C61—22no—
115C2×M16×C61—21no—
116C2×M16×C61—21no—
117C2×M16×C61—22no—
118C2×M16×C61—22no—
119C2×M16×C61D414yes—
120C2×M16×C61C2312yes—
121C2×M16×C61C2312yes—
122C22×C8×C61—21no—
123C22×C8×C61—21no—
124#101×C61—28no—
125C24×C61C2×D411yescharacteristic
126C23×D61—21no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—24no—
130C2×Q8×C61—24no—
131C2×Q8×C61—22no—
132C2×Q8×C61—22no—
133C2×Q8×C61—22no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—42no—
138C2×Q8×C61—42no—
139C2×Q8×C61—42no—
140C2×Q8×C61—42no—
141C2×Q8×C61—42no—
142C2×Q8×C61—42no—
143C2×Q8×C61—42no—
144C2×Q8×C61—42no—
145C22×C4×C61—22no—
146C22×C4×C61—24no—
147C22×C4×C61—22no—
148C22×C4×C61—24no—
149C22×C4×C61C2×D414yes—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61#4112yes—
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—22no—
161C2×C22⋊D61—22no—
162C2×C22⋊D61—42no—
163C2×C22⋊D61—42no—
164C22×Dic61—28no—
165C22×Dic61—24no—
166C22×Dic61—22no—
167C22×Dic61—21no—
168C22×Dic61—24no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—41no—
172C22×Dic61—41no—
173C22×Dic61—42no—
174#11758222—24no—
175#11758222—24no—
176#11758222—22no—
177#11758222—22no—
178#11758222—22no—
179#11758222—22no—
180#11758222—42no—
181#11758222—42no—
182#11758222—42no—
183#11758222—42no—
184C2×D244—22no—
185C2×D244—22no—
186C2×D244—21no—
187C2×D244C2×D412yes—
188C2×C4×D61—22no—
189C2×C4×D61—22no—
190C2×C4×D61—42no—
191C2×C4×D61—42no—
192C2×Dic122—24no—
193C2×Dic122—24no—
194C2×Dic122—22no—
195C2×Dic122—24no—
196C2×Dic122—24no—
197C2×Dic122—22no—
198C2×Dic122—22no—
199C2×Dic122—22no—
200C2×Dic122—22no—
201C2×Dic122—22no—
202C2×Dic122—24no—
203C2×Dic122—21no—
204C2×Dic122—22no—
205C2×Dic122—41no—
206C2×Dic122—41no—
207C2×Dic122—42no—
208C2×Dic122—42no—
209C2×Dic122—42no—
210C2×Dic122—42no—
211C2×Dic122—42no—
212C2×Dic122C2×D414yes—
213C2×Dic122C2×D412yes—
214M16×C61—24no—
215M16×C61—22no—
216M16×C61—22no—
217M16×C61—22no—
218M16×C61—42no—
219M16×C61C2×D414yes—
220C2×C8×C61—24no—
221C2×C8×C61—22no—
222C2×C8×C61—22no—
223C2×C8×C61—42no—
224C22⋊Dic61—22no—
225C22⋊Dic61—22no—
226#11758198—216no—
227C4⋊Dic61—22no—
228C4⋊Dic61C2×D414yes—
229#11758196—232no—
230#11758196—216no—
231C4×Dic61—24no—
232C4×Dic61—22no—
233C4×Dic61—22no—
234C4×Dic61—22no—
235C4×Dic61C2×D414yes—
236Dic244—216no—
237#11758190—216no—
238C23×C61—24no—
239C23×C61#12011yescharacteristic
240C23×C61#12012yes—
241C23×C61#12011yescharacteristic
242C23×C61#12112yes—
243C23×C61C22×D411yescharacteristic
244C22×D61—22no—
245C22×D61—42no—
246C22×D61—44no—
247Q8×C61—42no—
248Q8×C61—42no—
249Q8×C61—42no—
250Q8×C61—42no—
251Q8×C61—44no—
252Q8×C61—44no—
253C2×C4×C61—24no—
254C2×C4×C61—24no—
255C2×C4×C61—24no—
256C2×C4×C61—44no—
257C2×C4×C61—44no—
258C2×C4×C61#12012yes—
259C2×C4×C61#12012yes—
260C2×C4×C61#12012yes—
261C2×C4×C61#12011yescharacteristic
262C2×C4×C61#12011yescharacteristic
263C2×C4×C61#12112yes—
264C2×C4×C61#12112yes—
