Finite Group Databasebeta

← #29707612

Subgroups — #29707612

Order 15616.

1450 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459148C211yesmaximal, characteristic
2#21454408C211yesmaximal, characteristic, complemented
3C22×#14519704C211yesmaximal, characteristic, complemented
4C2×#16839744C211yesmaximal, characteristic, complemented
5#21448307C211yesmaximal, characteristic, complemented
6#21447034C214yesmaximal, complemented
7#21447027C214yesmaximal, complemented
8C2×#16839523C211yesmaximal, characteristic, complemented
9C61×#2896C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840340—22no—
13#16840336—21no—
14#16840333C2212yes—
15#16840329—21no—
16#16840326C2212yes—
17#16840321—22no—
18#16840308—22no—
19#16840273—22no—
20#16840269—22no—
21C2×#14519736C2211yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23#16839805—22no—
24#16839802C2214yescomplemented
25#16839798—22no—
26C2×#14519704—24no—
27C2×#14519704C2214yes—
28#16839744—24no—
29C2×#14519695C2212yescomplemented
30C2×#14519693C2212yescomplemented
31#16839667C2212yescomplemented
32#16839663C2212yescomplemented
33#16839639—22no—
34C4×#11758198—22no—
35#16839523C2214yescomplemented
36C22×#41×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C61×#419—21no—
40C61×#418—21no—
41C61×#417C2214yes—
42C61×#416—22no—
43C2×#11758222—22no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758222—21no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C22×Dic122C2311yescharacteristic
52C22×D244C2311yescharacteristic
53C22×C22⋊D61D412yes—
54C23×Dic61—21no—
55C22×C4×D61D412yes—
56C2×#11758194—21no—
57C2×#11758194—22no—
58C2×#11758194D414yescomplemented
59C2×#11758194C2312yes—
60C22×#92418—21no—
61#14519775—22no—
62#14519773—22no—
63#14519766—22no—
64#14519764—22no—
65#14519760—24no—
66#14519758—24no—
67#14519756—24no—
68#14519753—24no—
69#14519751—24no—
70#14519749—28no—
71#14519748—24no—
72#14519746—24no—
73#14519745—24no—
74C4×C22⋊D61—28no—
75#14519736—21no—
76#14519736—21no—
77#14519736D412yes—
78#14519736C2312yes—
79C2×C4⋊Dic61D412yes—
80C2×C22⋊Dic61—21no—
81C2×#11758198—24no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2312yes—
87#14519710—24no—
88#14519708—24no—
89#14519704—28no—
90#14519695—24no—
91#14519693—24no—
92#14519684—24no—
93C2×#41×C61—21no—
94C2×#41×C61—21no—
95C2×#41×C61—22no—
96C2×#41×C61—22no—
97C2×#41×C61—22no—
98C2×#41×C61—22no—
99C2×#41×C61D412yes—
100C2×#41×C61C2312yes—
101C22×Q8×C61D411yescharacteristic
102C22×D4×C61—21no—
103C22×D4×C61D411yescharacteristic
104C23×C4×C61—21no—
105C23×C4×C61C2311yescharacteristic
106C2×M16×C61—21no—
107C2×M16×C61—22no—
108C2×M16×C61—22no—
109C2×M16×C61C2312yes—
110C22×C8×C61—21no—
111#117×C61—21no—
112#117×C61—21no—
113#117×C61—21no—
114#117×C61—21no—
115C2×C4⋊C4×C61C2312yes—
116C2×C22⋊C4×C61—21no—
117C2×C42×C61—21no—
118#103×C61—24no—
119#102×C61—24no—
120C24×C61—21no—
121C24×C61C2×D411yescharacteristic
122C23×D61C2×D412yes—
123#41×C61—24no—
124#41×C61—24no—
125#41×C61—22no—
126#41×C61—22no—
127#41×C61—22no—
128#41×C61—22no—
129#41×C61—42no—
130#41×C61—42no—
131#41×C61—42no—
132#41×C61—42no—
133C2×Q8×C61—21no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C2×Q8×C61C2×D412yes—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—21no—
140C2×D4×C61—22no—
141C2×D4×C61—22no—
142C2×D4×C61—42no—
143C2×D4×C61—42no—
144C2×D4×C61C2×D412yes—
