Finite Group Databasebeta

← #29585342

Subgroups — #29585342

Order 15616.

1226 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840528C211yesmaximal, characteristic
2C2×#16839735C211yesmaximal, characteristic
3#21448407C211yesmaximal, characteristic
4#21445540C211yesmaximal, characteristic, complemented
5#21445539C211yesmaximal, characteristic, complemented
6C2×#16839518C211yesmaximal, characteristic, complemented
7C2×C61×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9#16840528—22no—
10#16840528—22no—
11#16840528—22no—
12#16840528C414yescomplemented
13C2×#14519785—22no—
14C2×#14519783—22no—
15C22×#11758194C2211yescharacteristic
16C2×M16×D61—22no—
17C2×#14519736C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19#16839735—22no—
20#16839735C414yescomplemented
21C2×#14519691—22no—
22#16839668C412yescomplemented
23#16839664C412yescomplemented
24#16839639—22no—
25#16839619—22no—
26#16839525—22no—
27#16839524—22no—
28#16839518—22no—
29C22×M16×C61C2211yescharacteristic
30C2×#117×C61C2211yescharacteristic
31C2×#117×C61C2211yescharacteristic
32C61×H(3,ℤ/4)—22no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222C2×C412yes—
42C2×#11758222C2×C412yes—
43C2×#11758222D412yes—
44C2×#11758222Q812yes—
45C22×Dic122C2×C411yescharacteristic
46C22×D244C2×C411yescharacteristic
47C22×C22⋊D61D412yes—
48C23×Dic61—21no—
49C22×C4×D61D412yes—
50#14519785—28no—
51#14519785—28no—
52#14519783—28no—
53#14519783—28no—
54C2×#11758194—21no—
55C2×#11758194—21no—
56C2×#11758194—21no—
57C2×#11758194—21no—
58C2×#11758194C2×C412yes—
59C2×#11758194D412yes—
60C2×#11758189—24no—
61M16×D61—28no—
62M16×D61—28no—
63C22×#92418D412yes—
64C2×C8×D61—24no—
65#14519736—21no—
66#14519736—21no—
67#14519736—21no—
68#14519736C2×C412yes—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2×C412yes—
76C2×C22⋊Dic61—21no—
77C2×#11758198—24no—
78C2×C4×Dic61—21no—
79#14519691—28no—
80#14519684—24no—
81C23×C4×C61C2311yescharacteristic
82C2×M16×C61—21no—
83C2×M16×C61—21no—
84C2×M16×C61—21no—
85C2×M16×C61—21no—
86C2×M16×C61C2×C412yes—
87C2×M16×C61D412yes—
88C22×C8×C61D412yes—
89#117×C61—21no—
90#117×C61—21no—
91#117×C61—21no—
92#117×C61—21no—
93#117×C61—21no—
94#117×C61—21no—
95#117×C61—21no—
96#117×C61C2×C412yes—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—21no—
101C2×C42×C61—21no—
102C2×C42×C61C2×C412yes—
103C24×C61C2×D411yescharacteristic
104C23×D61C22⋊C412yes—
105C22×C4×C61—22no—
106C22×C4×C61—41no—
107C22×C4×C61C22×C411yescharacteristic
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×Q811yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C22×C4×C61#4111yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C2×C22⋊D61—22no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—22no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61C22⋊C414yes—
126C2×C22⋊D61C4⋊C414yes—
127C22×Dic61—22no—
128C22×Dic61—41no—
129C22×Dic61—41no—
130C22×Dic61—41no—
131C22×Dic61—82no—
132C22×Dic61C22⋊C412yes—
133#11758222—22no—
134#11758222—22no—
135#11758222—22no—
136#11758222—22no—
137#11758222—22no—
138#11758222—22no—
139#11758222—22no—
140#11758222—22no—
141#11758222—22no—
142#11758222—22no—
143#11758222—22no—
144#11758222—22no—
145#11758222—22no—
146#11758222—22no—
147#11758222—22no—
148#11758222—22no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244C4212yes—
154C2×D244C4⋊C412yes—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61C22⋊C414yes—
160C2×C4×D61C4⋊C414yes—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122C4212yes—
166C2×Dic122C4⋊C412yes—
