Finite Group Databasebeta

← #29602841

Subgroups — #29602841

Order 15616.

1918 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839623C211yesmaximal, characteristic
2C2×#16839623C211yesmaximal, characteristic
3C2×C4×C22⋊Dic61C211yesmaximal, characteristic
4C22×#14519684C211yesmaximal, characteristic
5#21444080C212yesmaximal, complemented
6#21444078C212yesmaximal, complemented
7#21444051C212yesmaximal, complemented
8#21444045C212yesmaximal, complemented
9#21444041C211yesmaximal, characteristic
10#21444041C211yesmaximal, characteristic
11C2×C61×#385C211yesmaximal, characteristic
12C24×Dic61C2211yescharacteristic
13C24×Dic61C2211yescharacteristic
14C22×C22⋊Dic61C2211yescharacteristic
15C22×C22⋊Dic61C2211yescharacteristic
16C22×C4×Dic61C2212yes—
17#16839623—24no—
18#16839623C418yescomplemented
19C4×C22⋊Dic61—24no—
20C2×#14519684—24no—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C24×C4×C61C2211yescharacteristic
33C22×C4⋊C4×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2211yescharacteristic
35C22×C22⋊C4×C61C2211yescharacteristic
36C22×C22⋊C4×C61C2211yescharacteristic
37C61×#385—24no—
38C2×#95×C61C2211yescharacteristic
39C2×#95×C61C2211yescharacteristic
40C23×Dic61—22no—
41C23×Dic61—22no—
42C23×Dic61—22no—
43C23×Dic61—22no—
44C23×Dic61—22no—
45C23×Dic61—22no—
46C23×Dic61—22no—
47C23×Dic61—24no—
48C23×Dic61—22no—
49C23×Dic61—22no—
50C23×Dic61—42no—
51C23×Dic61C2×C414yes—
52C23×Dic61C2312yes—
53C23×Dic61C2312yes—
54C23×Dic61C2312yes—
55C23×Dic61C2312yes—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—22no—
58C2×C22⋊Dic61—22no—
59C2×C22⋊Dic61—22no—
60C2×C22⋊Dic61C2×C414yes—
61C2×C22⋊Dic61D414yes—
62C2×C4×Dic61—24no—
63C2×C4×Dic61—24no—
64C2×C4×Dic61—42no—
65#14519684—24no—
66#14519684—24no—
67#14519684—24no—
68#14519684—24no—
69#14519684—24no—
70#14519684—24no—
71#14519684—24no—
72#14519684—42no—
73#14519684—42no—
74#14519684C2×C418yescomplemented
75#14519684C2×C418yescomplemented
76C25×C61C2311yescharacteristic
77C23×C4×C61—22no—
78C23×C4×C61—22no—
79C23×C4×C61—24no—
80C23×C4×C61—42no—
81C23×C4×C61D414yes—
82C23×C4×C61C2311yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C23×C4×C61C2311yescharacteristic
85C23×C4×C61C2311yescharacteristic
86C23×C4×C61C2311yescharacteristic
87C23×C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—24no—
91C2×C4⋊C4×C61—42no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61C2×C414yes—
98#95×C61—22no—
99#95×C61—22no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61—22no—
103C24×C61—22no—
104C24×C61C22×C412yes—
105C24×C61C2×D412yes—
106C24×C61C2×D412yes—
107C24×C61#4112yes—
108C24×C61#4112yes—
109C24×C61#4112yes—
110C24×C61#4112yes—
111C24×C61C2411yescharacteristic
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—28no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—28no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—44no—
140C22×C4×C61—48no—
141C22×C4×C61—48no—
142C22×C4×C61C22×C412yes—
143C22×C4×C61C22×C412yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61#4114yes—
147C22×Dic61—28no—
148C22×Dic61—28no—
149C22×Dic61—28no—
