Finite Group Databasebeta

← #29683616

Subgroups — #29683616

Order 15616.

2010 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D61C211yesmaximal, characteristic
2C2×#16839785C211yesmaximal, characteristic, complemented
3C2×#16839726C211yesmaximal, characteristic, complemented
4C2×#16839725C211yesmaximal, characteristic, complemented
5C2×#16839725C211yesmaximal, characteristic, complemented
6C2×M16×Dic61C211yesmaximal, characteristic, complemented
7#21446615C214yesmaximal, complemented
8#21446611C214yesmaximal, complemented
9C2×C61×#407C211yesmaximal, characteristic, complemented
10C23×C22⋊D61C411yescharacteristic
11C23×C4×D61C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C2×C4×C22⋊D61—24no—
15C2×C4×C22⋊D61—24no—
16C2×C4×C22⋊D61C412yes—
17C2×C4×C22⋊D61C412yes—
18C2×C4×C22⋊D61C2212yes—
19C2×C4×C22⋊D61C2212yes—
20C22×#11758196C411yescharacteristic
21C22×C22⋊Dic61C411yescharacteristic
22C22×#11758198C411yescharacteristic
23C22×C4×Dic61C2211yescharacteristic
24#16839785—24no—
25C2×#14519706C2212yescomplemented
26C2×#14519706C2212yescomplemented
27#16839726—24no—
28#16839725—24no—
29#16839725—24no—
30C2×C22⋊#92418C2214yescomplemented
31C2×#14519689C2214yescomplemented
32C2×#14519689C2214yescomplemented
33M16×Dic61—24no—
34C2×#14519687C2212yescomplemented
35C2×C8×Dic61C2212yescomplemented
36C24×C4×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C22×M16×C61C2211yescharacteristic
39C61×#407—24no—
40C2×C22⋊C8×C61C2214yescomplemented
41C24×D61C2×C411yescharacteristic
42C22×C22⋊D61—28no—
43C22×C22⋊D61—24no—
44C22×C22⋊D61C2×C412yes—
45C22×C22⋊D61C2×C412yes—
46C23×Dic61C2×C411yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C23×Dic61C2×C411yescharacteristic
49C22×C4×D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61—22no—
55C22×C4×D61C2×C412yes—
56C22×C4×D61C2312yes—
57C2×#11758194—22no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194D414yescomplemented
63C22×#92418C2312yes—
64C22×#92418C2312yes—
65C4×C22⋊D61—28no—
66C4×C22⋊D61—28no—
67C4×C22⋊D61—24no—
68C4×C22⋊D61—24no—
69C4×C22⋊D61—24no—
70C4×C22⋊D61—24no—
71C2×#11758196—24no—
72C2×#11758196C2×C412yes—
73C2×#11758196C2×C412yes—
74C2×C22⋊Dic61—24no—
75C2×C22⋊Dic61C2×C412yes—
76C2×C22⋊Dic61C2×C412yes—
77C2×#11758198—24no—
78C2×#11758198C2×C412yes—
79C2×#11758198C2×C412yes—
80C2×C4×Dic61C2×C411yescharacteristic
81C2×C4×Dic61C2×C411yescharacteristic
82C2×C4×Dic61D414yes—
83C2×C4×Dic61D414yes—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519706—24no—
87#14519706—24no—
88C22⋊#92418—28no—
89#14519689—28no—
90#14519689—28no—
91#14519687—24no—
92C8×Dic61—24no—
93C25×C61C2×C411yescharacteristic
94C23×C4×C61—22no—
95C23×C4×C61—22no—
96C23×C4×C61—22no—
97C23×C4×C61—22no—
98C23×C4×C61—22no—
99C23×C4×C61—22no—
100C23×C4×C61C2×C411yescharacteristic
101C23×C4×C61C2×C412yes—
102C23×C4×C61C2312yes—
103C23×C4×C61C2311yescharacteristic
104C2×M16×C61—22no—
105C2×M16×C61—22no—
106C2×M16×C61—22no—
107C2×M16×C61—22no—
108C2×M16×C61—22no—
109C2×M16×C61D414yescomplemented
110C22×C8×C61C2312yes—
111C22×C8×C61C2312yes—
112C22⋊C8×C61—28no—
113C24×C61—22no—
114C24×C61—24no—
115C24×C61—22no—
116C24×C61—22no—
117C24×C61—42no—
118C24×C61C22×C412yes—
119C24×C61C22×C411yescharacteristic
120C23×D61—22no—
121C23×D61—24no—
122C23×D61—22no—
123C23×D61—22no—
124C23×D61—42no—
125C23×D61C22×C412yes—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—24no—
130C22×C4×C61—24no—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—44no—
139C22×C4×C61—44no—
140C22×C4×C61—44no—
