Finite Group Databasebeta

← C2×#21447113 #29707882

Subgroups — C2×#21447113 #29707882

Order 15616.

1490 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519805C211yesmaximal, characteristic
2C2×#16839764C211yesmaximal, characteristic, complemented
3C22×#14519699C211yesmaximal, characteristic, complemented
4C22×#14519693C211yesmaximal, characteristic, complemented
5C2×#16839728C211yesmaximal, characteristic, complemented
6#21447113C218yesmaximal, direct product factor
7C2×#16839530C211yesmaximal, characteristic, complemented
8C22×#108×C61C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×D244C2211yescharacteristic
11C2×#14519805—24no—
12C2×#14519805—22no—
13C2×#14519805—24no—
14C2×#14519805C2214yes—
15C22×#11758190—21no—
16C22×D488—21no—
17C22×#11758194C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19#16839764—22no—
20#16839764C2214yescomplemented
21C2×#14519702—21no—
22C2×#14519699—22no—
23C2×#14519699—22no—
24C2×#14519699—21no—
25C2×#14519699C2214yescomplemented
26C2×#14519693—24no—
27C2×#14519693C2214yescomplemented
28#16839728—22no—
29#16839728C2214yescomplemented
30C2×#14519689—22no—
31#16839530C2214yescomplemented
32C22×M16×C61C2211yescharacteristic
33C22×M16×C61C2211yescharacteristic
34C23×C8×C61C2211yescharacteristic
35C2×#108×C61—24no—
36C2×#108×C61C2214yescomplemented
37C24×D61—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222D412yes—
45C2×#11758222C2312yes—
46C22×Dic122D411yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—22no—
52C22×D244—22no—
53C22×D244—22no—
54C22×D244—21no—
55C22×D244D414yes—
56C22×D244D411yescharacteristic
57C22×D244C2312yes—
58C22×C22⋊D61—21no—
59C23×Dic61—21no—
60C22×C4×D61—21no—
61#14519805—216no—
62#14519805—28no—
63#14519805—28no—
64C2×#11758190—28no—
65C2×#11758190—24no—
66C2×D488—28no—
67C2×D488—24no—
68C2×#11758194—21no—
69C2×#11758194—22no—
70C2×#11758194—22no—
71C2×#11758194C2312yes—
72C22×#92418—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2312yes—
79#14519702—24no—
80#14519699—24no—
81#14519699—28no—
82#14519693—28no—
83#14519689—28no—
84C23×C4×C61C2311yescharacteristic
85C2×M16×C61—22no—
86C2×M16×C61—21no—
87C2×M16×C61—21no—
88C2×M16×C61—22no—
89C2×M16×C61—22no—
90C2×M16×C61D414yes—
91C2×M16×C61C2312yes—
92C2×M16×C61C2312yes—
93C22×C8×C61—22no—
94C22×C8×C61—21no—
95C22×C8×C61—22no—
96C22×C8×C61—21no—
97C22×C8×C61D412yes—
98C22×C8×C61D412yes—
99C22×C8×C61C2312yes—
100#108×C61—28no—
101C24×C61#4111yescharacteristic
102C23×D61—21no—
103C23×D61—24no—
104C23×D61—22no—
105C23×D61—41no—
106C23×D61—41no—
107C23×D61—42no—
108C23×D61—82no—
109C23×D61—82no—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61#4112yes—
117C22×C4×C61#4111yescharacteristic
118C22×C4×C61#4112yes—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—22no—
121C2×C22⋊D61—22no—
122C2×C22⋊D61—42no—
123C2×C22⋊D61—42no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61—41no—
127C22×Dic61—41no—
128C22×Dic61—82no—
129#11758222—24no—
130#11758222—24no—
131#11758222—22no—
132#11758222—22no—
133#11758222—22no—
134#11758222—22no—
135#11758222—42no—
136#11758222—42no—
137#11758222—42no—
138#11758222—42no—
139C2×D244—24no—
140C2×D244—24no—
141C2×D244—22no—
142C2×D244—24no—
143C2×D244—24no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—21no—
