Finite Group Databasebeta

← #29707640

Subgroups — #29707640

Order 15616.

1422 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459148C211yesmaximal, characteristic
2C2×#16839802C211yesmaximal, characteristic, complemented
3C2×#16839780C211yesmaximal, characteristic, complemented
4C2×#16839761C211yesmaximal, characteristic, complemented
5C2×#16839746C211yesmaximal, characteristic, complemented
6#21447036C214yesmaximal, complemented
7#21447029C214yesmaximal, complemented
8#21445529C211yesmaximal, characteristic, complemented
9C61×#2311C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13#16840340—21no—
14#16840340—21no—
15#16840336C2212yes—
16#16840333—21no—
17#16840329C2212yes—
18#16840326—21no—
19#16840321—22no—
20#16840308—22no—
21#16840273—22no—
22#16840269—22no—
23C2×#14519736C2211yescharacteristic
24#16839802—22no—
25#16839802C2214yescomplemented
26C2×#14519710—21no—
27C2×#14519708—21no—
28#16839780—24no—
29C2×#14519705C2214yes—
30#16839761—22no—
31#16839761C2214yes—
32C2×#14519700—21no—
33C2×#14519697—21no—
34#16839746—24no—
35C2×#14519696C2212yescomplemented
36C2×#14519694C2212yescomplemented
37#16839522C2214yescomplemented
38C22×#41×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C61×#391C2212yescomplemented
42C61×#390C2212yescomplemented
43C61×#386—22no—
44C61×#380—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—
56C22×C4×D61—21no—
57C2×#11758194—21no—
58C2×#11758194—22no—
59C2×#11758194—21no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194D414yescomplemented
63C2×#11758194C2312yes—
64C2×#11758194C2312yes—
65C22×#92418—21no—
66C22×#92418—21no—
67#14519775—22no—
68#14519773—22no—
69#14519766—22no—
70#14519764—22no—
71#14519760—24no—
72#14519758—24no—
73#14519756—22no—
74#14519756—22no—
75#14519753—22no—
76#14519753—22no—
77#14519751—22no—
78#14519751—22no—
79#14519749—24no—
80#14519749—24no—
81#14519748—22no—
82#14519748—22no—
83#14519746—24no—
84#14519745—24no—
85C4×C22⋊D61—24no—
86C4×C22⋊D61—24no—
87#14519736—21no—
88#14519736—21no—
89#14519736D412yes—
90#14519736C2312yes—
91C2×C4⋊Dic61D411yescharacteristic
92C2×C4⋊Dic61D411yescharacteristic
93C2×C22⋊Dic61—21no—
94C2×C4×Dic61—21no—
95#14519710—24no—
96#14519708—24no—
97#14519705—28no—
98#14519700—24no—
99#14519697—24no—
100#14519696—24no—
101#14519694—24no—
102C2×#41×C61—22no—
103C2×#41×C61—21no—
104C2×#41×C61—21no—
105C2×#41×C61—21no—
106C2×#41×C61—21no—
107C2×#41×C61—22no—
108C2×#41×C61—22no—
109C2×#41×C61—22no—
110C22×Q8×C61C2311yescharacteristic
111C22×D4×C61D412yes—
112C22×D4×C61C2311yescharacteristic
113C23×C4×C61—21no—
114C23×C4×C61D412yes—
115C23×C4×C61C2311yescharacteristic
116#117×C61—22no—
117#117×C61—22no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2312yes—
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—24no—
127C2×C42×C61C2312yes—
128#95×C61—24no—
129C24×C61C2×D412yes—
130C24×C61C2×D411yescharacteristic
131C23×D61—21no—
132#41×C61—42no—
133#41×C61—42no—
134#41×C61—42no—