150C22×Dic61—28no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—22no—
172C22×Dic61—22no—
173C22×Dic61—22no—
174C22×Dic61—22no—
175C22×Dic61—48no—
176C22×Dic61—48no—
177C22×Dic61—44no—
178C22×Dic61—44no—
179C22×Dic61—44no—
180C22×Dic61—44no—
181C22×Dic61—44no—
182C22×Dic61—44no—
183C22×Dic61C22×C414yes—
184C22×Dic61C22×C414yes—
185C22×Dic61C2×D414yes—
186C22×Dic61#4114yes—
187C4⋊C4×C61—44no—
188C4⋊C4×C61—44no—
189C22⋊C4×C61—28no—
190C22⋊C4×C61—44no—
191C22⋊Dic61—28no—
192C22⋊Dic61—28no—
193C4×Dic61—48no—
194C23×C61—24no—
195C23×C61—24no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—24no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—24no—
202C23×C61—24no—
203C23×C61—24no—
204C23×C61—24no—
205C23×C61—24no—
206C23×C61—24no—
207C23×C61—24no—
208C23×C61—24no—
209C23×C61—24no—
210C23×C61#11714yes—
211C23×C61C4×D414yes—
212C23×C61C4×D414yes—
213C23×C61#12114yes—
214C23×C61#12314yes—
215C23×C61#12414yes—
216C23×C61#12614yes—
217C23×C61C23×C411yescharacteristic
218C23×C61C22×D411yescharacteristic
219C23×C61C22×D411yescharacteristic
220C23×C61C2×#4111yescharacteristic
221C23×C61C2×#4111yescharacteristic
222C23×C61C2×#4111yescharacteristic
223C23×C61C2×#4111yescharacteristic
224C23×C61C2×#4112yes—
225C23×C61C2×#4112yes—
226C23×C61H(5,𝔽2)12yes—
227C23×C612−1+412yes—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—416no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61#12314yes—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—24no—
263C2×Dic61—24no—
264C2×Dic61—24no—
265C2×Dic61—24no—
266C2×Dic61—24no—
267C2×Dic61—24no—
268C2×Dic61—24no—
269C2×Dic61—24no—
270C2×Dic61—24no—
271C2×Dic61—416no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61—42no—
283C2×Dic61—42no—
284C2×Dic61—42no—
285C2×Dic61—42no—
286C2×Dic61—42no—
287C2×Dic61—42no—
288C2×Dic61—416no—
289#13972—611nomaximal, Sylow
290C22×C61—24no—
291C22×C61—24no—
292C22×C61—28no—
293C22×C61—28no—
294C22×C61—24no—
295C22×C61—24no—
296C22×C61—28no—
297C22×C61—28no—
298C22×C61—24no—
299C22×C61—24no—
300C22×C61#38818yes—
301C22×C61C2×#11711yescharacteristic
302C22×C61C2×C4×D411yescharacteristic
303C22×C61C2×C4×D411yescharacteristic
304C22×C61C4×#4112yes—
305C22×C61#52012yes—
306C22×C61C2×#12111yescharacteristic
307C22×C61C2×#12311yescharacteristic
308C22×C61C2×#12311yescharacteristic
309C22×C61#52511yescharacteristic
310C22×C61C2×#12411yescharacteristic
311C22×C61C2×#12611yescharacteristic
312C22×C61#52912yes—
313C22×C61#53411yescharacteristic
314C22×C61#53611yescharacteristic
315C22×C61#53912yes—
316C22×C61#54612yes—
317C22×C61#54712yes—
318C22×C61#55112yes—
319C22×C61#55112yes—
320C22×C61#55112yes—
321C22×C61#55212yes—
322C22×C61#55212yes—
323C22×C61#55212yes—
324C4×C61—24no—
325C4×C61—42no—
326C4×C61—42no—
327C4×C61—42no—
328C4×C61—81no—
329C4×C61—81no—
330Dic61—44no—
331Dic61—44no—
332Dic61—44no—
333Dic61—44no—
334C2×#385D6111yescharacteristic
335C2×#381—611no—
336C2×#381—611no—
337C2×C4×C22⋊C4—611no—
338C22×#95—611no—
339#1443—612no—
340#1442—612no—
341#1436—612no—
342#1434—612no—
343C22⋊#95—611no—
344C22⋊#95—611no—
