Finite Group Databasebeta

← #29706990

Subgroups — #29706990

Order 15616.

1946 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519808C211yesmaximal, characteristic, complemented
2#21459148C211yesmaximal, characteristic
3C2×#16839779C211yesmaximal, characteristic, complemented
4C2×#16839778C211yesmaximal, characteristic, complemented
5#21448405C211yesmaximal, characteristic, complemented
6#21446864C218yesmaximal, complemented
7#21445557C211yesmaximal, characteristic, complemented
8C22×#137×C61C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C2×#14519808—24no—
11C2×#14519808—24no—
12C2×#14519808—22no—
13C2×#14519808C2214yescomplemented
14C22×#11758202—21no—
15C22×#11758201—21no—
16C22×#11758194C2211yescharacteristic
17#16840340C2214yes—
18#16840336—21no—
19#16840333—21no—
20#16840329—21no—
21#16840326—21no—
22#16840321—22no—
23#16840308—22no—
24#16840273—22no—
25#16840269—22no—
26C2×#14519736C2211yescharacteristic
27C2×#14519736C2211yescharacteristic
28#16839779—24no—
29#16839778—24no—
30C2×#14519704C2214yescomplemented
31C2×#14519703C2214yescomplemented
32#16839669—22no—
33#16839669C2214yescomplemented
34#16839625—22no—
35#16839530C2214yescomplemented
36C22×#41×C61C2211yescharacteristic
37C23×Q8×C61C2211yescharacteristic
38C2×#137×C61—22no—
39C2×#137×C61—24no—
40C2×#137×C61—24no—
41C2×#137×C61C2214yescomplemented
42C22×Q16×C61—21no—
43C22×SD16×C61—21no—
44C22×M16×C61C2211yescharacteristic
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—
57#14519808—216no—
58#14519808—28no—
59#14519808—28no—
60C2×#11758202—28no—
61C2×#11758202—24no—
62C2×#11758201—28no—
63C2×#11758201—24no—
64C2×#11758194—22no—
65C2×#11758194—21no—
66C2×#11758194D414yescomplemented
67C2×#11758194C2312yes—
68C22×#92418—21no—
69#14519775—22no—
70#14519773—22no—
71#14519766—22no—
72#14519764—22no—
73#14519760—24no—
74#14519758—24no—
75#14519756—24no—
76#14519753—24no—
77#14519751—24no—
78#14519749—28no—
79#14519748—24no—
80#14519746—24no—
81#14519745—24no—
82C4×C22⋊D61—28no—
83#14519736—21no—
84#14519736—21no—
85#14519736—21no—
86#14519736—21no—
87#14519736—22no—
88#14519736—22no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61C2312yes—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61D412yes—
93C2×C4×Dic61D412yes—
94C2×C4×Dic61C2312yes—
95#14519704—28no—
96#14519703—28no—
97#14519684—28no—
98C2×#41×C61—21no—
99C2×#41×C61—21no—
100C2×#41×C61—22no—
101C2×#41×C61—22no—
102C2×#41×C61—22no—
103C2×#41×C61—22no—
104C2×#41×C61D412yes—
105C2×#41×C61C2312yes—
106C22×Q8×C61—21no—
107C22×Q8×C61—22no—
108C22×Q8×C61—21no—
109C22×Q8×C61—21no—