167M16×C61—22no—
168M16×C61—22no—
169M16×C61—22no—
170M16×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61C22⋊C414yescomplemented
174C4⋊C4×C61—22no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C42×C61—22no—
183C42×C61—22no—
184C42×C61—22no—
185C42×C61—41no—
186C22⋊Dic61—22no—
187C22⋊Dic61—41no—
188#11758198—84no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61—41no—
193C4⋊Dic61—41no—
194C4⋊Dic61—41no—
195C4×Dic61—22no—
196C4×Dic61—41no—
197#11758194—22no—
198#11758194—22no—
199#11758194—22no—
200#11758194—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418C22⋊C414yescomplemented
204#11758189—216no—
205C8×D61—216no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61C2×C22⋊C411yescharacteristic
209C23×C61C2×C22⋊C411yescharacteristic
210C23×C61C2×C4⋊C411yescharacteristic
211C23×C61#12111yescharacteristic
212C23×C61#12111yescharacteristic
213C23×C61#12111yescharacteristic
214C23×C61#12111yescharacteristic
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61#9514yes—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—81no—
225C2×C4×C61—81no—
226C2×C4×C61C2×C4211yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C4⋊C411yescharacteristic
231C2×C4×C61C2×C4⋊C411yescharacteristic
232C2×C4×C61#11711yescharacteristic
233C2×C4×C61#11711yescharacteristic
234C2×C4×C61C4×D411yescharacteristic
235C2×C4×C61C4×D411yescharacteristic
236C2×C4×C61C4×D411yescharacteristic
237C2×C4×C61C4×D411yescharacteristic
238C2×C4×C61C4×D411yescharacteristic
239C2×C4×C61C4×D411yescharacteristic
240C2×C4×C61C4×D411yescharacteristic
241C2×C4×C61C4×D411yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61C22⋊Q811yescharacteristic
247C2×C4×C61C22⋊Q811yescharacteristic
248C2×C4×C61C22⋊Q811yescharacteristic
249C2×C4×C61C22⋊Q811yescharacteristic
250C2×C4×C61#12311yescharacteristic
251C2×C4×C61#12311yescharacteristic
252C2×C4×C61#12311yescharacteristic
253C2×C4×C61#12311yescharacteristic
254C22⋊D61—24no—
255C22⋊D61—24no—
256C22⋊D61—24no—
257C22⋊D61—24no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—44no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—81no—
264C2×Dic61—81no—
265C2×Dic61—82no—
266C2×Dic61—82no—
267C2×Dic61#9514yes—
268D244—22no—
269D244—22no—
270D244—22no—
271D244—22no—
272C4×D61—24no—
273C4×D61—24no—
274C4×D61—24no—
275C4×D61—24no—
276Dic122—22no—
277Dic122—22no—
278Dic122—22no—
279Dic122—22no—
280C8×C61—24no—
281#92418—24no—
282#10742—611nomaximal, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61C2×#9511yescharacteristic
294C22×C61#38011yescharacteristic
295C22×C61#38011yescharacteristic
296C22×C61#38611yescharacteristic
297C22×C61#38611yescharacteristic
298C22×C61#38611yescharacteristic
299C22×C61#38611yescharacteristic
300C2×D61—24no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—44no—
304C2×D61—82no—
305C4×C61—82no—
306C4×C61—82no—
307C4×C61C4×C22⋊C411yescharacteristic
308C4×C61C4×C22⋊C411yescharacteristic
309C4×C61#37911yescharacteristic
310C4×C61#38111yescharacteristic
311C4×C61#38511yescharacteristic
312C4×C61#38511yescharacteristic
313C4×C61#38511yescharacteristic
314C4×C61#38511yescharacteristic
315Dic61—24no—
316Dic61—82no—
317Dic61—82no—
318C2×#568—611no—
319C2×#410—611no—
320#2311—611no—
321#1736—611no—
322#1736—611no—
323C2×H(3,ℤ/4)—611no—
324C2×H(3,ℤ/4)D6111yescharacteristic, complemented
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—21no—
328C2×C61—21no—
329C2×C61—21no—
330C2×C61—21no—
331C2×C61C22⋊#9511yescharacteristic
332C2×C61#190911yescharacteristic
333C2×C61#191011yescharacteristic
