Finite Group Databasebeta

← #29606775

Subgroups — #29606775

Order 15616.

2242 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519739C211yesmaximal, characteristic
2C2×#16839680C211yesmaximal, characteristic
3C2×#16839644C211yesmaximal, characteristic, complemented
4C2×#16839633C212yesmaximal, complemented
5#21444069C211yesmaximal, characteristic, complemented
6#21444067C212yesmaximal, complemented
7#21444061C212yesmaximal, complemented
8#21444059C211yesmaximal, characteristic
9#21444044C211yesmaximal, characteristic, complemented
10#21444040C212yesmaximal, complemented
11C61×#1444C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×C4×D61C2212yes—
14C2×#14519739—24no—
15C2×#14519739—28no—
16C22×#11758196C2211yescharacteristic
17C22×#11758196C2211yescharacteristic
18C22×#11758198C2212yes—
19C22×#11758198C2212yes—
20C22×#11758198C2212yes—
21C22×#11758198C2212yes—
22C22×#11758198C2212yes—
23C22×C4×Dic61C2211yescharacteristic
24#16839680C418yescomplemented
25#16839644—24no—
26#16839633—28no—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2211yescharacteristic
31C2×#14519684C2211yescharacteristic
32C2×#14519684C2211yescharacteristic
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C22×C4⋊C4×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2212yes—
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C2×#95×C61C2211yescharacteristic
40C24×D61C2312yes—
41C22×D244—24no—
42C22×D244—24no—
43C22×D244—22no—
44C22×D244—22no—
45C22×D244C2×C414yes—
46C23×Dic61C2312yes—
47C23×Dic61C2311yescharacteristic
48C23×Dic61C2311yescharacteristic
49C23×Dic61C2311yescharacteristic
50C23×Dic61C2311yescharacteristic
51C22×C4×D61—24no—
52C22×C4×D61—24no—
53C22×C4×D61—42no—
54C22×C4×D61—42no—
55C22×C4×D61—42no—
56C22×C4×D61—42no—
57C22×C4×D61D418yes—
58#14519739—232no—
59C2×#11758196—22no—
60C2×#11758196—22no—
61C2×#11758196—24no—
62C2×#11758196—22no—
63C2×#11758196C2×C414yescomplemented
64C2×#11758198—24no—
65C2×#11758198—24no—
66C2×#11758198—24no—
67C2×#11758198—24no—
68C2×#11758198—24no—
69C2×#11758198—24no—
70C2×#11758198—24no—
71C2×#11758198—42no—
72C2×#11758198—42no—
73C2×#11758198—42no—
74C2×#11758198—42no—
75C2×#11758198—42no—
76C2×#11758198—42no—
77C2×#11758198C2×C418yescomplemented
78C2×#11758198C2×C418yescomplemented
79C2×C4×Dic61—24no—
80C2×C4×Dic61D414yes—
81#14519684—24no—
82#14519684—22no—
83#14519684—24no—
84#14519684—22no—
85#14519684—22no—
86#14519684—22no—
87#14519684—42no—
88#14519684—42no—
89C23×C4×C61C2311yescharacteristic
90C23×C4×C61C2311yescharacteristic
91C23×C4×C61C2312yes—
92C23×C4×C61C2312yes—
93C23×C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61—24no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—24no—
97C2×C4⋊C4×C61—42no—
98C2×C4⋊C4×C61—42no—
99#95×C61—22no—
100#95×C61—42no—
101#95×C61C2×C414yescomplemented
102C24×C61C2411yescharacteristic
103C23×D61—22no—
104C23×D61—22no—
105C23×D61—22no—
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—22no—
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—22no—
114C23×D61—22no—
115C23×D61C22×C414yes—
116C23×D61C2×D414yes—
117C23×D61#4114yes—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
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×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C22×C414yes—
143C22×C4×C61C22×C412yes—
144C22×C4×C61C2×D414yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61#4112yes—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—22no—
152C22×Dic61—21no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—22no—
163C22×Dic61—21no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61C22×C412yes—
