Finite Group Databasebeta

← #29707919

Subgroups — #29707919

Order 15616.

1358 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840560C211yesmaximal, characteristic
2C2×#16839810C211yesmaximal, characteristic, complemented
3C2×#16839761C211yesmaximal, characteristic, complemented
4C2×#16839744C211yesmaximal, characteristic, complemented
5#21454264C211yesmaximal, characteristic, complemented
6#21447116C214yesmaximal, complemented
7#21447109C214yesmaximal, complemented
8C2×#16839531C211yesmaximal, characteristic, complemented
9C2×C61×#417C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840560—24no—
12#16840560—24no—
13C2×#14519808—21no—
14C2×#14519807C2212yes—
15C2×#14519803—21no—
16C2×#14519802C2212yes—
17C2×#14519797—22no—
18C22×#11758194C2211yescharacteristic
19C22×#11758194C2211yescharacteristic
20C22×#11758194C2211yescharacteristic
21#16839810—24no—
22C2×#14519712C2214yescomplemented
23#16839761—22no—
24#16839761—22no—
25C2×#14519700C2212yescomplemented
26C2×#14519697C2212yescomplemented
27#16839744—24no—
28C2×#14519695C2212yescomplemented
29C2×#14519693C2212yescomplemented
30#16839729—22no—
31#16839729C2214yes—
32#16839726—22no—
33#16839531C2214yescomplemented
34C22×#41×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C61×#417—22no—
38C61×#417C2214yes—
39C2×#103×C61—21no—
40C2×#102×C61—21no—
41C2×#11758222—22no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222D414yes—
49C22×Dic122C2311yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C4×D61—21no—
53#14519808—28no—
54#14519807—24no—
55#14519807—24no—
56#14519803—28no—
57#14519802—24no—
58#14519802—24no—
59#14519797—216no—
60C2×#11758202—22no—
61C2×#11758200—22no—
62C2×#11758201—22no—
63C2×#11758199—22no—
64C2×#11758194—21no—
65C2×#11758194—22no—
66C2×#11758194—22no—
67C2×#11758194—22no—
68C2×#11758194—21no—
69C2×#11758194—22no—
70C2×#11758194—22no—
71C2×#11758194D414yes—
72C2×#11758194D412yes—
73C2×#11758194D412yes—
74C2×#11758194C2312yes—
75C2×#11758194C2312yes—
76C22×#92418—21no—
77C22×#92418—21no—
78C22×#92418C2312yes—
79#14519712—28no—
80#14519700—24no—
81#14519697—24no—
82#14519695—24no—
83#14519693—24no—
84C22⋊#92418—28no—
85C2×#41×C61—21no—
86C2×#41×C61—21no—
87C2×#41×C61—22no—
88C2×#41×C61—22no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61D412yes—
92C2×#41×C61C2312yes—
93C22×Q8×C61D411yescharacteristic
94C22×D4×C61—21no—
95C22×D4×C61D411yescharacteristic
96C23×C4×C61—21no—
97C23×C4×C61—21no—
98C23×C4×C61C2311yescharacteristic
99C2×M16×C61—21no—
100C2×M16×C61—22no—
101C2×M16×C61—22no—
102C2×M16×C61C2312yes—
103C22×C8×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61C2312yes—
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—22no—
118#41×C61—22no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—42no—
122#41×C61—42no—
123#41×C61—42no—
124#41×C61—42no—
125C2×Q8×C61—21no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61C2×D412yes—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61—22no—
134C2×D4×C61—42no—
135C2×D4×C61—42no—
136C2×D4×C61C2×D412yes—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—41no—
144C22×C4×C61—41no—
145C22×C4×C61—82no—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61#4112yes—