141C22×C4×C61—44no—
142C22×C4×C61C22×C412yes—
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D412yes—
148C22×C4×C61#4112yes—
149C22×C4×C61#4112yes—
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—232no—
152C2×C22⋊D61—24no—
153C2×C22⋊D61—24no—
154C2×C22⋊D61—24no—
155C2×C22⋊D61—24no—
156C2×C22⋊D61—24no—
157C2×C22⋊D61M1618yescomplemented
158C22×Dic61—24no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—42no—
163C22×Dic61C22⋊C414yes—
164C22×Dic61C22⋊C414yes—
165C22×Dic61C22×C412yes—
166C22×Dic61C22×C411yescharacteristic
167C22×Dic61C22×C411yescharacteristic
168C22×Dic61C22×C411yescharacteristic
169C22×Dic61C22×C411yescharacteristic
170C2×C4×D61—24no—
171C2×C4×D61—24no—
172C2×C4×D61—24no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×C4×D61—44no—
180C2×C4×D61—44no—
181C2×C4×D61—44no—
182C2×C4×D61—44no—
183M16×C61—28no—
184M16×C61—44no—
185C2×C8×C61—22no—
186C2×C8×C61—22no—
187C2×C8×C61—22no—
188C2×C8×C61—22no—
189C2×C8×C61—22no—
190C2×C8×C61C2×D414yescomplemented
191C22⋊Dic61—24no—
192C22⋊Dic61—24no—
193#11758198—24no—
194#11758198—24no—
195#11758196—24no—
196#11758196—24no—
197C4×Dic61C22⋊C414yes—
198C4×Dic61C22⋊C414yes—
199C4×Dic61C2×D412yes—
200C4×Dic61C2×D412yes—
201C4×Dic61C2×D412yes—
202C4×Dic61C2×D412yes—
203#11758194—28no—
204#11758194—44no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—22no—
210C2×#92418C2×D414yescomplemented
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—28no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—22no—
220C23×C61—48no—
221C23×C61—48no—
222C23×C61—44no—
223C23×C61—44no—
224C23×C61C2×C22⋊C412yes—
225C23×C61C4×D412yes—
226C23×C61C4×D412yes—
227C23×C61C2×M1614yes—
228C23×C61C23×C411yescharacteristic
229C22×D61—28no—
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—22no—
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—48no—
237C22×D61—48no—
238C22×D61—44no—
239C22×D61—44no—
240C22×D61C2×M1614yes—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—44no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—48no—
250C2×C4×C61C2×C22⋊C412yes—
251C2×C4×C61C4×D412yes—
252C2×C4×C61C4×D412yes—
253C2×C4×C61#12114yes—
254C2×C4×C61#12114yes—
255C2×C4×C61#12414yes—
256C2×C4×C61#12714yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C22⋊D61—216no—
264C22⋊D61—216no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—28no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—44no—
273C2×Dic61—44no—
274C2×Dic61C2×C22⋊C412yes—
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C2×C22⋊C412yes—
277C2×Dic61C2×C22⋊C412yes—
278C2×Dic61C2×M1614yes—
279C4×D61—42no—
280C4×D61—42no—
281C4×D61—42no—
282C4×D61—42no—
283C4×D61—48no—
284C8×C61—24no—
285C8×C61—42no—
286#92418—24no—
287#92418—42no—
288#21190—611nomaximal, Sylow
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—24no—
292C22×C61—24no—
293C22×C61—24no—
294C22×C61—24no—
295C22×C61—24no—
296C22×C61—416no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61#39014yes—
302C22×C61C22×C22⋊C411yescharacteristic
303C22×C61C2×C4×D411yescharacteristic
304C22×C61C2×C4×D411yescharacteristic
305C22×C61C22×M1612yes—
306C22×C61#56812yes—
307C2×D61—24no—
308C2×D61—24no—
309C2×D61—24no—
310C2×D61—416no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—42no—
315C4×C61—42no—
316C4×C61#39014yes—
317C4×C61C2×#12111yescharacteristic
318C4×C61C2×#12111yescharacteristic
319C4×C61C2×#12411yescharacteristic
320C4×C61C2×#12711yescharacteristic
321Dic61—22no—
322Dic61—22no—
323Dic61—22no—
324Dic61—42no—
