Finite Group Databasebeta

← #29587135

Subgroups — #29587135

Order 15616.

1078 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
1C22×#14519703C211yesmaximal, characteristic
2C2×#16839764C211yesmaximal, characteristic
3#21448412C211yesmaximal, characteristic
4C2×#16839631C211yesmaximal, characteristic, complemented
5#21445539C211yesmaximal, characteristic, complemented
6C2×#16839523C211yesmaximal, characteristic, complemented
7C2×C61×#343C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C2×#14519736C2211yescharacteristic
10C22×C4⋊Dic61C2211yescharacteristic
11C22×#11758196—22no—
12C22×C4×Dic61C2211yescharacteristic
13C2×#14519703—22no—
14C2×#14519703—22no—
15C2×#14519703C414yescomplemented
16#16839764—22no—
17#16839764—22no—
18C2×#14519702—21no—
19C2×#14519699—21no—
20#16839667—21no—
21#16839663—21no—
22#16839641—22no—
23#16839631—22no—
24#16839631—22no—
25#16839619—22no—
26#16839525—22no—
27#16839523—22no—
28C22×M16×C61C2211yescharacteristic
29C22×M16×C61C2211yescharacteristic
30C22×C4⋊C4×C61C2211yescharacteristic
31C61×#343—22no—
32C2×#11758222—21no—
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×#11758222—21no—
42C2×#11758222C2×C412yes—
43C2×#11758222Q812yes—
44C22×Dic122D411yescharacteristic
45C22×D244D411yescharacteristic
46C22×C22⋊D61—21no—
47C23×Dic61—21no—
48C23×Dic61—21no—
49C22×C4×D61—21no—
50#14519736—21no—
51#14519736—21no—
52#14519736—21no—
53#14519736—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61D412yes—
60C2×C4⋊Dic61D412yes—
61C2×#11758196—28no—
62C2×#11758196—48no—
63C2×C22⋊Dic61—21no—
64C2×#11758198—24no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—21no—
69C2×C4×Dic61C2×C412yes—
70C2×C4×Dic61D412yes—
71C2×C4×Dic61D412yes—
72#14519703—28no—
73#14519702—24no—
74#14519699—24no—
75#14519684—24no—
76C23×C4×C61D412yes—
77C23×C4×C61C2311yescharacteristic
78C2×M16×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61—21no—
82C2×M16×C61—21no—
83C2×M16×C61—21no—
84C2×M16×C61—21no—
85C2×M16×C61—21no—
86C2×M16×C61—21no—
87C2×M16×C61—21no—
88C2×M16×C61C2×C412yes—
89C2×M16×C61D412yes—
90C22×C8×C61—21no—
91C22×C8×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C24×C61C2×D411yescharacteristic
99C23×D61—21no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—44no—
104C22×C4×C61C22×C411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×Q811yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C2×C22⋊D61—22no—
119C2×C22⋊D61—22no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—41no—
123C2×C22⋊D61—41no—
124C22×Dic61—24no—
125C22×Dic61—24no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—44no—
133C22×Dic61—41no—
134C22×Dic61—41no—
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—41no—
144#11758222—41no—
145#11758222—41no—
146#11758222—41no—
147#11758222—41no—
148#11758222—41no—
149#11758222—41no—
150#11758222—41no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244C4⋊C412yescomplemented
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122C4⋊C412yescomplemented
169M16×C61—22no—
170M16×C61—22no—
171M16×C61—22no—
172M16×C61—22no—
173M16×C61—41no—
174M16×C61—41no—
175M16×C61—41no—
176M16×C61—41no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C2×C8×C61—41no—
180C2×C8×C61—41no—
181C2×C8×C61—41no—
182C2×C8×C61—41no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61—41no—
