Finite Group Databasebeta

← #29707898

Subgroups — #29707898

Order 15616.

1350 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840560C211yesmaximal, characteristic
2C22×#14519713C211yesmaximal, characteristic, complemented
3#21454336C211yesmaximal, characteristic, complemented
4C2×#16839746C211yesmaximal, characteristic, complemented
5C2×#16839728C211yesmaximal, characteristic, complemented
6#21447119C214yesmaximal, complemented
7#21447112C214yesmaximal, complemented
8#21445558C211yesmaximal, characteristic, complemented
9C2×C61×#416C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840560—24no—
12#16840560—24no—
13C2×#14519808C2212yes—
14C2×#14519807—21no—
15C2×#14519803C2212yes—
16C2×#14519802—21no—
17C2×#14519797—22no—
18C22×#11758194C2211yescharacteristic
19C22×#11758194C2211yescharacteristic
20C22×#11758194C2211yescharacteristic
21C2×#14519713—24no—
22C2×#14519713C2214yescomplemented
23#16839770—21no—
24#16839766—21no—
25#16839761C2214yes—
26#16839756—22no—
27#16839746—24no—
28C2×#14519696C2212yescomplemented
29C2×#14519694C2212yescomplemented
30#16839728—22no—
31#16839728C2214yes—
32C2×#14519689—22no—
33#16839531C2214yescomplemented
34C22×#41×C61C2211yescharacteristic
35C23×C8×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#416—22no—
38C61×#416—22no—
39C2×#103×C61C2212yescomplemented
40C2×#102×C61C2212yescomplemented
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222D412yes—
48C2×#11758222C2312yes—
49C22×Dic122D411yescharacteristic
50C22×D244D411yescharacteristic
51C22×C22⋊D61—21no—
52C22×C4×D61—21no—
53#14519808—24no—
54#14519808—24no—
55#14519807—28no—
56#14519803—24no—
57#14519803—24no—
58#14519802—28no—
59#14519797—216no—
60C2×#11758202—22no—
61C2×#11758200—22no—
62C2×#11758201—22no—
63C2×#11758199—22no—
64C2×#11758194—21no—
65C2×#11758194—22no—
66C2×#11758194—21no—
67C2×#11758194—21no—
68C2×#11758194—22no—
69C2×#11758194—22no—
70C2×#11758194—22no—
71C2×#11758194—22no—
72C2×#11758194D414yes—
73C2×#11758194C2312yes—
74C2×#11758194C2312yes—
75C2×#11758194C2312yes—
76C22×#92418—21no—
77C22×#92418—21no—
78C22×#92418—21no—
79#14519713—28no—
80#14519700—24no—
81#14519697—24no—
82#14519696—24no—
83#14519694—24no—
84#14519689—28no—
85C2×#41×C61—22no—
86C2×#41×C61—21no—
87C2×#41×C61—21no—
88C2×#41×C61—22no—
89C2×#41×C61—21no—
90C2×#41×C61—21no—
91C2×#41×C61—22no—
92C2×#41×C61D414yes—
93C22×Q8×C61C2311yescharacteristic
94C22×D4×C61—21no—
95C22×D4×C61C2311yescharacteristic
96C23×C4×C61—21no—
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61—22no—
99C22×C8×C61—22no—
100C22×C8×C61D412yes—
101C22×C8×C61D412yes—
102C22×C8×C61C2312yes—
103#117×C61—21no—
104#117×C61—21no—
105#117×C61—21no—
106#117×C61—21no—
107C2×C4⋊C4×C61C2312yes—
108C2×C22⋊C4×C61—21no—
109C2×C42×C61—21no—
110#103×C61—24no—
111#102×C61—24no—
112C24×C61—21no—
113C24×C61#4111yescharacteristic
114C23×D61—21no—
115#41×C61—24no—
116#41×C61—24no—
117#41×C61—24no—
118#41×C61—24no—
119#41×C61—42no—
120#41×C61—42no—
121#41×C61—42no—
122#41×C61—42no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61C2×D412yes—
129C2×D4×C61—24no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—41no—
136C2×D4×C61—41no—
137C2×D4×C61—42no—
138C2×D4×C61C2×D412yes—