153C22×C4×C61#4111yescharacteristic
154C22×C4×C61#4112yes—
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—24no—
157C2×C22⋊D61—41no—
158C2×C22⋊D61—41no—
159C2×C22⋊D61—42no—
160C22×Dic61—21no—
161#11758222—24no—
162#11758222—24no—
163#11758222—24no—
164#11758222—24no—
165#11758222—42no—
166#11758222—42no—
167#11758222—42no—
168#11758222—42no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244C2×D412yes—
175C2×C4×D61—24no—
176C2×C4×D61—41no—
177C2×C4×D61—41no—
178C2×C4×D61—42no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122C2×D412yes—
185M16×C61—22no—
186M16×C61—22no—
187M16×C61—42no—
188C2×C8×C61—22no—
189C2×C8×C61—42no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—41no—
193C4⋊C4×C61—41no—
194#11758202—28no—
195#11758201—28no—
196#11758200—28no—
197#11758199—28no—
198#11758194—24no—
199#11758194—24no—
200#11758194—24no—
201#11758194—22no—
202#11758194—22no—
203#11758194—22no—
204#11758194—42no—
205#11758194C2×D414yes—
206C2×#92418—24no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—22no—
210C2×#92418—22no—
211C2×#92418—42no—
212C2×#92418C2×D414yes—
213C23×C61—22no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—42no—
217C23×C61—81no—
218C23×C61—81no—
219C23×C61#12111yescharacteristic
220C23×C61#12111yescharacteristic
221C23×C61#12312yes—
222C23×C61#12412yes—
223C23×C61C2×#4111yescharacteristic
224C22×D61—22no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—81no—
228C22×D61—81no—
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—42no—
232D4×C61—22no—
233D4×C61—22no—
234D4×C61—42no—
235D4×C61—42no—
236D4×C61—44no—
237D4×C61—42no—
238C2×C4×C61—22no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—44no—
243C2×C4×C61—42no—
244C2×C4×C61—82no—
245C2×C4×C61—82no—
246C2×C4×C61—82no—
247C2×C4×C61#12012yes—
248C2×C4×C61#12012yes—
249C2×C4×C61#12112yes—
250C2×C4×C61#12112yes—
251C2×C4×C61#12112yes—
252C2×C4×C61#12112yes—
253C2×C4×C61#12111yescharacteristic
254C2×C4×C61#12111yescharacteristic
255C2×C4×C61#12312yes—
256C2×C4×C61#12312yes—
257C2×C4×C61#12312yes—
258C2×C4×C61#12412yes—
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C22⋊D61—44no—
266C22⋊D61—44no—
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270D244—22no—
271D244—22no—
272D244—41no—
273D244—41no—
274C4×D61—44no—
275C4×D61—44no—
276Dic122—22no—
277Dic122—22no—
278Dic122—41no—
279Dic122—41no—
280C8×C61—42no—
281#92418—24no—
282#92418—24no—
283#92418—42no—
284#23533—611nomaximal, Sylow
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—84no—
297C22×C61#39412yes—
298C22×C61#39412yes—
299C22×C61C2×#12111yescharacteristic
300C22×C61C2×#12311yescharacteristic
301C22×C61C2×#12411yescharacteristic
302C2×D61—42no—
303C2×D61—82no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—82no—
307C4×C61—42no—
308C4×C61—82no—
309C4×C61#39412yes—
310C4×C61#39412yes—
311C4×C61C2×#12011yescharacteristic
312C4×C61C2×#12111yescharacteristic
313C4×C61C2×#12111yescharacteristic
314C4×C61C2×#12311yescharacteristic
315Dic61—42no—
316C2×#575—611no—
317C2×#430—611no—
318C2×#417—611no—
319C2×#417D6111yescharacteristic, complemented
320C2×#413—611no—
321#2879—611no—
322#2052—614no—
