Finite Group Databasebeta

← #29707659

Subgroups — #29707659

Order 15616.

1450 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459148C211yesmaximal, characteristic
2C22×#14519704C211yesmaximal, characteristic, complemented
3C22×#14519703C211yesmaximal, characteristic, complemented
4#21454336C211yesmaximal, characteristic, complemented
5#21448222C211yesmaximal, characteristic, complemented
6#21447046C214yesmaximal, complemented
7#21447041C214yesmaximal, complemented
8#21445557C211yesmaximal, characteristic, complemented
9C61×#2896C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840340C2214yes—
13#16840336—21no—
14#16840333—21no—
15#16840329—21no—
16#16840326—21no—
17#16840321—22no—
18#16840308—22no—
19#16840273—22no—
20#16840269—22no—
21C2×#14519736C2211yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23C2×#14519704—24no—
24C2×#14519704C2212yescomplemented
25C2×#14519704C2212yescomplemented
26C2×#14519703—24no—
27C2×#14519703C2212yescomplemented
28C2×#14519703C2212yescomplemented
29#16839770—21no—
30#16839766—21no—
31#16839761C2214yescomplemented
32#16839756—22no—
33#16839631C2212yescomplemented
34#16839627C2212yescomplemented
35#16839620—22no—
36C4×#11758196—22no—
37#16839530C2214yescomplemented
38C22×#41×C61C2211yescharacteristic
39C22×M16×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C61×#419—21no—
42C61×#418—21no—
43C61×#417C2214yescomplemented
44C61×#416—22no—
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222D412yes—
52C2×#11758222C2312yes—
53C22×Dic122D411yescharacteristic
54C22×D244D411yescharacteristic
55C22×C22⋊D61—21no—
56C23×Dic61—21no—
57C22×C4×D61—21no—
58C2×#11758194—21no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194C2312yes—
62C22×#92418—21no—
63#14519775—22no—
64#14519773—22no—
65#14519766—22no—
66#14519764—22no—
67#14519760—24no—
68#14519758—24no—
69#14519756—24no—
70#14519753—24no—
71#14519751—24no—
72#14519749—28no—
73#14519748—24no—
74#14519746—24no—
75#14519745—24no—
76C4×C22⋊D61—28no—
77#14519736—21no—
78#14519736—21no—
79#14519736—21no—
80#14519736—21no—
81C2×C4⋊Dic61C2312yes—
82C2×#11758196—24no—
83C2×C22⋊Dic61D412yes—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61D412yes—
87C2×C4×Dic61D414yescomplemented
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519704—24no—
91#14519704—24no—
92#14519703—24no—
93#14519703—24no—
94#14519700—24no—
95#14519697—24no—
96#14519684—24no—
97C2×#41×C61—21no—
98C2×#41×C61—21no—
99C2×#41×C61—22no—
100C2×#41×C61—22no—
101C2×#41×C61—22no—
102C2×#41×C61—22no—
103C2×#41×C61D412yes—
104C2×#41×C61C2312yes—
105C22×Q8×C61D411yescharacteristic
106C22×D4×C61—21no—
107C22×D4×C61D411yescharacteristic
108C23×C4×C61—21no—
109C23×C4×C61C2311yescharacteristic
110C2×M16×C61—21no—
111C2×M16×C61—22no—
112C2×M16×C61—22no—
113C2×M16×C61C2312yes—
114C22×C8×C61—21no—
115#117×C61—21no—
116#117×C61—21no—
117#117×C61—21no—
118#117×C61—21no—
119C2×C4⋊C4×C61C2312yes—
120C2×C22⋊C4×C61—21no—
121C2×C42×C61—21no—
122#103×C61—24no—
123#102×C61—24no—
124C24×C61—21no—
125C24×C61C2×D411yescharacteristic
126C23×D61—21no—
127#41×C61—24no—
128#41×C61—24no—
129#41×C61—22no—
130#41×C61—22no—
131#41×C61—22no—
132#41×C61—22no—
133#41×C61—42no—
134#41×C61—42no—
135#41×C61—42no—
136#41×C61—42no—
137C2×Q8×C61—21no—
138C2×Q8×C61—22no—
139C2×Q8×C61—22no—
140C2×Q8×C61C2×D412yes—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—21no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—42no—
147C2×D4×C61—42no—
148C2×D4×C61C2×D412yes—
149C22×C4×C61—21no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—22no—