334D61—44no—
335C22×#41—611no—
336#568—614no—
337#568—1222no—
338#568—1222no—
339#568—1222no—
340C2×#131—1222no—
341C2×#131—1222no—
342C22×M16—611no—
343C22×M16—1222no—
344C22×M16C2×D6111yescharacteristic
345C2×#117—611no—
346C2×#117C2×D6111yescharacteristic
347C2×#117C2×D6111yescharacteristic
348C22×C4⋊C4—611no—
349#410—614no—
350#410—1222no—
351C2×#99—1222no—
352#391—612no—
353#390—612no—
354#386—1222no—
355#380—1222no—
356#340—1222no—
357#340—1222no—
358H(3,ℤ/4)—22no—
359H(3,ℤ/4)—1222no—
360C61#1074211yescharacteristic, complemented, Sylow
361C2×#41—612no—
362C2×#41—612no—
363C2×#41—612no—
364C2×#41—612no—
365C2×#41—1221no—
366C2×#41—1221no—
367C2×#41—1221no—
368C2×#41—1221no—
369C2×#41—1221no—
370C2×#41—1221no—
371C2×#41—1221no—
372C2×#41—1221no—
373C22×Q8—611no—
374C22×D4—612no—
375C22×D4—611no—
376C23×C4—612no—
377C23×C4—1221no—
378C23×C4C22×D6111yescharacteristic
379#131—1228no—
380#131—1228no—
381#131—1228no—
382#131—1228no—
383C2×M16—21no—
384C2×M16—21no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—612no—
388C2×M16—612no—
389C2×M16—1224no—
390C2×M16—1228no—
391C2×M16—1228no—
392C2×M16—1221no—
393C2×M16—1221no—
394C2×M16—1221no—
395C2×M16—1221no—
396C2×M16C4×D6112yes—
397C2×M16C22⋊D6112yes—
398C22×C8—612no—
399C22×C8—1224no—
400C22×C8D24412yes—
401#117—21no—
402#117—21no—
403#117—21no—
404#117—21no—
405#117—21no—
406#117—21no—
407#117—21no—
408#117—612no—
409#117—1221no—
410#117—1221no—
411#117—1221no—
412#117C4×D6112yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—612no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1224no—
426C2×C42—21no—
427C2×C42—1221no—
428C2×C42C4×D6112yes—
429#99—1228no—
430#95—1224no—
431C24—612no—
432C24D4×D6111yescharacteristic
433#41—1222no—
434#41—1222no—
435#41—1222no—
436#41—1222no—
437#41—1222no—
438#41—1222no—
439#41—1222no—
440#41—1222no—
441#41—1222no—
442#41—1222no—
443#41—1222no—
444#41—1222no—
445#41—1222no—
446#41—1222no—
447#41—1222no—
448#41—1222no—
449C2×Q8—612no—
450C2×Q8—612no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×D4—614no—
456C2×D4—614no—
457C2×D4—612no—
458C2×D4—612no—
459C2×D4—1222no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C22×C4—22no—
468C22×C4—41no—
469C22×C4—612no—
470C22×C4—614no—
471C22×C4—614no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—4882no—
478C22×C4—2441no—
479C22×C4—2441no—
480C22×C4—2441no—
481C22×C4C2×C4×D6111yescharacteristic
482C22×C4C2×C4×D6111yescharacteristic
483C22×C4C2×C4×D6111yescharacteristic
484C22×C4C2×D24411yescharacteristic
485C22×C4#1175822211yescharacteristic
486C22×C4#1175822211yescharacteristic
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4D4×D6111yescharacteristic
490C22×C4D4×D6111yescharacteristic
491C22×C4#1175822411yescharacteristic
492C22×C4Q8×D6111yescharacteristic
493C22×C4#1175822611yescharacteristic
494C22×C4C2×C22⋊D6111yescharacteristic
495M16—22no—
496M16—22no—
497M16—22no—
498M16—22no—
499M16—12216no—
500M16—1222no—
501M16—1222no—
502M16—1222no—
503M16—1222no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—614no—
507C2×C8—12216no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8#1175819814yescomplemented
511C4⋊C4—22no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—2441no—
519C4⋊C4—2441no—
520C4⋊C4—2441no—
521C22⋊C4—22no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—1222no—
526C22⋊C4—4884no—
527C22⋊C4—2441no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—41no—
532C42—1222no—
533C42—2441no—
534C23—41no—
535C23—41no—
536C23—614no—
537C23—1222no—