145C22×C4×C61—21no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D412yes—
157C22×C4×C61#4112yes—
158C22×C4×C61#4112yes—
159C22×C4×C61#4112yes—
160C22×C4×C61C2411yescharacteristic
161C2×C22⋊D61—22no—
162C2×C22⋊D61—22no—
163C2×C22⋊D61—24no—
164C2×C22⋊D61—24no—
165C2×C22⋊D61—28no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—44no—
169C22×Dic61—42no—
170C22×Dic61C2×D412yes—
171#11758222—42no—
172#11758222—42no—
173#11758222—42no—
174#11758222—42no—
175#11758222—42no—
176#11758222—42no—
177#11758222—42no—
178#11758222—42no—
179C2×D244—21no—
180C2×D244—21no—
181C2×D244—21no—
182C2×D244—21no—
183C2×D244—21no—
184C2×D244—21no—
185C2×C4×D61—22no—
186C2×C4×D61—22no—
187C2×C4×D61—24no—
188C2×C4×D61—24no—
189C2×Dic122—21no—
190C2×Dic122—21no—
191C2×Dic122—21no—
192C2×Dic122—21no—
193C2×Dic122—21no—
194C2×Dic122—21no—
195M16×C61—22no—
196M16×C61—22no—
197M16×C61—42no—
198C2×C8×C61—22no—
199C2×C8×C61—42no—
200C4⋊C4×C61—22no—
201C4⋊C4×C61—22no—
202C22⋊C4×C61—41no—
203C22⋊C4×C61—41no—
204C42×C61—41no—
205C42×C61—41no—
206C22⋊Dic61—28no—
207C22⋊Dic61—22no—
208C22⋊Dic61—22no—
209#11758198—28no—
210#11758198—48no—
211C4⋊Dic61—22no—
212C4⋊Dic61C2×D414yes—
213#11758196—28no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—42no—
219#11758194—24no—
220#11758194—22no—
221#11758194C2×D414yes—
222C2×#92418—24no—
223C2×#92418—22no—
224C23×C61—22no—
225C23×C61—42no—
226C23×C61—42no—
227C23×C61—82no—
228C23×C61#12011yescharacteristic
229C23×C61#12011yescharacteristic
230C23×C61#12112yes—
231C23×C61#12312yes—
232C23×C61C22×D411yescharacteristic
233C22×D61—22no—
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—44no—
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—42no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—42no—
243D4×C61—42no—
244D4×C61—44no—
245D4×C61—42no—
246C2×C4×C61—22no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—44no—
251C2×C4×C61—44no—
252C2×C4×C61—42no—
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61#12112yes—
256C2×C4×C61#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61#12112yes—
259C2×C4×C61#12112yes—
260C2×C4×C61#12312yes—
261C2×C4×C61#12312yes—
262C2×C4×C61#12312yes—
263C2×C4×C61#12412yes—
264C2×C4×C61#12412yes—
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C22⋊D61—44no—
272C22⋊D61—44no—
273C22⋊D61—88no—
274C2×Dic61—24no—
275C2×Dic61—24no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—42no—
280C2×Dic61—48no—
281C2×Dic61—42no—
282C2×Dic61—82no—
283D244—41no—
284D244—41no—
285D244—41no—
286D244—41no—
287C4×D61—44no—
288C4×D61—44no—
289Dic122—41no—
290Dic122—41no—
291Dic122—41no—
292Dic122—41no—
293C8×C61—42no—
294#92418—24no—
295#23492—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—44no—
303C22×C61—84no—
304C22×C61#39412yes—
305C22×C61#39412yes—
306C22×C61C2×#12011yescharacteristic
307C22×C61C2×#12111yescharacteristic
308C22×C61C2×#12311yescharacteristic
309C2×D61—44no—
310C2×D61—42no—
311C2×D61—44no—
312C2×D61—82no—
313C4×C61—42no—
314C4×C61—82no—
315C4×C61#39412yes—
316C4×C61#39412yes—
317C4×C61C2×#12111yescharacteristic
318C4×C61C2×#12111yescharacteristic
319C4×C61C2×#12311yescharacteristic
320C4×C61C2×#12411yescharacteristic
321Dic61—44no—
322Dic61—42no—
323Dic61—44no—
324#3449—611no—
325#2911—611no—
326C22×#104—611no—
327#2896D6111yescharacteristic, complemented
328C2×#413—611no—
329#2297—611no—
330#2038—614no—
331#2037—614no—
