Finite Group Databasebeta

← #29736624

Subgroups — #29736624

Order 15616.

1446 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×C4⋊Q8×D61C211yesmaximal, characteristic, complemented
2C2×#16840107C212yesmaximal, complemented
3#21454163C211yesmaximal, characteristic
4#21446956C211yesmaximal, characteristic, complemented
5#21446955C211yesmaximal, characteristic, complemented
6#21446952C211yesmaximal, characteristic, complemented
7#21446778C212yesmaximal, complemented
8#21446135C212yesmaximal, complemented
9#21446133C212yesmaximal, complemented
10#21445576C211yesmaximal, characteristic, complemented
11C61×#2023C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C2×#14519777C2211yescharacteristic
14C2×#14519776—21no—
15C2×#14519776C2212yes—
16C2×#14519775—21no—
17C4⋊Q8×D61—28no—
18C2×#14519773C2211yescharacteristic
19C2×#14519752C2211yescharacteristic
20C2×#14519750—21no—
21C2×#14519750—21no—
22C2×#14519748C2211yescharacteristic
23C2×C4⋊C4×D61—21no—
24C2×C4⋊C4×D61C2212yes—
25C2×C42×D61C2211yescharacteristic
26#16840107—28no—
27C2×#14519699C2212yes—
28C2×#14519698C2212yes—
29C2×#14519697C2212yes—
30C2×#14519689C2212yes—
31C2×C8×Dic61C2212yes—
32#16839664C2211yescharacteristic
33#16839662—21no—
34#16839662—21no—
35#16839631C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839517C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C61×#390C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122—22no—
47C2×Q8×D61—28no—
48C2×Q8×D61—44no—
49C2×Q8×D61—44no—
50C22×D244—21no—
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C22⋊D61—21no—
55C22×C4×D61—22no—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C22×#92418C2312yes—
60#14519777—24no—
61#14519776—28no—
62#14519776—24no—
63#14519776D418yescomplemented
64#14519775—28no—
65#14519773—24no—
66#14519752—24no—
67#14519750—44no—
68#14519750—44no—
69#14519748—24no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×#11758196—22no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C4⋊C4×D61—28no—
78C4⋊C4×D61—24no—
79C4⋊C4×D61—24no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87C42×D61—24no—
88#14519699—24no—
89#14519698—24no—
90#14519697—24no—
91#14519689—24no—
92C8×Dic61—24no—
93C22×Q8×C61—21no—
94C22×D4×C61C2311yescharacteristic
95C22×C8×C61C2312yes—
96C4⋊Q8×C61—24no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#102×C61—24no—
104C24×C61—21no—
105C23×D61—21no—
106C23×D61C2×D411yescharacteristic
107C2×Q8×C61—24no—
108C2×Q8×C61—42no—
109C2×Q8×C61—42no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C2×C22⋊D61—42no—
122C2×C22⋊D61—42no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C2×C22⋊D61—42no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133Q8×D61—432no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—42no—
139C2×D244—42no—
140C2×D244—42no—
141C2×C4×D61—22no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—28no—
149C2×C4×D61—44no—
150C2×C4×D61—44no—
151C2×C4×D61D814yes—
152C2×C4×D61SD1614yes—
153C2×C4×D61#4114yes—
154C2×Dic122—24no—
155C2×Dic122—28no—
156C2×Dic122—42no—
157C2×Dic122—42no—
158C2×Dic122—44no—
159C2×Dic122—44no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61C2×D414yes—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—42no—
169C42×C61—22no—
170C22⋊Dic61—42no—
171#11758198—42no—
172#11758198—42no—
173#11758198—42no—
174C4⋊Dic61—24no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61C2×D414yes—
178#11758196—28no—
179#11758196—28no—
180#11758196—42no—
181#11758196—42no—
182C4×Dic61—22no—
183C4×Dic61—22no—
184C4×Dic61C2×D414yes—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61C22×D411yescharacteristic
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61#12412yes—
208C22×D61C2×D812yes—
209C22×D61C2×SD1612yes—
210Q8×C61—48no—
211D4×C61—42no—
