Finite Group Databasebeta

← #29682724

Subgroups — #29682724

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839763C211yesmaximal, characteristic
2C2×#16839763C211yesmaximal, characteristic
3C2×C4×#11758194C211yesmaximal, characteristic, complemented
4C2×#16839660C211yesmaximal, characteristic, complemented
5#21446519C212yesmaximal, complemented
6#21446512C212yesmaximal, complemented
7#21446505C212yesmaximal, complemented
8#21446498C212yesmaximal, complemented
9#21445540C211yesmaximal, characteristic, complemented
10#21445540C211yesmaximal, characteristic, complemented
11C61×#2297C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16C22×#11758196—21no—
17#16839763—24no—
18#16839763C418yescomplemented
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C4×#11758194—24no—
24C2×#14519688C2212yes—
25C2×C4×#92418C2212yes—
26#16839660—21no—
27#16839660—21no—
28#16839660C2212yes—
29#16839660C2212yes—
30C2×#14519684—21no—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C22×#41×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C22×C42×C61C2211yescharacteristic
38C61×#391C2211yescharacteristic
39C61×#391C2211yescharacteristic
40C61×#390C2211yescharacteristic
41C61×#390C2211yescharacteristic
42C61×#386—21no—
43C61×#386—21no—
44C4×C22⋊C4×C61—21no—
45C4×C22⋊C4×C61—21no—
46C23×Dic61—21no—
47C23×Dic61—21no—
48C2×#11758194—22no—
49C2×#11758194—22no—
50C2×#11758194—22no—
51C2×#11758194—22no—
52C2×#11758194C2×C414yes—
53C2×#11758194D414yes—
54C22×#92418C2312yes—
55C22×#92418C2312yes—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61C2×C414yes—
62C2×C4⋊Dic61C2312yes—
63C2×C4⋊Dic61C2312yes—
64C2×#11758196—22no—
65C2×#11758196—22no—
66C2×#11758196—44no—
67#14519701—24no—
68#14519701C2×C418yescomplemented
69#14519698—24no—
70#14519698C2×C418yescomplemented
71#14519688—24no—
72C4×#92418—24no—
73#14519684—22no—
74#14519684—22no—
75C2×#41×C61—22no—
76C2×#41×C61—22no—
77C2×#41×C61—22no—
78C2×#41×C61—22no—
79C2×#41×C61—22no—
80C2×#41×C61C2×C414yes—
81C22×Q8×C61C2311yescharacteristic
82C22×D4×C61—21no—
83C22×D4×C61C2311yescharacteristic
84C23×C4×C61—21no—
85C23×C4×C61D411yescharacteristic
86C23×C4×C61D411yescharacteristic
87C23×C4×C61C2311yescharacteristic
88#117×C61—22no—
89#117×C61—22no—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—21no—
97C2×C42×C61—22no—
98C2×C42×C61—22no—
99C2×C42×C61—21no—
100C2×C42×C61D411yescharacteristic
101C2×C42×C61D411yescharacteristic
102C2×C42×C61C2311yescharacteristic
103C2×C42×C61C2311yescharacteristic
104#95×C61—21no—
105#95×C61—21no—
106#95×C61—21no—
107#95×C61—21no—
108C24×C61—21no—
109C24×C61C2×D411yescharacteristic
110#41×C61—28no—
111#41×C61—28no—
112#41×C61—44no—
113#41×C61—44no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61C22×C412yes—
120C2×D4×C61—24no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—42no—
127C2×D4×C61—42no—
128C2×D4×C61C22×C412yes—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—21no—
132C22×C4×C61—24no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—41no—
138C22×C4×C61—41no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61C22×C412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61#4112yes—
149C22×C4×C61#4112yes—
150C22×C4×C61#4112yes—
151C22×C4×C61#4112yes—
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—24no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C22⋊C4×C61—42no—
166C22⋊C4×C61—42no—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—42no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—42no—
177C4⋊Dic61—24no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61—42no—
182C4⋊Dic61C22×C414yes—