139C22×C4×C61—21no—
140C22×C4×C61—24no—
141C22×C4×C61—22no—
142C22×C4×C61—41no—
143C22×C4×C61—41no—
144C22×C4×C61—42no—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61#4112yes—
152C22×C4×C61#4111yescharacteristic
153C22×C4×C61#4112yes—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—42no—
158C2×C22⋊D61—42no—
159C22×Dic61—21no—
160#11758222—24no—
161#11758222—24no—
162#11758222—22no—
163#11758222—22no—
164#11758222—22no—
165#11758222—22no—
166#11758222—42no—
167#11758222—42no—
168#11758222—42no—
169#11758222—42no—
170C2×D244—22no—
171C2×D244—22no—
172C2×D244—21no—
173C2×D244C2×D412yes—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×Dic122—22no—
179C2×Dic122—22no—
180C2×Dic122—21no—
181C2×Dic122C2×D412yes—
182C2×C8×C61—24no—
183C2×C8×C61—24no—
184C2×C8×C61—22no—
185C2×C8×C61—22no—
186C2×C8×C61C2×D414yes—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61—41no—
192C42×C61—41no—
193#11758202—28no—
194#11758201—28no—
195#11758200—28no—
196#11758199—28no—
197#11758194—24no—
198#11758194—22no—
199#11758194—22no—
200#11758194—22no—
201#11758194—22no—
202#11758194—22no—
203#11758194—42no—
204#11758194—42no—
205#11758194C2×D414yes—
206C2×#92418—24no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—22no—
210C2×#92418—42no—
211C2×#92418—42no—
212C23×C61—22no—
213C23×C61—42no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—82no—
217C23×C61#12111yescharacteristic
218C23×C61#12111yescharacteristic
219C23×C61#12312yes—
220C23×C61#12412yes—
221C23×C61C2×#4111yescharacteristic
222C22×D61—22no—
223C22×D61—44no—
224C22×D61—42no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—41no—
228Q8×C61—41no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—44no—
232D4×C61—44no—
233D4×C61—41no—
234D4×C61—41no—
235C2×C4×C61—22no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—44no—
239C2×C4×C61—44no—
240C2×C4×C61—42no—
241C2×C4×C61—82no—
242C2×C4×C61#12012yes—
243C2×C4×C61#12012yes—
244C2×C4×C61#12112yes—
245C2×C4×C61#12112yes—
246C2×C4×C61#12112yes—
247C2×C4×C61#12112yes—
248C2×C4×C61#12111yescharacteristic
249C2×C4×C61#12111yescharacteristic
250C2×C4×C61#12312yes—
251C2×C4×C61#12312yes—
252C2×C4×C61#12312yes—
253C2×C4×C61#12412yes—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C22⋊D61—42no—
261C22⋊D61—42no—
262C22⋊D61—44no—
263C2×Dic61—22no—
264C2×Dic61—42no—
265D244—22no—
266D244—22no—
267D244—42no—
268C4×D61—42no—
269C4×D61—42no—
270C4×D61—44no—
271Dic122—22no—
272Dic122—22no—
273Dic122—42no—
274C8×C61—24no—
275#92418—24no—
276#92418—42no—
277#92418—42no—
278#23530—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—82no—
286C22×C61—82no—
287C22×C61—82no—
288C22×C61#39412yes—
289C22×C61#39412yes—
290C22×C61C2×#12111yescharacteristic
291C22×C61C2×#12311yescharacteristic
292C22×C61C2×#12411yescharacteristic
293C2×D61—42no—
294C2×D61—44no—
295C2×D61—84no—
296C4×C61—42no—
297C4×C61—82no—
298C4×C61#39412yes—
299C4×C61#39412yes—
300C4×C61C2×#12011yescharacteristic
301C4×C61C2×#12111yescharacteristic
302C4×C61C2×#12111yescharacteristic
303C4×C61C2×#12311yescharacteristic
304Dic61—42no—
305C2×#575—611no—
306C22×#108—611no—
307#2896—611no—
308C2×#416D6111yescharacteristic, complemented
309C2×#413—611no—
310C2×#408—611no—
