Finite Group Databasebeta

← #29601814

Subgroups — #29601814

Order 15616.

2158 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519742C211yesmaximal, characteristic, complemented
2C2×#16839674C211yesmaximal, characteristic
3C2×C22⋊C4×Dic61C211yesmaximal, characteristic
4C2×C22⋊C4×Dic61C212yesmaximal
5#21444089C211yesmaximal, characteristic, complemented
6#21444078C211yesmaximal, characteristic, complemented
7#21444074C212yesmaximal, complemented
8#21444062C212yesmaximal
9#95×Dic61C212yesmaximal, complemented
10#95×Dic61C211yesmaximal, characteristic, complemented
11C61×#1439C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C24×Dic61C2211yescharacteristic
14C2×#14519742—24no—
15C2×#14519742—28no—
16C22×#11758196C2212yes—
17C22×C22⋊Dic61C2211yescharacteristic
18C22×C22⋊Dic61C2212yes—
19C22×C4×Dic61C2212yes—
20C22×C4×Dic61C2211yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22C22×C4×Dic61C2212yes—
23#16839674C418yescomplemented
24C22⋊C4×Dic61—28no—
25C22⋊C4×Dic61—24no—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2211yescharacteristic
33C2×#14519684C2212yes—
34C2×#14519684C2211yescharacteristic
35C22×C4⋊C4×C61C2211yescharacteristic
36C22×C22⋊C4×C61C2212yes—
37C22×C22⋊C4×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C2×#95×C61C2211yescharacteristic
40C22×Dic122—22no—
41C22×Dic122—22no—
42C22×Dic122—24no—
43C22×Dic122—24no—
44C22×Dic122C2×C414yescomplemented
45C23×Dic61—24no—
46C23×Dic61—22no—
47C23×Dic61—24no—
48C23×Dic61—22no—
49C23×Dic61Q814yescomplemented
50C23×Dic61C2312yes—
51C23×Dic61C2311yescharacteristic
52C23×Dic61C2311yescharacteristic
53C23×Dic61C2311yescharacteristic
54C23×Dic61C2312yes—
55C23×Dic61C2311yescharacteristic
56#14519742—232no—
57C2×#11758196—24no—
58C2×#11758196—24no—
59C2×#11758196—24no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—24no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—24no—
65C2×C22⋊Dic61C2×C418yes—
66C2×C4×Dic61—24no—
67C2×C4×Dic61—24no—
68C2×C4×Dic61—24no—
69C2×C4×Dic61—24no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—24no—
72C2×C4×Dic61—24no—
73C2×C4×Dic61—24no—
74C2×C4×Dic61D414yes—
75C2×C4×Dic61D418yes—
76#14519684—24no—
77#14519684—22no—
78#14519684—24no—
79#14519684—22no—
80#14519684—24no—
81#14519684—24no—
82#14519684—24no—
83#14519684C2×C414yescomplemented
84#14519684C2×C418yescomplemented
85C25×C61C2311yescharacteristic
86C23×C4×C61C2311yescharacteristic
87C23×C4×C61C2312yes—
88C23×C4×C61C2312yes—
89C23×C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61—24no—
91C2×C4⋊C4×C61—22no—
92C2×C22⋊C4×C61—24no—
93C2×C22⋊C4×C61—24no—
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61—24no—
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61C2×C414yes—
98#95×C61—24no—
99#95×C61—22no—
100C24×C61—22no—
101C24×C61—21no—
102C24×C61—22no—
103C24×C61—21no—
104C24×C61—21no—
105C24×C61—22no—
106C24×C61—22no—
107C24×C61—21no—
108C24×C61C22×C412yes—
109C24×C61C2×Q812yes—
110C24×C61#4112yes—
111C24×C61C2411yescharacteristic
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C2×D414yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61#4114yes—
135C22×C4×C61#4112yes—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—21no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—28no—
