Finite Group Databasebeta

← #29707674

Subgroups — #29707674

Order 15616.

1422 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459148C211yesmaximal, characteristic
2C2×#16839780C211yesmaximal, characteristic, complemented
3C2×#16839764C211yesmaximal, characteristic, complemented
4C2×#16839763C211yesmaximal, characteristic, complemented
5#21448442C211yesmaximal, characteristic, complemented
6#21447047C214yesmaximal, complemented
7#21447042C214yesmaximal, complemented
8#21445557C211yesmaximal, characteristic, complemented
9C2×C61×#421C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840340C2212yes—
13#16840340C2212yes—
14#16840336—21no—
15#16840333—21no—
16#16840329—21no—
17#16840326—21no—
18#16840321—22no—
19#16840308—22no—
20#16840273—22no—
21#16840269—22no—
22C2×#14519736C2211yescharacteristic
23C22×C4⋊Dic61C2211yescharacteristic
24#16839780—24no—
25C2×#14519705C2214yescomplemented
26#16839764—22no—
27#16839764C2214yescomplemented
28#16839763—22no—
29#16839763C2214yescomplemented
30C2×#14519702—21no—
31C2×#14519701—21no—
32C2×#14519699—21no—
33C2×#14519698—21no—
34#16839661—22no—
35#16839637—22no—
36#16839632C2212yescomplemented
37#16839628C2212yescomplemented
38#16839530C2212yescomplemented
39#16839530C2212yescomplemented
40C22×#41×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C61×#421—24no—
44C2×#104×C61C2214yescomplemented
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—22no—
70#14519756—22no—
71#14519753—22no—
72#14519753—22no—
73#14519751—22no—
74#14519751—22no—
75#14519749—24no—
76#14519749—24no—
77#14519748—22no—
78#14519748—22no—
79#14519746—24no—
80#14519745—24no—
81C4×C22⋊D61—24no—
82C4×C22⋊D61—24no—
83#14519736—21no—
84#14519736—21no—
85#14519736—21no—
86#14519736—21no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61—24no—
93C2×C4⋊Dic61C2312yes—
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C22⋊Dic61D412yes—
97C2×C4×Dic61D412yes—
98#14519705—28no—
99#14519702—24no—
100#14519701—24no—
101#14519699—24no—
102#14519698—24no—
103#14519684—24no—
104C2×#41×C61—21no—
105C2×#41×C61—21no—
106C2×#41×C61—22no—
107C2×#41×C61—22no—
108C2×#41×C61—22no—
109C2×#41×C61—22no—
110C2×#41×C61D412yes—
111C2×#41×C61C2312yes—
112C22×Q8×C61D411yescharacteristic
113C22×D4×C61—21no—
114C22×D4×C61D411yescharacteristic
115C23×C4×C61—21no—
116C23×C4×C61C2311yescharacteristic
117C2×M16×C61—21no—
118C2×M16×C61—22no—
119C2×M16×C61—22no—
120C2×M16×C61C2312yes—
121C22×C8×C61—21no—
122#117×C61—22no—
123#117×C61D414yescomplemented
124C2×C4⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C42×C61C2312yes—
127#104×C61—28no—
128C24×C61—21no—
129C24×C61C2×D411yescharacteristic
130C23×D61—21no—
131#41×C61—24no—
132#41×C61—24no—
133#41×C61—22no—
134#41×C61—22no—
135#41×C61—22no—
136#41×C61—22no—
137#41×C61—42no—
138#41×C61—42no—
139#41×C61—42no—
140#41×C61—42no—
141C2×Q8×C61—21no—
142C2×Q8×C61—22no—
143C2×Q8×C61—22no—
