Finite Group Databasebeta

← #29852893

Subgroups — #29852893

Order 15616.

2522 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840387C212yesmaximal
2#21454169C212yesmaximal, complemented
3#21454167C212yesmaximal, complemented
4#21451936C212yesmaximal, complemented
5#21451933C211yesmaximal, characteristic, complemented
6#21450826C214yesmaximal, complemented
7#21450824C214yesmaximal, complemented
8#21449516C214yesmaximal, complemented
9#21449514C214yesmaximal, complemented
10#21448432C212yesmaximal
11#380×D61C212yesmaximal
12C2×#16839609C211yesmaximal, characteristic
13C61×#2659C211yesmaximal, characteristic, complemented
14C23×C4×D61C2211yescharacteristic
15#16840387—216no—
16C2×#14519776C2212yescomplemented
17C2×#14519775C2212yescomplemented
18C2×#14519757C2214yes—
19C2×#14519756C2212yes—
20C2×#14519755C2214yes—
21C2×#14519755C2214yes—
22C2×#14519754C2214yes—
23C2×#14519753C2214yes—
24C2×C4⋊C4×D61C2212yes—
25C2×C4⋊C4×D61C2212yes—
26C22×#11758198C2211yescharacteristic
27C22×#11758198C2211yescharacteristic
28C2×C22⋊C4×D61C2214yes—
29C22×C4×Dic61C2211yescharacteristic
30C22×C4×Dic61C2211yescharacteristic
31#16839696C2214yescomplemented
32#16839695C2214yescomplemented
33#16839694C2214yescomplemented
34#16839683C2214yescomplemented
35#16839683C2214yescomplemented
36#16839682C2214yescomplemented
37#16839669C2212yescomplemented
38#16839667C2212yescomplemented
39#16839666C2214yescomplemented
40#16839661C2212yescomplemented
41#16839659C2214yescomplemented
42#16839658C2212yescomplemented
43#16839657C2214yescomplemented
44#16839657C2214yescomplemented
45#16839636C2214yes—
46#16839636C2214yes—
47#16839633C2214yes—
48#16839633C2214yes—
49#16839631C2212yescomplemented
50#16839631C2212yescomplemented
51#16839622C2212yes—
52#16839622C2212yes—
53#16839621C2214yes—
54#16839620C2214yes—
55#16839620C2214yes—
56#16839619C2212yes—
57#16839619C2212yes—
58#16839609C2214yes—
59#16839609C2214yes—
60C2×#14519684C2211yescharacteristic
61#95×D61C2214yes—
62C2×C22⋊Q8×C61C2212yes—
63C61×#398C2214yescomplemented
64C61×#395C2214yescomplemented
65C61×#380C2212yes—
66C61×#380C2212yes—
67C2×#95×C61C2211yescharacteristic
68C24×D61C2311yescharacteristic
69C22×Dic122C2314yes—
70C22×Dic122C2312yes—
71C23×Dic61C2311yescharacteristic
72C23×Dic61C2311yescharacteristic
73C23×Dic61C2311yescharacteristic
74C22×C4×D61—42no—
75C22×C4×D61—42no—
76C22×C4×D61Q818yescomplemented
77C22×C4×D61C2312yes—
78C22×C4×D61C2314yes—
79C22×C4×D61C2314yes—
80C22×C4×D61C2312yes—
81#14519776—44no—
82#14519775—44no—
83#14519757—216no—
84#14519756—28no—
85#14519755—216no—
86#14519755—216no—
87#14519754—216no—
88#14519753—216no—
89C2×C4⋊Dic61C2314yes—
90C2×C4⋊Dic61C2312yes—
91C2×C4⋊Dic61C2312yes—
92C2×C4⋊Dic61C2312yes—
93C2×C4⋊Dic61C2312yes—
94C2×C4⋊Dic61C2312yes—
95C2×#11758196C2314yes—
96C2×#11758196C2314yes—
97C2×#11758196C2314yes—
98C2×#11758196C2314yes—
99C2×#11758196C2314yes—
100C2×#11758196C2314yes—
101C2×#11758196C2314yes—
102C4⋊C4×D61—44no—
103C4⋊C4×D61—44no—
104C2×C22⋊Dic61C2314yes—
105C2×#11758198D418yes—
106C2×#11758198Q818yescomplemented
107C2×#11758198C2314yes—
108C2×#11758198C2314yes—
109C2×#11758198C2314yes—
110C2×#11758198C2314yes—
111C22⋊C4×D61—216no—
112C2×C4×Dic61—42no—
113C2×C4×Dic61—42no—
114C2×C4×Dic61C2312yes—
115C2×C4×Dic61C2312yes—
116C2×C4×Dic61C2312yes—
117C2×C4×Dic61C2312yes—
118#14519684C2314yes—
119#14519684C2314yes—
120#14519684C2314yes—
121#14519684C2314yes—
122#14519684C2314yes—
123#14519684C2314yes—
124#14519684C2314yes—
125C22×Q8×C61C2312yes—