311#2052—614no—
312#2051—614no—
313#1742—611no—
314C2×C61—22no—
315C2×C61—21no—
316C2×C61—21no—
317C2×C61—22no—
318C2×C61—82no—
319C2×C61C2×#39411yescharacteristic
320C2×C61#306712yes—
321D61—82no—
322C22×#41—611no—
323C22×#41C2×D6111yescharacteristic
324#575—1224no—
325#575—1224no—
326C2×#137—612no—
327C2×#137—1221no—
328C2×AGL(1,ℤ/8)—612no—
329C2×AGL(1,ℤ/8)—1221no—
330C2×#135—1222no—
331C22×M16—611no—
332C22×M16—611no—
333C22×M16—611no—
334C23×C8C2×D6111yescharacteristic
335C2×#117C2×D6111yescharacteristic
336C2×#108—614no—
337C2×#108—1224no—
338#419—1221no—
339#418—1221no—
340#417—614no—
341#416—22no—
342#416—22no—
343#416—1222no—
344C2×#103C2×D6112yescomplemented
345C2×#102C2×D6112yescomplemented
346#413—1224no—
347C2×#101—612no—
348C2×#100—612no—
349#408—614no—
350#408—1222no—
351C2×C22⋊C8—1222no—
352#343—614no—
353C61#2353011yescharacteristic, complemented, Sylow
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—21no—
357C2×#41—21no—
358C2×#41—22no—
359C2×#41—21no—
360C2×#41—21no—
361C2×#41—612no—
362C2×#41—612no—
363C2×#41—1221no—
364C2×#41—1221no—
365C2×#41—1222no—
366C2×#41—1222no—
367C2×#41—1222no—
368C2×#41—1222no—
369C2×#41D24414yes—
370C22×Q8—611no—
371C22×Q8C22×D6111yescharacteristic
372C22×D4—21no—
373C22×D4—611no—
374C22×D4—1221no—
375C22×D4C22×D6111yescharacteristic
376C23×C4—21no—
377C23×C4—1221no—
378C23×C4C22×D6111yescharacteristic
379#137—1224no—
380#137—1228no—
381#137—1224no—
382AGL(1,ℤ/8)—1224no—
383AGL(1,ℤ/8)—1228no—
384AGL(1,ℤ/8)—1224no—
385#135—12216no—
386C2×Q16—1222no—
387C2×SD16—1222no—
388C2×SD16—1222no—
389C2×D8—1222no—
390C2×M16—614no—
391C2×M16—612no—
392C2×M16—612no—
393C2×M16—612no—
394C2×M16—1222no—
395C2×M16—1221no—
396C2×M16—1221no—
397C2×M16—1221no—
398C2×M16—1222no—
399C2×M16—1222no—
400C2×M16—1222no—
401C2×M16—1222no—
402C22×C8—22no—
403C22×C8—22no—
404C22×C8—1221no—
405C22×C8—1221no—
406C22×C8—1221no—
407C22×C8C22⋊D6112yes—
408C22×C8C22⋊D6112yes—
409C22×C8C22×D6112yes—
410#117—21no—
411#117—21no—
412#117—21no—
413#117—21no—
414C2×C4⋊C4C22×D6112yes—
415C2×C22⋊C4—21no—
416C2×C42—21no—
417#108—1228no—
418#103—24no—
419#103—1224no—
420#102—24no—
421#102—1224no—
422#101—1224no—
423#100—1224no—
424C22⋊C8—1228no—
425C24—21no—
426C24—1221no—
427C24#1175822411yescharacteristic
428#41—24no—
429#41—24no—
430#41—24no—
431#41—24no—
432#41—42no—
433#41—42no—
434#41—42no—
435#41—42no—
436#41—1222no—
437#41—1222no—
438#41—1222no—
439#41—1222no—
440#41—1224no—
441#41—1224no—
442#41—2442no—
443#41—2442no—
444#41—2442no—
445#41—2442no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—612no—
452C2×Q8—1221no—
453C2×Q8—1222no—
454C2×Q8—1222no—
455C2×Q8C2×D24412yes—
456C2×D4—24no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—42no—
465C2×D4—612no—
466C2×D4—1222no—
467C2×D4—1222no—
468C2×D4—1222no—
469C2×D4—1221no—
470C2×D4—1222no—
471C2×D4—2442no—
472C2×D4—2442no—
473C2×D4C2×D24412yes—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—42no—
480C22×C4—1221no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4C2×D24412yes—
486C22×C4#1175822212yes—
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4D4×D6112yes—
490C22×C4#1175822412yes—
491C22×C4#1175822411yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C23×D6111yescharacteristic