187C22⋊Dic61—41no—
188C22⋊Dic61—41no—
189#11758198—84no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—41no—
195C4⋊Dic61—41no—
196#11758196—416no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—41no—
201C4×Dic61—41no—
202C4×Dic61C22⋊C414yescomplemented
203C23×C61—22no—
204C23×C61—22no—
205C23×C61C4×D411yescharacteristic
206C23×C61#12011yescharacteristic
207C23×C61#12011yescharacteristic
208C23×C61#12111yescharacteristic
209C23×C61#12111yescharacteristic
210C23×C61C22⋊Q811yescharacteristic
211C23×C61#12411yescharacteristic
212C22×D61—22no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—42no—
216C22×D61—42no—
217C2×C4×C61—44no—
218C2×C4×C61—42no—
219C2×C4×C61—44no—
220C2×C4×C61—82no—
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C4⋊C411yescharacteristic
223C2×C4×C61#11711yescharacteristic
224C2×C4×C61C4×D411yescharacteristic
225C2×C4×C61C4×D411yescharacteristic
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61C22⋊Q811yescharacteristic
235C2×C4×C61C22⋊Q811yescharacteristic
236C2×C4×C61C22⋊Q811yescharacteristic
237C2×C4×C61C22⋊Q811yescharacteristic
238C2×C4×C61C22⋊Q811yescharacteristic
239C2×C4×C61#12311yescharacteristic
240C2×C4×C61#12311yescharacteristic
241C2×C4×C61#12311yescharacteristic
242C2×C4×C61#12311yescharacteristic
243C2×C4×C61#12311yescharacteristic
244C2×C4×C61#12411yescharacteristic
245C2×C4×C61#12411yescharacteristic
246C2×C4×C61#12611yescharacteristic
247C2×C4×C61#12611yescharacteristic
248C2×C4×C61#12711yescharacteristic
249C22⋊D61—42no—
250C22⋊D61—42no—
251C22⋊D61—42no—
252C22⋊D61—42no—
253C2×Dic61—24no—
254C2×Dic61—28no—
255C2×Dic61—24no—
256C2×Dic61—28no—
257C2×Dic61—24no—
258C2×Dic61—22no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—44no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—82no—
267C2×Dic61—82no—
268D244—22no—
269D244—22no—
270D244—41no—
271D244—41no—
272C4×D61—42no—
273C4×D61—42no—
274C4×D61—42no—
275C4×D61—42no—
276Dic122—22no—
277Dic122—22no—
278Dic122—41no—
279Dic122—41no—
280C8×C61—42no—
281C8×C61—42no—
282#11078—611nomaximal, Sylow
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61#38511yescharacteristic
292C22×C61#38611yescharacteristic
293C22×C61#38811yescharacteristic
294C22×C61#39211yescharacteristic
295C22×C61#39311yescharacteristic
296C22×C61#39411yescharacteristic
297C22×C61#39611yescharacteristic
298C22×C61#45312yes—
299C22×C61#45612yes—
300C2×D61—44no—
301C2×D61—42no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—82no—
305C4×C61—82no—
306C4×C61#38011yescharacteristic
307C4×C61#38511yescharacteristic
308C4×C61#38811yescharacteristic
309C4×C61#39011yescharacteristic
310C4×C61#39411yescharacteristic
311C4×C61#39611yescharacteristic
312C4×C61#39711yescharacteristic
313C4×C61#39911yescharacteristic
314Dic61—28no—
315Dic61—42no—
316Dic61—82no—
317Dic61—82no—
318C22×#104—611no—
319C2×#417—611no—
320#2311—611no—
321C2×#384—611no—
322C2×#343D6111yescharacteristic, complemented
323#1736—611no—
324C2×#339—611no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—21no—
328C2×C61—21no—
329C2×C61#143911yescharacteristic
330C2×C61#185912yes—
331C2×C61#200211yescharacteristic
332C2×C61#200511yescharacteristic
333C2×C61#203912yes—
334D61—82no—
335C22×#41—611no—
336C22×M16C2×D6111yescharacteristic
337C22×M16C2×D6111yescharacteristic
