Finite Group Databasebeta

← #29707637

Subgroups — #29707637

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×#16839779C211yesmaximal, characteristic, complemented
4C2×#16839744C211yesmaximal, characteristic, complemented
5#21448412C211yesmaximal, characteristic, complemented
6#21447034C214yesmaximal, complemented
7#21447027C214yesmaximal, complemented
8#21445530C211yesmaximal, characteristic, complemented
9C2×C61×#417C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840340—21no—
13#16840340—21no—
14#16840336—21no—
15#16840333C2212yes—
16#16840329—21no—
17#16840326C2212yes—
18#16840321—22no—
19#16840308—22no—
20#16840273—22no—
21#16840269—22no—
22C2×#14519736C2211yescharacteristic
23C2×#14519736C2211yescharacteristic
24#16839802—22no—
25#16839802C2214yescomplemented
26C2×#14519710—21no—
27C2×#14519708—21no—
28#16839779—24no—
29C2×#14519704C2214yes—
30#16839744—24no—
31C2×#14519695C2212yescomplemented
32C2×#14519693C2212yescomplemented
33#16839667C2212yescomplemented
34#16839663C2212yescomplemented
35#16839641—22no—
36#16839619—22no—
37#16839523C2214yescomplemented
38C22×#41×C61C2211yescharacteristic
39C22×M16×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C61×#417—22no—
42C61×#417C2214yes—
43C2×#103×C61—21no—
44C2×#102×C61—21no—
45C2×#11758222—22no—
46C2×#11758222—21no—
47C2×#11758222—21no—
48C2×#11758222—21no—
49C2×#11758222—21no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C22×Dic122C2311yescharacteristic
54C22×D244C2311yescharacteristic
55C22×C22⋊D61D412yes—
56C22×C4×D61D412yes—
57C2×#11758194—21no—
58C2×#11758194—22no—
59C2×#11758194D414yescomplemented
60C2×#11758194C2312yes—
61C22×#92418—21no—
62#14519775—22no—
63#14519773—22no—
64#14519766—22no—
65#14519764—22no—
66#14519760—24no—
67#14519758—24no—
68#14519756—22no—
69#14519756—22no—
70#14519753—22no—
71#14519753—22no—
72#14519751—22no—
73#14519751—22no—
74#14519749—24no—
75#14519749—24no—
76#14519748—22no—
77#14519748—22no—
78#14519746—24no—
79#14519745—24no—
80C4×C22⋊D61—24no—
81C4×C22⋊D61—24no—
82#14519736—21no—
83#14519736—21no—
84#14519736—22no—
85#14519736—22no—
86#14519736D412yes—
87#14519736C2312yes—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61D411yescharacteristic
90C2×C4⋊Dic61D411yescharacteristic
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×#11758198—24no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2312yes—
96#14519710—24no—
97#14519708—24no—
98#14519704—28no—
99#14519695—24no—
100#14519693—24no—
101#14519684—24no—
102C2×#41×C61—21no—
103C2×#41×C61—21no—
104C2×#41×C61—22no—
105C2×#41×C61—22no—
106C2×#41×C61—22no—
107C2×#41×C61—22no—
108C2×#41×C61D412yes—
109C2×#41×C61C2312yes—
110C22×Q8×C61D411yescharacteristic
111C22×D4×C61—21no—
112C22×D4×C61D411yescharacteristic
113C23×C4×C61—21no—
114C23×C4×C61—21no—
115C23×C4×C61C2311yescharacteristic
116C2×M16×C61—21no—