495Q16—1228no—
496SD16—1228no—
497SD16—1228no—
498D8—1228no—
499M16—614no—
500M16—1222no—
501M16—1224no—
502M16—1222no—
503M16—1222no—
504M16—1222no—
505M16—1222no—
506M16—2442no—
507M16—2442no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—1222no—
513C2×C8—1224no—
514C2×C8—1222no—
515C2×C8—1222no—
516C2×C8—2442no—
517C2×C8—2442no—
518C2×C8C2×C22⋊D6114yes—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C42—41no—
524C42—41no—
525C23—22no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—82no—
530C23—1222no—
531C23—2444no—
532C23—2442no—
533C23#1451974811yescharacteristic
534C23#1451974811yescharacteristic
535C23#1451976112yes—
536C23#1451976312yes—
537C23C2×#1175822411yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—41no—
541Q8—41no—
542Q8—1222no—
543Q8—1222no—
544Q8—2442no—
545D4—22no—
546D4—22no—
547D4—44no—
548D4—44no—
549D4—41no—
550D4—41no—
551D4—1222no—
552D4—1222no—
553D4—2442no—
554D4—2442no—
555D4—2444no—
556D4—2442no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—82no—
564C2×C4—1222no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2444no—
569C2×C4#1451974712yes—
570C2×C4#1451974712yes—
571C2×C4#1451974811yescharacteristic
572C2×C4#1451974811yescharacteristic
573C2×C4#1451974912yes—
574C2×C4#1451974912yes—
575C2×C4#1451975112yes—
576C2×C4#1451975112yes—
577C2×C4#1451975912yes—
578C2×C4#1451975912yes—
579C2×C4#1451976112yes—
580C2×C4#1451976312yes—
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4C22×D24411yescharacteristic
585C2×C4C2×#1175822411yescharacteristic
586C2×C4C2×#1175822211yescharacteristic
587C8—24no—
588C8—1224no—
589C8—2442no—
590C8—2442no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—4884no—
601C22—2442no—
602C22—2444no—
603C22#1683967812yes—
604C22#1683967812yes—
605C22C2×#1451974811yescharacteristic
606C22C2×#1451976111yescharacteristic
607C22C2×#1451976311yescharacteristic
608C4—42no—
609C4—82no—
610C4—2442no—
611C4#1683967812yes—
612C4#1683967812yes—
613C4C2×#1451974711yescharacteristic
614C4C2×#1451974911yescharacteristic
615C4C2×#1451975111yescharacteristic
616C4C2×#1451975911yescharacteristic
617C2—21no—
618C2—21no—
619C2—22no—
620C2—22no—
621C2—82no—
622C2—4882no—
623C2C2×#1683967811yescharacteristic
624C2#2145553512yes—
subgroupG/H
centerC2×C4#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC22×C61C2×#121
prime radicalC2×#16840560C2
hypercenterC2×#416D61
hypocenterC61#23530
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C22×#14519713C2semidirect
3. #21454336C2semidirect
4. C2×#16839746C2semidirect
5. C2×#16839728C2semidirect
6. #21447119C2semidirect
7. #21447112C2semidirect
8. #21445558C2semidirect
9. C2×C61×#416C2semidirect
22. C2×#14519713C22semidirect
28. C2×#14519696C22semidirect
29. C2×#14519694C22semidirect
33. #16839531C22semidirect
39. C2×#103×C61C22semidirect
40. C2×#102×C61C22semidirect
308. C2×#416D61semidirect
344. C2×#103C2×D61semidirect
345. C2×#102C2×D61semidirect
353. C61#23530semidirect

Maximal subgroups

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

subgroupindex
C2×#168405602
C22×#145197132
#214543362
C2×#168397462
C2×#168397282
#214471192
#214471122
#214455582
C2×C61×#4162
#2353061