323#2051—614no—
324C2×#343—611no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—22no—
328C2×C61—22no—
329C2×C61—82no—
330C2×C61C2×#39411yescharacteristic
331C2×C61#306811yescharacteristic
332C2×C61#306911yescharacteristic
333D61—82no—
334C22×#41—611no—
335C22×#41C2×D6111yescharacteristic
336#575—1224no—
337#575—1224no—
338C2×#137—612no—
339C2×#137—1221no—
340C2×AGL(1,ℤ/8)—612no—
341C2×AGL(1,ℤ/8)—1221no—
342C2×#135—1222no—
343C22×M16—611no—
344C22×M16—611no—
345C22×M16—611no—
346C22×M16C2×D6111yescharacteristic
347C22×C4⋊C4C2×D6111yescharacteristic
348#430—1224no—
349C2×#108—614no—
350#417—22no—
351#417—1222no—
352#417—1222no—
353#417C2×D6114yes—
354C2×#103—21no—
355C2×#103—612no—
356C2×#102—21no—
357C2×#102—612no—
358#413—1224no—
359C2×#101—612no—
360C2×#100—612no—
361#408—614no—
362#408—1222no—
363#407—1222no—
364#343—614no—
365C61#2353311yescharacteristic, complemented, Sylow
366C2×#41—21no—
367C2×#41—21no—
368C2×#41—22no—
369C2×#41—22no—
370C2×#41—22no—
371C2×#41—22no—
372C2×#41—614no—
373C2×#41—1222no—
374C2×#41—1222no—
375C2×#41—1222no—
376C2×#41—1221no—
377C2×#41—1221no—
378C2×#41—1221no—
379C2×#41—1221no—
380C2×#41C22⋊D6112yes—
381C2×#41C22×D6112yes—
382C22×Q8—611no—
383C22×Q8C22⋊D6111yescharacteristic
384C22×D4—21no—
385C22×D4—611no—
386C22×D4—1221no—
387C22×D4C22⋊D6111yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392#137—1224no—
393#137—1228no—
394#137—1224no—
395AGL(1,ℤ/8)—1224no—
396AGL(1,ℤ/8)—1228no—
397AGL(1,ℤ/8)—1224no—
398#135—12216no—
399C2×Q16—1222no—
400C2×SD16—1222no—
401C2×SD16—1222no—
402C2×D8—1222no—
403C2×M16—21no—
404C2×M16—22no—
405C2×M16—22no—
406C2×M16—612no—
407C2×M16—612no—
408C2×M16—614no—
409C2×M16—612no—
410C2×M16—612no—
411C2×M16—1222no—
412C2×M16—1222no—
413C2×M16—1222no—
414C2×M16—1221no—
415C2×M16—1221no—
416C2×M16—1222no—
417C2×M16—1222no—
418C2×M16C22×D6112yes—
419C22×C8—21no—
420C22×C8—612no—
421C22×C8—1221no—
422C22×C8—1221no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4C22×D6112yes—
429#108—1228no—
430#103—24no—
431#103—1224no—
432#102—24no—
433#102—1224no—
434#101—1224no—
435#100—1224no—
436C22⋊C8—1228no—
437C24—21no—
438C24—1221no—
439C24#1175822411yescharacteristic
440#41—22no—
441#41—22no—
442#41—22no—
443#41—22no—
444#41—24no—
445#41—24no—
446#41—42no—
447#41—42no—
448#41—42no—
449#41—42no—
450#41—1224no—
451#41—1224no—
452#41—1224no—
453#41—1224no—
454#41—2442no—
455#41—2442no—
456#41—2442no—
457#41—2442no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—612no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8C2×C22⋊D6112yes—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—22no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—612no—
476C2×D4—1224no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—2441no—
483C2×D4—2441no—
484C2×D4—2442no—
485C2×D4C2×C22⋊D6112yes—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—82no—
495C22×C4—1221no—
496C22×C4—1224no—
497C22×C4—2441no—
498C22×C4—2441no—
499C22×C4—2442no—
500C22×C4#1175822212yes—
501C22×C4#1175822212yes—
502C22×C4D4×D6112yes—
503C22×C4D4×D6112yes—
504C22×C4D4×D6111yescharacteristic
505C22×C4D4×D6111yescharacteristic
506C22×C4#1175822411yescharacteristic