150C2×D244—24no—
151C2×D244—22no—
152C2×D244—42no—
153C2×D244—42no—
154C2×D244—41no—
155C2×D244—41no—
156C2×D244—42no—
157C2×D244—42no—
158C2×D244—42no—
159C2×D244C2×D414yes—
160C2×D244C2×D412yes—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—21no—
168C2×Dic122C2×D412yes—
169M16×C61—24no—
170M16×C61—22no—
171M16×C61—22no—
172M16×C61—22no—
173M16×C61—42no—
174M16×C61C2×D414yes—
175C2×C8×C61—24no—
176C2×C8×C61—24no—
177C2×C8×C61—24no—
178C2×C8×C61—22no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C2×C8×C61—42no—
183C2×C8×C61C2×D414yes—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61—41no—
187C4⋊Dic61—41no—
188#11758194—22no—
189#11758194—22no—
190#11758194—42no—
191C2×#92418—22no—
192C2×#92418—42no—
193D488—216no—
194#11758190—216no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61#12111yescharacteristic
198C23×C61#12111yescharacteristic
199C23×C61#12312yes—
200C23×C61#12412yes—
201C23×C61C2×#4111yescharacteristic
202C22×D61—22no—
203C22×D61—24no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—44no—
208C22×D61—44no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—88no—
212C22×D61—82no—
213C22×D61—82no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—88no—
217C22×D61—82no—
218C22×D61—82no—
219C2×C4×C61#12012yes—
220C2×C4×C61#12012yes—
221C2×C4×C61#12112yes—
222C2×C4×C61#12112yes—
223C2×C4×C61#12112yes—
224C2×C4×C61#12112yes—
225C2×C4×C61#12111yescharacteristic
226C2×C4×C61#12111yescharacteristic
227C2×C4×C61#12312yes—
228C2×C4×C61#12312yes—
229C2×C4×C61#12312yes—
230C2×C4×C61#12412yes—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C22⋊D61—42no—
238C22⋊D61—42no—
239C22⋊D61—44no—
240C2×Dic61—22no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—82no—
244C2×Dic61—82no—
245C2×Dic61—82no—
246D244—24no—
247D244—22no—
248D244—24no—
249D244—22no—
250D244—42no—
251D244—42no—
252D244—44no—
253D244—44no—
254D244—42no—
255C4×D61—42no—
256C4×D61—42no—
257C4×D61—44no—
258Dic122—22no—
259Dic122—22no—
260Dic122—42no—
261C8×C61—24no—
262C8×C61—24no—
263C8×C61—42no—
264#92418—42no—
265C2×#2051—611nomaximal, Sylow
266C22×C61—24no—
267C22×C61—24no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61#39412yes—
273C22×C61#39412yes—
274C22×C61#47112yes—
275C22×C61#47212yes—
276C22×C61C2×#12111yescharacteristic
277C22×C61C2×#12311yescharacteristic
278C22×C61C2×#12411yescharacteristic
279C2×D61—42no—
280C2×D61—44no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—84no—
284C2×D61—88no—
285C2×D61—84no—
286C2×D61—88no—
287C2×D61—82no—
288C2×D61—82no—
289C2×D61—84no—
290C4×C61#39412yes—
291C4×C61#39412yes—
292C4×C61C2×#12011yescharacteristic
293C4×C61C2×#12111yescharacteristic
294C4×C61C2×#12111yescharacteristic
295C4×C61C2×#12311yescharacteristic
296Dic61—42no—
297Dic61—82no—
298C22×AGL(1,ℤ/8)—611no—
299C22×#108D6111yescharacteristic, complemented
300C2×#417—611no—
301C22×#102—611no—
302C22×#100—611no—
303C2×#408—611no—
304#2051—618no—
305C2×#343—611no—
306C2×C61—21no—
307C2×C61—21no—
308C2×C61—22no—
309C2×C61#205114yescomplemented
310C2×C61C2×#39411yescharacteristic
311C2×C61C2×#47111yescharacteristic
312C2×C61C2×#47211yescharacteristic
313D61—82no—
314D61—84no—
315C22×#41—611no—
316C23×D4—611no—