183#11758196—44no—
184#11758196—44no—
185#11758194—28no—
186#11758194—28no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418C22×C414yes—
192C2×#92418C2×D414yes—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—22no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61C4×D412yes—
205C23×C61#12112yes—
206C23×C61#12412yes—
207C23×C61C22×D411yescharacteristic
208Q8×C61—22no—
209Q8×C61—22no—
210Q8×C61—41no—
211Q8×C61—41no—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—48no—
215D4×C61—41no—
216D4×C61—41no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—44no—
229C2×C4×C61—44no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—48no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61#11714yes—
240C2×C4×C61C4×D414yes—
241C2×C4×C61C4×D412yes—
242C2×C4×C61#12112yes—
243C2×C4×C61#12314yes—
244C2×C4×C61#12412yes—
245C2×C4×C61#12614yes—
246C2×C4×C61AGL(1,ℤ/8)12yes—
247C2×C4×C61#13712yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×Dic61—24no—
255C2×Dic61—48no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—84no—
262#92418—24no—
263#92418—24no—
264#21093—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61#38814yes—
279C22×C61C2×C4×D411yescharacteristic
280C22×C61C2×#12111yescharacteristic
281C22×C61C2×#12411yescharacteristic
282C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C61C2×#13711yescharacteristic
284C22×C61#57511yescharacteristic
285C22×C61#57511yescharacteristic
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—42no—
291C4×C61—81no—
292C4×C61—81no—
293C4×C61#38814yes—
294C4×C61C2×#11711yescharacteristic
295C4×C61C2×C4×D411yescharacteristic
296C4×C61C2×#12311yescharacteristic
297C4×C61C2×#12611yescharacteristic
298Dic61—44no—
299Dic61—82no—
300C2×#417—611no—
301C2×#417—611no—
302C2×C4×M16—611no—
303#2297D6111yescharacteristic
304C2×#389—611no—
305#1936—612no—
306#1935—612no—
307#1934—612no—
308#1933—612no—
309#1736—611no—
310#1736—611no—
311C2×C61—24no—
312C2×C61—82no—
313C2×C61C2×#38811yescharacteristic
314C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
315C2×C61C4×#13711yescharacteristic
316C2×C61#305511yescharacteristic
317C2×C61#305611yescharacteristic
318C2×C61#314111yescharacteristic
319C2×C61#314211yescharacteristic
320C22×#41C2×D6111yescharacteristic
321C22×M16—611no—
322C22×M16—611no—
323C2×#117C2×D6111yescharacteristic
324C22×C4⋊C4—611no—
325C22×C4⋊C4—611no—
326C22×C4⋊C4—1221no—
327C22×C42C2×D6111yescharacteristic
328#417—618no—
329#417—1224no—
330C2×#103—612no—
331C2×#103—612no—
332C2×#102—612no—
333C2×#102—612no—
334C4×M16—1224no—
335C2×#97—612no—
336C2×C4×C8—612no—
337#391C2×D6111yescharacteristic
338#391C2×D6111yescharacteristic
339#390C2×D6111yescharacteristic
340#390C2×D6111yescharacteristic
341#389—612no—
342#389—612no—
343#389—1221no—
344#389—1221no—
345#386—21no—
346#386—21no—
347C4×C22⋊C4—21no—
348C4×C22⋊C4—21no—
349C2×#95—1221no—
350#340—612no—
351#340—612no—
352#340—612no—
353#340—612no—
354C61#2109311yescharacteristic, complemented, Sylow
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41C4×D6114yes—
361C22×Q8C22×D6111yescharacteristic
362C22×D4—21no—
363C22×D4C22×D6111yescharacteristic
364C23×C4—21no—
365C23×C4—1221no—
366C23×C4—1221no—
367C23×C4C22⋊D6111yescharacteristic
368C23×C4C22⋊D6111yescharacteristic
369C23×C4C22×D6111yescharacteristic
370C2×M16—614no—
371C2×M16—614no—
372C2×M16—1222no—
373C2×M16—1222no—
374C2×M16—1222no—
375C2×M16—1222no—
376C22×C8—612no—
377C22×C8—612no—
378#117—22no—
379#117—22no—
380C2×C4⋊C4—614no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—2444no—
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—21no—