172C22×Dic61C22×C412yes—
173C22×Dic61C2×D412yes—
174C22×Dic61C2×D412yes—
175C22×Dic61C2×D414yes—
176C2×D244—28no—
177C2×D244—216no—
178C2×D244—28no—
179C2×D244—216no—
180C2×D244—28no—
181C2×C4×D61—216no—
182C2×C4×D61—48no—
183C2×C4×D61—48no—
184C2×C4×D61—48no—
185C2×C4×D61—48no—
186C2×C4×D61—48no—
187C2×C4×D61—48no—
188C4⋊C4×C61—28no—
189C4⋊C4×C61—48no—
190#11758198—216no—
191#11758198—216no—
192#11758198—48no—
193#11758198—48no—
194#11758198—48no—
195#11758196—28no—
196#11758196—44no—
197C4×Dic61—28no—
198C23×C61C23×C411yescharacteristic
199C23×C61C22×D412yes—
200C23×C61C22×D411yescharacteristic
201C23×C61C2×#4112yes—
202C23×C61C2×#4111yescharacteristic
203C23×C61H(5,𝔽2)11yescharacteristic
204C23×C61H(5,𝔽2)11yescharacteristic
205C23×C61H(5,𝔽2)12yes—
206C23×C61H(5,𝔽2)11yescharacteristic
207C23×C612−1+412yes—
208C23×C612−1+411yescharacteristic
209C22×D61—24no—
210C22×D61—24no—
211C22×D61—24no—
212C22×D61—24no—
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—24no—
216C22×D61—24no—
217C22×D61—24no—
218C22×D61—24no—
219C22×D61—24no—
220C22×D61—24no—
221C22×D61—24no—
222C22×D61—24no—
223C22×D61—24no—
224C22×D61—24no—
225C22×D61—24no—
226C22×D61—24no—
227C22×D61—42no—
228C22×D61—42no—
229C22×D61—42no—
230C22×D61—42no—
231C22×D61—42no—
232C22×D61—42no—
233C22×D61—42no—
234C22×D61—42no—
235C22×D61—42no—
236C22×D61—42no—
237C22×D61—42no—
238C22×D61—42no—
239C22×D61—42no—
240C22×D61—42no—
241C22×D61—42no—
242C22×D61—42no—
243C22×D61C4×D418yes—
244C2×C4×C61—24no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—24no—
248C2×C4×C61—24no—
249C2×C4×C61—24no—
250C2×C4×C61—24no—
251C2×C4×C61—24no—
252C2×C4×C61—24no—
253C2×C4×C61—24no—
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×C61—42no—
261C2×C4×C61—42no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61C4×D414yes—
273C2×Dic61—24no—
274C2×Dic61—24no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—24no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—22no—
284C2×Dic61—22no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—42no—
288C2×Dic61—41no—
289C2×Dic61—42no—
290C2×Dic61—41no—
291C2×Dic61—42no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—42no—
295C2×Dic61—42no—
296C2×Dic61—42no—
297C2×Dic61—42no—
298C2×Dic61—42no—
299C2×Dic61—42no—
300C2×Dic61—42no—
301C2×Dic61#12014yes—
302D244—232no—
303C4×D61—432no—
304#14300—611nomaximal, Sylow
305C22×C61C2×C4×D412yes—
306C22×C61C2×C4×D411yescharacteristic
307C22×C61#51811yescharacteristic
308C22×C61#52012yes—
309C22×C61#52011yescharacteristic
310C22×C61C2×#12011yescharacteristic
311C22×C61#52511yescharacteristic
312C22×C61#52512yes—
313C22×C61#53412yes—
314C22×C61#53412yes—
315C22×C61#53511yescharacteristic
316C22×C61#53511yescharacteristic
317C22×C61#53612yes—
318C22×C61#53611yescharacteristic
319C22×C61#53711yescharacteristic
320C22×C61#53712yes—
321C22×C61#53812yes—
322C22×C61#53912yes—
323C22×C61#53912yes—
324C22×C61#53912yes—
325C22×C61#54011yescharacteristic
326C22×C61#54211yescharacteristic
327C22×C61#54212yes—
328C2×D61—28no—
329C2×D61—28no—
330C2×D61—28no—
331C2×D61—44no—
332C2×D61—44no—
333C2×D61—44no—
334C2×D61—44no—
335C2×D61—44no—
336C2×D61—44no—
337C2×D61—44no—
338C2×D61—44no—
339C2×D61—44no—
340C2×D61—44no—
341C2×D61—44no—
342C2×D61—44no—
343C4×C61—24no—
344C4×C61—44no—
345C4×C61—44no—
346C4×C61—81no—
347C4×C61—81no—
348Dic61—24no—
349Dic61—42no—
350Dic61—42no—
351Dic61—42no—
352Dic61—82no—
353C22×C4×D4—611no—
354C2×#394—611no—
355C2×#387—611no—
356C2×#385—612no—
357#1444D6111yescharacteristic, complemented
358#1441—611no—