317C2×AGL(1,ℤ/8)—614no—
318C2×AGL(1,ℤ/8)—1224no—
319C2×AGL(1,ℤ/8)—1222no—
320C2×AGL(1,ℤ/8)—1224no—
321C22×SD16—1221no—
322C22×D8—1221no—
323C22×M16—611no—
324C22×M16C2×D6111yescharacteristic
325C22×M16C2×D6111yescharacteristic
326C23×C8C2×D6111yescharacteristic
327C22×C4⋊C4—611no—
328C2×#108—24no—
329C2×#108C2×D6114yescomplemented
330#417—614no—
331#417—1222no—
332C2×#103—1221no—
333C2×#102—614no—
334C2×#102—1221no—
335C2×#102—1222no—
336C2×#102—1222no—
337C2×#100—614no—
338C2×#100—1224no—
339#408—614no—
340#408—1222no—
341C2×C22⋊C8—1222no—
342#343—614no—
343C61C2×#205111yescharacteristic, complemented, Sylow
344C25—1221no—
345C2×#41—612no—
346C2×#41—612no—
347C2×#41—1221no—
348C2×#41—1221no—
349C2×#41—1222no—
350C2×#41—1222no—
351C2×#41—1222no—
352C2×#41—1222no—
353C22×Q8—611no—
354C22×D4—611no—
355C22×D4—614no—
356C22×D4—612no—
357C22×D4—1221no—
358C22×D4—1221no—
359C22×D4—1221no—
360C22×D4—1222no—
361C22×D4—1221no—
362C22×D4—1221no—
363C22×D4—1222no—
364C22×D4—1221no—
365C22×D4—1222no—
366C23×C4—1221no—
367C23×C4—1221no—
368C23×C4C22×D6111yescharacteristic
369AGL(1,ℤ/8)—1228no—
370AGL(1,ℤ/8)—12216no—
371AGL(1,ℤ/8)—1228no—
372C2×SD16—1224no—
373C2×SD16—1228no—
374C2×D8—1224no—
375C2×D8—1228no—
376C2×M16—21no—
377C2×M16—22no—
378C2×M16—21no—
379C2×M16—22no—
380C2×M16—22no—
381C2×M16—612no—
382C2×M16—1221no—
383C2×M16—1222no—
384C2×M16—1222no—
385C2×M16C22⋊D6114yes—
386C2×M16C22×D6112yes—
387C2×M16C22×D6112yes—
388C22×C8—22no—
389C22×C8—21no—
390C22×C8—22no—
391C22×C8—21no—
392C22×C8—1221no—
393C22×C8D24412yes—
394C22×C8D24412yes—
395C22×C8C22×D6112yes—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402#108—28no—
403#103—1224no—
404#102—1224no—
405#102—1228no—
406#100—1228no—
407C22⋊C8—1228no—
408C24—1222no—
409C24—1224no—
410C24—1221no—
411C24—4882no—
412C24—4882no—
413C24—2441no—
414C24—2441no—
415C24—2442no—
416C24#1175822611yescharacteristic
417#41—1222no—
418#41—1222no—
419#41—1222no—
420#41—1222no—
421#41—1224no—
422#41—1224no—
423#41—2442no—
424#41—2442no—
425#41—2442no—
426#41—2442no—
427C2×Q8—612no—
428C2×Q8—1221no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×D4—614no—
432C2×D4—612no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1222no—
436C2×D4—1222no—
437C2×D4—1222no—
438C2×D4—1222no—
439C2×D4—1222no—
440C2×D4—1222no—
441C2×D4—1224no—
442C2×D4—1224no—
443C2×D4—1224no—
444C2×D4—1224no—
445C2×D4—1221no—
446C2×D4—1222no—
447C2×D4—1224no—
448C2×D4—2442no—
449C2×D4—2442no—
450C2×D4—2442no—
451C2×D4—2442no—
452C2×D4—2441no—
453C2×D4—2441no—
454C2×D4—2442no—
455C2×D4—2442no—
456C2×D4—2442no—
457C22×C4—1221no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—4882no—
462C22×C4—2441no—
463C22×C4—2441no—
464C22×C4—2442no—
465C22×C4—2442no—
466C22×C4C2×D24411yescharacteristic
467C22×C4C2×D24411yescharacteristic
468C22×C4#1175822212yes—
469C22×C4D4×D6112yes—
470C22×C4D4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4#1175822612yes—
473C22×C4#1175822611yescharacteristic
474C22×C4C2×C22⋊D6112yes—
475C22×C4C23×D6111yescharacteristic
476SD16—12216no—
477D8—12216no—
478M16—22no—
479M16—24no—
480M16—22no—
481M16—22no—
482M16—42no—
483M16—1222no—
484M16—1222no—
485M16—2442no—
486M16C2×C22⋊D6114yes—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—42no—
