Finite Group Databasebeta

← #29606918

Subgroups — #29606918

Order 15616.

1766 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839676C211yesmaximal, characteristic
2C2×#16839665C211yesmaximal, characteristic
3C2×#16839656C212yesmaximal, complemented
4C2×#16839613C211yesmaximal, characteristic, complemented
5#21444083C211yesmaximal, characteristic, complemented
6#21444073C211yesmaximal, characteristic, complemented
7#21444062C212yesmaximal, complemented
8#21444062C212yesmaximal, complemented
9#21444056C212yesmaximal, complemented
10#21444044C211yesmaximal, characteristic, complemented
11C61×#1438C211yesmaximal, characteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×#11758196C2211yescharacteristic
14C22×#11758196C2211yescharacteristic
15C22×#11758196C2212yes—
16C22×C22⋊Dic61C2212yes—
17C22×C22⋊Dic61C2212yes—
18C22×C22⋊Dic61C2212yes—
19C22×C22⋊Dic61C2212yes—
20C22×C4×Dic61C2211yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23#16839676C418yescomplemented
24#16839665—24no—
25#16839656—28no—
26#16839613—24no—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2211yescharacteristic
34C23×D4×C61C2211yescharacteristic
35C22×C22⋊C4×C61C2212yes—
36C22×C22⋊C4×C61C2212yes—
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2211yescharacteristic
39C23×Dic61C2312yes—
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2312yes—
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2311yescharacteristic
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—42no—
47C2×#11758196—22no—
48C2×#11758196—22no—
49C2×#11758196—22no—
50C2×#11758196—22no—
51C2×#11758196—22no—
52C2×#11758196—24no—
53C2×#11758196—42no—
54C2×#11758196—42no—
55C2×#11758196C2×C414yes—
56C2×C22⋊Dic61—24no—
57C2×C22⋊Dic61—24no—
58C2×C22⋊Dic61—24no—
59C2×C22⋊Dic61—24no—
60C2×C22⋊Dic61—24no—
61C2×C22⋊Dic61—24no—
62C2×C22⋊Dic61—42no—
63C2×C22⋊Dic61—42no—
64C2×C22⋊Dic61—42no—
65C2×C22⋊Dic61—42no—
66C2×C22⋊Dic61C2×C418yes—
67C2×C22⋊Dic61C2×C418yes—
68C2×C4×Dic61—22no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—42no—
72C2×C4×Dic61—42no—
73C2×C4×Dic61D414yes—
74C2×C4×Dic61D414yescomplemented
75#14519684—22no—
76#14519684—24no—
77#14519684—24no—
78#14519684—42no—
79#14519684—42no—
80#14519684—42no—
81#14519684C2×C414yes—
82C25×C61C2312yes—
83C22×D4×C61—22no—
84C22×D4×C61—22no—
85C22×D4×C61—42no—
86C22×D4×C61—42no—
87C22×D4×C61C2×C414yes—
88C23×C4×C61C2312yes—
89C23×C4×C61C2311yescharacteristic
90C23×C4×C61C2311yescharacteristic
91C23×C4×C61C2311yescharacteristic
92C2×C22⋊C4×C61—24no—
93C2×C22⋊C4×C61—24no—
94C2×C22⋊C4×C61—42no—
95C2×C22⋊C4×C61—42no—
96C2×C22⋊C4×C61—42no—
97C2×C22⋊C4×C61—42no—
98#95×C61—22no—
99#95×C61—22no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61—22no—
103C24×C61—22no—
104C24×C61—22no—
105C24×C61—22no—
106C24×C61—22no—
107C24×C61—22no—
108C24×C61—22no—
109C24×C61—22no—
110C24×C61—22no—
111C24×C61—22no—
112C24×C61—22no—
113C24×C61—22no—
114C24×C61C22×C414yes—
115C24×C61C2411yescharacteristic
116C2×D4×C61—28no—
117C2×D4×C61—48no—
118C2×D4×C61—44no—
119C2×D4×C61—44no—
120C2×D4×C61—48no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