338C2×#117—611no—
339C22×C4⋊C4—611no—
340C22×C4⋊C4—1222no—
341C22×C4⋊C4C2×D6111yescharacteristic
342C22×C42—611no—
343C2×#104—614no—
344C2×#104—1222no—
345C2×#104—1222no—
346#417—1222no—
347#417—1222no—
348C2×#103—1221no—
349C2×#102—1221no—
350#391—1221no—
351#390—1221no—
352#386—1222no—
353#384—1222no—
354#384—1222no—
355#380—1222no—
356#343—22no—
357#340—1222no—
358#339—1222no—
359C61#1107811yescharacteristic, complemented, Sylow
360C2×#41—612no—
361C2×#41—612no—
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1221no—
365C2×#41—1221no—
366C2×#41—1221no—
367C2×#41—1221no—
368C2×#41—1221no—
369C2×#41—1221no—
370C2×#41—1221no—
371C2×#41—1221no—
372C22×Q8—611no—
373C22×D4—611no—
374C22×D4—1221no—
375C23×C4—1221no—
376C23×C4—1221no—
377C23×C4—1221no—
378C23×C4D24412yes—
379C23×C4C22×D6111yescharacteristic
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—21no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—21no—
389C2×M16—21no—
390C2×M16C4×D6112yes—
391C2×M16C22⋊D6112yes—
392C22×C8—21no—
393C22×C8—21no—
394#117—1221no—
395#117—1221no—
396#117—1221no—
397#117—1221no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1228no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—2448no—
413C2×C22⋊C4—1224no—
414C2×C22⋊C4—1221no—
415C2×C42—612no—
416C2×C42—612no—
417C2×C42—612no—
418C2×C42—1221no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42—1221no—
422#104—1228no—
423#103—1224no—
424#102—1224no—
425#95—1224no—
426C24—1221no—
427C24C2×D24411yescharacteristic
428#41—1222no—
429#41—1222no—
430#41—1222no—
431#41—1222no—
432#41—1222no—
433#41—1222no—
434#41—1222no—
435#41—1222no—
436#41—2441no—
437#41—2441no—
438#41—2441no—
439#41—2441no—
440#41—2441no—
441#41—2441no—
442#41—2441no—
443#41—2441no—
444C2×Q8—612no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×D4—612no—
451C2×D4—1222no—
452C2×D4—1222no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—2441no—
459C2×D4—2441no—
460C2×D4—2441no—
461C2×D4—2441no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—44no—
466C22×C4—1224no—
467C22×C4—1224no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—2444no—
477C22×C4—2441no—
478C22×C4—2441no—
479C22×C4—2441no—
480C22×C4—2441no—
481C22×C4—2441no—
482C22×C4—2441no—
483C22×C4C2×C4×D6111yescharacteristic
484C22×C4#1175822211yescharacteristic
485C22×C4#1175822211yescharacteristic
486C22×C4#1175822211yescharacteristic
487C22×C4#1175822211yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4D4×D6111yescharacteristic
490C22×C4D4×D6111yescharacteristic
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4#1175822411yescharacteristic
494C22×C4Q8×D6111yescharacteristic
495C22×C4#1175822611yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497M16—22no—
498M16—22no—
499M16—22no—
500M16—22no—
501M16—41no—
502M16—41no—
503M16—41no—
504M16—41no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—24416no—
520C4⋊C4—2441no—
521C4⋊C4—2441no—
522C22⋊C4—4884no—
523C22⋊C4—2441no—
524C22⋊C4—2441no—
525C42—614no—
526C42—1222no—
527C42—1222no—
528C42—1222no—
529C42—2441no—
530C42—2441no—
531C23—22no—
532C23—22no—
533C23—1222no—
534C23—2441no—
535C23—2442no—
536C23—2441no—
537C23—2442no—
538C23C4×D24411yescharacteristic
539C23#1451974711yescharacteristic
540C23#1451974711yescharacteristic