332C2×#339—611no—
333C2×C61—21no—
334C2×C61—21no—
335C2×C61—22no—
336C2×C61—22no—
337C2×C61—82no—
338C2×C61C2×#39411yescharacteristic
339C2×C61#301812yes—
340D61—82no—
341C22×#41—611no—
342C22×#41C2×D6111yescharacteristic
343C22×M16—611no—
344C22×M16C2×D6111yescharacteristic
345#537—1222no—
346#536—612no—
347#536—1221no—
348#535—612no—
349#535—1221no—
350#534—1222no—
351#532—1222no—
352#525—1222no—
353#525—1222no—
354C2×#117—611no—
355C2×#117C2×D6111yescharacteristic
356C22×C42—611no—
357#428—614no—
358#428—1222no—
359#427—1222no—
360C2×#104—614no—
361C2×#104—1224no—
362#419—21no—
363#418—21no—
364#417C2×D6114yes—
365#416—22no—
366#413—1224no—
367C2×#101—612no—
368C2×#100—612no—
369#391—612no—
370#390—612no—
371#386—1222no—
372C4×C22⋊C4—1222no—
373#339—614no—
374C61#2349211yescharacteristic, complemented, Sylow
375C2×#41—21no—
376C2×#41—21no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41—22no—
380C2×#41—22no—
381C2×#41—1222no—
382C2×#41—1222no—
383C2×#41—1221no—
384C2×#41—1221no—
385C2×#41—1222no—
386C2×#41—1222no—
387C2×#41—1221no—
388C2×#41—1221no—
389C2×#41C22⋊D6112yes—
390C2×#41C22×D6112yes—
391C22×Q8—611no—
392C22×Q8C22⋊D6111yescharacteristic
393C22×D4—21no—
394C22×D4—612no—
395C22×D4—611no—
396C22×D4C22⋊D6111yescharacteristic
397C23×C4—21no—
398C23×C4—612no—
399C23×C4—1221no—
400C23×C4C22×D6111yescharacteristic
401C2×M16—21no—
402C2×M16—22no—
403C2×M16—22no—
404C2×M16—614no—
405C2×M16—612no—
406C2×M16—1222no—
407C2×M16—1221no—
408C2×M16C22×D6112yes—
409C22×C8—21no—
410C22×C8—1221no—
411C4⋊Q8—1222no—
412#127—1222no—
413#124—1222no—
414#124—1222no—
415#123—1224no—
416#123—1224no—
417C22⋊Q8—1224no—
418C22⋊Q8—1224no—
419#121—1224no—
420#121—1224no—
421#121—1228no—
422#120—1224no—
423#120—1224no—
424C4×D4—1228no—
425#117—21no—
426#117—21no—
427#117—21no—
428#117—21no—
429#117—612no—
430#117—612no—
431#117—1221no—
432#117—1221no—
433C2×C4⋊C4—612no—
434C2×C4⋊C4C22×D6112yes—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1224no—
438C2×C42—21no—
439C2×C42—612no—
440C2×C42—1221no—
441C2×C42—1222no—
442C2×C42—1222no—
443C2×C42—1221no—
444#107—1224no—
445#106—1224no—
446#104—1228no—
447#103—24no—
448#102—24no—
449#101—1224no—
450#100—1224no—
451#95—1224no—
452C24—21no—
453C24—612no—
454C24D4×D6111yescharacteristic
455#41—22no—
456#41—22no—
457#41—22no—
458#41—22no—
459#41—24no—
460#41—24no—
461#41—42no—
462#41—42no—
463#41—42no—
464#41—42no—
465#41—2442no—
466#41—2442no—
467#41—2442no—
468#41—2442no—
469#41—2442no—
470#41—2442no—
471#41—2442no—
472#41—2442no—
473C2×Q8—21no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8C2×C22⋊D6112yes—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—1228no—
491C2×D4—1222no—
492C2×D4—1222no—
493C2×D4—1224no—
494C2×D4—1224no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—1221no—
500C2×D4—1221no—
501C2×D4C2×C22⋊D6112yes—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—612no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1224no—
514C22×C4—1224no—
515C22×C4—2444no—
516C22×C4—2442no—
517C22×C4#1175822212yes—
518C22×C4#1175822212yes—
519C22×C4D4×D6111yescharacteristic
520C22×C4D4×D6111yescharacteristic
521C22×C4D4×D6112yes—
522C22×C4D4×D6111yescharacteristic
523C22×C4#1175822412yes—
524C22×C4C2×C22⋊D6111yescharacteristic
525C22×C4C2×C22⋊D6111yescharacteristic
526C22×C4C23×D6111yescharacteristic
527M16—22no—
528M16—22no—
529M16—42no—