345C2×C61—28no—
346C2×C61C2×#38811yescharacteristic
347C2×C61C4×#12311yescharacteristic
348C2×C61#232211yescharacteristic
349C2×C61#234312yes—
350C2×C61#235912yes—
351C2×C61#241211yescharacteristic
352C2×C61#241311yescharacteristic
353C2×C61#241912yes—
354C2×C61#249312yes—
355C2×C61#250311yescharacteristic
356C2×C61#256111yescharacteristic
357C24×C4—611no—
358C24×C4—611no—
359C24×C4C2×D6111yescharacteristic
360C22×C4⋊C4C2×D6111yescharacteristic
361C22×C4⋊C4C2×D6111yescharacteristic
362C22×C22⋊C4—611no—
363C22×C22⋊C4—611no—
364C22×C22⋊C4C2×D6111yescharacteristic
365C22×C22⋊C4C2×D6111yescharacteristic
366C22×C42—612no—
367#385—24no—
368#381—618no—
369#381—1224no—
370C4×C22⋊C4—1224no—
371C2×#95—612no—
372C2×#95—612no—
373C2×#95—612no—
374C2×#95—612no—
375C2×#95—612no—
376C2×#95—612no—
377C2×#95—612no—
378C2×#95—612no—
379C2×#95—612no—
380C2×#95—612no—
381C2×#95—612no—
382C2×#95—1224no—
383C2×#95C2×D6111yescharacteristic
384C2×#95C2×D6111yescharacteristic
385C61#1397211yescharacteristic, complemented, Sylow
386C25C22×D6111yescharacteristic
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—24no—
390C23×C4—42no—
391C23×C4—614no—
392C23×C4—612no—
393C23×C4—612no—
394C23×C4—612no—
395C23×C4—612no—
396C23×C4—1224no—
397C23×C4—1222no—
398C23×C4—1222no—
399C23×C4—1222no—
400C23×C4—1222no—
401C23×C4—1222no—
402C23×C4—1222no—
403C23×C4—1222no—
404C23×C4—1222no—
405C23×C4—1222no—
406C23×C4—2442no—
407C23×C4C22⋊D6114yes—
408C23×C4C22×D6111yescharacteristic
409C23×C4C22×D6111yescharacteristic
410C23×C4C22×D6111yescharacteristic
411C23×C4C22×D6111yescharacteristic
412C23×C4C22×D6111yescharacteristic
413C23×C4C22×D6111yescharacteristic
414C2×C4⋊C4—24no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—22no—
417C2×C4⋊C4—42no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—614no—
424C2×C22⋊C4—614no—
425C2×C22⋊C4—1222no—
426C2×C22⋊C4—1222no—
427C2×C22⋊C4—1222no—
428C2×C22⋊C4—1222no—
429C2×C22⋊C4C4×D6114yes—
430C2×C42—1224no—
431C2×C42—1224no—
432C2×C42—2442no—
433#95—22no—
434#95—22no—
435#95—618no—
436#95—618no—
437#95—1224no—
438#95—1224no—
439#95—1224no—
440#95—1224no—
441#95—1224no—
442#95—1224no—
443#95—1224no—
444#95—2442no—
445#95—2442no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24C2×C4×D6112yes—
451C24#1175822212yes—
452C24D4×D6112yes—
453C24#1175822412yes—
454C24#1175822612yes—
455C24#1175822612yes—
456C24C2×C22⋊D6112yes—
457C24C23×D6111yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—44no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—614no—
489C22×C4—614no—
490C22×C4—614no—
491C22×C4—614no—
492C22×C4—1228no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1222no—
517C22×C4—1228no—
518C22×C4—1228no—
519C22×C4—1228no—
520C22×C4—2448no—
521C22×C4—2448no—
522C22×C4—2444no—
523C22×C4—2444no—
524C22×C4—2444no—
525C22×C4—2444no—
526C22×C4—2444no—
527C22×C4—2444no—
528C22×C4C2×C4×D6112yes—
529C22×C4C2×C4×D6112yes—
530C22×C4#1175822414yes—
531C22×C4C2×C22⋊D6112yes—
532C22×C4C2×C22⋊D6112yes—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C22⋊C4—28no—
536C22⋊C4—44no—
537C22⋊C4—1228no—
