Finite Group Databasebeta

← #29611425

Subgroups — #29611425

Order 15616.

2982 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519751C211yesmaximal, characteristic
2C2×#16839679C212yesmaximal, complemented
3C2×#16839641C212yesmaximal, complemented
4C2×#16839641C212yesmaximal, complemented
5#21444072C212yesmaximal, complemented
6#21444061C214yesmaximal, complemented
7#21444042C211yesmaximal, characteristic, complemented
8C23×C4⋊C4×C61C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C23×C22⋊D61C2212yes—
11C2×#14519751—24no—
12C2×#14519751—28no—
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C22⋊Dic61C2212yes—
15C22×#11758198C2214yes—
16C22×#11758198C2212yes—
17C22×#11758198C2214yes—
18#16839679—28no—
19#16839641—28no—
20#16839641—28no—
21C2×#14519684C2214yes—
22C2×#14519684C2214yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C24×C4×C61C2211yescharacteristic
26C24×C4×C61C2211yescharacteristic
27C24×C4×C61C2211yescharacteristic
28C22×C4⋊C4×C61—24no—
29C22×C4⋊C4×C61—28no—
30C22×C4⋊C4×C61C2212yes—
31C22×C4⋊C4×C61C2212yes—
32C24×D61C2312yes—
33C22×D244—24no—
34C22×D244—22no—
35C22×D244—24no—
36C22×D244—42no—
37C22×C22⋊D61—24no—
38C22×C22⋊D61—24no—
39C22×C22⋊D61—24no—
40C22×C22⋊D61—24no—
41C22×C22⋊D61—24no—
42C22×C22⋊D61—24no—
43C22×C22⋊D61D418yes—
44C23×Dic61C2314yes—
45C23×Dic61C2312yes—
46#14519751—232no—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—24no—
49C2×C22⋊Dic61—24no—
50C2×C22⋊Dic61—24no—
51C2×C22⋊Dic61—24no—
52C2×#11758198—24no—
53C2×#11758198—24no—
54C2×#11758198—24no—
55C2×#11758198—28no—
56C2×#11758198—28no—
57C2×#11758198—28no—
58C2×#11758198—28no—
59C2×#11758198—28no—
60C2×#11758198—44no—
61#14519684—28no—
62#14519684—24no—
63#14519684—28no—
64#14519684—42no—
65C25×C61C2311yescharacteristic
66C23×C4×C61—22no—
67C23×C4×C61—22no—
68C23×C4×C61—24no—
69C23×C4×C61—24no—
70C23×C4×C61—24no—
71C23×C4×C61—22no—
72C23×C4×C61—24no—
73C23×C4×C61—24no—
74C23×C4×C61—24no—
75C23×C4×C61—42no—
76C23×C4×C61—42no—
77C23×C4×C61D414yes—
78C23×C4×C61D414yescomplemented
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2312yes—
83C2×C4⋊C4×C61—232no—
84C2×C4⋊C4×C61—24no—
85C2×C4⋊C4×C61—24no—
86C2×C4⋊C4×C61—24no—
87C2×C4⋊C4×C61—24no—
88C2×C4⋊C4×C61—24no—
89C2×C4⋊C4×C61D418yescomplemented
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—22no—
94C24×C61C2×D414yes—
95C24×C61C2×D412yes—
96C24×C61C2×D412yes—
97C24×C61#4112yes—
98C24×C61#4114yes—
99C24×C61C2411yescharacteristic
100C23×D61—22no—
101C23×D61—22no—
102C23×D61—22no—
103C23×D61—24no—
104C23×D61—22no—
105C23×D61—22no—
106C23×D61—22no—
107C23×D61—24no—
108C23×D61—24no—
109C23×D61—24no—
110C23×D61C2×D414yes—
111C22×C4×C61—28no—
112C22×C4×C61—216no—
113C22×C4×C61—28no—
114C22×C4×C61—28no—
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—216no—
122C22×C4×C61—28no—
123C22×C4×C61—28no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—24no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—24no—
131C22×C4×C61—28no—
132C22×C4×C61—44no—
133C22×C4×C61—416no—
134C22×C4×C61—416no—
135C22×C4×C61—44no—
136C22×C4×C61—44no—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D414yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D414yes—
141C22×C4×C61C2×D414yes—