159C22×Dic61—216no—
160C22×Dic61—44no—
161C22×Dic61—44no—
162C22×Dic61—48no—
163C22×Dic61C22×C414yes—
164C22×Dic61C22×C414yes—
165C22×Dic61C2×D412yes—
166C22×Dic61C2×D414yes—
167C22×Dic61C2×D414yes—
168C22×Dic61C2×D412yes—
169C22×Dic61C2×Q812yes—
170C22×Dic61C2×Q812yes—
171C22×Dic61#4112yes—
172C22×Dic61#4114yes—
173C22×Dic61#4114yes—
174C22×Dic61#4112yes—
175C2×Dic122—216no—
176C2×Dic122—28no—
177C2×Dic122—28no—
178C2×Dic122—28no—
179C2×Dic122—216no—
180C4⋊C4×C61—28no—
181C22⋊C4×C61—216no—
182C22⋊C4×C61—28no—
183C22⋊Dic61—28no—
184C22⋊Dic61—216no—
185#11758196—216no—
186C4×Dic61—216no—
187C4×Dic61—28no—
188C4×Dic61—44no—
189C4×Dic61—48no—
190C23×C61—24no—
191C23×C61—22no—
192C23×C61—24no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—24no—
196C23×C61—22no—
197C23×C61—24no—
198C23×C61—24no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—24no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—42no—
205C23×C61—41no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61—41no—
212C23×C61C4×Q814yescomplemented
213C23×C61C23×C411yescharacteristic
214C23×C61C22×D412yes—
215C23×C61C22×D411yescharacteristic
216C23×C61C22×Q811yescharacteristic
217C23×C61C2×#4112yes—
218C23×C61C2×#4111yescharacteristic
219C23×C61C2×#4111yescharacteristic
220C23×C61C2×#4112yes—
221C23×C61C2×#4111yescharacteristic
222C23×C61C2×#4111yescharacteristic
223C23×C61C2×#4112yes—
224C2×C4×C61—24no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—41no—
247C2×C4×C61C4×D414yes—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—24no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—24no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—42no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—42no—
269C2×Dic61—41no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—416no—
277C2×Dic61C4×D418yes—
278C2×Dic61#12014yes—
279C2×Dic61C22⋊Q814yes—
280C2×Dic61C22⋊Q818yes—
281C2×Dic61#12414yes—
282C2×Dic61#12418yes—
283Dic122—232no—
284C4×#393—611nomaximal, Sylow
285C22×C61—24no—
286C22×C61—28no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—44no—
291C22×C61—42no—
292C22×C61—44no—
293C22×C61—42no—
294C22×C61—44no—
295C22×C61—42no—
296C22×C61C2×C4×D412yes—
297C22×C61C2×C4×D411yescharacteristic
298C22×C61C2×C4×Q811yescharacteristic
299C22×C61C4×#4112yes—
300C22×C61C4×#4111yescharacteristic
301C22×C61C2×#12011yescharacteristic
302C22×C61C2×C22⋊Q811yescharacteristic
303C22×C61C2×C22⋊Q812yes—
304C22×C61#52512yes—
305C22×C61#52511yescharacteristic
306C22×C61#52511yescharacteristic
307C22×C61#52512yes—
308C22×C61C2×#12411yescharacteristic
309C22×C61C2×#12412yes—
310C22×C61#52912yes—
311C22×C61#52912yes—
312C22×C61#52912yes—
313C22×C61#52911yescharacteristic
314C22×C61#52912yes—
315C22×C61#53212yes—
316C22×C61#53211yescharacteristic
317C22×C61#53312yes—
318C22×C61#53311yescharacteristic
319C4×C61—24no—
320C4×C61—44no—
321C4×C61—44no—
322C4×C61—42no—
323Dic61—28no—
324Dic61—42no—
325Dic61—42no—
326Dic61—42no—
327Dic61—44no—
328Dic61#39318yescomplemented
329C22×C4×Q8—611no—
330C2×#393—611no—
331C2×C4×C22⋊C4—612no—
332C2×C4×C22⋊C4—611no—
333#1446—611no—
334#1442—612no—
335#1442—611no—
336#1439—612no—
337#1439D6111yescharacteristic
338C4×#95—611no—