538C23—1222no—
539C23—2442no—
540C23—2442no—
541C23C22⋊C4×D6111yescharacteristic
542C23C22⋊C4×D6111yescharacteristic
543C23C4⋊C4×D6111yescharacteristic
544C23#1451974811yescharacteristic
545C23#1451975011yescharacteristic
546C23#1451975011yescharacteristic
547C23#1451975211yescharacteristic
548Q8—1222no—
549Q8—1222no—
550Q8—1222no—
551Q8—1222no—
552D4—1224no—
553D4—1224no—
554D4—1224no—
555D4—1224no—
556D4—1222no—
557D4—1222no—
558D4—1222no—
559D4—1222no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—614no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1224no—
570C2×C4—1224no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—4882no—
574C2×C4—4882no—
575C2×C4—4881no—
576C2×C4—4881no—
577C2×C4—2444no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4C42×D6111yescharacteristic
581C2×C4C22⋊C4×D6111yescharacteristic
582C2×C4C22⋊C4×D6111yescharacteristic
583C2×C4C2×#1175819811yescharacteristic
584C2×C4C4⋊C4×D6111yescharacteristic
585C2×C4C4⋊C4×D6111yescharacteristic
586C2×C4#1451973411yescharacteristic
587C2×C4#1451973511yescharacteristic
588C2×C4#1451973711yescharacteristic
589C2×C4#1451973711yescharacteristic
590C2×C4C4×C22⋊D6111yescharacteristic
591C2×C4C4×C22⋊D6111yescharacteristic
592C2×C4#1451973911yescharacteristic
593C2×C4#1451973911yescharacteristic
594C2×C4C4×D24411yescharacteristic
595C2×C4C4×D24411yescharacteristic
596C2×C4#1451974511yescharacteristic
597C2×C4#1451974511yescharacteristic
598C2×C4#1451974711yescharacteristic
599C2×C4#1451974711yescharacteristic
600C2×C4#1451975311yescharacteristic
601C2×C4#1451975511yescharacteristic
602C2×C4#1451975511yescharacteristic
603C2×C4#1451975711yescharacteristic
604C2×C4#1451975911yescharacteristic
605C2×C4#1451975911yescharacteristic
606C2×C4#1451976211yescharacteristic
607C2×C4#1451976211yescharacteristic
608C8—24no—
609C8—1224no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—1224no—
621C22—4882no—
622C22—2444no—
623C22—2442no—
624C22—2442no—
625C22#95×D6111yescharacteristic
626C22#1683961911yescharacteristic
627C22#1683962111yescharacteristic
628C22#1683963811yescharacteristic
629C22#1683963811yescharacteristic
630C22#1683964011yescharacteristic
631C22#1683964011yescharacteristic
632C4—82no—
633C4—82no—
634C4—1224no—
635C4—4882no—
636C4—4882no—
637C4#1683960911yescharacteristic
638C4C4×#1175819811yescharacteristic
639C4#1683961711yescharacteristic
640C4#1683962411yescharacteristic
641C4#1683963311yescharacteristic
642C4#1683963311yescharacteristic
643C4#1683963611yescharacteristic
644C4#1683963611yescharacteristic
645C2—21no—
646C2—21no—
647C2—21no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2—2444no—
652C2#2144404011yescharacteristic
653C2#2144634011yescharacteristic
654C2#2144634511yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC2×C61×H(3,ℤ/4)C2
socleC22×C61#386
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×H(3,ℤ/4)D61
hypocenterC61#10742
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
4. #21445540C2semidirect
5. #21445539C2semidirect
6. C2×#16839518C2semidirect
7. C2×C61×H(3,ℤ/4)C2semidirect
12. #16840528C4semidirect
20. #16839735C4semidirect
22. #16839668C4semidirect
23. #16839664C4semidirect
173. C2×C8×C61C22⋊C4semidirect
203. C2×#92418C22⋊C4semidirect
324. C2×H(3,ℤ/4)D61semidirect
360. C61#10742semidirect
510. C2×C8#11758198semidirect

Maximal subgroups

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

subgroupindex
C2×#168405282
C2×#168397352
#214484072
#214455402
#214455392
C2×#168395182
C2×C61×H(3,ℤ/4)2
#1074261