495C2×C8—1222no—
496C2×C8—2442no—
497C2×C8C2×D24414yes—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—2441no—
501C4⋊C4—2441no—
502C23—22no—
503C23—22no—
504C23—1224no—
505C23—1222no—
506C23—4882no—
507C23—4882no—
508C23—4888no—
509C23—4882no—
510C23—4882no—
511C23—4888no—
512C23—4882no—
513C23—4882no—
514C23—2442no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23—2442no—
519C23—2444no—
520C23—2444no—
521C23#1451975211yescharacteristic
522C23#1451975211yescharacteristic
523C23#1451976212yes—
524C23#1451976612yes—
525C23C2×#1175822611yescharacteristic
526Q8—1222no—
527Q8—1222no—
528Q8—2442no—
529D4—1222no—
530D4—1224no—
531D4—1224no—
532D4—1222no—
533D4—2442no—
534D4—2442no—
535D4—2444no—
536D4—2442no—
537D4—2442no—
538D4—2444no—
539D4—2444no—
540D4—2442no—
541C2×C4—1222no—
542C2×C4—4882no—
543C2×C4—4882no—
544C2×C4—4882no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2444no—
550C2×C4#1451974512yes—
551C2×C4#1451974512yes—
552C2×C4#1451975112yes—
553C2×C4#1451975112yes—
554C2×C4#1451975212yes—
555C2×C4#1451975212yes—
556C2×C4#1451975211yescharacteristic
557C2×C4#1451975211yescharacteristic
558C2×C4#1451976212yes—
559C2×C4#1451976212yes—
560C2×C4#1451976212yes—
561C2×C4#1451976612yes—
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4C22×D24411yescharacteristic
566C2×C4C2×#1175822211yescharacteristic
567C2×C4C2×#1175822611yescharacteristic
568C8—24no—
569C8—24no—
570C8—42no—
571C8—2442no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—24no—
577C22—24no—
578C22—4884no—
579C22—4884no—
580C22—4888no—
581C22—4882no—
582C22—4882no—
583C22—4888no—
584C22—4884no—
585C22—2442no—
586C22—2444no—
587C22—2442no—
588C22—2442no—
589C22#1683967912yes—
590C22#1683967912yes—
591C22#1684002612yes—
592C22#1684003312yes—
593C22C2×#1451975211yescharacteristic
594C22C2×#1451976211yescharacteristic
595C22C2×#1451976611yescharacteristic
596C4—4882no—
597C4—2442no—
598C4#1683967912yes—
599C4#1683967912yes—
600C4C2×#1451974511yescharacteristic
601C4C2×#1451975111yescharacteristic
602C4C2×#1451975211yescharacteristic
603C4C2×#1451976211yescharacteristic
604C2—21no—
605C2—21no—
606C2—22no—
607C2—4884no—
608C2—4882no—
609C2#2144711314yesdirect product factor
610C2C2×#1683967911yescharacteristic
611C2C2×#1684002611yescharacteristic
612C2C2×#1684003311yescharacteristic
subgroupG/H
centerC23#14519752
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#108×C61C2
socleC23×C61#121
prime radicalC22×#14519805C2
hypercenterC22×#108D61
hypocenterC61C2×#2051
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
2. C2×#16839764C2semidirect
3. C22×#14519699C2semidirect
4. C22×#14519693C2semidirect
5. C2×#16839728C2semidirect
6. #21447113C2direct (N × H)
7. C2×#16839530C2semidirect
8. C22×#108×C61C2semidirect
20. #16839764C22semidirect
25. C2×#14519699C22semidirect
27. C2×#14519693C22semidirect
29. #16839728C22semidirect
31. #16839530C22semidirect
36. C2×#108×C61C22semidirect
299. C22×#108D61semidirect
309. C2×C61#2051semidirect
329. C2×#108C2×D61semidirect
343. C61C2×#2051semidirect
609. C2#21447113direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198052
C2×#168397642
C22×#145196992
C22×#145196932
C2×#168397282
#214471132
C2×#168395302
C22×#108×C612
C2×#205161