135#41×C61—42no—
136#41×C61—42no—
137#41×C61—42no—
138#41×C61—42no—
139#41×C61—42no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—24no—
149C2×D4×C61—24no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—24no—
159C22×C4×C61—24no—
160C22×C4×C61—22no—
161C22×C4×C61—22no—
162C22×C4×C61—41no—
163C22×C4×C61—41no—
164C22×C4×C61—82no—
165C22×C4×C61C2×D412yes—
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D412yes—
172C22×C4×C61#4112yes—
173C22×C4×C61#4112yes—
174C22×C4×C61#4112yes—
175C22×C4×C61C2411yescharacteristic
176C2×C22⋊D61—22no—
177C2×C22⋊D61—22no—
178C2×C22⋊D61—24no—
179C2×C22⋊D61—24no—
180C2×C22⋊D61—42no—
181C2×C22⋊D61—42no—
182C22×Dic61—21no—
183C22×Dic61—21no—
184C22×Dic61—21no—
185#11758222—24no—
186#11758222—24no—
187#11758222—22no—
188#11758222—22no—
189#11758222—22no—
190#11758222—22no—
191#11758222—42no—
192#11758222—42no—
193#11758222—42no—
194#11758222—42no—
195C2×D244—22no—
196C2×D244—22no—
197C2×D244—21no—
198C2×D244C2×D412yes—
199C2×C4×D61—22no—
200C2×C4×D61—22no—
201C2×C4×D61—42no—
202C2×C4×D61—42no—
203C2×Dic122—22no—
204C2×Dic122—22no—
205C2×Dic122—21no—
206C2×Dic122C2×D412yes—
207C4⋊C4×C61—22no—
208C4⋊C4×C61—22no—
209C4⋊C4×C61—41no—
210C4⋊C4×C61—41no—
211C4⋊C4×C61—42no—
212C22⋊C4×C61—42no—
213C22⋊C4×C61—84no—
214C42×C61—22no—
215C42×C61—22no—
216C22⋊Dic61—24no—
217C22⋊Dic61—24no—
218C22⋊Dic61—22no—
219C22⋊Dic61—22no—
220#11758198—24no—
221#11758198—24no—
222C4⋊Dic61—21no—
223C4⋊Dic61—21no—
224C4⋊Dic61C2×D412yes—
225C4⋊Dic61C2×D412yes—
226#11758196—24no—
227#11758196—24no—
228C4×Dic61—22no—
229C4×Dic61—22no—
230#11758194—24no—
231#11758194—22no—
232#11758194—22no—
233#11758194—22no—
234#11758194—42no—
235#11758194C2×D414yes—
236C2×#92418—24no—
237C2×#92418—22no—
238C2×#92418—22no—
239C2×#92418—42no—
240C23×C61—22no—
241C23×C61—22no—
242C23×C61—22no—
243C23×C61—41no—
244C23×C61—41no—
245C23×C61—42no—
246C23×C61—42no—
247C23×C61#12011yescharacteristic
248C23×C61#12011yescharacteristic
249C23×C61#12112yes—
250C23×C61#12312yes—
251C23×C61C22×D411yescharacteristic
252C22×D61—22no—
253C22×D61—42no—
254C22×D61—81no—
255C22×D61—81no—
256Q8×C61—41no—
257Q8×C61—41no—
258Q8×C61—41no—
259Q8×C61—41no—
260D4×C61—44no—
261D4×C61—44no—
262D4×C61—41no—
263D4×C61—41no—
264D4×C61—41no—
265D4×C61—41no—
266C2×C4×C61—22no—
267C2×C4×C61—22no—
268C2×C4×C61—22no—
269C2×C4×C61—44no—
270C2×C4×C61—44no—
271C2×C4×C61—42no—
272C2×C4×C61—44no—
273C2×C4×C61—42no—
274C2×C4×C61—42no—
275C2×C4×C61—42no—
276C2×C4×C61—82no—
277C2×C4×C61—82no—
278C2×C4×C61#12011yescharacteristic
279C2×C4×C61#12011yescharacteristic
280C2×C4×C61#12112yes—
281C2×C4×C61#12112yes—
282C2×C4×C61#12112yes—
283C2×C4×C61#12112yes—
284C2×C4×C61#12112yes—
285C2×C4×C61#12312yes—
286C2×C4×C61#12312yes—
287C2×C4×C61#12312yes—
288C2×C4×C61#12412yes—
289C2×C4×C61#12412yes—
290C2×C4×C61C22×D411yescharacteristic
291C2×C4×C61C22×D411yescharacteristic
292C2×C4×C61C22×D411yescharacteristic
293C2×C4×C61C2×#4111yescharacteristic