153C22×C4×C61—42no—
154C22×C4×C61—42no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D412yes—
161C22×C4×C61#4112yes—
162C22×C4×C61#4112yes—
163C22×C4×C61#4112yes—
164C22×C4×C61C2411yescharacteristic
165C2×C22⋊D61—22no—
166C2×C22⋊D61—22no—
167C2×C22⋊D61—28no—
168C2×C22⋊D61—42no—
169C2×C22⋊D61—42no—
170C22×Dic61—21no—
171C22×Dic61—21no—
172C22×Dic61—24no—
173C22×Dic61—21no—
174C22×Dic61—44no—
175C22×Dic61C2×D414yes—
176#11758222—24no—
177#11758222—24no—
178#11758222—22no—
179#11758222—22no—
180#11758222—22no—
181#11758222—22no—
182#11758222—42no—
183#11758222—42no—
184#11758222—42no—
185#11758222—42no—
186C2×D244—22no—
187C2×D244—22no—
188C2×D244—21no—
189C2×D244C2×D412yes—
190C2×C4×D61—22no—
191C2×C4×D61—22no—
192C2×C4×D61—42no—
193C2×C4×D61—42no—
194C2×Dic122—22no—
195C2×Dic122—22no—
196C2×Dic122—21no—
197C2×Dic122C2×D412yes—
198M16×C61—22no—
199M16×C61—22no—
200M16×C61—42no—
201C2×C8×C61—22no—
202C2×C8×C61—42no—
203C4⋊C4×C61—22no—
204C4⋊C4×C61—22no—
205C22⋊C4×C61—41no—
206C22⋊C4×C61—41no—
207C42×C61—41no—
208C42×C61—41no—
209C22⋊Dic61—28no—
210C22⋊Dic61—22no—
211C22⋊Dic61—22no—
212#11758198—28no—
213C4⋊Dic61—22no—
214C4⋊Dic61—22no—
215#11758196—28no—
216#11758196—48no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—24no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61C2×D412yes—
223C4×Dic61C2×D412yes—
224#11758194—22no—
225#11758194—22no—
226#11758194—42no—
227C2×#92418—22no—
228C2×#92418—42no—
229C23×C61—22no—
230C23×C61—42no—
231C23×C61—42no—
232C23×C61—82no—
233C23×C61#12011yescharacteristic
234C23×C61#12011yescharacteristic
235C23×C61#12112yes—
236C23×C61#12412yes—
237C23×C61C22×D411yescharacteristic
238C22×D61—22no—
239C22×D61—42no—
240C22×D61—82no—
241Q8×C61—22no—
242Q8×C61—22no—
243Q8×C61—42no—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—42no—
247D4×C61—42no—
248D4×C61—44no—
249D4×C61—42no—
250C2×C4×C61—22no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—44no—
255C2×C4×C61—44no—
256C2×C4×C61—42no—
257C2×C4×C61#12011yescharacteristic
258C2×C4×C61#12011yescharacteristic
259C2×C4×C61#12112yes—
260C2×C4×C61#12112yes—
261C2×C4×C61#12112yes—
262C2×C4×C61#12112yes—
263C2×C4×C61#12112yes—
264C2×C4×C61#12312yes—
265C2×C4×C61#12312yes—
266C2×C4×C61#12312yes—
267C2×C4×C61#12312yes—
268C2×C4×C61#12412yes—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C22⋊D61—42no—
276C22⋊D61—42no—
277C22⋊D61—44no—
278C22⋊D61—88no—
279C2×Dic61—24no—
280C2×Dic61—22no—
281C2×Dic61—24no—
282C2×Dic61—22no—
283C2×Dic61—22no—
284C2×Dic61—48no—
285C2×Dic61—42no—
286C2×Dic61—42no—
287C2×Dic61—44no—
288D244—22no—
289D244—22no—
290D244—42no—
291C4×D61—42no—
292C4×D61—42no—
293C4×D61—44no—
294Dic122—22no—
295Dic122—22no—
296Dic122—42no—
297C8×C61—42no—
298#92418—42no—
299#23498—611nomaximal, Sylow
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—44no—
307C22×C61—84no—
308C22×C61#39412yes—
309C22×C61#39412yes—
310C22×C61C2×#12011yescharacteristic
311C22×C61C2×#12111yescharacteristic
312C22×C61C2×#12411yescharacteristic
313C2×D61—42no—
314C2×D61—44no—
315C2×D61—84no—
316C4×C61—42no—
317C4×C61—82no—
318C4×C61#39412yes—
319C4×C61#39412yes—
320C4×C61C2×#12111yescharacteristic
321C4×C61C2×#12111yescharacteristic
322C4×C61C2×#12311yescharacteristic
323C4×C61C2×#12311yescharacteristic
324Dic61—24no—
325Dic61—42no—
326Dic61—42no—
327Dic61—44no—
328#3449—611no—
329C22×#104—611no—
330C22×#104—611no—
331#2896—611no—
332#2896D6111yescharacteristic, complemented
333#2285—611no—
334#2040—614no—
335#2039—614no—
336#1742—611no—
337C2×C61—21no—
338C2×C61—22no—
339C2×C61—22no—