359#1440—612no—
360#1439—612no—
361#1438—611no—
362#1434—611no—
363C22⋊#95—612no—
364C2×C61C4×#12011yescharacteristic
365C2×C61#232212yes—
366C2×C61#232411yescharacteristic
367C2×C61#232512yes—
368C2×C61#232611yescharacteristic
369C2×C61#261311yescharacteristic
370C2×C61#261611yescharacteristic
371C2×C61#263012yes—
372C2×C61#263311yescharacteristic
373C2×C61#263611yescharacteristic
374C2×C61#263812yes—
375D61—48no—
376C23×D4—611no—
377C24×C4—612no—
378C2×C4×D4—1224no—
379C2×C4×D4—1228no—
380C22×C4⋊C4—611no—
381C22×C4⋊C4—611no—
382C22×C4⋊C4C2×D6111yescharacteristic
383C22×C4⋊C4C2×D6112yes—
384C22×C22⋊C4—612no—
385C22×C22⋊C4—612no—
386C22×C22⋊C4—612no—
387C22×C22⋊C4—612no—
388C22×C22⋊C4—612no—
389C22×C42—611no—
390#394—618no—
391#387—1224no—
392#385—1228no—
393C2×#95—611no—
394C2×#95—612no—
395C2×#95—611no—
396C2×#95—611no—
397C2×#95—611no—
398C2×#95—612no—
399C2×#95—612no—
400C2×#95—612no—
401C2×#95C2×D6111yescharacteristic
402C2×#95C2×D6112yes—
403C2×#95C2×D6111yescharacteristic
404C61#1430011yescharacteristic, complemented, Sylow
405C25—612no—
406C22×D4—614no—
407C22×D4—1224no—
408C22×D4—1224no—
409C22×D4—1222no—
410C22×D4—1222no—
411C23×C4—611no—
412C23×C4—611no—
413C23×C4—611no—
414C23×C4—611no—
415C23×C4—618no—
416C23×C4—612no—
417C23×C4—1224no—
418C23×C4—1224no—
419C23×C4—2442no—
420C23×C4—2442no—
421C23×C4—2442no—
422C23×C4—2442no—
423C23×C4C22×D6111yescharacteristic
424C23×C4C22×D6111yescharacteristic
425C23×C4C22×D6112yes—
426C23×C4C22×D6112yes—
427C23×C4C22×D6111yescharacteristic
428C4×D4—12232no—
429C2×C4⋊C4—24no—
430C2×C4⋊C4—22no—
431C2×C4⋊C4—24no—
432C2×C4⋊C4—42no—
433C2×C4⋊C4—42no—
434C2×C4⋊C4—614no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4—1222no—
437C2×C4⋊C4—1224no—
438C2×C4⋊C4—1222no—
439C2×C22⋊C4—618no—
440C2×C22⋊C4—618no—
441C2×C22⋊C4—1224no—
442C2×C22⋊C4—1224no—
443C2×C22⋊C4—1224no—
444C2×C22⋊C4—1224no—
445C2×C22⋊C4—1224no—
446C2×C22⋊C4—1224no—
447C2×C22⋊C4—1224no—
448C2×C22⋊C4—2442no—
449C2×C22⋊C4—2442no—
450C2×C22⋊C4—2442no—
451C2×C22⋊C4—2442no—
452C2×C22⋊C4—2442no—
453C2×C22⋊C4—2442no—
454C2×C42—614no—
455C2×C42—1224no—
456#95—22no—
457#95—42no—
458#95—1224no—
459#95—1222no—
460#95—1222no—
461#95—1222no—
462#95—1224no—
463#95—1222no—
464#95—2442no—
465#95—2442no—
466#95C4×D6114yescomplemented
467C24—614no—
468C24—614no—
469C24—614no—
470C24—1222no—
471C24—1222no—
472C24—1222no—
473C24—1222no—
474C24—1222no—
475C24—1222no—
476C24—1222no—
477C24—1222no—
478C24—1222no—
479C24—1222no—
480C24—1222no—
481C24—1222no—
482C24C23×D6111yescharacteristic
483C2×D4—12216no—
484C2×D4—1228no—
485C2×D4—1228no—
486C2×D4—12216no—
487C2×D4—1228no—
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—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—612no—
513C22×C4—614no—
514C22×C4—612no—
515C22×C4—612no—
516C22×C4—612no—
517C22×C4—12216no—
518C22×C4—1222no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1222no—
523C22×C4—1221no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1222no—
534C22×C4—1221no—
535C22×C4—1222no—
536C22×C4—1222no—
537C22×C4—1222no—
538C22×C4—1222no—
539C22×C4—1222no—
540C22×C4—1222no—
541C22×C4—1222no—
542C22×C4—2448no—
543C22×C4—2448no—
544C22×C4—2448no—
545C22×C4—2448no—
546C22×C4—2448no—
547C22×C4—2448no—
548C22×C4C2×C4×D6114yes—
549C22×C4C2×C4×D6112yes—
550C22×C4D4×D6114yes—
551C22×C4D4×D6112yes—
552C22×C4#1175822612yes—
553C4⋊C4—28no—
554C4⋊C4—48no—
555C4⋊C4—1228no—
556C4⋊C4—2444no—
557C22⋊C4—12216no—
558C22⋊C4—12216no—
559C22⋊C4—2448no—
560C22⋊C4—2448no—
561C22⋊C4—2448no—
562C42—1228no—
563C23—618no—
564C23—1224no—
565C23—1224no—