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×C61C22×C412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61#4112yes—
142C22×C4×C61#4112yes—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—21no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
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×Dic61C22×C414yes—
171C22×Dic61C22×C412yes—
172C22×Dic61C22×C412yes—
173C22×Dic61C2×D412yes—
174C22×Dic61C2×D412yes—
175C22×Dic61C2×D412yes—
176C22×Dic61C2×D412yes—
177C22×Dic61#4112yes—
178C22×Dic61#4112yes—
179C22⋊C4×C61—48no—
180C22⋊C4×C61—48no—
181C22⋊Dic61—216no—
182C22⋊Dic61—48no—
183C22⋊Dic61—48no—
184C22⋊Dic61—48no—
185C4⋊Dic61—44no—
186#11758196—28no—
187#11758196—28no—
188#11758196—48no—
189C4×Dic61—28no—
190C4×Dic61—44no—
191C4×Dic61—44no—
192C23×C61—24no—
193C23×C61—24no—
194C23×C61—24no—
195C23×C61—24no—
196C23×C61—24no—
197C23×C61—24no—
198C23×C61—24no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61—42no—
222C23×C61—42no—
223C23×C61—42no—
224C23×C61—42no—
225C23×C61—42no—
226C23×C61—42no—
227C23×C61C23×C411yescharacteristic
228C23×C61C22×D411yescharacteristic
229C23×C61C22×D411yescharacteristic
230C23×C61C2×#4111yescharacteristic
231C23×C61C2×#4111yescharacteristic
232C23×C61C2×#4111yescharacteristic
233C23×C61C2×#4111yescharacteristic
234C23×C61H(5,𝔽2)12yes—
235C23×C61H(5,𝔽2)12yes—
236C23×C61H(5,𝔽2)12yes—
237C23×C612−1+412yes—
238D4×C61—416no—
239C2×C4×C61—24no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—24no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—24no—
246C2×C4×C61—24no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61C4×D414yes—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—24no—
270C2×Dic61—24no—
271C2×Dic61—22no—
272C2×Dic61—24no—
273C2×Dic61—24no—
274C2×Dic61—24no—
275C2×Dic61—24no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—41no—
280C2×Dic61—42no—
281C2×Dic61—41no—
282C2×Dic61—42no—
283C2×Dic61—42no—
284C2×Dic61—42no—
285C2×Dic61—42no—
286C2×Dic61—42no—
287C2×Dic61—42no—
288C2×Dic61—42no—
289C2×Dic61—42no—
290C2×Dic61—42no—
291C2×Dic61—42no—
292C2×Dic61—42no—
293C2×Dic61—42no—
294C2×Dic61—42no—
295C2×Dic61C4×D414yes—
296C2×Dic61#12114yes—
297#14312—611nomaximal, Sylow
298C22×C61—44no—
299C22×C61—44no—
300C22×C61—44no—
301C22×C61—44no—
302C22×C61—44no—
303C22×C61—44no—
304C22×C61—44no—
305C22×C61—82no—
306C22×C61—82no—
307C22×C61—82no—
308C22×C61—82no—
309C22×C61—82no—
310C22×C61—82no—
311C22×C61—82no—
312C22×C61—82no—
313C22×C61C2×C4×D411yescharacteristic
314C22×C61C2×C4×D411yescharacteristic
315C22×C61C4×#4111yescharacteristic
316C22×C61#51812yes—
317C22×C61#52012yes—
318C22×C61C2×#12111yescharacteristic
319C22×C61#52511yescharacteristic
320C22×C61#52911yescharacteristic
321C22×C61#53211yescharacteristic
322C22×C61#53411yescharacteristic
323C22×C61#53511yescharacteristic
324C22×C61#53511yescharacteristic
325C22×C61#53512yes—
326C22×C61#53611yescharacteristic
327C22×C61#53712yes—
328C22×C61#53812yes—
329C22×C61#53812yes—
330C22×C61#53912yes—
331C22×C61#54012yes—
332C22×C61#54012yes—
333C22×C61#54212yes—
334C22×C61#54212yes—
335C22×C61#54212yes—
336C4×C61—24no—
337C4×C61—42no—
338C4×C61—42no—
339C4×C61—82no—
340Dic61—24no—