142C22×C4×C61C2×D414yes—
143C22×C4×C61#4114yes—
144C2×C22⋊D61—216no—
145C2×C22⋊D61—216no—
146C2×C22⋊D61—216no—
147C2×C22⋊D61—48no—
148C2×C22⋊D61—48no—
149C2×C22⋊D61—48no—
150C2×C22⋊D61—48no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—24no—
158C22×Dic61—24no—
159C22×Dic61—24no—
160C22×Dic61—24no—
161C22×Dic61—24no—
162C22×Dic61—24no—
163C22×Dic61—24no—
164C22×Dic61C2×D414yes—
165C2×D244—28no—
166C2×D244—416no—
167C2×D244—44no—
168C2×D244—44no—
169C4⋊C4×C61—216no—
170C4⋊C4×C61—216no—
171C22⋊Dic61—48no—
172#11758198—216no—
173#11758198—416no—
174#11758198—416no—
175C4⋊Dic61—28no—
176C23×C61—22no—
177C23×C61—24no—
178C23×C61—22no—
179C23×C61—24no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—24no—
183C23×C61—24no—
184C23×C61—28no—
185C23×C61—28no—
186C23×C61—28no—
187C23×C61—28no—
188C23×C61#12014yes—
189C23×C61#12114yes—
190C23×C61#12118yes—
191C23×C61#12318yes—
192C23×C61#12414yes—
193C23×C61C22×D411yescharacteristic
194C23×C61C22×D412yes—
195C23×C61C22×D412yes—
196C23×C61C22×D411yescharacteristic
197C23×C61C2×#4111yescharacteristic
198C23×C61C2×#4112yes—
199C23×C61C2×#4112yes—
200C23×C61H(5,𝔽2)12yes—
201C23×C612−1+412yes—
202C22×D61—24no—
203C22×D61—24no—
204C22×D61—24no—
205C22×D61—28no—
206C22×D61—28no—
207C22×D61—42no—
208C22×D61—44no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—44no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—44no—
218C22×D61—44no—
219C22×D61—44no—
220C22×D61—44no—
221C22×D61—44no—
222C22×D61—44no—
223C22×D61—44no—
224C22×D61—44no—
225C2×C4×C61—232no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—22no—
234C2×C4×C61—28no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—22no—
239C2×C4×C61—28no—
240C2×C4×C61—416no—
241C2×C4×C61—416no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—44no—
245C2×C4×C61—44no—
246C2×C4×C61#12014yes—
247C2×C4×C61#12118yescomplemented
248C22⋊D61—432no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—44no—
257C2×Dic61—44no—
258C2×Dic61—44no—
259C2×Dic61—44no—
260C2×Dic61—44no—
261C2×Dic61—44no—
262C2×Dic61—44no—
263D244—416no—
264#14719—611nomaximal, Sylow
265C22×C61—24no—
266C22×C61—24no—
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—24no—
270C22×C61—216no—
271C22×C61—24no—
272C22×C61—216no—
273C22×C61#39418yescomplemented
274C22×C61C2×#12011yescharacteristic
275C22×C61C2×#12011yescharacteristic
276C22×C61C2×#12111yescharacteristic
277C22×C61C2×#12112yes—
278C22×C61C2×#12112yes—
279C22×C61C2×#12312yes—
280C22×C61#52511yescharacteristic
281C22×C61#52512yes—
282C22×C61C2×#12411yescharacteristic
283C22×C61#53411yescharacteristic
284C22×C61#53611yescharacteristic
285C22×C61#53712yes—
286C22×C61#53912yes—
287C22×C61D4214yes—
288C22×C61#54714yes—
289C22×C61#54814yes—
290C22×C61#55314yes—
291C2×D61—44no—
292C2×D61—44no—
293C2×D61—44no—
294C2×D61—48no—
295C2×D61—48no—
296C2×D61—82no—
297C2×D61—82no—
298C2×D61—82no—
299C2×D61—84no—
300C2×D61—82no—
301C2×D61—84no—
302C4×C61—28no—
303C4×C61—24no—
304C4×C61—42no—
305C4×C61—44no—
306Dic61—44no—
307Dic61—84no—
308C22×#121—611no—
309C23×C4⋊C4D6111yescharacteristic, complemented
310C2×#394—612no—
311C2×#386—612no—
312C2×#386—612no—
313#1441—612no—
314#1439—614no—
315C22⋊#95—611no—
316C2×C61—28no—