110C22×Q8×C61—22no—
111C22×Q8×C61—21no—
112C22×Q8×C61—21no—
113C22×Q8×C61—22no—
114C22×Q8×C61D411yescharacteristic
115C22×Q8×C61D414yescomplemented
116C22×Q8×C61C2312yes—
117C22×D4×C61—21no—
118C22×D4×C61D411yescharacteristic
119C23×C4×C61—21no—
120C23×C4×C61—21no—
121C23×C4×C61C2311yescharacteristic
122#137×C61—216no—
123#137×C61—28no—
124#137×C61—28no—
125C2×Q16×C61—28no—
126C2×Q16×C61—24no—
127C2×SD16×C61—28no—
128C2×SD16×C61—24no—
129C2×M16×C61—22no—
130C2×M16×C61—21no—
131C2×M16×C61D414yescomplemented
132C2×M16×C61C2312yes—
133C22×C8×C61—21no—
134C24×C61—21no—
135C24×C61C2×D411yescharacteristic
136C23×D61—21no—
137#41×C61—24no—
138#41×C61—24no—
139#41×C61—22no—
140#41×C61—22no—
141#41×C61—22no—
142#41×C61—22no—
143#41×C61—42no—
144#41×C61—42no—
145#41×C61—42no—
146#41×C61—42no—
147C2×Q8×C61—21no—
148C2×Q8×C61—24no—
149C2×Q8×C61—22no—
150C2×Q8×C61—24no—
151C2×Q8×C61—24no—
152C2×Q8×C61—22no—
153C2×Q8×C61—24no—
154C2×Q8×C61—24no—
155C2×Q8×C61—22no—
156C2×Q8×C61—22no—
157C2×Q8×C61—22no—
158C2×Q8×C61—22no—
159C2×Q8×C61—22no—
160C2×Q8×C61—41no—
161C2×Q8×C61—41no—
162C2×Q8×C61—42no—
163C2×Q8×C61—42no—
164C2×Q8×C61—42no—
165C2×Q8×C61—42no—
166C2×Q8×C61—42no—
167C2×Q8×C61C2×D414yes—
168C2×Q8×C61C2×D412yes—
169C2×D4×C61—22no—
170C2×D4×C61—22no—
171C2×D4×C61—22no—
172C2×D4×C61—21no—
173C2×D4×C61—22no—
174C2×D4×C61—42no—
175C2×D4×C61—42no—
176C2×D4×C61C2×D412yes—
177C22×C4×C61—21no—
178C22×C4×C61—24no—
179C22×C4×C61—22no—
180C22×C4×C61—22no—
181C22×C4×C61—22no—
182C22×C4×C61—41no—
183C22×C4×C61—41no—
184C22×C4×C61—42no—
185C22×C4×C61—42no—
186C22×C4×C61—42no—
187C22×C4×C61C2×D412yes—
188C22×C4×C61C2×D411yescharacteristic
189C22×C4×C61C2×D411yescharacteristic
190C22×C4×C61C2×D411yescharacteristic
191C22×C4×C61C2×D411yescharacteristic
192C22×C4×C61C2×D411yescharacteristic
193C22×C4×C61C2×D412yes—
194C22×C4×C61C2×D412yes—
195C22×C4×C61#4112yes—
196C22×C4×C61C2411yescharacteristic
197C2×C22⋊D61—22no—
198C2×C22⋊D61—22no—
199C2×C22⋊D61—28no—
200C2×C22⋊D61—42no—
201C2×C22⋊D61—42no—
202C22×Dic61—21no—
203C22×Dic61—22no—
204C22×Dic61C2×D414yes—
205#11758222—24no—
206#11758222—24no—
207#11758222—22no—
208#11758222—22no—
209#11758222—22no—
210#11758222—22no—
211#11758222—42no—
212#11758222—42no—
213#11758222—42no—
214#11758222—42no—
215C2×D244—22no—
216C2×D244—22no—
217C2×D244—21no—
218C2×D244C2×D412yes—
219C2×C4×D61—22no—
220C2×C4×D61—22no—
221C2×C4×D61—42no—
222C2×C4×D61—42no—
223C2×Dic122—22no—
224C2×Dic122—22no—
225C2×Dic122—21no—
226C2×Dic122C2×D412yes—
227Q16×C61—216no—
228SD16×C61—216no—
229M16×C61—24no—
230M16×C61—22no—
231M16×C61C2×D414yes—