126C23×C4×C61C2311yescharacteristic
127C23×C4×C61C2311yescharacteristic
128C23×C4×C61C2311yescharacteristic
129C22⋊Q8×C61—28no—
130C2×C4⋊C4×C61C2312yes—
131C2×C4⋊C4×C61C2314yes—
132C2×C4⋊C4×C61C2312yes—
133C2×C4⋊C4×C61C2312yes—
134C2×C4⋊C4×C61C2312yes—
135C2×C4⋊C4×C61C2312yes—
136C2×C22⋊C4×C61C2314yes—
137#95×C61C2314yes—
138#95×C61C2314yes—
139#95×C61C2314yes—
140C24×C61C2411yescharacteristic
141C23×D61—41no—
142C23×D61—41no—
143C23×D61—41no—
144C23×D61—41no—
145C23×D61C2×D414yes—
146C23×D61C2×Q814yes—
147C23×D61C2×Q814yes—
148C23×D61C2412yes—
149C2×Q8×C61—24no—
150C2×Q8×C61—42no—
151C2×Q8×C61—42no—
152C22×C4×C61—41no—
153C22×C4×C61—41no—
154C22×C4×C61—41no—
155C22×C4×C61—41no—
156C22×C4×C61—41no—
157C22×C4×C61—41no—
158C22×C4×C61C2×D414yes—
159C22×C4×C61C2×Q814yes—
160C22×C4×C61C2×Q814yes—
161C22×C4×C61C2414yes—
162C22×C4×C61C2414yes—
163C22×C4×C61C2412yes—
164C22×C4×C61C2412yes—
165C22×C4×C61C2412yes—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—41no—
169C22×Dic61—41no—
170C22×Dic61—41no—
171C22×Dic61—41no—
172C22×Dic61C2×D414yes—
173C22×Dic61C2×Q814yes—
174C22×Dic61C2×Q814yes—
175C22×Dic61C2414yes—
176C22×Dic61C2414yes—
177C22×Dic61C2412yes—
178C22×Dic61C2411yescharacteristic
179C22×Dic61C2412yes—
180C22×Dic61C2411yescharacteristic
181C2×C4×D61—24no—
182C2×C4×D61—24no—
183C2×C4×D61—28no—
184C2×C4×D61—28no—
185C2×C4×D61—416no—
186C2×C4×D61—42no—
187C2×C4×D61—42no—
188C2×C4×D61—44no—
189C2×C4×D61—44no—
190C2×C4×D61—44no—
191C2×C4×D61—44no—
192C2×C4×D61C2×Q818yes—
193C2×C4×D61C2×Q818yes—
194C2×Dic122—24no—
195C2×Dic122—28no—
196C2×Dic122—44no—
197C2×Dic122—42no—
198C2×Dic122—42no—
199C2×Dic122—44no—
200C4⋊C4×C61—24no—
201C4⋊C4×C61—28no—
202C4⋊C4×C61—42no—
203C4⋊C4×C61—42no—
204C4⋊C4×C61—42no—
205C4⋊C4×C61—42no—
206C22⋊C4×C61—28no—
207C22⋊Dic61—28no—
208#11758198—28no—
209#11758198—28no—
210#11758198C2×D4116yes—
211#11758198C2×Q8116yes—
212C4⋊Dic61—24no—
213C4⋊Dic61—28no—
214C4⋊Dic61—42no—
215C4⋊Dic61—42no—
216C4⋊Dic61—42no—
217C4⋊Dic61—42no—
218#11758196—28no—
219#11758196—28no—
220#11758196—28no—
221#11758196—28no—
222#11758196—28no—
223#11758196—44no—
224#11758196—44no—
225C4×Dic61—24no—
226C4×Dic61—24no—
227C4×Dic61—42no—
228C4×Dic61—42no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61C22×D412yes—
234C23×C61C22×Q812yes—
235C23×C61C22×Q812yes—
236C23×C61C2511yescharacteristic
237C22×D61—22no—
238C22×D61—22no—
239C22×D61—22no—
240C22×D61—22no—
241C22×D61—44no—
242C22×D61—44no—
243C22×D61—44no—
244C22×D61—44no—
245C22×D61—44no—
246C22×D61—44no—
247C22×D61C4⋊Q8116yescomplemented
248C22×D61C22×D414yes—
249C22×D61C22×Q814yes—
250C22×D61C22×Q814yes—
251Q8×C61—84no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—24no—
259C2×C4×C61—24no—
260C2×C4×C61—24no—
261C2×C4×C61—24no—
262C2×C4×C61—24no—
263C2×C4×C61—24no—
264C2×C4×C61—44no—
265C2×C4×C61—44no—
266C2×C4×C61—44no—
267C2×C4×C61C22×D414yes—
268C2×C4×C61C22×Q814yes—
269C2×C4×C61C22×Q814yes—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—28no—
277C2×Dic61—24no—
278C2×Dic61—24no—
279C2×Dic61—24no—
280C2×Dic61—24no—
281C2×Dic61—24no—
282C2×Dic61—24no—
283C2×Dic61—44no—
284C2×Dic61—44no—
285C2×Dic61C22×D414yes—
286C2×Dic61C22×Q814yes—
287C2×Dic61C22×Q812yes—
288C2×Dic61C22×Q812yes—
289C2×Dic61H(5,𝔽2)12yes—
290C2×Dic61H(5,𝔽2)12yes—
291C4×D61—216no—
292C4×D61—48no—
293C4×D61—88no—
294C4×D61—88no—
295Dic122—88no—
296Dic122—84no—
297#35002—611nomaximal, Sylow
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—22no—
304C22×C61C2×C4⋊Q814yes—