144C2×Q8×C61C2×D412yes—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—21no—
148C2×D4×C61—22no—
149C2×D4×C61—22no—
150C2×D4×C61—42no—
151C2×D4×C61—42no—
152C2×D4×C61C2×D412yes—
153C22×C4×C61—21no—
154C22×C4×C61—22no—
155C22×C4×C61—22no—
156C22×C4×C61—22no—
157C22×C4×C61—42no—
158C22×C4×C61—42no—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D412yes—
165C22×C4×C61#4112yes—
166C22×C4×C61#4112yes—
167C22×C4×C61#4112yes—
168C22×C4×C61C2411yescharacteristic
169C2×C22⋊D61—22no—
170C2×C22⋊D61—22no—
171C2×C22⋊D61—24no—
172C2×C22⋊D61—24no—
173C2×C22⋊D61—42no—
174C2×C22⋊D61—42no—
175C22×Dic61—21no—
176C22×Dic61—22no—
177C22×Dic61—41no—
178C22×Dic61—41no—
179C22×Dic61—82no—
180C22×Dic61C2×D412yes—
181C22×Dic61C2×D412yes—
182#11758222—24no—
183#11758222—24no—
184#11758222—22no—
185#11758222—22no—
186#11758222—22no—
187#11758222—22no—
188#11758222—42no—
189#11758222—42no—
190#11758222—42no—
191#11758222—42no—
192C2×D244—22no—
193C2×D244—22no—
194C2×D244—21no—
195C2×D244C2×D412yes—
196C2×C4×D61—22no—
197C2×C4×D61—22no—
198C2×C4×D61—42no—
199C2×C4×D61—42no—
200C2×Dic122—22no—
201C2×Dic122—22no—
202C2×Dic122—21no—
203C2×Dic122C2×D412yes—
204M16×C61—22no—
205M16×C61—22no—
206M16×C61—42no—
207C2×C8×C61—22no—
208C2×C8×C61—42no—
209C4⋊C4×C61—24no—
210C22⋊C4×C61—24no—
211C42×C61—22no—
212C42×C61C2×D414yes—
213C22⋊Dic61—24no—
214C22⋊Dic61—24no—
215C22⋊Dic61—22no—
216C22⋊Dic61—22no—
217#11758198—24no—
218#11758198—24no—
219C4⋊Dic61—21no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222C4⋊Dic61—21no—
223C4⋊Dic61—22no—
224C4⋊Dic61—22no—
225C4⋊Dic61—41no—
226C4⋊Dic61—41no—
227C4⋊Dic61—84no—
228#11758196—24no—
229#11758196—24no—
230C4×Dic61—22no—
231C4×Dic61—22no—
232#11758194—22no—
233#11758194—22no—
234#11758194—42no—
235C2×#92418—22no—
236C2×#92418—42no—
237C23×C61—22no—
238C23×C61—41no—
239C23×C61—41no—
240C23×C61—42no—
241C23×C61—81no—
242C23×C61—81no—
243C23×C61#12011yescharacteristic
244C23×C61#12011yescharacteristic
245C23×C61#12112yes—
246C23×C61#12412yes—
247C23×C61C22×D411yescharacteristic
248C22×D61—22no—
249C22×D61—42no—
250C22×D61—81no—
251C22×D61—81no—
252Q8×C61—22no—
253Q8×C61—22no—
254Q8×C61—42no—
255D4×C61—22no—
256D4×C61—22no—
257D4×C61—42no—
258D4×C61—42no—
259D4×C61—44no—
260D4×C61—42no—
261C2×C4×C61—22no—
262C2×C4×C61—24no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61—44no—
269C2×C4×C61#12011yescharacteristic
270C2×C4×C61#12011yescharacteristic
271C2×C4×C61#12112yes—
272C2×C4×C61#12112yes—
273C2×C4×C61#12112yes—
274C2×C4×C61#12112yes—
275C2×C4×C61#12112yes—
276C2×C4×C61#12312yes—
277C2×C4×C61#12312yes—
278C2×C4×C61#12312yes—
279C2×C4×C61#12312yes—
280C2×C4×C61#12412yes—
281C2×C4×C61C22×D411yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C2×#4111yescharacteristic
285C2×C4×C61C2×#4111yescharacteristic
286C2×C4×C61C2×#4111yescharacteristic
287C22⋊D61—42no—
288C22⋊D61—42no—
289C22⋊D61—44no—
290C22⋊D61—84no—