317C2×C61C2×#39411yescharacteristic
318C2×C61#240311yescharacteristic
319C2×C61#240611yescharacteristic
320C2×C61#241614yes—
321C2×C61#243611yescharacteristic
322C2×C61#244314yes—
323C2×C61#259111yescharacteristic
324C2×C61#261811yescharacteristic
325C2×C61#284711yescharacteristic
326D61—84no—
327C23×D4—612no—
328C23×D4—611no—
329C24×C4C2×D6111yescharacteristic
330C24×C4C2×D6111yescharacteristic
331C24×C4C2×D6111yescharacteristic
332C2×#121—1228no—
333C2×#121—1224no—
334C22×C4⋊C4—24no—
335C22×C4⋊C4—28no—
336C22×C4⋊C4—611no—
337C22×C4⋊C4C2×D6112yes—
338C22×C4⋊C4C2×D6112yes—
339C22×C22⋊C4—612no—
340C22×C22⋊C4—614no—
341C22×C22⋊C4—612no—
342C22×C22⋊C4—614no—
343#394—1228no—
344#386—1228no—
345#386—1228no—
346C2×#95—612no—
347C2×#95—614no—
348C2×#95—614no—
349C2×#95—612no—
350C61#1471911yescharacteristic, complemented, Sylow
351C25—612no—
352C25C22×D6111yescharacteristic
353C22×D4—618no—
354C22×D4—1224no—
355C22×D4—1224no—
356C22×D4—1224no—
357C22×D4—1224no—
358C22×D4—1224no—
359C22×D4—1224no—
360C22×D4—1224no—
361C22×D4—1222no—
362C22×D4—1224no—
363C22×D4—2442no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—24no—
367C23×C4—24no—
368C23×C4—24no—
369C23×C4—22no—
370C23×C4—24no—
371C23×C4—24no—
372C23×C4—24no—
373C23×C4—42no—
374C23×C4—42no—
375C23×C4—612no—
376C23×C4—614no—
377C23×C4D24414yescomplemented
378C23×C4C22⋊D6114yes—
379C23×C4C22×D6112yes—
380C23×C4C22×D6111yescharacteristic
381C23×C4C22×D6112yes—
382C23×C4C22×D6111yescharacteristic
383#121—12232no—
384C2×C4⋊C4—232no—
385C2×C4⋊C4—24no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—1224no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4C22⋊D6118yescomplemented
393C2×C22⋊C4—1224no—
394C2×C22⋊C4—1224no—
395C2×C22⋊C4—1224no—
396C2×C22⋊C4—1228no—
397C2×C22⋊C4—1224no—
398C2×C22⋊C4—1224no—
399C2×C22⋊C4—1224no—
400C2×C22⋊C4—1228no—
401C2×C22⋊C4—1228no—
402C2×C22⋊C4—1228no—
403C2×C22⋊C4—1228no—
404C2×C22⋊C4—2444no—
405#95—1228no—
406#95—1228no—
407#95—1224no—
408#95—2442no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—614no—
414C24—1222no—
415C24—1224no—
416C24—1222no—
417C24—1222no—
418C24—1224no—
419C24—1222no—
420C24—1222no—
421C24—1222no—
422C24—1224no—
423C24—1224no—
424C24C2×D24412yes—
425C24#1175822212yes—
426C24D4×D6114yes—
427C24#1175822614yes—
428C24C2×C22⋊D6112yes—
429C24C23×D6111yescharacteristic
430C2×D4—1228no—
431C2×D4—12216no—
432C2×D4—12216no—
433C2×D4—12216no—
434C2×D4—24416no—
435C2×D4—2444no—
436C2×D4—2448no—
437C2×D4—2448no—
438C2×D4—2448no—
439C2×D4—2448no—
440C2×D4—2444no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—216no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—28no—
451C22×C4—216no—
452C22×C4—28no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—44no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—614no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1224no—
475C22×C4—1224no—
476C22×C4—1224no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4C2×D24414yes—
482C22×C4#1175822214yes—
483C22×C4C2×C22⋊D6112yes—
484C22×C4C2×C22⋊D6114yes—
485C22×C4C2×C22⋊D6114yes—
486C22×C4C2×C22⋊D6112yes—
487C22×C4C2×C22⋊D6114yes—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—1228no—
491C22⋊C4—12216no—
492C22⋊C4—24416no—