265C2×C4×C61#12112yes—
266C2×C4×C61#12112yes—
267C2×C4×C61#12112yes—
268C2×C4×C61#12712yes—
269C2×C4×C61#12712yes—
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C22⋊D61—42no—
277C22⋊D61—42no—
278C22⋊D61—44no—
279C2×Dic61—28no—
280C2×Dic61—216no—
281C2×Dic61—24no—
282C2×Dic61—24no—
283C2×Dic61—24no—
284C2×Dic61—22no—
285C2×Dic61—42no—
286C2×Dic61—42no—
287C2×Dic61—44no—
288C2×Dic61—44no—
289C2×Dic61—42no—
290C2×Dic61—44no—
291D244—22no—
292D244—22no—
293D244—42no—
294C4×D61—42no—
295C4×D61—42no—
296C4×D61—44no—
297Dic122—24no—
298Dic122—22no—
299Dic122—24no—
300Dic122—22no—
301Dic122—42no—
302Dic122—42no—
303Dic122—44no—
304Dic122—44no—
305Dic122—42no—
306C8×C61—24no—
307C8×C61—42no—
308C2×#2005—611nomaximal, Sylow
309C22×C61—22no—
310C22×C61—22no—
311C22×C61—24no—
312C22×C61—24no—
313C22×C61#39212yes—
314C22×C61#39212yes—
315C22×C61#45614yes—
316C22×C61C2×#12011yescharacteristic
317C22×C61C2×#12011yescharacteristic
318C22×C61C2×#12111yescharacteristic
319C2×D61—42no—
320C2×D61—44no—
321C2×D61—84no—
322C4×C61—44no—
323C4×C61#39212yes—
324C4×C61#39212yes—
325C4×C61C2×#12011yescharacteristic
326C4×C61C2×#12111yescharacteristic
327C4×C61C2×#12111yescharacteristic
328C4×C61C2×#12711yescharacteristic
329Dic61—28no—
330Dic61—44no—
331Dic61—42no—
332Dic61—44no—
333C22×#137—611no—
334C2×#536—611no—
335C22×C4⋊Q8—611no—
336C22×#104—611no—
337C2×#419—611no—
338C22×#101D6111yescharacteristic, complemented
339#2005—618no—
340C2×#339—611no—
341C2×C61—21no—
342C2×C61—21no—
343C2×C61—22no—
344C2×C61#200514yescomplemented
345C2×C61C2×#39211yescharacteristic
346C2×C61C2×#45612yes—
347D61—82no—
348C22×#41—611no—
349C23×Q8—611no—
350C23×Q8C2×D6111yescharacteristic
351C2×#137—614no—
352C2×#137—1224no—
353C2×#137—1222no—
354C2×#137—1224no—
355C22×Q16—1221no—
356C22×SD16—1221no—
357C22×M16C2×D6111yescharacteristic
358C22×M16C2×D6111yescharacteristic
359#536—614no—
360#536—1222no—
361#536—1222no—
362#536—1222no—
363C2×C4⋊Q8—614no—
364C2×C4⋊Q8—1224no—
365C2×C4⋊Q8—1224no—
366C2×C4⋊Q8—1221no—
367C2×C4⋊Q8—1222no—
368C2×#124—1221no—
369C2×#123—1222no—
370C2×C22⋊Q8—1222no—
371C2×#117—611no—
372C22×C4⋊C4—1222no—
373C22×C42—611no—
374C2×#104—614no—
375C2×#104—1224no—
376#419—614no—
377#419—1222no—
378C2×#103—1222no—
379C2×#101—24no—
380C2×#101C2×D6114yescomplemented
381#339—614no—
382C61C2×#200511yescharacteristic, complemented, Sylow
383C2×#41—612no—
384C2×#41—612no—
385C2×#41—1221no—
386C2×#41—1221no—
387C2×#41—1222no—
388C2×#41—1222no—
389C2×#41—1222no—
390C2×#41—1222no—
391C22×Q8—22no—
392C22×Q8—21no—
393C22×Q8—21no—
394C22×Q8—22no—
395C22×Q8—22no—
396C22×Q8—21no—
397C22×Q8—21no—
398C22×Q8—22no—
399C22×Q8—611no—
400C22×Q8—614no—
401C22×Q8—612no—
402C22×Q8—1221no—
403C22×Q8—1222no—
404C22×Q8—1222no—
405C22×Q8—1221no—
406C22×Q8—1221no—
407C22×Q8—1221no—
408C22×Q8—1221no—
409C22×Q8—1222no—
410C22×Q8D24412yes—
411C22×Q8C22×D6112yes—
412C22×D4—611no—
413C22×D4—1221no—