291C22⋊D61—84no—
292C2×Dic61—22no—
293C2×Dic61—22no—
294C2×Dic61—22no—
295C2×Dic61—42no—
296C2×Dic61—42no—
297C2×Dic61—42no—
298C2×Dic61—44no—
299C2×Dic61—42no—
300C2×Dic61—82no—
301C2×Dic61—82no—
302D244—22no—
303D244—22no—
304D244—42no—
305C4×D61—42no—
306C4×D61—42no—
307C4×D61—44no—
308Dic122—22no—
309Dic122—22no—
310Dic122—42no—
311C8×C61—42no—
312#92418—42no—
313#23500—611nomaximal, Sylow
314C22×C61—21no—
315C22×C61—21no—
316C22×C61—21no—
317C22×C61—21no—
318C22×C61—21no—
319C22×C61—21no—
320C22×C61—42no—
321C22×C61—42no—
322C22×C61—42no—
323C22×C61—42no—
324C22×C61—42no—
325C22×C61—84no—
326C22×C61#39412yes—
327C22×C61#39412yes—
328C22×C61C2×#12011yescharacteristic
329C22×C61C2×#12111yescharacteristic
330C22×C61C2×#12411yescharacteristic
331C2×D61—42no—
332C2×D61—42no—
333C2×D61—42no—
334C2×D61—84no—
335C4×C61—44no—
336C4×C61—42no—
337C4×C61#39412yes—
338C4×C61#39412yes—
339C4×C61C2×#12111yescharacteristic
340C4×C61C2×#12111yescharacteristic
341C4×C61C2×#12311yescharacteristic
342C4×C61C2×#12311yescharacteristic
343Dic61—42no—
344Dic61—42no—
345Dic61—42no—
346Dic61—82no—
347#3449—611no—
348C2×#421—611no—
349C2×#421D6111yescharacteristic, complemented
350C2×#417—611no—
351C2×#417—611no—
352#2317—611no—
353#2040—614no—
354#2039—614no—
355#1742—611no—
356C2×C61—21no—
357C2×C61—22no—
358C2×C61—22no—
359C2×C61—21no—
360C2×C61—82no—
361C2×C61C2×#39411yescharacteristic
362C2×C61#302111yescharacteristic
363C2×C61#302211yescharacteristic
364D61—82no—
365C22×#41—611no—
366C22×#41C2×D6111yescharacteristic
367C22×M16—611no—
368C22×M16C2×D6111yescharacteristic
369#537—612no—
370#537—612no—
371#536—1221no—
372#536—1221no—
373#535—1221no—
374#535—1221no—
375#534—1222no—
376#532—1222no—
377#525—1222no—
378#525—1222no—
379C2×#117—611no—
380C2×#117C2×D6111yescharacteristic
381C22×C4⋊C4—611no—
382#421—24no—
383#421—1224no—
384C2×#104—614no—
385C2×#104C2×D6114yescomplemented
386#417—614no—
387#417—614no—
388#417—1222no—
389#417—1222no—
390C2×#103—1221no—
391C2×#103—1221no—
392C2×#102—1221no—
393C2×#102—1221no—
394#389—1222no—
395#385—1222no—
396#384—612no—
397#382—612no—
398#343—612no—
399#343—612no—
400C61#2350011yescharacteristic, complemented, Sylow
401C2×#41—21no—
402C2×#41—21no—
403C2×#41—22no—
404C2×#41—22no—
405C2×#41—22no—
406C2×#41—22no—
407C2×#41—612no—
408C2×#41—612no—
409C2×#41—1221no—
410C2×#41—1221no—
411C2×#41—1222no—
412C2×#41—1222no—
413C2×#41—1222no—
414C2×#41—1222no—
415C2×#41D24412yes—
416C2×#41C22×D6112yes—
417C22×Q8—611no—
418C22×Q8D24411yescharacteristic
419C22×D4—21no—
420C22×D4—611no—
421C22×D4—1221no—
422C22×D4D24411yescharacteristic
423C23×C4—21no—
424C23×C4—1221no—
425C23×C4—1221no—
426C23×C4C22×D6111yescharacteristic
427C2×M16—21no—
428C2×M16—22no—
429C2×M16—22no—
430C2×M16—612no—
431C2×M16—1221no—
432C2×M16—1222no—
433C2×M16—1222no—
434C2×M16C22×D6112yes—
435C22×C8—21no—
436C22×C8—1221no—
437C4⋊Q8—1222no—
