Finite Group Databasebeta

← #29602913

Subgroups — #29602913

Order 15616.

2358 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D244C211yesmaximal, characteristic
2C2×#16839664C211yesmaximal, characteristic
3C2×#16839664C211yesmaximal, characteristic
4C2×#16839660C211yesmaximal, characteristic, complemented
5C2×#16839628C211yesmaximal, characteristic, complemented
6#21444063C214yesmaximal, complemented
7#21444049C211yesmaximal, characteristic, complemented
8#21444040C214yesmaximal, complemented
9C4×#95×C61C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C23×C4×D61C2212yes—
12C2×C4×D244—24no—
13C2×C4×D244—24no—
14C2×C4×D244—24no—
15C22×C4⋊Dic61C2211yescharacteristic
16C22×C4⋊Dic61C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic
18C22×#11758196C2212yes—
19C22×#11758198C2212yes—
20C22×#11758198C2214yes—
21C22×#11758198C2214yes—
22#16839664—24no—
23#16839664C418yescomplemented
24#16839660—24no—
25#16839628—24no—
26C2×#14519684C2214yes—
27C2×#14519684C2212yes—
28C2×#14519684C2214yes—
29C22×C42×C61C2211yescharacteristic
30C22×C42×C61C2211yescharacteristic
31C22×C42×C61C2211yescharacteristic
32C2×#95×C61C2212yes—
33C2×#95×C61C2212yes—
34C24×D61C2312yes—
35C22×D244—22no—
36C22×D244—22no—
37C22×D244—22no—
38C22×D244—22no—
39C22×D244C2×C414yes—
40C22×D244D414yes—
41C22×D244Q814yes—
42C23×Dic61C2312yes—
43C23×Dic61C2312yes—
44C23×Dic61C2312yes—
45C22×C4×D61—24no—
46C22×C4×D61—24no—
47C22×C4×D61—24no—
48C22×C4×D61—28no—
49C22×C4×D61—44no—
50C4×D244—232no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61C2×C414yescomplemented
59C2×C4⋊Dic61D414yes—
60C2×#11758196—24no—
61C2×#11758196—44no—
62C2×#11758198—24no—
63C2×#11758198—24no—
64C2×#11758198—24no—
65C2×#11758198—28no—
66C2×#11758198—28no—
67C2×#11758198—44no—
68C2×#11758198—44no—
69C2×#11758198—44no—
70C2×#11758198C2×C4116yescomplemented
71#14519684—24no—
72#14519684—28no—
73#14519684—44no—
74C23×C4×C61C2312yes—
75C23×C4×C61C2311yescharacteristic
76C23×C4×C61C2312yes—
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2311yescharacteristic
79C2×C42×C61—22no—
80C2×C42×C61—22no—
81C2×C42×C61—22no—
82C2×C42×C61—22no—
83C2×C42×C61—22no—
84C2×C42×C61—22no—
85C2×C42×C61—22no—
86C2×C42×C61C2×C414yescomplemented
87C2×C42×C61D414yes—
88#95×C61—24no—
89#95×C61—24no—
90C24×C61C2411yescharacteristic
91C23×D61—22no—
92C23×D61—22no—
93C23×D61—24no—
94C23×D61—22no—
95C23×D61—22no—
96C23×D61—24no—
97C23×D61—24no—
98C23×D61—24no—
99C23×D61C22×C414yes—
100C23×D61C2×D414yes—
101C23×D61C2×Q814yes—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
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×C61C22×C414yes—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×Q812yes—
129C22×C4×C61#4114yes—
130C22×C4×C61#4114yes—
131C22×C4×C61#4112yes—
132C22×C4×C61#4112yes—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—24no—
137C22×Dic61—22no—
138C22×Dic61—24no—
139C22×Dic61—24no—
140C22×Dic61—24no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—24no—
145C22×Dic61—24no—