414C23×C4—1221no—
415C23×C4—1221no—
416C23×C4—1221no—
417C23×C4D24412yes—
418C23×C4C22×D6111yescharacteristic
419#137—1228no—
420#137—12216no—
421#137—1228no—
422C2×Q16—1224no—
423C2×Q16—1228no—
424C2×SD16—1224no—
425C2×SD16—1228no—
426C2×M16—21no—
427C2×M16—22no—
428C2×M16—21no—
429C2×M16—22no—
430C2×M16—22no—
431C2×M16C22⋊D6114yes—
432C2×M16C22×D6112yes—
433C2×M16C22×D6112yes—
434C22×C8—21no—
435C22×C8—21no—
436C4⋊Q8—1228no—
437C4⋊Q8—1228no—
438C4⋊Q8—1224no—
439C4⋊Q8—12216no—
440C4⋊Q8—1224no—
441#124—1224no—
442#124—1224no—
443#123—1228no—
444#123—1228no—
445C22⋊Q8—1228no—
446C22⋊Q8—1228no—
447#117—612no—
448#117—612no—
449#117—1221no—
450#117—1221no—
451C2×C4⋊C4—612no—
452C2×C4⋊C4—12216no—
453C2×C4⋊C4—1224no—
454C2×C4⋊C4—1228no—
455C2×C22⋊C4—1224no—
456C2×C22⋊C4—1221no—
457C2×C42—614no—
458C2×C42—612no—
459C2×C42—1221no—
460C2×C42—1221no—
461C2×C42—1222no—
462#104—1228no—
463#103—1228no—
464#101—28no—
465C24—1221no—
466C24C2×D24411yescharacteristic
467#41—1222no—
468#41—1222no—
469#41—1222no—
470#41—1222no—
471#41—1224no—
472#41—1224no—
473#41—2442no—
474#41—2442no—
475#41—2442no—
476#41—2442no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—22no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—24no—
486C2×Q8—24no—
487C2×Q8—42no—
488C2×Q8—42no—
489C2×Q8—42no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—612no—
496C2×Q8—614no—
497C2×Q8—1222no—
498C2×Q8—1224no—
499C2×Q8—1221no—
500C2×Q8—1222no—
501C2×Q8—1222no—
502C2×Q8—1222no—
503C2×Q8—1222no—
504C2×Q8—1224no—
505C2×Q8—1222no—
506C2×Q8—1224no—
507C2×Q8—1224no—
508C2×Q8—1224no—
509C2×Q8—1222no—
510C2×Q8—2441no—
511C2×Q8—2441no—
512C2×Q8—2442no—
513C2×Q8—2442no—
514C2×Q8—2442no—
515C2×Q8—2442no—
516C2×Q8—2442no—
517C2×D4—612no—
518C2×D4—1222no—
519C2×D4—1222no—
520C2×D4—1222no—
521C2×D4—1221no—
522C2×D4—1222no—
523C2×D4—2442no—
524C2×D4—2442no—
525C22×C4—24no—
526C22×C4—22no—
527C22×C4—22no—
528C22×C4—24no—
529C22×C4—1228no—
530C22×C4—1222no—
531C22×C4—1224no—
532C22×C4—1222no—
533C22×C4—1224no—
534C22×C4—1222no—
535C22×C4—1221no—
536C22×C4—1222no—
537C22×C4—1222no—
538C22×C4—2441no—
539C22×C4—2441no—
540C22×C4—2442no—
541C22×C4—2442no—
542C22×C4—2442no—
543C22×C4C2×D24414yes—
544C22×C4C2×D24411yescharacteristic
545C22×C4#1175822212yes—
546C22×C4D4×D6112yes—
547C22×C4D4×D6111yescharacteristic
548C22×C4D4×D6111yescharacteristic
549C22×C4D4×D6112yes—
550C22×C4D4×D6111yescharacteristic
551C22×C4D4×D6111yescharacteristic
552C22×C4C2×C22⋊D6112yes—
553C22×C4C23×D6111yescharacteristic
554Q16—12216no—
555SD16—12216no—
556M16—22no—
557M16—24no—
558M16—22no—
559M16—22no—
560M16—42no—
561M16C2×C22⋊D6114yes—
562C2×C8—22no—
563C2×C8—24no—
564C2×C8—22no—
565C2×C8—42no—
566C4⋊C4—614no—
567C4⋊C4—12232no—
568C4⋊C4—12216no—
569C4⋊C4—1222no—
570C22⋊C4—12216no—
571C22⋊C4—1222no—
572C22⋊C4—1222no—
573C42—614no—
574C42—1222no—
575C42—1222no—
576C42—1224no—
577C42—1222no—
578C23—24no—
579C23—1222no—