341Dic61—24no—
342Dic61—42no—
343Dic61—44no—
344Dic61—42no—
345Dic61—82no—
346C2×#394—611no—
347C2×#390—611no—
348C2×#388—612no—
349C2×C4×C4⋊C4—611no—
350#1444—611no—
351#1441—611no—
352#1439—612no—
353#1439—612no—
354#1438D6111yescharacteristic
355#1437—612no—
356#1434—611no—
357C2×C61—84no—
358C2×C61C4×#12111yescharacteristic
359C2×C61#232111yescharacteristic
360C2×C61#232412yes—
361C2×C61#232611yescharacteristic
362C2×C61#233512yes—
363C2×C61#261811yescharacteristic
364C2×C61#261911yescharacteristic
365C2×C61#263211yescharacteristic
366C2×C61#263511yescharacteristic
367C2×C61#265112yes—
368C2×C61#266212yes—
369C23×D4C2×D6111yescharacteristic
370C22×C4⋊C4—611no—
371C22×C4⋊C4—611no—
372C22×C4⋊C4—612no—
373C22×C4⋊C4—611no—
374C22×C22⋊C4—612no—
375C22×C22⋊C4—612no—
376C22×C22⋊C4—612no—
377C22×C22⋊C4—612no—
378C22×C22⋊C4C2×D6112yes—
379C22×C22⋊C4C2×D6112yes—
380C22×C42—611no—
381C22×C42—611no—
382C22×C42—611no—
383#394—618no—
384#390—1224no—
385#388—1228no—
386C4×C4⋊C4—1224no—
387C2×#95—611no—
388C2×#95—612no—
389C2×#95—611no—
390C2×#95—612no—
391C2×#95—612no—
392C2×#95—612no—
393C2×#95—612no—
394C2×#95C2×D6111yescharacteristic
395C2×#95C2×D6111yescharacteristic
396C61#1431211yescharacteristic, complemented, Sylow
397C25C22×D6112yes—
398C22×D4—22no—
399C22×D4—22no—
400C22×D4—42no—
401C22×D4—42no—
402C22×D4C4×D6114yes—
403C23×C4—611no—
404C23×C4—611no—
405C23×C4—612no—
406C23×C4—611no—
407C23×C4—611no—
408C23×C4—612no—
409C23×C4C22×D6112yes—
410C23×C4C22×D6111yescharacteristic
411C23×C4C22×D6111yescharacteristic
412C23×C4C22×D6111yescharacteristic
413C2×C4⋊C4—614no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1224no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4—2442no—
422C2×C4⋊C4—2442no—
423C2×C4⋊C4—2442no—
424C2×C22⋊C4—24no—
425C2×C22⋊C4—24no—
426C2×C22⋊C4—42no—
427C2×C22⋊C4—42no—
428C2×C22⋊C4—42no—
429C2×C22⋊C4—42no—
430C2×C22⋊C4—618no—
431C2×C22⋊C4—618no—
432C2×C22⋊C4—1224no—
433C2×C22⋊C4—1224no—
434C2×C22⋊C4—1224no—
435C2×C22⋊C4—1224no—
436C2×C22⋊C4—1224no—
437C2×C22⋊C4—1224no—
438C2×C22⋊C4—2442no—
439C2×C22⋊C4—2442no—
440C2×C22⋊C4—2442no—
441C2×C22⋊C4—2442no—
442C2×C42—614no—
443C2×C42—614no—
444C2×C42—1222no—
445C2×C42—1222no—
446C2×C42—1222no—
447C2×C42—2442no—
448C2×C42—2442no—
449#95—22no—
450#95—22no—
451#95—614no—
452#95—1222no—
453#95—1224no—
454#95—1224no—
455#95—2442no—
456#95—2442no—
457#95—2442no—
458C24—22no—
459C24—22no—
460C24—22no—
461C24—22no—
462C24—22no—
463C24—22no—
464C24—22no—
465C24—22no—
466C24—22no—
467C24—22no—
468C24—22no—
469C24—22no—
470C24—22no—
471C24—22no—
472C24C2×C4×D6114yes—
473C24C23×D6111yescharacteristic
474C2×D4—28no—
475C2×D4—48no—
476C2×D4—44no—
477C2×D4—44no—
478C2×D4—48no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—612no—
497C22×C4—612no—
498C22×C4—612no—
499C22×C4—612no—
500C22×C4—612no—
501C22×C4—612no—
502C22×C4—614no—
503C22×C4—612no—
504C22×C4—612no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1221no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—1222no—
529C22×C4—1222no—
530C22×C4—1222no—
531C22×C4—1222no—
532C22×C4C2×C4×D6112yes—
533C22×C4D4×D6112yes—
534C22×C4D4×D6112yes—
535C22×C4#1175822412yes—