232C2×C8×C61—24no—
233C2×C8×C61—22no—
234C22⋊Dic61—28no—
235C22⋊Dic61—22no—
236C22⋊Dic61—22no—
237C22⋊Dic61—42no—
238#11758202—216no—
239#11758201—216no—
240#11758198—28no—
241C4⋊Dic61—22no—
242C4⋊Dic61—22no—
243C4⋊Dic61—42no—
244#11758196—28no—
245C4×Dic61—22no—
246C4×Dic61—22no—
247C4×Dic61—22no—
248C4×Dic61—22no—
249#11758194—24no—
250#11758194—22no—
251#11758194C2×D414yes—
252C2×#92418—24no—
253C2×#92418—22no—
254C23×C61—22no—
255C23×C61—42no—
256C23×C61—42no—
257C23×C61—82no—
258C23×C61#12011yescharacteristic
259C23×C61#12011yescharacteristic
260C23×C61#12112yes—
261C23×C61#12712yes—
262C23×C61C22×D411yescharacteristic
263C22×D61—22no—
264C22×D61—42no—
265C22×D61—82no—
266Q8×C61—24no—
267Q8×C61—24no—
268Q8×C61—22no—
269Q8×C61—22no—
270Q8×C61—42no—
271Q8×C61—42no—
272Q8×C61—44no—
273Q8×C61—44no—
274Q8×C61—42no—
275D4×C61—22no—
276D4×C61—22no—
277D4×C61—42no—
278D4×C61—42no—
279D4×C61—44no—
280D4×C61—42no—
281C2×C4×C61—22no—
282C2×C4×C61—24no—
283C2×C4×C61—42no—
284C2×C4×C61—42no—
285C2×C4×C61—42no—
286C2×C4×C61—44no—
287C2×C4×C61—44no—
288C2×C4×C61—42no—
289C2×C4×C61—42no—
290C2×C4×C61—44no—
291C2×C4×C61#12012yes—
292C2×C4×C61#12012yes—
293C2×C4×C61#12012yes—
294C2×C4×C61#12012yes—
295C2×C4×C61#12011yescharacteristic
296C2×C4×C61#12011yescharacteristic
297C2×C4×C61#12112yes—
298C2×C4×C61#12112yes—
299C2×C4×C61#12112yes—
300C2×C4×C61#12112yes—
301C2×C4×C61#12112yes—
302C2×C4×C61#12712yes—
303C2×C4×C61C22×D411yescharacteristic
304C2×C4×C61C22×D411yescharacteristic
305C2×C4×C61C22×D411yescharacteristic
306C2×C4×C61C22×D411yescharacteristic
307C2×C4×C61C22×D411yescharacteristic
308C2×C4×C61C2×#4111yescharacteristic
309C22⋊D61—42no—
310C22⋊D61—42no—
311C22⋊D61—44no—
312C22⋊D61—88no—
313C2×Dic61—24no—
314C2×Dic61—22no—
315C2×Dic61—42no—
316C2×Dic61—44no—
317C2×Dic61—44no—
318C2×Dic61—42no—
319D244—22no—
320D244—22no—
321D244—42no—
322C4×D61—42no—
323C4×D61—42no—
324C4×D61—44no—
325Dic122—22no—
326Dic122—22no—
327Dic122—42no—
328C8×C61—24no—
329#92418—24no—
330#23408—611nomaximal, Sylow
331C22×C61—22no—
332C22×C61—22no—
333C22×C61—22no—
334C22×C61—42no—
335C22×C61—42no—
336C22×C61—42no—
337C22×C61—44no—
338C22×C61—84no—
339C22×C61#39212yes—
340C22×C61#39212yes—
341C22×C61C2×#12011yescharacteristic
342C22×C61C2×#12111yescharacteristic
343C22×C61C2×#12711yescharacteristic
344C2×D61—42no—
345C2×D61—44no—
346C2×D61—84no—
347C4×C61—42no—
348C4×C61—44no—
349C4×C61#39212yes—
350C4×C61#39212yes—
351C4×C61C2×#12011yescharacteristic
352C4×C61C2×#12011yescharacteristic
353C4×C61C2×#12111yescharacteristic
354C4×C61C2×#12111yescharacteristic