117C2×M16×C61—22no—
118C2×M16×C61—22no—
119C2×M16×C61C2312yes—
120C22×C8×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2312yes—
127#103×C61—24no—
128#102×C61—24no—
129C24×C61—21no—
130C24×C61C2×D411yescharacteristic
131C23×D61C2×D412yes—
132#41×C61—24no—
133#41×C61—24no—
134#41×C61—22no—
135#41×C61—22no—
136#41×C61—22no—
137#41×C61—22no—
138#41×C61—42no—
139#41×C61—42no—
140#41×C61—42no—
141#41×C61—42no—
142C2×Q8×C61—21no—
143C2×Q8×C61—22no—
144C2×Q8×C61—22no—
145C2×Q8×C61C2×D412yes—
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—21no—
149C2×D4×C61—22no—
150C2×D4×C61—22no—
151C2×D4×C61—42no—
152C2×D4×C61—42no—
153C2×D4×C61C2×D412yes—
154C22×C4×C61—21no—
155C22×C4×C61—22no—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—42no—
159C22×C4×C61—42no—
160C22×C4×C61—41no—
161C22×C4×C61—41no—
162C22×C4×C61—82no—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D412yes—
169C22×C4×C61#4112yes—
170C22×C4×C61#4112yes—
171C22×C4×C61#4112yes—
172C22×C4×C61C2411yescharacteristic
173C2×C22⋊D61—22no—
174C2×C22⋊D61—22no—
175C2×C22⋊D61—24no—
176C2×C22⋊D61—24no—
177C2×C22⋊D61—24no—
178C2×C22⋊D61—24no—
179C22×Dic61—21no—
180C22×Dic61—21no—
181C22×Dic61—22no—
182C22×Dic61C2×D412yes—
183#11758222—42no—
184#11758222—42no—
185#11758222—42no—
186#11758222—42no—
187#11758222—42no—
188#11758222—42no—
189#11758222—42no—
190#11758222—42no—
191C2×D244—21no—
192C2×D244—21no—
193C2×D244—21no—
194C2×D244—21no—
195C2×D244—21no—
196C2×D244—21no—
197C2×C4×D61—22no—
198C2×C4×D61—22no—
199C2×C4×D61—24no—
200C2×C4×D61—24no—
201C2×Dic122—21no—
202C2×Dic122—21no—
203C2×Dic122—21no—
204C2×Dic122—21no—
205C2×Dic122—21no—
206C2×Dic122—21no—
207M16×C61—22no—
208M16×C61—22no—
209M16×C61—42no—
210C2×C8×C61—22no—
211C2×C8×C61—42no—
212C4⋊C4×C61—22no—
213C4⋊C4×C61—22no—
214C4⋊C4×C61—41no—
215C4⋊C4×C61—41no—
216C22⋊Dic61—24no—
217C22⋊Dic61—24no—
218C22⋊Dic61—22no—
219C22⋊Dic61—22no—
220C22⋊Dic61—42no—
221#11758198—24no—
222#11758198—24no—
223#11758198—84no—
224C4⋊Dic61—21no—
225C4⋊Dic61—21no—
226C4⋊Dic61—42no—
227C4⋊Dic61C2×D412yes—
228C4⋊Dic61C2×D412yes—
229#11758196—24no—
230#11758196—24no—
231C4×Dic61—22no—
232C4×Dic61—22no—
233C4×Dic61—22no—
234C4×Dic61—22no—
235#11758194—24no—
236#11758194—22no—
237#11758194C2×D414yes—
238C2×#92418—24no—
239C2×#92418—22no—
240C23×C61—22no—
241C23×C61—41no—
242C23×C61—41no—
243C23×C61—42no—
244C23×C61—81no—
245C23×C61—81no—
246C23×C61#12011yescharacteristic
247C23×C61#12011yescharacteristic
248C23×C61#12112yes—
249C23×C61#12312yes—
250C23×C61C22×D411yescharacteristic
251C22×D61—22no—
252C22×D61—22no—
253C22×D61—22no—
254C22×D61—42no—
255C22×D61—42no—
256Q8×C61—22no—
257Q8×C61—22no—
258Q8×C61—42no—
259D4×C61—22no—
260D4×C61—22no—
261D4×C61—42no—
262D4×C61—42no—
263D4×C61—44no—
264D4×C61—42no—
265C2×C4×C61—22no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61—42no—
269C2×C4×C61—44no—