294C2×C4×C61C2×#4111yescharacteristic
295C2×C4×C61C2×#4111yescharacteristic
296C22⋊D61—42no—
297C22⋊D61—42no—
298C22⋊D61—44no—
299C22⋊D61—84no—
300C22⋊D61—84no—
301C2×Dic61—22no—
302C2×Dic61—22no—
303C2×Dic61—22no—
304C2×Dic61—42no—
305C2×Dic61—81no—
306C2×Dic61—81no—
307D244—22no—
308D244—22no—
309D244—42no—
310C4×D61—42no—
311C4×D61—42no—
312C4×D61—44no—
313Dic122—22no—
314Dic122—22no—
315Dic122—42no—
316#92418—24no—
317#92418—42no—
318#23495—611nomaximal, Sylow
319C22×C61—21no—
320C22×C61—21no—
321C22×C61—21no—
322C22×C61—21no—
323C22×C61—21no—
324C22×C61—21no—
325C22×C61—42no—
326C22×C61—42no—
327C22×C61—44no—
328C22×C61—42no—
329C22×C61—42no—
330C22×C61—42no—
331C22×C61—82no—
332C22×C61#39412yes—
333C22×C61#39412yes—
334C22×C61C2×#12011yescharacteristic
335C22×C61C2×#12111yescharacteristic
336C22×C61C2×#12311yescharacteristic
337C2×D61—42no—
338C2×D61—42no—
339C2×D61—42no—
340C2×D61—84no—
341C4×C61—42no—
342C4×C61—82no—
343C4×C61—82no—
344C4×C61#39412yes—
345C4×C61#39412yes—
346C4×C61C2×#12111yescharacteristic
347C4×C61C2×#12111yescharacteristic
348C4×C61C2×#12311yescharacteristic
349C4×C61C2×#12411yescharacteristic
350Dic61—42no—
351Dic61—42no—
352Dic61—42no—
353#3449—611no—
354C2×#428—611no—
355C2×#421—611no—
356C2×#417—611no—
357C2×#413—611no—
358#2311D6111yescharacteristic, complemented
359#2038—614no—
360#2037—614no—
361#1733—611no—
362C2×C61—21no—
363C2×C61—22no—
364C2×C61—21no—
365C2×C61—22no—
366C2×C61—82no—
367C2×C61C2×#39411yescharacteristic
368C2×C61#302111yescharacteristic
369C2×C61#302211yescharacteristic
370D61—82no—
371C22×#41—611no—
372C22×#41C2×D6111yescharacteristic
373C22×M16—611no—
374C22×M16—611no—
375#537—1221no—
376#537—1221no—
377#536—612no—
378#536—1221no—
379#535—612no—
380#535—1221no—
381#534—1222no—
382#532—1222no—
383#525—1222no—
384#525—1222no—
385C2×#117—611no—
386C2×#117C2×D6111yescharacteristic
387C22×C4⋊C4C2×D6111yescharacteristic
388#428—614no—
389#428—1222no—
390C2×#107—1221no—
391C2×#106—1221no—
392#421—1224no—
393C2×#104—614no—
394#417—614no—
395#417—1222no—
396C2×#103—1221no—
397C2×#102—1221no—
398#413—1224no—
399C2×#101—612no—
400C2×#100—612no—
401#391C2×D6112yescomplemented
402#390C2×D6112yescomplemented
403#386—22no—
404#380—22no—
405#339—614no—
406C61#2349511yescharacteristic, complemented, Sylow
407C2×#41—22no—
408C2×#41—22no—
409C2×#41—21no—
410C2×#41—21no—
411C2×#41—21no—
412C2×#41—21no—
413C2×#41—22no—
414C2×#41—22no—
415C2×#41—612no—
416C2×#41—612no—
417C2×#41—1221no—
418C2×#41—1221no—
419C2×#41—1222no—
420C2×#41—1222no—
421C2×#41—1222no—
422C2×#41—1222no—
423C22×Q8—611no—
424C22×Q8C22×D6111yescharacteristic
425C22×D4—611no—
426C22×D4—1221no—
427C22×D4C22⋊D6112yes—
428C22×D4C22×D6111yescharacteristic
429C23×C4—21no—
430C23×C4—1221no—
431C23×C4C22⋊D6112yes—
432C23×C4C22×D6111yescharacteristic
433C2×M16—614no—
434C2×M16—612no—
435C2×M16—612no—
436C2×M16—1222no—
437C2×M16—1221no—
438C2×M16—1221no—
439C2×M16—1222no—
440C2×M16—1222no—