566C23—1224no—
567C23—1224no—
568C23—1224no—
569C23—1224no—
570C23—1224no—
571C23—1224no—
572C23—1224no—
573C23—1224no—
574C23—1224no—
575C23—1224no—
576C23—1224no—
577C23—1224no—
578C23—1224no—
579C23—1224no—
580C23—1224no—
581C23—1224no—
582C23—2442no—
583C23—2442no—
584C23—2442no—
585C23—2442no—
586C23—2442no—
587C23—2442no—
588C23—2442no—
589C23—2442no—
590C23—2442no—
591C23—2442no—
592C23—2442no—
593C23—2442no—
594C23—2442no—
595C23—2442no—
596C23—2442no—
597C23—2442no—
598C23C22×C4×D6111yescharacteristic
599C23C2×D4×D6112yes—
600C23C2×D4×D6111yescharacteristic
601C23#41×D6112yes—
602C23C2×#1175822611yescharacteristic
603C23#1451982211yescharacteristic
604C23#1451982212yes—
605C23#1451982311yescharacteristic
606C23#1451982311yescharacteristic
607C23#1451982512yes—
608C23#1451982511yescharacteristic
609D4—12232no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—614no—
639C2×C4—1224no—
640C2×C4—1224no—
641C2×C4—1222no—
642C2×C4—1222no—
643C2×C4—1224no—
644C2×C4—1222no—
645C2×C4—1222no—
646C2×C4—1222no—
647C2×C4—1222no—
648C2×C4—1222no—
649C2×C4—1222no—
650C2×C4—1222no—
651C2×C4—24432no—
652C2×C4—2442no—
653C2×C4—2442no—
654C2×C4—2442no—
655C2×C4—2442no—
656C2×C4—2442no—
657C2×C4—2442no—
658C2×C4—2442no—
659C2×C4—2441no—
660C2×C4—2441no—
661C2×C4—2442no—
662C2×C4—2441no—
663C2×C4—2442no—
664C2×C4—2442no—
665C2×C4—2441no—
666C2×C4—2441no—
667C2×C4—2441no—
668C2×C4#1451973914yes—
669C22—1228no—
670C22—1228no—
671C22—1228no—
672C22—2444no—
673C22—2444no—
674C22—2444no—
675C22—2444no—
676C22—2444no—
677C22—2444no—
678C22—2444no—
679C22—2444no—
680C22—2444no—
681C22—2444no—
682C22—2444no—
683C22—2444no—
684C22C4×D4×D6112yes—
685C22C2×#1451973911yescharacteristic
686C22#1684023411yescharacteristic
687C22#1684024312yes—
688C22#1684024411yescharacteristic
689C22#120×D6111yescharacteristic
690C22#1684027011yescharacteristic
691C22#1684027512yes—
692C22#1684032012yes—
693C22#1684032212yes—
694C22#1684032511yescharacteristic
695C22#1684032811yescharacteristic
696C22#1684033712yes—
697C22#1684033711yescharacteristic
698C22#1684034211yescharacteristic
699C22#1684034212yes—
700C22#1684034812yes—
701C22#1684035812yes—
702C22#1684035812yes—
703C22#1684036012yes—
704C22#1684036811yescharacteristic
705C22#1684038611yescharacteristic
706C22#1684038612yes—
707C4—24no—
708C4—44no—
709C4—44no—
710C4—81no—
711C4—81no—
712C4—1224no—
713C4—4882no—
714C4—2442no—
715C4—2442no—
716C4—2442no—
717C2—2448no—
718C2#2144832011yescharacteristic
719C2#2144848312yes—
720C2#2144851011yescharacteristic
721C2#2144851612yes—
722C2#2144852711yescharacteristic
723C2#2145153011yescharacteristic
724C2#2145155111yescharacteristic
725C2#2145166912yes—
726C2#2145168911yescharacteristic
727C2#2145171711yescharacteristic
728C2#2145174512yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1444C2
socle, cosocleC24×C61C24
prime radicalC23×D244C22
hypercenter#1444D61
hypocenterC61#14300
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
3. C2×#16839644C2semidirect
4. C2×#16839633C2semidirect
5. #21444069C2semidirect
6. #21444067C2semidirect
7. #21444061C2semidirect
9. #21444044C2semidirect
10. #21444040C2semidirect
11. C61×#1444C2semidirect
24. #16839680C4semidirect
63. C2×#11758196C2×C4semidirect
77. C2×#11758198C2×C4semidirect
78. C2×#11758198C2×C4semidirect
101. #95×C61C2×C4semidirect
357. #1444D61semidirect
404. C61#14300semidirect
466. #95C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197392
C2×#168396802
C2×#168396442
C2×#168396332
#214440692
#214440672
#214440612
#214440592
#214440442
#214440402
C61×#14442
#1430061