438#127—1222no—
439#124—1222no—
440#124—1222no—
441#123—1224no—
442#123—1224no—
443C22⋊Q8—1222no—
444C22⋊Q8—1222no—
445C22⋊Q8—1222no—
446C22⋊Q8—1222no—
447#121—1222no—
448#121—1222no—
449#121—1224no—
450#121—1224no—
451#121—1222no—
452#121—1222no—
453#120—1224no—
454#120—1224no—
455C4×D4—1224no—
456C4×D4—1224no—
457#117—22no—
458#117—1221no—
459#117—1221no—
460#117—1221no—
461#117—1221no—
462#117C22⋊D6114yescomplemented
463C2×C4⋊C4—21no—
464C2×C4⋊C4—611no—
465C2×C4⋊C4—611no—
466C2×C4⋊C4—612no—
467C2×C4⋊C4—1221no—
468C2×C4⋊C4—1221no—
469C2×C4⋊C4—1221no—
470C2×C4⋊C4—1221no—
471C2×C4⋊C4—1224no—
472C2×C4⋊C4—1221no—
473C2×C22⋊C4—21no—
474C2×C22⋊C4—612no—
475C2×C42—612no—
476C2×C42C22×D6112yes—
477#104—28no—
478#104—1228no—
479#103—1224no—
480#103—1224no—
481#102—1224no—
482#102—1224no—
483#95—1224no—
484C24—21no—
485C24—1221no—
486C24D4×D6111yescharacteristic
487#41—22no—
488#41—22no—
489#41—22no—
490#41—22no—
491#41—24no—
492#41—24no—
493#41—42no—
494#41—42no—
495#41—42no—
496#41—42no—
497#41—1222no—
498#41—1222no—
499#41—1222no—
500#41—1222no—
501#41—1224no—
502#41—1224no—
503#41—2442no—
504#41—2442no—
505#41—2442no—
506#41—2442no—
507C2×Q8—21no—
508C2×Q8—22no—
509C2×Q8—22no—
510C2×Q8—612no—
511C2×Q8—1221no—
512C2×Q8—1222no—
513C2×Q8—1222no—
514C2×Q8C2×D24412yes—
515C2×D4—22no—
516C2×D4—22no—
517C2×D4—21no—
518C2×D4—22no—
519C2×D4—22no—
520C2×D4—42no—
521C2×D4—42no—
522C2×D4—612no—
523C2×D4—1224no—
524C2×D4—1224no—
525C2×D4—1222no—
526C2×D4—1222no—
527C2×D4—1221no—
528C2×D4—1222no—
529C2×D4—1222no—
530C2×D4—2442no—
531C2×D4—2442no—
532C2×D4C2×D24412yes—
533C22×C4—22no—
534C22×C4—21no—
535C22×C4—22no—
536C22×C4—22no—
537C22×C4—42no—
538C22×C4—42no—
539C22×C4—612no—
540C22×C4—612no—
541C22×C4—1221no—
542C22×C4—1222no—
543C22×C4—1222no—
544C22×C4—1222no—
545C22×C4—4882no—
546C22×C4—2441no—
547C22×C4—2441no—
548C22×C4—2442no—
549C22×C4—2442no—
550C22×C4C2×D24411yescharacteristic
551C22×C4C2×D24411yescharacteristic
552C22×C4#1175822212yes—
553C22×C4D4×D6111yescharacteristic
554C22×C4D4×D6111yescharacteristic
555C22×C4D4×D6111yescharacteristic
556C22×C4#1175822412yes—
557C22×C4#1175822412yes—
558C22×C4C2×C22⋊D6112yes—
559C22×C4C23×D6111yescharacteristic
560M16—22no—
561M16—22no—
562M16—42no—
563M16—1222no—
564M16—1222no—
565M16—2442no—
566C2×C8—22no—
567C2×C8—42no—
568C2×C8—1222no—
569C2×C8—2442no—
570C4⋊C4—24no—
571C4⋊C4—1224no—
572C4⋊C4—1224no—
573C4⋊C4—1221no—
574C4⋊C4—1221no—
575C4⋊C4—1221no—
576C4⋊C4—1221no—
577C4⋊C4—1222no—
578C4⋊C4—1222no—
579C4⋊C4—4884no—
580C4⋊C4—2441no—
581C4⋊C4—2441no—
582C22⋊C4—24no—
583C22⋊C4—1224no—
584C22⋊C4—1224no—
585C22⋊C4—1224no—
586C22⋊C4—1224no—
587C22⋊C4—1222no—
588C22⋊C4—1222no—
589C42—22no—
590C42—1222no—
591C42—1222no—
592C42C2×C22⋊D6114yes—
593C23—22no—
594C23—41no—
595C23—41no—
596C23—42no—
597C23—81no—
598C23—81no—
599C23—1222no—
600C23—4881no—
601C23—4881no—
602C23—2442no—