146C22×Dic61C22×C414yes—
147C22×Dic61C2×D414yes—
148C2×D244—28no—
149C2×D244—28no—
150C2×D244—28no—
151C2×D244—28no—
152C2×D244—28no—
153C2×D244—28no—
154C2×D244C4⋊C4116yescomplemented
155C2×C4×D61—216no—
156C2×C4×D61—416no—
157C2×C4×D61—416no—
158C2×C4×D61—48no—
159C2×C4×D61—48no—
160C42×C61—28no—
161C42×C61—28no—
162C42×C61—44no—
163#11758198—216no—
164#11758198—416no—
165#11758198—416no—
166C4⋊Dic61—28no—
167C4⋊Dic61—28no—
168C4⋊Dic61—44no—
169#11758196—48no—
170C23×C61C23×C411yescharacteristic
171C23×C61C22×D411yescharacteristic
172C23×C61C22×D411yescharacteristic
173C23×C61C22×D411yescharacteristic
174C23×C61C22×Q811yescharacteristic
175C23×C61C2×#4111yescharacteristic
176C23×C61C2×#4111yescharacteristic
177C23×C61C2×#4112yes—
178C23×C61C2×#4112yes—
179C23×C61H(5,𝔽2)12yes—
180C23×C61H(5,𝔽2)12yes—
181C22×D61—24no—
182C22×D61—24no—
183C22×D61—28no—
184C22×D61—28no—
185C22×D61—24no—
186C22×D61—24no—
187C22×D61—24no—
188C22×D61—24no—
189C22×D61—28no—
190C22×D61—28no—
191C22×D61—28no—
192C22×D61—28no—
193C22×D61—42no—
194C22×D61—44no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—44no—
199C22×D61—44no—
200C22×D61—44no—
201C22×D61—44no—
202C22×D61—44no—
203C22×D61C2×C4⋊C418yes—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61C2×C4⋊C414yes—
228C2×C4×C61C4×D414yes—
229C2×C4×C61C4×D414yes—
230C2×C4×C61#12014yes—
231C2×C4×C61C22⋊Q814yes—
232C2×C4×C61C22⋊Q814yes—
233C2×C4×C61#12314yes—
234C2×C4×C61#12414yes—
235C2×Dic61—24no—
236C2×Dic61—28no—
237C2×Dic61—24no—
238C2×Dic61—28no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—44no—
244C2×Dic61—44no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—44no—
249C2×Dic61—44no—
250D244—232no—
251C4×D61—432no—
252#13979—611nomaximal, Sylow
253C22×C61C22×C4⋊C411yescharacteristic
254C22×C61C2×C4×D411yescharacteristic
255C22×C61C2×C4×D411yescharacteristic
256C22×C61C4×#4112yes—
257C22×C61#51812yes—
258C22×C61C2×#12011yescharacteristic
259C22×C61C2×C22⋊Q811yescharacteristic
260C22×C61C2×C22⋊Q811yescharacteristic
261C22×C61C2×#12311yescharacteristic
262C22×C61C2×#12411yescharacteristic
263C22×C61#52912yes—
264C22×C61#53211yescharacteristic
265C22×C61#53511yescharacteristic
266C22×C61#53511yescharacteristic
267C22×C61#54012yes—
268C22×C61#54612yes—
269C22×C61#54612yes—
270C22×C61#54612yes—
271C22×C61#55012yes—
272C22×C61#55312yes—
273C22×C61#55412yes—
274C22×C61#55412yes—
275C22×C61#55512yes—
276C2×D61—216no—
277C2×D61—28no—
278C2×D61—44no—
279C2×D61—44no—
280C2×D61—44no—
281C2×D61—48no—
282C2×D61—48no—
283C2×D61—44no—
284C2×D61—44no—
285C2×D61—48no—
286C2×D61—44no—
287C4×C61—24no—
288C4×C61—44no—
289C4×C61—44no—
290C4×C61—42no—
291C4×C61#38518yes—
292Dic61—44no—
293Dic61—44no—
294Dic61—82no—
295C22×C4×D4—611no—
296C2×#390—611no—
297C2×#390—611no—
298C2×#389—611no—
299C2×#382—611no—
300#1439—614no—
301C4⋊#95—611no—
302C22⋊#95—614no—
303C4×#95D6111yescharacteristic, complemented
304C2×C61C2×#38511yescharacteristic
305C2×C61C4×#12411yescharacteristic
306C2×C61#232811yescharacteristic