270C2×C4×C61—44no—
271C2×C4×C61—42no—
272C2×C4×C61—82no—
273C2×C4×C61—82no—
274C2×C4×C61#12011yescharacteristic
275C2×C4×C61#12011yescharacteristic
276C2×C4×C61#12112yes—
277C2×C4×C61#12112yes—
278C2×C4×C61#12112yes—
279C2×C4×C61#12112yes—
280C2×C4×C61#12112yes—
281C2×C4×C61#12312yes—
282C2×C4×C61#12312yes—
283C2×C4×C61#12312yes—
284C2×C4×C61#12412yes—
285C2×C4×C61#12412yes—
286C2×C4×C61C22×D411yescharacteristic
287C2×C4×C61C22×D411yescharacteristic
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61C2×#4111yescharacteristic
290C2×C4×C61C2×#4111yescharacteristic
291C2×C4×C61C2×#4111yescharacteristic
292C22⋊D61—44no—
293C22⋊D61—44no—
294C22⋊D61—84no—
295C22⋊D61—84no—
296C2×Dic61—22no—
297C2×Dic61—22no—
298C2×Dic61—22no—
299C2×Dic61—22no—
300C2×Dic61—22no—
301C2×Dic61—42no—
302C2×Dic61—42no—
303C2×Dic61—42no—
304C2×Dic61—81no—
305C2×Dic61—81no—
306D244—41no—
307D244—41no—
308D244—41no—
309D244—41no—
310C4×D61—44no—
311C4×D61—44no—
312Dic122—41no—
313Dic122—41no—
314Dic122—41no—
315Dic122—41no—
316C8×C61—42no—
317#92418—24no—
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—42no—
328C22×C61—42no—
329C22×C61—42no—
330C22×C61—84no—
331C22×C61#39412yes—
332C22×C61#39412yes—
333C22×C61C2×#12011yescharacteristic
334C22×C61C2×#12111yescharacteristic
335C22×C61C2×#12311yescharacteristic
336C2×D61—44no—
337C2×D61—42no—
338C2×D61—42no—
339C2×D61—42no—
340C2×D61—82no—
341C4×C61—42no—
342C4×C61—82no—
343C4×C61#39412yes—
344C4×C61#39412yes—
345C4×C61C2×#12111yescharacteristic
346C4×C61C2×#12111yescharacteristic
347C4×C61C2×#12311yescharacteristic
348C4×C61C2×#12411yescharacteristic
349Dic61—42no—
350Dic61—42no—
351Dic61—42no—
352Dic61—82no—
353#3449—611no—
354C2×#428—611no—
355C2×#421—611no—
356C2×#417D6111yescharacteristic, complemented
357C2×#413—611no—
358#2311—611no—
359#2038—614no—
360#2037—614no—
361#1733—611no—
362C2×C61—21no—
363C2×C61—21no—
364C2×C61—22no—
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×M16C2×D6111yescharacteristic
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×#117—611no—
387C22×C4⋊C4C2×D6111yescharacteristic
388#428—614no—
389#428—1222no—
390C2×#107—1221no—
391C2×#106—1221no—
392#421—1224no—
393C2×#104—614no—
394#417—22no—
395#417C2×D6114yes—
396C2×#103—21no—
397C2×#102—21no—
398#413—1224no—
399C2×#101—612no—
400C2×#100—612no—
401#391—612no—
402#390—612no—
403#386—1222no—
404#380—1222no—
405#339—614no—
406C61#2349511yescharacteristic, complemented, Sylow
407C2×#41—21no—
408C2×#41—21no—
409C2×#41—22no—
410C2×#41—22no—
411C2×#41—22no—
412C2×#41—22no—
413C2×#41—1222no—
414C2×#41—1222no—
415C2×#41—1221no—
416C2×#41—1221no—
417C2×#41—1222no—
418C2×#41—1222no—
419C2×#41—1221no—
420C2×#41—1221no—
421C2×#41C22⋊D6112yes—
422C2×#41C22×D6112yes—
423C22×Q8—611no—
424C22×Q8C22⋊D6111yescharacteristic
425C22×D4—21no—
426C22×D4—612no—
427C22×D4—611no—