538C22⋊C4—1228no—
539C42—2448no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23#1451973514yes—
557C23C4×C22⋊D6114yes—
558C23#1451973914yes—
559C23#1451974914yes—
560C23#1451976214yes—
561C23#1451976614yes—
562C23#1451977214yes—
563C23C22×C4×D6111yescharacteristic
564C23C2×D4×D6111yescharacteristic
565C23C22×C22⋊D6111yescharacteristic
566C23#41×D6112yes—
567C23C2×#1175822411yescharacteristic
568C23C2×#1175822412yes—
569C23C2×#1175822211yescharacteristic
570C23C2×#1175822611yescharacteristic
571C23C2×#1175822611yescharacteristic
572C23#1451982212yes—
573C23#1451982412yes—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—416no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—1224no—
605C2×C4—1224no—
606C2×C4—1224no—
607C2×C4—1224no—
608C2×C4—1224no—
609C2×C4—1224no—
610C2×C4—1224no—
611C2×C4—1224no—
612C2×C4—1224no—
613C2×C4—1224no—
614C2×C4—1224no—
615C2×C4—1224no—
616C2×C4—24416no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4—2442no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2442no—
628C2×C4—2442no—
629C2×C4—24416no—
630C2×C4—2442no—
631C2×C4—2442no—
632C2×C4—2442no—
633C2×C4—2442no—
634C2×C4#1451975814yes—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—28no—
640C22—28no—
641C22—24no—
642C22—28no—
643C22—24no—
644C22—28no—
645C22#1683965218yes—
646C22C2×#1451973511yescharacteristic
647C22C2×C4×C22⋊D6111yescharacteristic
648C22C2×#1451973911yescharacteristic
649C22C4×#1175822412yes—
650C22#1684024112yes—
651C22C2×#1451974911yescharacteristic
652C22C2×#1451975811yescharacteristic
653C22C2×#1451976211yescharacteristic
654C22#1684026911yescharacteristic
655C22C2×#1451976611yescharacteristic
656C22C2×#1451977211yescharacteristic
657C22#1684029212yes—
658C22#1684032111yescharacteristic
659C22#1684033311yescharacteristic
660C22#1684035412yes—
661C22#1684039912yes—
662C22#1684041012yes—
663C22#1684043912yes—
664C22#1684043912yes—
665C22#1684044312yes—
666C22#1684044912yes—
667C22#1684045412yes—
668C22#1684045412yes—
669C4—24no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—81no—
674C4—81no—
675C4—2444no—
676C4—2444no—
677C4—2444no—
678C4—2444no—
679C2—28no—
680C2C2×#1683965211yescharacteristic
681C2C4×#1451975811yescharacteristic
682C2#2144848411yescharacteristic
683C2#2144867712yes—
684C2#2144880912yes—
685C2#2144921611yescharacteristic
686C2#2144922711yescharacteristic
687C2#2144929812yes—
688C2#2145020612yes—
689C2#2145030511yescharacteristic
690C2#2145097811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#385C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#385D61
hypocenterC61#13972
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
5. #21444080C2semidirect
6. #21444078C2semidirect
7. #21444051C2semidirect
8. #21444045C2semidirect
18. #16839623C4semidirect
74. #14519684C2×C4semidirect
75. #14519684C2×C4semidirect
385. C61#13972semidirect

Maximal subgroups

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

subgroupindex
C2×#168396232
C2×C4×C22⋊Dic612
C22×#145196842
#214440802
#214440782
#214440512
#214440452
#214440412
C2×C61×#3852
#1397261