307C2×C61#234912yes—
308C2×C61#236312yes—
309C2×C61#242012yes—
310C2×C61#243111yescharacteristic
311C2×C61#243511yescharacteristic
312C2×C61#245412yes—
313C2×C61#256311yescharacteristic
314C2×C61#256611yescharacteristic
315D61—48no—
316C23×D4—611no—
317C24×C4—612no—
318C2×C4×D4—1224no—
319C2×C4×D4—1224no—
320C2×C4×D4—1224no—
321C22×C4⋊C4—611no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4—611no—
324C22×C4⋊C4—612no—
325C22×C22⋊C4—614no—
326C22×C22⋊C4—612no—
327C22×C22⋊C4—614no—
328C22×C42C2×D6111yescharacteristic
329C22×C42C2×D6111yescharacteristic
330C22×C42C2×D6111yescharacteristic
331#390—618no—
332#390—1224no—
333#389—1224no—
334#382—1224no—
335C2×#95—612no—
336C2×#95—614no—
337C2×#95—614no—
338C2×#95C2×D6112yes—
339C2×#95C2×D6112yes—
340C61#1397911yescharacteristic, complemented, Sylow
341C25—612no—
342C22×D4—614no—
343C22×D4—614no—
344C22×D4—614no—
345C22×D4—1222no—
346C22×D4—1222no—
347C22×D4—1222no—
348C22×D4—1222no—
349C23×C4—612no—
350C23×C4—612no—
351C23×C4—612no—
352C23×C4—1224no—
353C23×C4—1224no—
354C23×C4—1224no—
355C23×C4—1228no—
356C23×C4—2444no—
357C23×C4C22×D6112yes—
358C23×C4C22×D6111yescharacteristic
359C23×C4C22×D6112yes—
360C23×C4C22×D6111yescharacteristic
361C23×C4C22×D6111yescharacteristic
362C4×D4—12232no—
363C2×C4⋊C4—614no—
364C2×C4⋊C4—614no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1224no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—2444no—
374C2×C22⋊C4—6116no—
375C2×C22⋊C4—1224no—
376C2×C22⋊C4—1224no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—1228no—
379C2×C22⋊C4—1228no—
380C2×C22⋊C4—2444no—
381C2×C22⋊C4—2444no—
382C2×C22⋊C4—2444no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42C4×D6114yescomplemented
391C2×C42D24414yes—
392#95—24no—
393#95—24no—
394#95—1228no—
395#95—1224no—
396#95—2444no—
397C24—614no—
398C24—614no—
399C24—614no—
400C24—1222no—
401C24—1222no—
402C24—1222no—
403C24—1224no—
404C24—1224no—
405C24—1222no—
406C24—1224no—
407C24—1224no—
408C24C23×D6111yescharacteristic
409C2×D4—6116no—
410C2×D4—1228no—
411C2×D4—1228no—
412C2×D4—1228no—
413C2×D4—1228no—
414C2×D4—1228no—
415C2×D4—1228no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—614no—
436C22×C4—614no—
437C22×C4—12216no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1224no—
450C22×C4—1224no—
451C22×C4—24416no—
452C22×C4—24416no—
453C22×C4—2448no—
454C22×C4—2448no—
455C22×C4C2×C4×D6114yes—
456C22×C4C2×C4×D6112yes—
457C22×C4C2×D24412yes—
458C22×C4C2×D24412yes—
459C22×C4C2×D24412yes—
460C22×C4#1175822214yes—
461C22×C4#1175822214yes—
462C22×C4#1175822212yes—
463C22×C4D4×D6112yes—
464C22×C4D4×D6112yes—
465C22×C4#1175822412yes—
466C22×C4Q8×D6112yes—
467C4⋊C4—1228no—
468C4⋊C4—1228no—
469C4⋊C4—2448no—
470C4⋊C4—2444no—
471C22⋊C4—12216no—
472C22⋊C4—24416no—
473C22⋊C4—24416no—
474C42—28no—
475C42—28no—
476C42—44no—
477C23—618no—
478C23—1228no—
479C23—1228no—
480C23—1224no—
481C23—1224no—
482C23—1228no—
483C23—1228no—
484C23—1224no—
485C23—1224no—
486C23—1224no—
487C23—1224no—
488C23—1228no—
489C23—1228no—