493C22⋊C4—2448no—
494C22⋊C4—24416no—
495C23—28no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—1224no—
508C23—1224no—
509C23—1224no—
510C23—1228no—
511C23—1228no—
512C23—2444no—
513C23—2444no—
514C23—2442no—
515C23—2444no—
516C23—2442no—
517C23—2442no—
518C23—2442no—
519C23—2442no—
520C23—2444no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23—2444no—
525C23—2444no—
526C23—2444no—
527C23—2444no—
528C23—2444no—
529C23—2444no—
530C23#1451974514yes—
531C23#1451975114yes—
532C23#1451975218yes—
533C23#1451976218yes—
534C23#1451976614yes—
535C23C2×D4×D6112yes—
536C23C22×C22⋊D6111yescharacteristic
537C23C22×C22⋊D6112yes—
538C23C22×D24411yescharacteristic
539C23C2×#1175822212yes—
540C23C2×#1175822211yescharacteristic
541C23C2×#1175822612yes—
542C23#1451982212yes—
543C23#1451982512yes—
544D4—24416no—
545D4—24432no—
546C2×C4—232no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—1224no—
570C2×C4—2444no—
571C2×C4—2444no—
572C2×C4—2444no—
573C2×C4—2444no—
574C2×C4—2444no—
575C2×C4—2444no—
576C2×C4—2444no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4#1451974614yes—
582C2×C4#1451975118yescomplemented
583C22—24no—
584C22—24no—
585C22—24no—
586C22—216no—
587C22—24no—
588C22—216no—
589C22—24no—
590C22—24no—
591C22—4882no—
592C22—4882no—
593C22—4882no—
594C22—4884no—
595C22—4882no—
596C22—4884no—
597C22—2448no—
598C22—2444no—
599C22—2448no—
600C22—2444no—
601C22—2444no—
602C22#1683967918yescomplemented
603C22C2×#1451974511yescharacteristic
604C22C2×#1451974611yescharacteristic
605C22C2×#1451975112yes—
606C22C2×#1451975111yescharacteristic
607C22C2×#1451975212yes—
608C22C2×#1451976212yes—
609C22#1684027311yescharacteristic
610C22#1684027312yes—
611C22C2×#1451976611yescharacteristic
612C22#1684032111yescharacteristic
613C22#1684033611yescharacteristic
614C22#1684034312yes—
615C22#1684035912yes—
616C22D4×C22⋊D6114yes—
617C22#1684041314yes—
618C22#1684042314yes—
619C22#1684046014yes—
620C4—28no—
621C4—24no—
622C4—42no—
623C4—44no—
624C4—4884no—
625C4—2444no—
626C2—28no—
627C2—4884no—
628C2C2×#1683967911yescharacteristic
629C2#2144912911yescharacteristic
630C2#2144915411yescharacteristic
631C2#2144925314yes—
632C2#2144947311yescharacteristic
633C2#2144955414yes—
634C2#2145129911yescharacteristic
635C2#2145156111yescharacteristic
636C2#2145410111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutator, socle, cosocleC24×C61C24
FittingC23×C4⋊C4×C61C2
prime radicalC22×#14519751C2
hypercenterC23×C4⋊C4D61
hypocenterC61#14719
LCSG ▷ C24×C61 ▷ C61
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16839679C2semidirect
3. C2×#16839641C2semidirect
4. C2×#16839641C2semidirect
5. #21444072C2semidirect
6. #21444061C2semidirect
7. #21444042C2semidirect
8. C23×C4⋊C4×C61C2semidirect
78. C23×C4×C61D4semidirect
89. C2×C4⋊C4×C61D4semidirect
247. C2×C4×C61#121semidirect
273. C22×C61#394semidirect
309. C23×C4⋊C4D61semidirect
350. C61#14719semidirect
377. C23×C4D244semidirect
392. C2×C4⋊C4C22⋊D61semidirect
582. C2×C4#14519751semidirect
602. C22#16839679semidirect

Maximal subgroups

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

subgroupindex
C22×#145197512
C2×#168396792
C2×#168396412
#214440722
#214440612
#214440422
C23×C4⋊C4×C612
#1471961