340C2×C61—21no—
341C2×C61—82no—
342C2×C61C2×#39411yescharacteristic
343C2×C61#301812yes—
344D61—82no—
345C22×#41—611no—
346C22×#41C2×D6111yescharacteristic
347C22×M16—611no—
348C22×M16C2×D6111yescharacteristic
349#537—614no—
350#536—1221no—
351#536—1221no—
352#535—1221no—
353#535—1221no—
354#534—1222no—
355#532—1222no—
356#525—1222no—
357#525—1222no—
358C2×#117—611no—
359C2×#117C2×D6111yescharacteristic
360C22×C42—611no—
361C2×#104—612no—
362C2×#104—612no—
363C2×#104—612no—
364C2×#104—612no—
365C2×#104—1224no—
366C2×#104—1224no—
367#419—21no—
368#419—1221no—
369#418—21no—
370#418—1221no—
371#417—614no—
372#417C2×D6114yescomplemented
373#416—22no—
374#416—1222no—
375#384—612no—
376#382—612no—
377#380—1222no—
378C4×C4⋊C4—1222no—
379#343—614no—
380C61#2349811yescharacteristic, complemented, Sylow
381C2×#41—21no—
382C2×#41—21no—
383C2×#41—22no—
384C2×#41—22no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—612no—
388C2×#41—612no—
389C2×#41—1221no—
390C2×#41—1221no—
391C2×#41—1222no—
392C2×#41—1222no—
393C2×#41—1222no—
394C2×#41—1222no—
395C2×#41C22⋊D6112yes—
396C2×#41C22×D6112yes—
397C22×Q8—611no—
398C22×Q8C22⋊D6111yescharacteristic
399C22×D4—21no—
400C22×D4—611no—
401C22×D4—1221no—
402C22×D4C22⋊D6111yescharacteristic
403C23×C4—21no—
404C23×C4—1221no—
405C23×C4—1221no—
406C23×C4C22×D6111yescharacteristic
407C2×M16—21no—
408C2×M16—22no—
409C2×M16—22no—
410C2×M16—612no—
411C2×M16—1221no—
412C2×M16—1222no—
413C2×M16—1222no—
414C2×M16C22×D6112yes—
415C22×C8—21no—
416C22×C8—1221no—
417C4⋊Q8—1222no—
418#127—1222no—
419#124—1222no—
420#124—1222no—
421#123—1224no—
422#123—1224no—
423C22⋊Q8—1224no—
424C22⋊Q8—1224no—
425#121—1224no—
426#121—1228no—
427#121—1224no—
428#120—1224no—
429#120—1224no—
430C4×D4—1228no—
431#117—21no—
432#117—21no—
433#117—21no—
434#117—21no—
435#117—1221no—
436#117—1221no—
437#117—1221no—
438#117—1221no—
439C2×C4⋊C4—612no—
440C2×C4⋊C4—1224no—
441C2×C4⋊C4C22×D6112yes—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—612no—
444C2×C42—21no—
445C2×C42—614no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—612no—
449C2×C42—1222no—
450C2×C42—1221no—
451#104—1224no—
452#104—1224no—
453#104—1224no—
454#104—1224no—
455#103—24no—
456#103—1224no—
457#102—24no—
458#102—1224no—
459#95—1224no—
460C24—21no—
461C24—1221no—
462C24D4×D6111yescharacteristic
463#41—22no—
464#41—22no—
465#41—22no—
466#41—22no—
467#41—24no—
468#41—24no—
469#41—42no—
470#41—42no—
471#41—42no—
472#41—42no—
473#41—1222no—
474#41—1222no—
475#41—1222no—
476#41—1222no—
477#41—1224no—
478#41—1224no—
479#41—2442no—
480#41—2442no—
481#41—2442no—
482#41—2442no—
483C2×Q8—21no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—612no—
487C2×Q8—1221no—
488C2×Q8—1222no—
489C2×Q8—1222no—
490C2×Q8C2×C22⋊D6112yes—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—21no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—42no—
497C2×D4—42no—
498C2×D4—612no—
499C2×D4—1228no—
500C2×D4—1222no—
501C2×D4—1222no—
502C2×D4—1221no—
503C2×D4—1222no—
504C2×D4—1222no—
505C2×D4—2442no—
506C2×D4—2442no—
507C2×D4C2×C22⋊D6112yes—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—614no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1224no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—2444no—
522C22×C4—2442no—
523C22×C4—2442no—
524C22×C4#1175822212yes—
525C22×C4#1175822212yes—
526C22×C4D4×D6111yescharacteristic
527C22×C4D4×D6111yescharacteristic
528C22×C4D4×D6112yes—
529C22×C4D4×D6111yescharacteristic
530C22×C4#1175822412yes—
531C22×C4C2×C22⋊D6111yescharacteristic
532C22×C4C2×C22⋊D6111yescharacteristic