355Dic61—42no—
356Dic61—44no—
357Dic61—82no—
358C22×#137—611no—
359C22×#137D6111yescharacteristic, complemented
360#3449—611no—
361C2×#421—611no—
362C2×#421—611no—
363#2310—611no—
364#2007—618no—
365#1742—611no—
366C2×C61—21no—
367C2×C61—22no—
368C2×C61—22no—
369C2×C61—21no—
370C2×C61—82no—
371C2×C61C2×#39211yescharacteristic
372C2×C61#302212yes—
373D61—82no—
374C22×#41—611no—
375C22×#41C2×D6111yescharacteristic
376C23×Q8C2×D6111yescharacteristic
377C2×#137—22no—
378C2×#137—24no—
379C2×#137—24no—
380C2×#137—614no—
381C2×#137—1224no—
382C2×#137—1222no—
383C2×#137—1224no—
384C2×#137C2×D6114yescomplemented
385C22×Q16—21no—
386C22×Q16—1221no—
387C22×SD16—21no—
388C22×SD16—1221no—
389C22×M16—611no—
390C22×M16C2×D6111yescharacteristic
391#537—614no—
392#536—1221no—
393#536—1221no—
394#535—1221no—
395#535—1221no—
396#534—1222no—
397#532—1222no—
398#525—1222no—
399#525—1222no—
400C2×#117—611no—
401C2×#117—611no—
402#421—1224no—
403#421—1224no—
404C2×#104—614no—
405C2×#104—614no—
406#391—614no—
407#391—1222no—
408#381—1222no—
409#343—614no—
410C61#2340811yescharacteristic, complemented, Sylow
411C2×#41—21no—
412C2×#41—21no—
413C2×#41—22no—
414C2×#41—22no—
415C2×#41—22no—
416C2×#41—22no—
417C2×#41—612no—
418C2×#41—612no—
419C2×#41—1221no—
420C2×#41—1221no—
421C2×#41—1222no—
422C2×#41—1222no—
423C2×#41—1222no—
424C2×#41—1222no—
425C2×#41C22⋊D6112yes—
426C2×#41C22×D6112yes—
427C22×Q8—21no—
428C22×Q8—22no—
429C22×Q8—21no—
430C22×Q8—21no—
431C22×Q8—22no—
432C22×Q8—21no—
433C22×Q8—21no—
434C22×Q8—22no—
435C22×Q8—611no—
436C22×Q8C22⋊D6111yescharacteristic
437C22×Q8C22⋊D6114yescomplemented
438C22×Q8C22×D6112yes—
439C22×D4—21no—
440C22×D4—611no—
441C22×D4—1221no—
442C22×D4C22⋊D6111yescharacteristic
443C23×C4—21no—
444C23×C4—21no—
445C23×C4—1221no—
446C23×C4C22×D6111yescharacteristic
447#137—216no—
448#137—28no—
449#137—28no—
450#137—1228no—
451#137—12216no—
452#137—1228no—
453C2×Q16—28no—
454C2×Q16—24no—
455C2×Q16—1224no—
456C2×Q16—1228no—
457C2×SD16—28no—
458C2×SD16—24no—
459C2×SD16—1224no—
460C2×SD16—1228no—
461C2×M16—21no—
462C2×M16—22no—
463C2×M16—614no—
464C2×M16—612no—
465C2×M16—1221no—
466C2×M16—1222no—
467C2×M16C22⋊D6114yescomplemented
468C2×M16C22×D6112yes—
469C22×C8—21no—
470C22×C8—1221no—
471C4⋊Q8—1222no—
472#127—1222no—
473#124—1222no—
474#124—1222no—
475#123—1224no—
476#123—1224no—
477C22⋊Q8—1224no—
478C22⋊Q8—1224no—
479#121—1224no—
480#121—1228no—
481#121—1224no—
482#120—1224no—
483#120—1224no—
484C4×D4—1228no—
485#117—1221no—
486#117—1221no—
487#117—1221no—
488#117—1221no—
489#117—1222no—
490#117—1222no—
491C2×C4⋊C4—612no—
492C2×C4⋊C4—1221no—