401C2×C42C22⋊D6111yescharacteristic
402C2×C42C22⋊D6111yescharacteristic
403C2×C42C22×D6111yescharacteristic
404C2×C42C22×D6111yescharacteristic
405#103—618no—
406#103—1224no—
407#102—618no—
408#102—1224no—
409#97—1224no—
410C4×C8—1224no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1222no—
416#95—1222no—
417C24—21no—
418C24C2×C22⋊D6111yescharacteristic
419#41—28no—
420#41—28no—
421#41—44no—
422#41—44no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8C2×C4×D6112yes—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C2×C4×D6112yes—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—1222no—
451C22×C4—1224no—
452C22×C4—1222no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4C2×C4×D6112yes—
459C22×C4#1175822212yes—
460C22×C4D4×D6112yes—
461C22×C4#1175822412yes—
462C22×C4#1175822412yes—
463C22×C4#1175822412yes—
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470M16—1228no—
471M16—1228no—
472C2×C8—614no—
473C2×C8—614no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—614no—
483C4⋊C4—1222no—
484C4⋊C4—1224no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—2444no—
488C4⋊C4—2442no—
489C4⋊C4—2444no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—22no—
500C42—22no—
501C42—42no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C4×C22⋊D6112yes—
514C23#1451974912yes—
515C23#1451976412yes—
516C23C22×C22⋊D6111yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—1224no—
549C2×C4—4884no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2448no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4#1451973314yes—
557C2×C4#1451973714yes—
558C2×C4C4×C22⋊D6112yes—
559C2×C4#1451974912yes—
560C2×C4#1451975914yes—
561C2×C4#1451976412yes—
562C2×C4#1451977014yes—
563C2×C4#1451980212yes—
564C2×C4#1451980712yes—
565C2×C4C22×C4×D6111yescharacteristic
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4C2×#1175822411yescharacteristic
568C2×C4C2×#1175822411yescharacteristic
569C2×C4C2×#1175822411yescharacteristic
570C2×C4C2×#1175822211yescharacteristic
571C8—1224no—
572C8—1224no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#1683965614yes—
587C22C2×C4×C22⋊D6111yescharacteristic
588C22C2×#1451974911yescharacteristic
589C22C2×#1451976411yescharacteristic
590C22C2×#1451980211yescharacteristic
591C22C2×#1451980711yescharacteristic
592C22#1684056011yescharacteristic
593C22#1684056011yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—4882no—
602C4—2444no—
603C4#1683965614yes—
604C4C2×#1451973311yescharacteristic
605C4C2×#1451973711yescharacteristic
606C4C2×#1451975911yescharacteristic
607C4C2×#1451977011yescharacteristic
608C2—24no—
609C2—82no—
610C2C2×#1683965611yescharacteristic
611C2C4×#1451980211yescharacteristic
612C2C4×#1451980711yescharacteristic
613C2#2145544011yescharacteristic
614C2#2145545111yescharacteristic
615C2#2145621211yescharacteristic
616C2#2145621911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2297C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenter#2297D61
hypocenterC61#21093
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
3. C2×C4×#11758194C2semidirect
4. C2×#16839660C2semidirect
5. #21446519C2semidirect
6. #21446512C2semidirect
7. #21446505C2semidirect
8. #21446498C2semidirect
9. #21445540C2semidirect
10. #21445540C2semidirect
18. #16839763C4semidirect
68. #14519701C2×C4semidirect
70. #14519698C2×C4semidirect
354. C61#21093semidirect

Maximal subgroups

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

subgroupindex
C2×#168397632
C2×C4×#117581942
C2×#168396602
#214465192
#214465122
#214465052
#214464982
#214455402
C61×#22972
#2109361