441C22×C8—1221no—
442C22×C8—1221no—
443C4⋊Q8—1222no—
444#127—1222no—
445#124—1222no—
446#124—1222no—
447#123—1224no—
448#123—1224no—
449C22⋊Q8—1222no—
450C22⋊Q8—1222no—
451C22⋊Q8—1222no—
452C22⋊Q8—1222no—
453#121—1222no—
454#121—1222no—
455#121—1222no—
456#121—1222no—
457#121—1224no—
458#121—1224no—
459#120—1224no—
460#120—1224no—
461C4×D4—1224no—
462C4×D4—1224no—
463#117—22no—
464#117—22no—
465#117—612no—
466#117—612no—
467#117—1221no—
468#117—1221no—
469C2×C4⋊C4—21no—
470C2×C4⋊C4—21no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—21no—
473C2×C4⋊C4—21no—
474C2×C4⋊C4—21no—
475C2×C4⋊C4—611no—
476C2×C4⋊C4—611no—
477C2×C4⋊C4C22×D6112yes—
478C2×C22⋊C4—21no—
479C2×C22⋊C4—24no—
480C2×C22⋊C4—1221no—
481C2×C42—1221no—
482C2×C42C22×D6112yes—
483#107—1224no—
484#106—1224no—
485#104—1228no—
486#103—1224no—
487#102—1224no—
488#101—1224no—
489#100—1224no—
490#95—24no—
491C24—1221no—
492C24C2×C22⋊D6112yes—
493C24C2×C22⋊D6111yescharacteristic
494#41—42no—
495#41—42no—
496#41—42no—
497#41—42no—
498#41—42no—
499#41—42no—
500#41—42no—
501#41—42no—
502#41—1222no—
503#41—1222no—
504#41—1222no—
505#41—1222no—
506#41—1224no—
507#41—1224no—
508#41—2442no—
509#41—2442no—
510#41—2442no—
511#41—2442no—
512C2×Q8—21no—
513C2×Q8—21no—
514C2×Q8—21no—
515C2×Q8—21no—
516C2×Q8—21no—
517C2×Q8—21no—
518C2×Q8—612no—
519C2×Q8—1221no—
520C2×Q8—1222no—
521C2×Q8—1222no—
522C2×D4—22no—
523C2×D4—22no—
524C2×D4—24no—
525C2×D4—24no—
526C2×D4—21no—
527C2×D4—21no—
528C2×D4—21no—
529C2×D4—21no—
530C2×D4—21no—
531C2×D4—21no—
532C2×D4—612no—
533C2×D4—1224no—
534C2×D4—1224no—
535C2×D4—1222no—
536C2×D4—1222no—
537C2×D4—1221no—
538C2×D4—1222no—
539C2×D4—1222no—
540C2×D4—2442no—
541C2×D4—2442no—
542C22×C4—22no—
543C22×C4—22no—
544C22×C4—24no—
545C22×C4—24no—
546C22×C4—22no—
547C22×C4—22no—
548C22×C4—41no—
549C22×C4—41no—
550C22×C4—82no—
551C22×C4—1221no—
552C22×C4—1221no—
553C22×C4—1221no—
554C22×C4—1222no—
555C22×C4—1222no—
556C22×C4—2442no—
557C22×C4—2442no—
558C22×C4#1175822212yes—
559C22×C4#1175822212yes—
560C22×C4D4×D6111yescharacteristic
561C22×C4D4×D6111yescharacteristic
562C22×C4D4×D6111yescharacteristic
563C22×C4D4×D6111yescharacteristic
564C22×C4D4×D6112yes—
565C22×C4#1175822412yes—
566C22×C4C2×C22⋊D6112yes—
567C22×C4C2×C22⋊D6111yescharacteristic
568C22×C4C23×D6111yescharacteristic
569M16—614no—
570M16—1222no—
571M16—1224no—
572M16—1222no—
573M16—1222no—
574M16—2442no—
575C2×C8—1222no—
576C2×C8—1224no—
577C2×C8—1222no—
578C2×C8—2442no—
579C4⋊C4—22no—
580C4⋊C4—22no—
581C4⋊C4—41no—
582C4⋊C4—41no—
583C4⋊C4—42no—
584C4⋊C4—612no—
585C4⋊C4—612no—
586C4⋊C4—1224no—
587C4⋊C4—1224no—
588C4⋊C4—1221no—
589C4⋊C4—1221no—
590C22⋊C4—42no—
591C22⋊C4—84no—
592C22⋊C4—1224no—
593C22⋊C4—1224no—
594C22⋊C4—1224no—
595C22⋊C4—1224no—
596C22⋊C4—1222no—
597C22⋊C4—1222no—
598C42—22no—
599C42—22no—
600C42—1222no—
601C42—1222no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—41no—
606C23—41no—
607C23—42no—
608C23—42no—
609C23—1222no—