493C2×C22⋊C4—612no—
494C2×C22⋊C4—1221no—
495C2×C42—612no—
496C2×C42—612no—
497#104—1228no—
498#104—1228no—
499#95—1228no—
500C24—21no—
501C24—1221no—
502C24D4×D6111yescharacteristic
503#41—22no—
504#41—22no—
505#41—22no—
506#41—22no—
507#41—24no—
508#41—24no—
509#41—42no—
510#41—42no—
511#41—42no—
512#41—42no—
513#41—1222no—
514#41—1222no—
515#41—1222no—
516#41—1222no—
517#41—1224no—
518#41—1224no—
519#41—2442no—
520#41—2442no—
521#41—2442no—
522#41—2442no—
523C2×Q8—22no—
524C2×Q8—21no—
525C2×Q8—24no—
526C2×Q8—22no—
527C2×Q8—22no—
528C2×Q8—22no—
529C2×Q8—22no—
530C2×Q8—24no—
531C2×Q8—24no—
532C2×Q8—24no—
533C2×Q8—22no—
534C2×Q8—24no—
535C2×Q8—22no—
536C2×Q8—41no—
537C2×Q8—41no—
538C2×Q8—42no—
539C2×Q8—42no—
540C2×Q8—42no—
541C2×Q8—42no—
542C2×Q8—42no—
543C2×Q8—612no—
544C2×Q8—1221no—
545C2×Q8—1222no—
546C2×Q8—1222no—
547C2×Q8C2×C22⋊D6112yes—
548C2×Q8C2×C22⋊D6114yes—
549C2×D4—22no—
550C2×D4—22no—
551C2×D4—21no—
552C2×D4—22no—
553C2×D4—22no—
554C2×D4—42no—
555C2×D4—42no—
556C2×D4—612no—
557C2×D4—1228no—
558C2×D4—1222no—
559C2×D4—1222no—
560C2×D4—1222no—
561C2×D4—1221no—
562C2×D4—1222no—
563C2×D4—2442no—
564C2×D4—2442no—
565C2×D4C2×C22⋊D6112yes—
566C22×C4—22no—
567C22×C4—21no—
568C22×C4—24no—
569C22×C4—22no—
570C22×C4—22no—
571C22×C4—41no—
572C22×C4—41no—
573C22×C4—42no—
574C22×C4—42no—
575C22×C4—42no—
576C22×C4—614no—
577C22×C4—1221no—
578C22×C4—1222no—
579C22×C4—1222no—
580C22×C4—1222no—
581C22×C4—2442no—
582C22×C4—2442no—
583C22×C4D4×D6111yescharacteristic
584C22×C4D4×D6111yescharacteristic
585C22×C4D4×D6112yes—
586C22×C4D4×D6111yescharacteristic
587C22×C4#1175822412yes—
588C22×C4C2×C22⋊D6112yes—
589C22×C4C2×C22⋊D6111yescharacteristic
590C22×C4C2×C22⋊D6111yescharacteristic
591C22×C4C2×C22⋊D6112yes—
592C22×C4C23×D6111yescharacteristic
593Q16—216no—
594Q16—12216no—
595SD16—216no—
596SD16—12216no—
597M16—22no—
598M16—24no—
599M16—614no—
600M16—1222no—
601M16—1224no—
602M16C2×C22⋊D6114yes—
603C2×C8—22no—
604C2×C8—24no—
605C2×C8—1222no—
606C2×C8—1224no—
607C4⋊C4—1228no—
608C4⋊C4—1222no—
609C4⋊C4—1222no—
610C4⋊C4—2442no—
611C22⋊C4—1228no—
612C22⋊C4—1228no—
613C22⋊C4—1222no—
614C22⋊C4—1222no—
615C22⋊C4—2442no—
616C42—1222no—
617C42—1222no—
618C42—1222no—
619C42—1222no—
620C23—22no—
621C23—42no—
622C23—42no—
623C23—82no—
624C23—1222no—
625C23—4882no—
626C23—2442no—
627C23#1451974511yescharacteristic
628C23#1451974511yescharacteristic
629C23#1451974912yes—
630C23#1451977312yes—
631C23C2×D4×D6111yescharacteristic
632Q8—24no—
633Q8—24no—
634Q8—22no—
635Q8—22no—
636Q8—42no—
637Q8—42no—
638Q8—44no—
639Q8—44no—
640Q8—42no—
641Q8—1222no—