428C22×D4C22⋊D6111yescharacteristic
429C23×C4—21no—
430C23×C4—21no—
431C23×C4—612no—
432C23×C4C22×D6111yescharacteristic
433C2×M16—21no—
434C2×M16—22no—
435C2×M16—22no—
436C2×M16—614no—
437C2×M16—612no—
438C2×M16—1222no—
439C2×M16—1221no—
440C2×M16C22×D6112yes—
441C22×C8—21no—
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—612no—
464#117—612no—
465#117—1221no—
466#117—1221no—
467#117—1222no—
468#117—1222no—
469C2×C4⋊C4—21no—
470C2×C4⋊C4—21no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—21no—
473C2×C4⋊C4—21no—
474C2×C4⋊C4—611no—
475C2×C4⋊C4—611no—
476C2×C4⋊C4—1221no—
477C2×C4⋊C4C22×D6112yes—
478C2×C22⋊C4—1221no—
479C2×C22⋊C4—1224no—
480C2×C22⋊C4—1221no—
481C2×C42—612no—
482C2×C42—1221no—
483#107—1224no—
484#106—1224no—
485#104—1228no—
486#103—24no—
487#102—24no—
488#101—1224no—
489#100—1224no—
490#95—1224no—
491C24—21no—
492C24—612no—
493C24D4×D6111yescharacteristic
494#41—22no—
495#41—22no—
496#41—22no—
497#41—22no—
498#41—24no—
499#41—24no—
500#41—42no—
501#41—42no—
502#41—42no—
503#41—42no—
504#41—2442no—
505#41—2442no—
506#41—2442no—
507#41—2442no—
508#41—2442no—
509#41—2442no—
510#41—2442no—
511#41—2442no—
512C2×Q8—21no—
513C2×Q8—22no—
514C2×Q8—22no—
515C2×Q8—1221no—
516C2×Q8—1221no—
517C2×Q8—1221no—
518C2×Q8—1221no—
519C2×Q8—1221no—
520C2×Q8—1221no—
521C2×Q8C2×C22⋊D6112yes—
522C2×D4—22no—
523C2×D4—22no—
524C2×D4—21no—
525C2×D4—22no—
526C2×D4—22no—
527C2×D4—42no—
528C2×D4—42no—
529C2×D4—1224no—
530C2×D4—1224no—
531C2×D4—1222no—
532C2×D4—1222no—
533C2×D4—1224no—
534C2×D4—1224no—
535C2×D4—1221no—
536C2×D4—1221no—
537C2×D4—1221no—
538C2×D4—1221no—
539C2×D4—1221no—
540C2×D4—1221no—
541C2×D4C2×C22⋊D6112yes—
542C22×C4—21no—
543C22×C4—22no—
544C22×C4—22no—
545C22×C4—22no—
546C22×C4—42no—
547C22×C4—42no—
548C22×C4—41no—
549C22×C4—41no—
550C22×C4—82no—
551C22×C4—612no—
552C22×C4—1221no—
553C22×C4—1221no—
554C22×C4—1222no—
555C22×C4—1222no—
556C22×C4—1224no—
557C22×C4—1224no—
558C22×C4—1222no—
559C22×C4#1175822212yes—
560C22×C4#1175822212yes—
561C22×C4D4×D6111yescharacteristic
562C22×C4D4×D6111yescharacteristic
563C22×C4D4×D6112yes—
564C22×C4D4×D6111yescharacteristic
565C22×C4#1175822412yes—
566C22×C4C2×C22⋊D6111yescharacteristic
567C22×C4C2×C22⋊D6111yescharacteristic
568C22×C4C23×D6111yescharacteristic
569M16—22no—
570M16—22no—
571M16—42no—
572M16—614no—
573M16—1222no—
574M16—1224no—
575C2×C8—22no—
576C2×C8—42no—
577C2×C8—1222no—
578C2×C8—1224no—
579C4⋊C4—22no—
580C4⋊C4—22no—
581C4⋊C4—41no—
582C4⋊C4—41no—
583C4⋊C4—612no—
584C4⋊C4—612no—
585C4⋊C4—1224no—
586C4⋊C4—1224no—
587C4⋊C4—1221no—
588C4⋊C4—1221no—
589C4⋊C4—2442no—
590C22⋊C4—1224no—
591C22⋊C4—1224no—
592C22⋊C4—1224no—
593C22⋊C4—1224no—
594C22⋊C4—1222no—
595C22⋊C4—1222no—
596C22⋊C4—4884no—
597C22⋊C4—2442no—
598C42—1222no—
599C42—1222no—
600C42—1222no—
601C42—1222no—
602C23—22no—
603C23—41no—
604C23—41no—
605C23—42no—
606C23—81no—