325Dic61#40714yescomplemented
326C22×C4×D4—611no—
327C2×#423—611no—
328C2×#407—611no—
329C2×#407—611no—
330C2×#407—611no—
331C2×#407D6111yescharacteristic, complemented
332C2×C4×M16—611no—
333#1955—614no—
334#1954—614no—
335C2×C61—24no—
336C2×C61—44no—
337C2×C61C2×#39011yescharacteristic
338C2×C61C2×#40711yescharacteristic
339C2×C61#287911yescharacteristic
340C2×C61D4×M1612yes—
341C2×C61D4×M1612yes—
342D61—44no—
343C23×D4—611no—
344C24×C4—611no—
345C24×C4C2×D6111yescharacteristic
346C22×M16—611no—
347C22×M16—611no—
348C22×M16C2×D6111yescharacteristic
349C22×M16C2×D6111yescharacteristic
350C2×C4×D4—612no—
351C2×C4×D4—612no—
352C2×C4×D4—612no—
353C2×C4×D4—612no—
354C2×C4×D4—1224no—
355C2×C4×D4—1224no—
356C22×C4⋊C4—611no—
357C22×C22⋊C4—611no—
358C22×C22⋊C4—611no—
359C22×C42—611no—
360#423—1224no—
361C2×C4⋊C8—612no—
362C2×C4⋊C8—612no—
363#407—24no—
364#407—1224no—
365#407—1224no—
366#407—1224no—
367C2×C22⋊C8—614no—
368C2×C22⋊C8—614no—
369C2×C22⋊C8—614no—
370C2×C22⋊C8C2×D6114yescomplemented
371C4×M16—1224no—
372C2×#97—612no—
373C2×C4×C8—612no—
374C61#2119011yescharacteristic, complemented, Sylow
375C25—611no—
376C25C4×D6111yescharacteristic
377C22×D4—612no—
378C22×D4—612no—
379C22×D4—1224no—
380C22×D4—1228no—
381C23×C4—22no—
382C23×C4—22no—
383C23×C4—22no—
384C23×C4—22no—
385C23×C4—22no—
386C23×C4—22no—
387C23×C4—611no—
388C23×C4—611no—
389C23×C4—611no—
390C23×C4—612no—
391C23×C4—612no—
392C23×C4—1222no—
393C23×C4—1222no—
394C23×C4—1222no—
395C23×C4—1222no—
396C23×C4—1222no—
397C23×C4—1222no—
398C23×C4C4×D6111yescharacteristic
399C23×C4C4×D6112yes—
400C23×C4C22×D6112yes—
401C23×C4C22×D6111yescharacteristic
402C2×M16—22no—
403C2×M16—22no—
404C2×M16—22no—
405C2×M16—22no—
406C2×M16—22no—
407C2×M16—614no—
408C2×M16—1222no—
409C2×M16—1222no—
410C2×M16—1222no—
411C2×M16—1222no—
412C2×M16—1222no—
413C2×M16C22⋊D6114yescomplemented
414C22×C8—612no—
415C22×C8—612no—
416C22×C8C22×D6112yes—
417C22×C8C22×D6112yes—
418C4×D4—1224no—
419C4×D4—1224no—
420C4×D4—1224no—
421C4×D4—1224no—
422C4×D4—1228no—
423C4×D4—1228no—
424C2×C4⋊C4—612no—
425C2×C4⋊C4—612no—
426C2×C4⋊C4—1224no—
427C2×C22⋊C4—612no—
428C2×C22⋊C4—612no—
429C2×C22⋊C4—612no—
430C2×C22⋊C4—612no—
431C2×C22⋊C4—1224no—
432C2×C22⋊C4—1224no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—614no—
436C2×C42—614no—
437C2×C42—611no—
438C2×C42—611no—
439C4⋊C8—1224no—
440C4⋊C8—1224no—
441C22⋊C8—28no—
442C22⋊C8—1228no—
443C22⋊C8—1228no—
444C22⋊C8—1228no—
445#97—1224no—
446C4×C8—1224no—
447C24—22no—
448C24—24no—
449C24—22no—
450C24—22no—
451C24—42no—
452C24—612no—
453C24—1224no—
454C24—1222no—
455C24—1222no—
456C24—1222no—
457C24—2442no—
458C24C2×C4×D6112yes—
459C24C2×C4×D6111yescharacteristic
460C2×D4—618no—
461C2×D4—1224no—
462C2×D4—1224no—
463C2×D4—1224no—
464C2×D4—1224no—
465C2×D4—12232no—
466C2×D4—1224no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—614no—
488C22×C4—614no—
489C22×C4—612no—
490C22×C4—1224no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1224no—
501C22×C4—1224no—
502C22×C4—1224no—
503C22×C4—2442no—
504C22×C4—2444no—
505C22×C4—2444no—
506C22×C4—2444no—
507C22×C4—2444no—
508C22×C4C2×C4×D6112yes—
509C22×C4C2×C4×D6111yescharacteristic
510C22×C4#1175822212yes—
511C22×C4D4×D6112yes—
512C22×C4D4×D6112yes—
513C22×C4D4×D6112yes—
514C22×C4#1175822412yes—
515C22×C4C2×C22⋊D6112yes—
516C22×C4C23×D6111yescharacteristic
517M16—28no—
518M16—44no—
519M16—1228no—
520M16—2444no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—614no—
527C2×C8—1222no—
528C2×C8—1222no—