536C22×C4#1175822412yes—
537C4⋊C4—1228no—
538C4⋊C4—1228no—
539C4⋊C4—2448no—
540C4⋊C4—2444no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—12216no—
544C22⋊C4—2448no—
545C22⋊C4—2448no—
546C22⋊C4—2448no—
547C42—1228no—
548C42—2444no—
549C42—2444no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23C22×C4×D6111yescharacteristic
586C23C2×D4×D6111yescharacteristic
587C23C2×D4×D6111yescharacteristic
588C23#41×D6111yescharacteristic
589C23#41×D6111yescharacteristic
590C23C2×#1175822411yescharacteristic
591C23C2×#1175822411yescharacteristic
592C23#1451982212yes—
593C23#1451982212yes—
594C23#1451982212yes—
595C23#1451982412yes—
596D4—416no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—614no—
619C2×C4—614no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1224no—
629C2×C4—1224no—
630C2×C4—1224no—
631C2×C4—1222no—
632C2×C4—1224no—
633C2×C4—1224no—
634C2×C4—1224no—
635C2×C4—1222no—
636C2×C4—1222no—
637C2×C4—1222no—
638C2×C4—2442no—
639C2×C4—2442no—
640C2×C4—2442no—
641C2×C4—2442no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4—2442no—
645C2×C4—2441no—
646C2×C4—2442no—
647C2×C4—2441no—
648C2×C4—2442no—
649C2×C4—2442no—
650C2×C4—2442no—
651C2×C4—2442no—
652C2×C4—2442no—
653C2×C4—2442no—
654C2×C4#1451973714yes—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22C4×D4×D6111yescharacteristic
671C22C2×#1451973711yescharacteristic
672C22C4×#1175822411yescharacteristic
673C22#1684023512yes—
674C22#1684024112yes—
675C22#121×D6111yescharacteristic
676C22#1684027211yescharacteristic
677C22#1684029111yescharacteristic
678C22#1684031011yescharacteristic
679C22#1684032211yescharacteristic
680C22#1684032511yescharacteristic
681C22#1684032711yescharacteristic
682C22#1684032712yes—
683C22#1684033211yescharacteristic
684C22#1684034112yes—
685C22#1684034712yes—
686C22#1684034712yes—
687C22#1684035512yes—
688C22#1684036212yes—
689C22#1684036212yes—
690C22#1684037912yes—
691C22#1684037912yes—
692C22#1684038012yes—
693C4—24no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—1224no—
698C4—1224no—
699C4—4882no—
700C4—2442no—
701C4—2444no—
702C4—2442no—
703C2—84no—
704C2#2144833511yescharacteristic
705C2#2144847411yescharacteristic
706C2#2144850912yes—
707C2#2144852911yescharacteristic
708C2#2144861012yes—
709C2#2145156011yescharacteristic
710C2#2145156811yescharacteristic
711C2#2145168411yescharacteristic
712C2#2145170811yescharacteristic
713C2#2145184912yes—
714C2#2145196112yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1438C2
socle, cosocleC24×C61C24
prime radicalC23×D4×C61C22
hypercenter#1438D61
hypocenterC61#14312
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
3. C2×#16839656C2semidirect
4. C2×#16839613C2semidirect
5. #21444083C2semidirect
6. #21444073C2semidirect
7. #21444062C2semidirect
8. #21444062C2semidirect
9. #21444056C2semidirect
10. #21444044C2semidirect
23. #16839676C4semidirect
74. C2×C4×Dic61D4semidirect
396. C61#14312semidirect

Maximal subgroups

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

subgroupindex
C2×#168396762
C2×#168396652
C2×#168396562
C2×#168396132
#214440832
#214440732
#214440622
#214440562
#214440442
C61×#14382
#1431261