541C23#1451975111yescharacteristic
542C23#1451975211yescharacteristic
543C23#1451975711yescharacteristic
544C23#1451976511yescharacteristic
545Q8—1222no—
546Q8—1222no—
547Q8—2441no—
548Q8—2441no—
549D4—1222no—
550D4—1222no—
551D4—2442no—
552D4—2442no—
553D4—2442no—
554D4—2442no—
555D4—2441no—
556D4—2441no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—82no—
561C2×C4—1224no—
562C2×C4—1228no—
563C2×C4—1228no—
564C2×C4—1224no—
565C2×C4—1224no—
566C2×C4—1222no—
567C2×C4—4882no—
568C2×C4—4882no—
569C2×C4—2444no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4C22⋊C4×D6111yescharacteristic
581C2×C4C4⋊C4×D6111yescharacteristic
582C2×C4#1451973411yescharacteristic
583C2×C4#1451973711yescharacteristic
584C2×C4C4×C22⋊D6111yescharacteristic
585C2×C4#1451973911yescharacteristic
586C2×C4#1451974511yescharacteristic
587C2×C4#1451974911yescharacteristic
588C2×C4#1451975011yescharacteristic
589C2×C4#1451975011yescharacteristic
590C2×C4#1451975011yescharacteristic
591C2×C4#1451975011yescharacteristic
592C2×C4#1451975011yescharacteristic
593C2×C4#1451975311yescharacteristic
594C2×C4#1451975511yescharacteristic
595C2×C4#1451975511yescharacteristic
596C2×C4#1451975511yescharacteristic
597C2×C4#1451975511yescharacteristic
598C2×C4#1451975911yescharacteristic
599C2×C4#1451976011yescharacteristic
600C2×C4#1451976211yescharacteristic
601C2×C4#1451976211yescharacteristic
602C2×C4#1451976211yescharacteristic
603C2×C4#1451976311yescharacteristic
604C2×C4#1451976311yescharacteristic
605C2×C4#1451977111yescharacteristic
606C2×C4#1451977211yescharacteristic
607C2×C4#1451977311yescharacteristic
608C8—42no—
609C8—42no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—4882no—
619C22—4881no—
620C22—4881no—
621C22—2444no—
622C22—2442no—
623C22#1683963611yescharacteristic
624C22#1683964011yescharacteristic
625C22#1683965311yescharacteristic
626C22#1683967111yescharacteristic
627C22#1683967511yescharacteristic
628C22#1683968011yescharacteristic
629C22#1683968411yescharacteristic
630C22#1683993212yes—
631C22#1683994712yes—
632C4—82no—
633C4—1228no—
634C4—4882no—
635C4—4882no—
636C4—2442no—
637C4#1683962111yescharacteristic
638C4#1683963311yescharacteristic
639C4#1683965111yescharacteristic
640C4#1683966211yescharacteristic
641C4#1683967711yescharacteristic
642C4#1683968611yescharacteristic
643C4#1683969111yescharacteristic
644C4#1683969711yescharacteristic
645C2—21no—
646C2—21no—
647C2—21no—
648C2—21no—
649C2—4882no—
650C2#2144406311yescharacteristic
651C2#2144604812yes—
652C2#2144683611yescharacteristic
653C2#2144685711yescharacteristic
654C2#2144704412yes—
subgroupG/H
centerC23#14519747
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#343C2
socleC23×C61#120
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×#343D61
hypocenterC61#11078
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
4. C2×#16839631C2semidirect
5. #21445539C2semidirect
6. C2×#16839523C2semidirect
7. C2×C61×#343C2semidirect
15. C2×#14519703C4semidirect
156. C2×D244C4⋊C4semidirect
168. C2×Dic122C4⋊C4semidirect
202. C4×Dic61C22⋊C4semidirect
322. C2×#343D61semidirect
359. C61#11078semidirect

Maximal subgroups

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

subgroupindex
C22×#145197032
C2×#168397642
#214484122
C2×#168396312
#214455392
C2×#168395232
C2×C61×#3432
#1107861