339C4×#95—612no—
340C2×C61—44no—
341C2×C61C4×#12011yescharacteristic
342C2×C61C4×C22⋊Q811yescharacteristic
343C2×C61C4×C22⋊Q812yes—
344C2×C61C4×#12412yes—
345C2×C61C4×#12411yescharacteristic
346C2×C61C2×#39311yescharacteristic
347C2×C61#238811yescharacteristic
348C2×C61#239611yescharacteristic
349C2×C61#239612yes—
350C2×C61#239811yescharacteristic
351C2×C61#239812yes—
352C23×Q8—611no—
353C24×C4—611no—
354C2×C4×Q8—1224no—
355C2×C4×Q8—1228no—
356C22×C4⋊C4—612no—
357C22×C4⋊C4C2×D6111yescharacteristic
358C22×C22⋊C4—612no—
359C22×C22⋊C4—611no—
360C22×C22⋊C4C2×D6112yes—
361C22×C22⋊C4C2×D6111yescharacteristic
362C22×C42—612no—
363C22×C42—611no—
364C22×C42—611no—
365C22×C42—612no—
366#393—618no—
367C4×C22⋊C4—1228no—
368C4×C22⋊C4—1224no—
369C2×#95—612no—
370C2×#95—611no—
371C2×#95—612no—
372C2×#95—612no—
373C2×#95—612no—
374C2×#95—611no—
375C2×#95—612no—
376C2×#95—612no—
377C2×#95—611no—
378C2×#95C2×D6112yes—
379C2×#95C2×D6111yescharacteristic
380C61C4×#39311yescharacteristic, complemented, Sylow
381C25C22×D6111yescharacteristic
382C22×Q8—614no—
383C22×Q8—1224no—
384C22×Q8—1224no—
385C22×Q8—1222no—
386C22×Q8—1222no—
387C23×C4—611no—
388C23×C4—612no—
389C23×C4—614no—
390C23×C4—611no—
391C23×C4—612no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—1224no—
395C23×C4—1222no—
396C23×C4—1224no—
397C23×C4—1222no—
398C23×C4C22×D6111yescharacteristic
399C23×C4C22×D6112yes—
400C23×C4C22×D6112yes—
401C23×C4C22×D6111yescharacteristic
402C4×Q8—12232no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—1224no—
406C2×C4⋊C4—1224no—
407C2×C4⋊C4—1224no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—618no—
414C2×C22⋊C4—1224no—
415C2×C22⋊C4—1222no—
416C2×C22⋊C4—1224no—
417C2×C22⋊C4—1222no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4C4×D6114yes—
420C2×C42—618no—
421C2×C42—614no—
422C2×C42—1224no—
423C2×C42—1224no—
424C2×C42—1224no—
425C2×C42—1222no—
426C2×C42—1224no—
427C2×C42—1224no—
428C2×C42—1224no—
429C2×C42—1224no—
430#95—24no—
431#95—22no—
432#95—618no—
433#95—614no—
434#95—1224no—
435#95—1222no—
436#95—1224no—
437#95—1224no—
438#95—1224no—
439#95—1222no—
440#95—1224no—
441C24—22no—
442C24—21no—
443C24—22no—
444C24—21no—
445C24—22no—
446C24—21no—
447C24—21no—
448C24—22no—
449C24C2×C4×D6112yes—
450C24#1175822412yes—
451C24Q8×D6112yes—
452C24C23×D6111yescharacteristic
453C2×Q8—12216no—
454C2×Q8—1228no—
455C2×Q8—1228no—
456C2×Q8—12216no—
457C2×Q8—1228no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—614no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—614no—
480C22×C4—612no—
481C22×C4—614no—
482C22×C4—614no—
483C22×C4—614no—
484C22×C4—614no—
485C22×C4—612no—
486C22×C4—612no—
487C22×C4—612no—
488C22×C4—1228no—
489C22×C4—12216no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1221no—
496C22×C4—1222no—
497C22×C4—1221no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1221no—
506C22×C4—1222no—
507C22×C4—1221no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—2444no—
513C22×C4—2444no—
514C22×C4—2448no—
515C22×C4C2×C4×D6112yes—
516C22×C4C2×C4×D6112yes—
517C22×C4D4×D6114yes—
518C22×C4D4×D6112yes—
519C22×C4#1175822414yes—
520C22×C4#1175822412yes—
521C4⋊C4—28no—
522C4⋊C4—12216no—
523C22⋊C4—216no—
524C22⋊C4—28no—
525C22⋊C4—12216no—