603C23#1451974711yescharacteristic
604C23#1451974711yescharacteristic
605C23#1451974912yes—
606C23#1451976312yes—
607C23C2×D4×D6111yescharacteristic
608Q8—22no—
609Q8—22no—
610Q8—42no—
611Q8—1222no—
612Q8—1222no—
613Q8—2442no—
614D4—22no—
615D4—22no—
616D4—42no—
617D4—42no—
618D4—44no—
619D4—42no—
620D4—1222no—
621D4—1222no—
622D4—4884no—
623D4—4884no—
624D4—2442no—
625D4—2442no—
626D4—2444no—
627D4—2442no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—1222no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—4882no—
640C2×C4—4882no—
641C2×C4—2444no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4—2442no—
645C2×C4—2442no—
646C2×C4—2444no—
647C2×C4—2442no—
648C2×C4—2442no—
649C2×C4#1451974711yescharacteristic
650C2×C4#1451974711yescharacteristic
651C2×C4#1451974812yes—
652C2×C4#1451974812yes—
653C2×C4#1451974912yes—
654C2×C4#1451975112yes—
655C2×C4#1451975112yes—
656C2×C4#1451975912yes—
657C2×C4#1451975912yes—
658C2×C4#1451976112yes—
659C2×C4#1451976112yes—
660C2×C4#1451976312yes—
661C2×C4C2×D4×D6111yescharacteristic
662C2×C4C22×C22⋊D6111yescharacteristic
663C2×C4C22×D24411yescharacteristic
664C2×C4C2×#1175822411yescharacteristic
665C2×C4C2×#1175822411yescharacteristic
666C2×C4C2×#1175822211yescharacteristic
667C8—42no—
668C8—2442no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—84no—
681C22—4884no—
682C22—2442no—
683C22—2442no—
684C22—2442no—
685C22#1683967812yes—
686C22#1683967812yes—
687C22C2×#1451974711yescharacteristic
688C22C2×#1451974911yescharacteristic
689C22C2×#1451976311yescharacteristic
690C4—44no—
691C4—42no—
692C4—4882no—
693C4—2442no—
694C4—2442no—
695C4—2442no—
696C4#1683967812yes—
697C4#1683967812yes—
698C4C2×#1451974811yescharacteristic
699C4C2×#1451975111yescharacteristic
700C4C2×#1451975911yescharacteristic
701C4C2×#1451976111yescharacteristic
702C2—21no—
703C2—22no—
704C2—22no—
705C2—21no—
706C2—82no—
707C2—4882no—
708C2C2×#1683967811yescharacteristic
709C2#2145519211yescharacteristic
710C2#2145519911yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#421C2
socleC22×C61C2×#120
prime radical#21459148C2
hypercenterC2×#421D61
hypocenterC61#23500
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. C2×#16839780C2semidirect
3. C2×#16839764C2semidirect
4. C2×#16839763C2semidirect
5. #21448442C2semidirect
6. #21447047C2semidirect
7. #21447042C2semidirect
8. #21445557C2semidirect
9. C2×C61×#421C2semidirect
25. C2×#14519705C22semidirect
27. #16839764C22semidirect
29. #16839763C22semidirect
36. #16839632C22semidirect
37. #16839628C22semidirect
38. #16839530C22semidirect
39. #16839530C22semidirect
44. C2×#104×C61C22semidirect
123. #117×C61D4semidirect
349. C2×#421D61semidirect
385. C2×#104C2×D61semidirect
400. C61#23500semidirect
462. #117C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214591482
C2×#168397802
C2×#168397642
C2×#168397632
#214484422
#214470472
#214470422
#214455572
C2×C61×#4212
#2350061