607C23—81no—
608C23—1222no—
609C23—1222no—
610C23—1222no—
611C23—2442no—
612C23—2442no—
613C23#1451974511yescharacteristic
614C23#1451974511yescharacteristic
615C23#1451975012yes—
616C23#1451975912yes—
617C23C2×D4×D6111yescharacteristic
618Q8—22no—
619Q8—22no—
620Q8—42no—
621Q8—2441no—
622Q8—2441no—
623Q8—2441no—
624Q8—2441no—
625D4—22no—
626D4—22no—
627D4—42no—
628D4—42no—
629D4—44no—
630D4—42no—
631D4—4884no—
632D4—4884no—
633D4—2444no—
634D4—2444no—
635D4—2441no—
636D4—2441no—
637D4—2441no—
638D4—2441no—
639C2×C4—22no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—1222no—
649C2×C4—1222no—
650C2×C4—1222no—
651C2×C4—1222no—
652C2×C4—1222no—
653C2×C4—4881no—
654C2×C4—4881no—
655C2×C4—2442no—
656C2×C4—2442no—
657C2×C4—2442no—
658C2×C4—2444no—
659C2×C4—2444no—
660C2×C4#1451974511yescharacteristic
661C2×C4#1451974511yescharacteristic
662C2×C4#1451974912yes—
663C2×C4#1451974912yes—
664C2×C4#1451975012yes—
665C2×C4#1451975012yes—
666C2×C4#1451975012yes—
667C2×C4#1451975912yes—
668C2×C4#1451976012yes—
669C2×C4#1451976012yes—
670C2×C4#1451976312yes—
671C2×C4#1451976312yes—
672C2×C4C2×D4×D6111yescharacteristic
673C2×C4C2×D4×D6111yescharacteristic
674C2×C4C22×C22⋊D6111yescharacteristic
675C2×C4C2×#1175822411yescharacteristic
676C2×C4C2×#1175822211yescharacteristic
677C2×C4C2×#1175822211yescharacteristic
678C8—42no—
679C8—1224no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—84no—
692C22—4882no—
693C22—2444no—
694C22—2442no—
695C22—2442no—
696C22—2442no—
697C22#1683967712yes—
698C22#1683967712yes—
699C22C2×#1451974511yescharacteristic
700C22C2×#1451975011yescharacteristic
701C22C2×#1451975911yescharacteristic
702C4—42no—
703C4—82no—
704C4—4882no—
705C4—2442no—
706C4—2442no—
707C4—2442no—
708C4#1683967712yes—
709C4#1683967712yes—
710C4C2×#1451974911yescharacteristic
711C4C2×#1451975011yescharacteristic
712C4C2×#1451976011yescharacteristic
713C4C2×#1451976311yescharacteristic
714C2—21no—
715C2—22no—
716C2—21no—
717C2—22no—
718C2—82no—
719C2—4882no—
720C2C2×#1683967711yescharacteristic
721C2#2145518811yescharacteristic
722C2#2145519511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC22×C61C2×#120
prime radical#21459148C2
hypercenterC2×#417D61
hypocenterC61#23495
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×#16839802C2semidirect
3. C2×#16839779C2semidirect
4. C2×#16839744C2semidirect
5. #21448412C2semidirect
6. #21447034C2semidirect
7. #21447027C2semidirect
8. #21445530C2semidirect
9. C2×C61×#417C2semidirect
25. #16839802C22semidirect
31. C2×#14519695C22semidirect
32. C2×#14519693C22semidirect
33. #16839667C22semidirect
34. #16839663C22semidirect
37. #16839523C22semidirect
59. C2×#11758194D4semidirect
356. C2×#417D61semidirect
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×#168397792
C2×#168397442
#214484122
#214470342
#214470272
#214455302
C2×C61×#4172
#2349561