305C22×C61C23×D411yescharacteristic
306C22×C61C23×Q811yescharacteristic
307C22×C61C23×Q811yescharacteristic
308C22×C61C2×H(5,𝔽2)11yescharacteristic
309C22×C61C2×H(5,𝔽2)11yescharacteristic
310C22×C61#58511yescharacteristic
311C22×C61#58511yescharacteristic
312C2×D61—24no—
313C2×D61—24no—
314C2×D61—24no—
315C2×D61—24no—
316C2×D61—24no—
317C2×D61—24no—
318C2×D61—416no—
319C2×D61C2×C4⋊Q818yes—
320C4×C61—24no—
321C4×C61—24no—
322C4×C61—24no—
323C4×C61—44no—
324C4×C61—44no—
325Dic61—22no—
326Dic61—24no—
327Dic61—24no—
328Dic61—44no—
329Dic61—44no—
330Dic61#54314yescomplemented
331C2×#543—612no—
332#2857—612no—
333#2857—612no—
334#2659—612no—
335#2659—611no—
336#2659D6111yescharacteristic, complemented
337#2549—614no—
338#2549—614no—
339#2440—614no—
340#2440—614no—
341#2315—612no—
342C2×#380—612no—
343C2×C4×C22⋊C4—611no—
344C2×C61—24no—
345C2×C61C22×C4⋊Q811yescharacteristic
346C2×C61C2×#54311yescharacteristic
347C2×C61#345811yescharacteristic
348C2×C61#349611yescharacteristic
349C2×C61#349611yescharacteristic
350C2×C61#350111yescharacteristic
351C2×C61#350111yescharacteristic
352D61—28no—
353C24×C4—611no—
354#543—12216no—
355C2×C4⋊Q8—612no—
356C2×C4⋊Q8—612no—
357C2×C22⋊Q8—614no—
358C2×C22⋊Q8—612no—
359C2×C22⋊Q8—614no—
360C2×C22⋊Q8—614no—
361C2×C22⋊Q8—614no—
362C2×C22⋊Q8—614no—
363C2×C22⋊Q8C2×D6112yes—
364C22×C4⋊C4—612no—
365C22×C4⋊C4—612no—
366C22×C22⋊C4—611no—
367C22×C22⋊C4—611no—
368C22×C22⋊C4—614no—
369C22×C42—611no—
370C22×C42—611no—
371#398—614no—
372#398—614no—
373#398—614no—
374#398C2×D6114yescomplemented
375#395—614no—
376#395—614no—
377#395—614no—
378#395C2×D6114yescomplemented
379#391—612no—
380#391—612no—
381#391—614no—
382#389—614no—
383#389—614no—
384#389—612no—
385#389—612no—
386#389—614no—
387#385—614no—
388#385—614no—
389#385—614no—
390#385—614no—
391#384—612no—
392#384—612no—
393#380—612no—
394#380—612no—
395#380—612no—
396#380—612no—
397#380—614no—
398#380—614no—
399#380—614no—
400#380C2×D6112yes—
401#380C2×D6112yes—
402C4×C22⋊C4—614no—
403C4×C22⋊C4—614no—
404C2×#95—611no—
405C2×#95—614no—
406C2×#95C2×D6111yescharacteristic
407C61#3500211yescharacteristic, complemented, Sylow
408C25—611no—
409C22×Q8—614no—
410C22×Q8—612no—
411C22×Q8C22×D6112yes—
412C23×C4—611no—
413C23×C4—611no—
414C23×C4—611no—
415C23×C4—612no—
416C23×C4—618no—
417C23×C4—612no—
418C23×C4—614no—
419C23×C4—614no—
420C23×C4—2442no—
421C23×C4—2442no—
422C23×C4C22×D6111yescharacteristic
423C23×C4C22×D6111yescharacteristic
424C23×C4C22×D6111yescharacteristic
425C4⋊Q8—2444no—
426C4⋊Q8—2444no—
427C22⋊Q8—28no—
428C22⋊Q8—12216no—
429C22⋊Q8—12216no—
430C22⋊Q8—12216no—
431C22⋊Q8—12216no—
432C22⋊Q8—12216no—
433C22⋊Q8—1228no—
434C2×C4⋊C4—614no—
435C2×C4⋊C4—614no—
436C2×C4⋊C4—614no—
437C2×C4⋊C4—612no—
438C2×C4⋊C4—614no—
439C2×C4⋊C4—614no—
440C2×C4⋊C4—614no—
441C2×C4⋊C4—612no—
442C2×C4⋊C4—612no—
443C2×C4⋊C4—614no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—614no—
446C2×C4⋊C4—612no—
447C2×C4⋊C4—2444no—
448C2×C4⋊C4—2444no—
449C2×C4⋊C4C22×D6112yes—
450C2×C4⋊C4C22×D6114yes—
451C2×C4⋊C4C22×D6112yes—
452C2×C4⋊C4C22×D6112yes—
453C2×C4⋊C4C22×D6112yes—
454C2×C4⋊C4C22×D6112yes—
455C2×C22⋊C4—618no—
456C2×C22⋊C4—618no—
457C2×C22⋊C4—614no—
458C2×C22⋊C4—614no—
459C2×C22⋊C4—614no—
460C2×C22⋊C4—614no—
461C2×C22⋊C4—614no—
462C2×C22⋊C4—12216no—
463C2×C22⋊C4C22×D6114yes—
464C2×C42—612no—
465C2×C42—612no—
466C2×C42—612no—
467C2×C42—612no—
468C2×C42—2442no—
469C2×C42—2442no—
470#95—614no—
471#95—614no—
472#95—614no—
473#95—614no—
474#95—614no—