526C22⋊C4—1228no—
527C42—1228no—
528C42—12216no—
529C42—2448no—
530C42—2444no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—41no—
553C23#1451974214yescomplemented
554C23C22×C4×D6111yescharacteristic
555C23C2×D4×D6112yes—
556C23C2×D4×D6111yescharacteristic
557C23C2×Q8×D6111yescharacteristic
558C23#41×D6112yes—
559C23#41×D6112yes—
560C23#41×D6111yescharacteristic
561C23#41×D6111yescharacteristic
562C23C2×#1175822411yescharacteristic
563C23C2×#1175822411yescharacteristic
564C23C2×#1175822412yes—
565Q8—12232no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—614no—
590C2×C4—614no—
591C2×C4—618no—
592C2×C4—614no—
593C2×C4—618no—
594C2×C4—618no—
595C2×C4—1224no—
596C2×C4—1224no—
597C2×C4—1224no—
598C2×C4—1224no—
599C2×C4—1224no—
600C2×C4—1224no—
601C2×C4—1224no—
602C2×C4—1224no—
603C2×C4—1224no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1224no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1224no—
610C2×C4—1222no—
611C2×C4—24416no—
612C2×C4—2441no—
613C2×C4—2442no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2442no—
617C2×C4—2441no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4#1451973714yes—
625C22—24no—
626C22—28no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—44no—
632C22—42no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22C4×D4×D6112yes—
637C22C2×#1451973711yescharacteristic
638C22C2×#1451974211yescharacteristic
639C22C4×#1175822412yes—
640C22#1684023011yescharacteristic
641C22#120×D6111yescharacteristic
642C22C22⋊Q8×D6111yescharacteristic
643C22C22⋊Q8×D6112yes—
644C22#1684026812yes—
645C22#1684027011yescharacteristic
646C22#1684027211yescharacteristic
647C22#1684027212yes—
648C22#124×D6111yescharacteristic
649C22#124×D6112yes—
650C22#1684028912yes—
651C22#1684028912yes—
652C22#1684029312yes—
653C22#1684029311yescharacteristic
654C22#1684029512yes—
655C22#1684030712yes—
656C22#1684031011yescharacteristic
657C22#1684031412yes—
658C22#1684031411yescharacteristic
659C4—24no—
660C4—44no—
661C4—44no—
662C4—42no—
663C4—618no—
664C4—1228no—
665C4—2442no—
666C4—2442no—
667C4—2442no—
668C4—2444no—
669C2—44no—
670C2#2144832411yescharacteristic
671C2#2144834711yescharacteristic
672C2#2144835112yes—
673C2#2144836012yes—
674C2#2144836611yescharacteristic
675C2#393×D6111yescharacteristic
676C2#2144903111yescharacteristic
677C2#2144908811yescharacteristic
678C2#2144909412yes—
679C2#2144910411yescharacteristic
680C2#2144910712yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1439C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenter#1439D61
hypocenterC61C4×#393
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#14519742C2semidirect
5. #21444089C2semidirect
6. #21444078C2semidirect
7. #21444074C2semidirect
9. #95×Dic61C2semidirect
10. #95×Dic61C2semidirect
23. #16839674C4semidirect
44. C22×Dic122C2×C4semidirect
49. C23×Dic61Q8semidirect
83. #14519684C2×C4semidirect
84. #14519684C2×C4semidirect
212. C23×C61C4×Q8semidirect
328. Dic61#393semidirect
380. C61C4×#393semidirect
553. C23#14519742semidirect

Maximal subgroups

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

subgroupindex
C22×#145197422
C2×#168396742
C2×C22⋊C4×Dic612
#214440892
#214440782
#214440742
#214440622
#95×Dic612
C61×#14392
C4×#39361