580C23—2444no—
581C23—2442no—
582C23#1451974711yescharacteristic
583C23#1451974712yes—
584C23#1451974711yescharacteristic
585C23#1451975112yes—
586C23C22×D24411yescharacteristic
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—44no—
592Q8—44no—
593Q8—1224no—
594Q8—1222no—
595Q8—1224no—
596Q8—1222no—
597Q8—2442no—
598Q8—2442no—
599Q8—2444no—
600Q8—2444no—
601Q8—2442no—
602D4—1222no—
603D4—1222no—
604D4—2442no—
605D4—2442no—
606D4—2444no—
607D4—2442no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—1228no—
614C2×C4—12216no—
615C2×C4—1224no—
616C2×C4—1224no—
617C2×C4—1222no—
618C2×C4—1224no—
619C2×C4—2444no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2444no—
624C2×C4—2444no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2444no—
628C2×C4#1451974512yes—
629C2×C4#1451974512yes—
630C2×C4#1451974711yescharacteristic
631C2×C4#1451974712yes—
632C2×C4#1451974711yescharacteristic
633C2×C4#1451975012yes—
634C2×C4#1451975012yes—
635C2×C4#1451975012yes—
636C2×C4#1451975012yes—
637C2×C4#1451975112yes—
638C2×C4#1451977312yes—
639C2×C4#1451977312yes—
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4C2×D4×D6111yescharacteristic
642C2×C4C2×D4×D6111yescharacteristic
643C2×C4C2×D4×D6111yescharacteristic
644C2×C4C22×C22⋊D6111yescharacteristic
645C2×C4C2×#1175822211yescharacteristic
646C8—24no—
647C8—42no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—4884no—
653C22—2442no—
654C22—2444no—
655C22#1683967112yes—
656C22#1683967112yes—
657C22#1683994714yes—
658C22C2×#1451974711yescharacteristic
659C22C2×#1451974711yescharacteristic
660C22C2×#1451975111yescharacteristic
661C4—44no—
662C4—1228no—
663C4—2442no—
664C4—2444no—
665C4—2444no—
666C4#1683967112yes—
667C4#1683967112yes—
668C4C2×#1451974511yescharacteristic
669C4C2×#1451975011yescharacteristic
670C4C2×#1451975011yescharacteristic
671C4C2×#1451977311yescharacteristic
672C2—21no—
673C2—21no—
674C2—22no—
675C2—4882no—
676C2#2144685714yesdirect product factor
677C2C2×#1683967111yescharacteristic
678C2C2×#1683994712yes—
subgroupG/H
centerC23#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#101×C61C2
socleC23×C61#120
prime radicalC22×#11758222C22
hypercenterC22×#101D61
hypocenterC61C2×#2005
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#101

Complements

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

NH ≅ G/Ntype
3. C22×#14519775C2semidirect
5. C2×#16839772C2semidirect
6. #21446857C2direct (N × H)
7. C2×#16839523C2semidirect
8. C22×#101×C61C2semidirect
25. C2×#14519775C22semidirect
35. #16839772C22semidirect
37. #16839523C22semidirect
42. C2×#101×C61C22semidirect
338. C22×#101D61semidirect
344. C2×C61#2005semidirect
380. C2×#101C2×D61semidirect
382. C61C2×#2005semidirect
676. C2#21446857direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198102
C2×#168403362
C22×#145197752
C22×#145197032
C2×#168397722
#214468572
C2×#168395232
C22×#101×C612
C2×#200561