642Q8—1222no—
643Q8—2442no—
644D4—22no—
645D4—22no—
646D4—42no—
647D4—42no—
648D4—44no—
649D4—42no—
650D4—1222no—
651D4—1222no—
652D4—4888no—
653D4—2442no—
654D4—2442no—
655D4—2444no—
656D4—2442no—
657C2×C4—22no—
658C2×C4—24no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—1224no—
668C2×C4—1222no—
669C2×C4—2444no—
670C2×C4—2442no—
671C2×C4—2442no—
672C2×C4—2442no—
673C2×C4—2442no—
674C2×C4—2444no—
675C2×C4—2444no—
676C2×C4#1451974512yes—
677C2×C4#1451974512yes—
678C2×C4#1451974511yescharacteristic
679C2×C4#1451974511yescharacteristic
680C2×C4#1451974612yes—
681C2×C4#1451974612yes—
682C2×C4#1451974812yes—
683C2×C4#1451974812yes—
684C2×C4#1451974912yes—
685C2×C4#1451974912yes—
686C2×C4#1451974912yes—
687C2×C4#1451977312yes—
688C2×C4C2×D4×D6111yescharacteristic
689C2×C4C2×D4×D6111yescharacteristic
690C2×C4C22×C22⋊D6111yescharacteristic
691C2×C4C22×C22⋊D6111yescharacteristic
692C2×C4C22×C22⋊D6111yescharacteristic
693C2×C4C2×#1175822411yescharacteristic
694C8—24no—
695C8—1224no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—44no—
703C22—84no—
704C22—4884no—
705C22—2442no—
706C22—2444no—
707C22#1683967012yes—
708C22#1683967012yes—
709C22C2×#1451974511yescharacteristic
710C22C2×#1451974911yescharacteristic
711C22C2×#1451977311yescharacteristic
712C4—42no—
713C4—44no—
714C4—4882no—
715C4—2442no—
716C4—2444no—
717C4#1683967012yes—
718C4#1683967012yes—
719C4C2×#1451974511yescharacteristic
720C4C2×#1451974611yescharacteristic
721C4C2×#1451974811yescharacteristic
722C4C2×#1451974911yescharacteristic
723C2—21no—
724C2—22no—
725C2—22no—
726C2—21no—
727C2—82no—
728C2—4882no—
729C2C2×#1683967011yescharacteristic
730C2#2145519512yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#137×C61C2
socleC22×C61C2×#120
prime radical#21459148C2
hypercenterC22×#137D61
hypocenterC61#23408
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
1. C22×#14519808C2semidirect
3. C2×#16839779C2semidirect
4. C2×#16839778C2semidirect
5. #21448405C2semidirect
6. #21446864C2semidirect
7. #21445557C2semidirect
8. C22×#137×C61C2semidirect
13. C2×#14519808C22semidirect
30. C2×#14519704C22semidirect
31. C2×#14519703C22semidirect
33. #16839669C22semidirect
35. #16839530C22semidirect
41. C2×#137×C61C22semidirect
66. C2×#11758194D4semidirect
115. C22×Q8×C61D4semidirect
131. C2×M16×C61D4semidirect
359. C22×#137D61semidirect
384. C2×#137C2×D61semidirect
410. C61#23408semidirect
437. C22×Q8C22⋊D61semidirect
467. C2×M16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
#214591482
C2×#168397792
C2×#168397782
#214484052
#214468642
#214455572
C22×#137×C612
#2340861