490C23—2444no—
491C23—2444no—
492C23—2442no—
493C23—2444no—
494C23—2442no—
495C23—2442no—
496C23—2442no—
497C23—2444no—
498C23—2444no—
499C23—2444no—
500C23C22×C4×D6111yescharacteristic
501C23C2×D4×D6111yescharacteristic
502C23C2×D4×D6111yescharacteristic
503C23C22×D24411yescharacteristic
504C23C2×Q8×D6111yescharacteristic
505C23C2×#1175822411yescharacteristic
506C23C2×#1175822212yes—
507C23C2×#1175822212yes—
508C23C2×#1175822211yescharacteristic
509C23#1451982312yes—
510C23#1451982312yes—
511D4—12232no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—1224no—
536C2×C4—1228no—
537C2×C4—1224no—
538C2×C4—1228no—
539C2×C4—24432no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2444no—
544C2×C4—2444no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2444no—
550C2×C4—2444no—
551C2×C4C4⋊C4×D6114yes—
552C2×C4#1451973714yes—
553C2×C4C4×D24414yes—
554C2×C4#1451974714yes—
555C2×C4#1451975514yes—
556C2×C4#1451975714yes—
557C2×C4#1451975914yes—
558C2×C4#1451976514yes—
559C22—12216no—
560C22—1228no—
561C22—2444no—
562C22—2448no—
563C22—2444no—
564C22—2448no—
565C22—2444no—
566C22—2448no—
567C22—2444no—
568C22—2444no—
569C22—2444no—
570C22C2×C4⋊C4×D6111yescharacteristic
571C22C2×#1451973711yescharacteristic
572C22C2×C4×D24411yescharacteristic
573C22C4×#1175822212yes—
574C22#1684023612yes—
575C22C2×#1451974711yescharacteristic
576C22C2×#1451975511yescharacteristic
577C22C2×#1451975711yescharacteristic
578C22C2×#1451975911yescharacteristic
579C22C2×#1451976511yescharacteristic
580C22#1684029612yes—
581C22#1684031211yescharacteristic
582C22#1684033111yescharacteristic
583C22#1684033111yescharacteristic
584C22#1684036712yes—
585C22#1684040812yes—
586C22#1684040812yes—
587C22#1684040812yes—
588C22#1684043312yes—
589C22#1684045812yes—
590C22#1684047212yes—
591C22#1684047212yes—
592C22#1684047712yes—
593C4—24no—
594C4—44no—
595C4—44no—
596C4—42no—
597C4—4882no—
598C4—2444no—
599C4—2444no—
600C4#1683963618yes—
601C2—2448no—
602C2C2×#1683963611yescharacteristic
603C2C4×#1451976511yescharacteristic
604C2#2144854411yescharacteristic
605C2#2144873012yes—
606C2#2144885512yes—
607C2#2144931112yes—
608C2#2144943811yescharacteristic
609C2#2144947011yescharacteristic
610C2#2144969112yes—
611C2#2145098811yescharacteristic
612C2#2145101911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC4×#95×C61C2
socle, cosocleC24×C61C24
prime radicalC23×D244C22
hypercenterC4×#95D61
hypocenterC61#13979
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
4. C2×#16839660C2semidirect
5. C2×#16839628C2semidirect
6. #21444063C2semidirect
7. #21444049C2semidirect
8. #21444040C2semidirect
9. C4×#95×C61C2semidirect
23. #16839664C4semidirect
58. C2×C4⋊Dic61C2×C4semidirect
70. C2×#11758198C2×C4semidirect
86. C2×C42×C61C2×C4semidirect
154. C2×D244C4⋊C4semidirect
303. C4×#95D61semidirect
340. C61#13979semidirect
390. C2×C42C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D2442
C2×#168396642
C2×#168396602
C2×#168396282
#214440632
#214440492
#214440402
C4×#95×C612
#1397961