475#95—614no—
476#95—614no—
477#95C22×D6114yes—
478#95C22×D6114yes—
479#95C22×D6114yes—
480C24—612no—
481C24—614no—
482C24—614no—
483C24—614no—
484C24—2441no—
485C24—2441no—
486C24—2441no—
487C24—2441no—
488C24C23×D6111yescharacteristic
489C2×Q8—24no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8—1228no—
493C2×Q8—1224no—
494C2×Q8—2442no—
495C2×Q8—2444no—
496C2×Q8—2442no—
497C2×Q8—2444no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—611no—
505C22×C4—614no—
506C22×C4—611no—
507C22×C4—612no—
508C22×C4—612no—
509C22×C4—614no—
510C22×C4—614no—
511C22×C4—618no—
512C22×C4—618no—
513C22×C4—614no—
514C22×C4—614no—
515C22×C4—1224no—
516C22×C4—1224no—
517C22×C4—1228no—
518C22×C4—1228no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—2444no—
522C22×C4—2444no—
523C22×C4—2444no—
524C22×C4—2444no—
525C22×C4—2442no—
526C22×C4—2442no—
527C22×C4—24416no—
528C22×C4—2441no—
529C22×C4—2441no—
530C22×C4—2441no—
531C22×C4—2441no—
532C22×C4D4×D6114yes—
533C22×C4Q8×D6114yes—
534C22×C4Q8×D6114yes—
535C22×C4C23×D6114yes—
536C22×C4C23×D6114yes—
537C22×C4C23×D6112yes—
538C22×C4C23×D6112yes—
539C22×C4C23×D6112yes—
540C4⋊C4—24no—
541C4⋊C4—28no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—1228no—
547C4⋊C4—1228no—
548C4⋊C4—1228no—
549C4⋊C4—1228no—
550C4⋊C4—1228no—
551C4⋊C4—1228no—
552C4⋊C4—1224no—
553C4⋊C4—2442no—
554C4⋊C4—2442no—
555C4⋊C4—2444no—
556C4⋊C4—2444no—
557C4⋊C4—2442no—
558C4⋊C4—2442no—
559C22⋊C4—28no—
560C22⋊C4—6116no—
561C22⋊C4—6116no—
562C22⋊C4—1228no—
563C22⋊C4—1228no—
564C22⋊C4—1228no—
565C42—1224no—
566C42—1224no—
567C42—2442no—
568C42—2442no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—614no—
574C23—614no—
575C23—614no—
576C23—6116no—
577C23—1222no—
578C23—1222no—
579C23—1222no—
580C23—1222no—
581C23—2444no—
582C23—2444no—
583C23—2444no—
584C23—2444no—
585C23—2444no—
586C23—2444no—
587C23C2×D4×D6112yes—
588C23C2×Q8×D6112yes—
589C23C2×Q8×D6112yes—
590C23C24×D6111yescharacteristic
591Q8—84no—
592Q8—4884no—
593Q8—4888no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—612no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—614no—
613C2×C4—614no—
614C2×C4—612no—
615C2×C4—12216no—
616C2×C4—1228no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1224no—
624C2×C4—1224no—
625C2×C4—1224no—
626C2×C4—1224no—
627C2×C4—1224no—
628C2×C4—1224no—
629C2×C4—4888no—
630C2×C4—4888no—
631C2×C4—2448no—
632C2×C4—2444no—
633C2×C4—2444no—
634C2×C4C2×D4×D6114yes—
635C2×C4C2×Q8×D6114yes—
636C2×C4C2×Q8×D6114yes—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—618no—
644C22—1224no—
645C22—1224no—
646C22—1224no—
647C22—1224no—
648C22—1224no—
649C22—1224no—
650C22—24416no—
651C22C4⋊Q8×D6114yes—
652C22C22×D4×D6111yescharacteristic
653C22C22×Q8×D6111yescharacteristic
654C22C22×Q8×D6111yescharacteristic
655C22H(5,𝔽2)×D6111yescharacteristic
656C22H(5,𝔽2)×D6111yescharacteristic
657C22#1684059711yescharacteristic
658C22#1684059911yescharacteristic
659C4—24no—
660C4—24no—
661C4—24no—
662C4—44no—
663C4—44no—
664C4—614no—
665C4—1222no—
666C4—1224no—
667C4—1224no—
668C4—2444no—
669C4—2444no—
670C2—24no—
671C2—1228no—
672C2C2×C4⋊Q8×D6111yescharacteristic
673C2#543×D6111yescharacteristic
674C2#2145922311yescharacteristic
675C2#2145958011yescharacteristic
676C2#2145958211yescharacteristic
677C2#2145961911yescharacteristic
678C2#2145962111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2659C2
prime radicalC24×D61C23
hypercenter#2659D61
hypocenterC61#35002
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2659