529C2×C8—1222no—
530C2×C8—1222no—
531C2×C8—1222no—
532C2×C8C2×C22⋊D6114yescomplemented
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C22⋊C4—1224no—
536C22⋊C4—1224no—
537C22⋊C4—1224no—
538C22⋊C4—1224no—
539C42—612no—
540C42—612no—
541C42—612no—
542C42—612no—
543C42—614no—
544C42—614no—
545C23—22no—
546C23—22no—
547C23—28no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—48no—
555C23—48no—
556C23—44no—
557C23—44no—
558C23—614no—
559C23—1228no—
560C23—1222no—
561C23—1222no—
562C23—1222no—
563C23—1222no—
564C23—1222no—
565C23—1222no—
566C23—2448no—
567C23—2448no—
568C23—2444no—
569C23—2444no—
570C23C22⋊C4×D6112yes—
571C23#1451973712yes—
572C23C4×C22⋊D6112yes—
573C23M16×D6114yes—
574C23C22×C4×D6111yescharacteristic
575D4—12216no—
576D4—12216no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—612no—
587C2×C4—614no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—1228no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—2444no—
599C2×C4—2444no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2448no—
605C2×C4C22⋊C4×D6112yes—
606C2×C4#1451973712yes—
607C2×C4C4×C22⋊D6112yes—
608C2×C4#1451974914yes—
609C2×C4#1451975014yes—
610C2×C4#1451976314yes—
611C2×C4#1451977314yes—
612C2×C4C2×D4×D6111yescharacteristic
613C2×C4C2×D4×D6111yescharacteristic
614C2×C4C2×D4×D6111yescharacteristic
615C2×C4C22×C22⋊D6111yescharacteristic
616C2×C4C2×#1175822411yescharacteristic
617C2×C4C2×#1175822211yescharacteristic
618C8—24no—
619C8—42no—
620C8—1224no—
621C8—2442no—
622C22—22no—
623C22—22no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—416no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—1224no—
635C22—1224no—
636C22—1224no—
637C22—24416no—
638C22—2442no—
639C22—2442no—
640C22—2442no—
641C22—2442no—
642C22#1683966214yes—
643C22C2×C22⋊C4×D6111yescharacteristic
644C22C2×#1451973711yescharacteristic
645C22C2×C4×C22⋊D6111yescharacteristic
646C22C2×M16×D6112yes—
647C22#1684052812yes—
648C4—42no—
649C4—614no—
650C4—1222no—
651C4—1222no—
652C4—1222no—
653C4—2442no—
654C4#1683966214yes—
655C4C2×#1451974911yescharacteristic
656C4C2×#1451975011yescharacteristic
657C4C2×#1451976311yescharacteristic
658C4C2×#1451977311yescharacteristic
659C2—24no—
660C2—44no—
661C2—2444no—
662C2C2×#1683966211yescharacteristic
663C2#407×D6111yescharacteristic
664C2#2145426211yescharacteristic
665C2#2145455112yes—
666C2M16×C22⋊D6112yes—
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC2×C61×#407C2
prime radicalC23×C22⋊D61C4
cosocleC22×C4×C61C24
hypercenterC2×#407D61
hypocenterC61#21190
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C2×#16839785C2semidirect
3. C2×#16839726C2semidirect
4. C2×#16839725C2semidirect
5. C2×#16839725C2semidirect
6. C2×M16×Dic61C2semidirect
7. #21446615C2semidirect
8. #21446611C2semidirect
9. C2×C61×#407C2semidirect
25. C2×#14519706C22semidirect
26. C2×#14519706C22semidirect
30. C2×C22⋊#92418C22semidirect
31. C2×#14519689C22semidirect
32. C2×#14519689C22semidirect
34. C2×#14519687C22semidirect
35. C2×C8×Dic61C22semidirect
40. C2×C22⋊C8×C61C22semidirect
62. C2×#11758194D4semidirect
109. C2×M16×C61D4semidirect
157. C2×C22⋊D61M16semidirect
190. C2×C8×C61C2×D4semidirect
210. C2×#92418C2×D4semidirect
325. Dic61#407semidirect
331. C2×#407D61semidirect
370. C2×C22⋊C8C2×D61semidirect
374. C61#21190semidirect
413. C2×M16C22⋊D61semidirect
532. C2×C8C2×C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D612
C2×#168397852
C2×#168397262
C2×#168397252
C2×M16×Dic612
#214466152
#214466112
C2×C61×#4072
#2119061