533C22×C4C23×D6111yescharacteristic
534M16—22no—
535M16—22no—
536M16—42no—
537M16—1222no—
538M16—1222no—
539M16—2442no—
540C2×C8—22no—
541C2×C8—42no—
542C2×C8—1222no—
543C2×C8—2442no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—1228no—
547C4⋊C4—1222no—
548C4⋊C4—1222no—
549C4⋊C4—2448no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—1228no—
553C22⋊C4—1228no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C42—41no—
557C42—41no—
558C42—612no—
559C42—612no—
560C42—1221no—
561C42—1221no—
562C42—1222no—
563C42—1222no—
564C42—1224no—
565C23—22no—
566C23—42no—
567C23—42no—
568C23—82no—
569C23—1222no—
570C23—4882no—
571C23—2442no—
572C23#1451974511yescharacteristic
573C23#1451974511yescharacteristic
574C23#1451975012yes—
575C23#1451976312yes—
576C23C2×D4×D6111yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—42no—
580Q8—1222no—
581Q8—1222no—
582Q8—2442no—
583D4—22no—
584D4—22no—
585D4—42no—
586D4—42no—
587D4—44no—
588D4—42no—
589D4—1222no—
590D4—1222no—
591D4—4888no—
592D4—2442no—
593D4—2442no—
594D4—2444no—
595D4—2442no—
596C2×C4—22no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—1224no—
604C2×C4—1222no—
605C2×C4—1224no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—2448no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2444no—
614C2×C4—2444no—
615C2×C4#1451974511yescharacteristic
616C2×C4#1451974511yescharacteristic
617C2×C4#1451974912yes—
618C2×C4#1451974912yes—
619C2×C4#1451975012yes—
620C2×C4#1451975012yes—
621C2×C4#1451975012yes—
622C2×C4#1451975912yes—
623C2×C4#1451975912yes—
624C2×C4#1451976012yes—
625C2×C4#1451976012yes—
626C2×C4#1451976312yes—
627C2×C4C2×D4×D6111yescharacteristic
628C2×C4C2×D4×D6111yescharacteristic
629C2×C4C22×C22⋊D6111yescharacteristic
630C2×C4C2×#1175822411yescharacteristic
631C2×C4C2×#1175822211yescharacteristic
632C2×C4C2×#1175822211yescharacteristic
633C8—42no—
634C8—2442no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—42no—
642C22—84no—
643C22—4884no—
644C22—2444no—
645C22—2442no—
646C22#1683967712yes—
647C22#1683967712yes—
648C22C2×#1451974511yescharacteristic
649C22C2×#1451975011yescharacteristic
650C22C2×#1451976311yescharacteristic
651C4—42no—
652C4—82no—
653C4—1224no—
654C4—2442no—
655C4—2444no—
656C4—2442no—
657C4#1683967712yes—
658C4#1683967712yes—
659C4C2×#1451974911yescharacteristic
660C4C2×#1451975011yescharacteristic
661C4C2×#1451975911yescharacteristic
662C4C2×#1451976011yescharacteristic
663C2—21no—
664C2—22no—
665C2—22no—
666C2—21no—
667C2—82no—
668C2—4882no—
669C2C2×#1683967711yescharacteristic
670C2#2145516512yes—
subgroupG/H
centerC2×C4#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2896C2
socleC22×C61C2×#120
prime radical#21459148C2
hypercenter#2896D61
hypocenterC61#23498
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. C22×#14519704C2semidirect
3. C22×#14519703C2semidirect
4. #21454336C2semidirect
5. #21448222C2semidirect
6. #21447046C2semidirect
7. #21447041C2semidirect
8. #21445557C2semidirect
9. C61×#2896C2semidirect
24. C2×#14519704C22semidirect
25. C2×#14519704C22semidirect
27. C2×#14519703C22semidirect
28. C2×#14519703C22semidirect
31. #16839761C22semidirect
33. #16839631C22semidirect
34. #16839627C22semidirect
37. #16839530C22semidirect
43. C61×#417C22semidirect
87. C2×C4×Dic61D4semidirect
332. #2896D61semidirect
372. #417C2×D61semidirect
380. C61#23498semidirect

Maximal subgroups

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

subgroupindex
#214591482
C22×#145197042
C22×#145197032
#214543362
#214482222
#214470462
#214470412
#214455572
C61×#28962
#2349861