Complements

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

NH ≅ G/Ntype
2. #21454169C2semidirect
3. #21454167C2semidirect
4. #21451936C2semidirect
5. #21451933C2semidirect
6. #21450826C2semidirect
7. #21450824C2semidirect
8. #21449516C2semidirect
9. #21449514C2semidirect
13. C61×#2659C2semidirect
16. C2×#14519776C22semidirect
17. C2×#14519775C22semidirect
31. #16839696C22semidirect
32. #16839695C22semidirect
33. #16839694C22semidirect
34. #16839683C22semidirect
35. #16839683C22semidirect
36. #16839682C22semidirect
37. #16839669C22semidirect
38. #16839667C22semidirect
39. #16839666C22semidirect
40. #16839661C22semidirect
41. #16839659C22semidirect
42. #16839658C22semidirect
43. #16839657C22semidirect
44. #16839657C22semidirect
49. #16839631C22semidirect
50. #16839631C22semidirect
63. C61×#398C22semidirect
64. C61×#395C22semidirect
76. C22×C4×D61Q8semidirect
106. C2×#11758198Q8semidirect
247. C22×D61C4⋊Q8semidirect
330. Dic61#543semidirect
336. #2659D61semidirect
374. #398C2×D61semidirect
378. #395C2×D61semidirect
407. C61#35002semidirect

Maximal subgroups

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

subgroupindex
C2×#168403872
#214541692
#214541672
#214519362
#214519332
#214508262
#214508242
#214495162
#214495142
#214484322
#380×D612
C2×#168396092
C61×#26592
#3500261