610C23—4881no—
611C23—4881no—
612C23—2442no—
613C23#1451974511yescharacteristic
614C23#1451974511yescharacteristic
615C23#1451974912yes—
616C23#1451976012yes—
617C23C22×C22⋊D6111yescharacteristic
618Q8—41no—
619Q8—41no—
620Q8—41no—
621Q8—41no—
622Q8—1222no—
623Q8—1222no—
624Q8—2442no—
625D4—44no—
626D4—44no—
627D4—41no—
628D4—41no—
629D4—41no—
630D4—41no—
631D4—1222no—
632D4—1222no—
633D4—4884no—
634D4—4884no—
635D4—2442no—
636D4—2442no—
637D4—2444no—
638D4—2442no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—1222no—
652C2×C4—1222no—
653C2×C4—1222no—
654C2×C4—4881no—
655C2×C4—4881no—
656C2×C4—2442no—
657C2×C4—2442no—
658C2×C4—2442no—
659C2×C4—2444no—
660C2×C4#1451974511yescharacteristic
661C2×C4#1451974511yescharacteristic
662C2×C4#1451974912yes—
663C2×C4#1451975012yes—
664C2×C4#1451975012yes—
665C2×C4#1451975012yes—
666C2×C4#1451975012yes—
667C2×C4#1451975912yes—
668C2×C4#1451975912yes—
669C2×C4#1451976012yes—
670C2×C4#1451976312yes—
671C2×C4#1451976312yes—
672C2×C4C2×D4×D6111yescharacteristic
673C2×C4C2×D4×D6111yescharacteristic
674C2×C4C2×D4×D6111yescharacteristic
675C2×C4C2×#1175822411yescharacteristic
676C2×C4C2×#1175822211yescharacteristic
677C2×C4C2×#1175822211yescharacteristic
678C8—1224no—
679C8—2442no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—42no—
687C22—42no—
688C22—44no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—82no—
693C22—4884no—
694C22—2442no—
695C22—2442no—
696C22—2442no—
697C22#1683967712yes—
698C22#1683967712yes—
699C22C2×#1451974511yescharacteristic
700C22C2×#1451974911yescharacteristic
701C22C2×#1451976011yescharacteristic
702C4—42no—
703C4—82no—
704C4—82no—
705C4—2442no—
706C4—2442no—
707C4—2442no—
708C4#1683967712yes—
709C4#1683967712yes—
710C4C2×#1451975011yescharacteristic
711C4C2×#1451975011yescharacteristic
712C4C2×#1451975911yescharacteristic
713C4C2×#1451976311yescharacteristic
714C2—21no—
715C2—21no—
716C2—22no—
717C2—22no—
718C2—82no—
719C2—4882no—
720C2C2×#1683967711yescharacteristic
721C2#2145519011yescharacteristic
722C2#2145519711yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2311C2
socleC22×C61C2×#120
prime radical#21459148C2
hypercenter#2311D61
hypocenterC61#23495
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#16839802C2semidirect
3. C2×#16839780C2semidirect
4. C2×#16839761C2semidirect
5. C2×#16839746C2semidirect
6. #21447036C2semidirect
7. #21447029C2semidirect
8. #21445529C2semidirect
9. C61×#2311C2semidirect
25. #16839802C22semidirect
35. C2×#14519696C22semidirect
36. C2×#14519694C22semidirect
37. #16839522C22semidirect
41. C61×#391C22semidirect
42. C61×#390C22semidirect
62. C2×#11758194D4semidirect
358. #2311D61semidirect
401. #391C2×D61semidirect
402. #390C2×D61semidirect
406. C61#23495semidirect

Maximal subgroups

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

subgroupindex
#214591482
C2×#168398022
C2×#168397802
C2×#168397612
C2×#168397462
#214470362
#214470292
#214455292
C61×#23112
#2349561