530M16—614no—
531M16—1222no—
532M16—1224no—
533C2×C8—22no—
534C2×C8—42no—
535C2×C8—1222no—
536C2×C8—1224no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—614no—
540C4⋊C4—1228no—
541C4⋊C4—1222no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—1228no—
545C22⋊C4—1228no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—2448no—
549C42—41no—
550C42—41no—
551C42—1222no—
552C42—1222no—
553C42—1222no—
554C42—1222no—
555C42—2442no—
556C23—22no—
557C23—42no—
558C23—42no—
559C23—82no—
560C23—1222no—
561C23—1222no—
562C23—1222no—
563C23—2444no—
564C23#1451974511yescharacteristic
565C23#1451974511yescharacteristic
566C23#1451975012yes—
567C23#1451975912yes—
568C23C2×D4×D6111yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—42no—
572Q8—2441no—
573Q8—2441no—
574Q8—2441no—
575Q8—2441no—
576D4—22no—
577D4—22no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—42no—
582D4—4888no—
583D4—2444no—
584D4—2444no—
585D4—2441no—
586D4—2441no—
587D4—2441no—
588D4—2441no—
589C2×C4—22no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—1224no—
597C2×C4—1224no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—4882no—
602C2×C4—2442no—
603C2×C4—2448no—
604C2×C4—2442no—
605C2×C4—2444no—
606C2×C4—2444no—
607C2×C4#1451974511yescharacteristic
608C2×C4#1451974511yescharacteristic
609C2×C4#1451974912yes—
610C2×C4#1451974912yes—
611C2×C4#1451975012yes—
612C2×C4#1451975012yes—
613C2×C4#1451975012yes—
614C2×C4#1451975912yes—
615C2×C4#1451976012yes—
616C2×C4#1451976012yes—
617C2×C4#1451976312yes—
618C2×C4#1451976312yes—
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C2×D4×D6111yescharacteristic
621C2×C4C22×C22⋊D6111yescharacteristic
622C2×C4C2×#1175822411yescharacteristic
623C2×C4C2×#1175822211yescharacteristic
624C2×C4C2×#1175822211yescharacteristic
625C8—42no—
626C8—1224no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—84no—
635C22—4882no—
636C22—2444no—
637C22—2444no—
638C22—2442no—
639C22#1683967712yes—
640C22#1683967712yes—
641C22C2×#1451974511yescharacteristic
642C22C2×#1451975011yescharacteristic
643C22C2×#1451975911yescharacteristic
644C4—42no—
645C4—82no—
646C4—2444no—
647C4—2444no—
648C4—2442no—
649C4#1683967712yes—
650C4#1683967712yes—
651C4C2×#1451974911yescharacteristic
652C4C2×#1451975011yescharacteristic
653C4C2×#1451976011yescharacteristic
654C4C2×#1451976311yescharacteristic
655C2—21no—
656C2—22no—
657C2—21no—
658C2—22no—
659C2—82no—
660C2—4882no—
661C2C2×#1683967711yescharacteristic
662C2#2145516512yes—
subgroupG/H
centerC2×C4#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2896C2
socleC22×C61C2×#120
prime radical#21459148C2
hypercenter#2896D61
hypocenterC61#23492
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. #21454408C2semidirect
3. C22×#14519704C2semidirect
4. C2×#16839744C2semidirect
5. #21448307C2semidirect
6. #21447034C2semidirect
7. #21447027C2semidirect
8. C2×#16839523C2semidirect
9. C61×#2896C2semidirect
24. #16839802C22semidirect
29. C2×#14519695C22semidirect
30. C2×#14519693C22semidirect
31. #16839667C22semidirect
32. #16839663C22semidirect
35. #16839523C22semidirect
58. C2×#11758194D4semidirect
327. #2896D61semidirect
374. C61#23492semidirect

Maximal subgroups

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

subgroupindex
#214591482
#214544082
C22×#145197042
C2×#168397442
#214483072
#214470342
#214470272
C2×#168395232
C61×#28962
#2349261