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510Q16—1228no—
511SD16—1228no—
512SD16—1228no—
513D8—1228no—
514M16—22no—
515M16—22no—
516M16—42no—
517M16—614no—
518M16—1222no—
519M16—1224no—
520M16—1224no—
521M16—1224no—
522M16—1222no—
523M16—1222no—
524M16—2442no—
525C2×C8—22no—
526C2×C8—42no—
527C2×C8—614no—
528C2×C8—1222no—
529C2×C8—1222no—
530C2×C8—1222no—
531C2×C8—1224no—
532C2×C8—1222no—
533C2×C8—2442no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—81no—
543C23—81no—
544C23—1222no—
545C23—4881no—
546C23—4881no—
547C23—2441no—
548C23—2441no—
549C23#1451974911yescharacteristic
550C23#1451974911yescharacteristic
551C23#1451975912yes—
552C23#1451976312yes—
553C23C2×#1175822411yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—42no—
557Q8—1222no—
558Q8—1222no—
559Q8—2441no—
560Q8—2441no—
561D4—22no—
562D4—22no—
563D4—42no—
564D4—42no—
565D4—44no—
566D4—42no—
567D4—1222no—
568D4—1222no—
569D4—2444no—
570D4—2444no—
571D4—2441no—
572D4—2441no—
573C2×C4—22no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—1222no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2444no—
586C2×C4—2444no—
587C2×C4#1451974512yes—
588C2×C4#1451974512yes—
589C2×C4#1451974911yescharacteristic
590C2×C4#1451974911yescharacteristic
591C2×C4#1451975012yes—
592C2×C4#1451975012yes—
593C2×C4#1451975012yes—
594C2×C4#1451975012yes—
595C2×C4#1451975912yes—
596C2×C4#1451976012yes—
597C2×C4#1451976012yes—
598C2×C4#1451976312yes—
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4C22×C22⋊D6111yescharacteristic
603C2×C4C2×#1175822211yescharacteristic
604C2×C4C2×#1175822211yescharacteristic
605C8—42no—
606C8—1224no—
607C8—1224no—
608C8—2442no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—84no—
621C22—4882no—
622C22—4882no—
623C22—4881no—
624C22—4881no—
625C22—2442no—
626C22#1683967712yes—
627C22#1683967712yes—
628C22C2×#1451974911yescharacteristic
629C22C2×#1451975911yescharacteristic
630C22C2×#1451976311yescharacteristic
631C4—42no—
632C4—82no—
633C4—2442no—
634C4#1683967712yes—
635C4#1683967712yes—
636C4C2×#1451974511yescharacteristic
637C4C2×#1451975011yescharacteristic
638C4C2×#1451975011yescharacteristic
639C4C2×#1451976011yescharacteristic
640C2—22no—
641C2—21no—
642C2—21no—
643C2—22no—
644C2—82no—
645C2—4882no—
646C2C2×#1683967711yescharacteristic
647C2#2145554411yescharacteristic
648C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC22×C61C2×#121
prime radicalC2×#16840560C2
hypercenterC2×#417D61
hypocenterC61#23533
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#16839810C2semidirect
3. C2×#16839761C2semidirect
4. C2×#16839744C2semidirect
5. #21454264C2semidirect
6. #21447116C2semidirect
7. #21447109C2semidirect
8. C2×#16839531C2semidirect
9. C2×C61×#417C2semidirect
22. C2×#14519712C22semidirect
25. C2×#14519700C22semidirect
26. C2×#14519697C22semidirect
28. C2×#14519695C22semidirect
29. C2×#14519693C22semidirect
33. #16839531C22semidirect
319. C2×#417D61semidirect
365. C61#23533semidirect

Maximal subgroups

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

subgroupindex
C2×#168405602
C2×#168398102
C2×#168397612
C2×#168397442
#214542642
#214471162
#214471092
C2×#168395312
C2×C61×#4172
#2353361