212D4×C61—42no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—24no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61C2×D812yes—
224C2×C4×C61C2×SD1612yes—
225C2×C4×C61C22×D412yes—
226C2×C4×C61C2×#4112yes—
227C2×C4×C612−1+411yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C22⋊D61—84no—
230C22⋊D61—84no—
231C22⋊D61—84no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—24no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61#12012yes—
241C2×Dic61#12012yes—
242C2×Dic61#12412yes—
243C2×Dic61C2×D812yes—
244C2×Dic61C2×SD1612yes—
245C2×Dic61AGL(1,ℤ/8)12yes—
246C2×Dic61#13712yes—
247D244—42no—
248D244—42no—
249D244—84no—
250C4×D61—24no—
251C4×D61—24no—
252C4×D61—28no—
253C4×D61—28no—
254C4×D61—416no—
255Dic122—48no—
256Dic122—416no—
257C8×C61—42no—
258#92418—42no—
259#25903—611nomaximal, Sylow
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61C2×#12011yescharacteristic
268C22×C61C2×#12411yescharacteristic
269C22×C61#53611yescharacteristic
270C22×C61C22×D811yescharacteristic
271C22×C61C22×SD1611yescharacteristic
272C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C61C2×#13711yescharacteristic
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61#48614yescomplemented
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—44no—
291C4×C61—81no—
292C4×C61#54812yes—
293C4×C61#54812yes—
294Dic61—22no—
295Dic61—44no—
296Dic61—81no—
297Dic61#44714yes—
298Dic61#44814yes—
299C22×C4⋊Q8—611no—
300C2×#486—612no—
301#2856—611no—
302#2023—611no—
303#2023—611no—
304#2023—611no—
305#2023D6111yescharacteristic, complemented
306#1990—612no—
307#1874—612no—
308#1874—612no—
309C4×C22⋊C8—611no—
310C2×C61—161no—
311C2×C61#243611yescharacteristic
312C2×C61C2×#44711yescharacteristic
313C2×C61C2×#44811yescharacteristic
314C2×C61C2×#48611yescharacteristic
315C2×C61#314111yescharacteristic
316C2×C61#322611yescharacteristic
317C2×C61#322711yescharacteristic
318D61—24no—
319D61—161no—
320C23×Q8—1221no—
321C2×C4⋊Q8—612no—
322C2×C4⋊Q8—611no—
323C2×C4⋊Q8—1221no—
324C2×C4⋊Q8—1221no—
325C2×C4⋊Q8—1228no—
326C2×C4⋊Q8C2×D6111yescharacteristic
327C2×#127—611no—
328C2×#121—611no—
329C2×#121—611no—
330C2×#121—1221no—
331C2×#121—1221no—
332C22×C4⋊C4—612no—
333C22×C4⋊C4—1221no—
334C22×C42—611no—
335#486—1228no—
336C2×#102—612no—
337C2×#102—612no—
338C2×#102—612no—
339C2×#102C2×D6112yes—
340C2×C22⋊C8—612no—
341C2×C4×C8—612no—
342#390—611no—
343#390—1221no—
344#390—1221no—
345#390C2×D6111yescharacteristic
346#384—611no—
347#340—612no—
348#340—612no—
349#340—612no—
350#340C2×D6112yes—
351#336—611no—
352#336C2×D6111yescharacteristic
353C61#2590311yescharacteristic, complemented, Sylow
354C22×Q8—21no—
355C22×Q8—1222no—
356C22×Q8—1228no—
357C22×Q8—1221no—
358C22×Q8—2444no—
359C22×Q8—2444no—
360C22×D4—611no—
361C22×D4—1221no—
362C22×D4—1221no—
363C22×D4—1221no—
364C22×D4—1221no—
365C22×D4C22×D6111yescharacteristic
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—611no—
369C23×C4—1222no—
370C22×C8—612no—
371C22×C8C22×D6112yes—
372C4⋊Q8—24no—
373C4⋊Q8—618no—
374C4⋊Q8—1224no—
375C4⋊Q8—1224no—
376C4⋊Q8—1228no—
377C4⋊Q8—1228no—
378#127—1224no—
379#121—1224no—
380#121—1224no—
381#121—2444no—
382#121—2444no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4—1228no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1224no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6112yes—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—1224no—
406C2×C42C22×D6111yescharacteristic
407#102—24no—
408#102—1224no—
409#102—1224no—
410#102—1224no—
411C22⋊C8—1224no—
412C4×C8—1224no—
413C24—21no—
414C24—611no—
415C24—1221no—
416C2×Q8—24no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—1228no—
420C2×Q8—1224no—
421C2×Q8—2442no—
422C2×Q8—2444no—
423C2×Q8—2444no—
424C2×Q8—24432no—
425C2×Q8—2442no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—2442no—
435C2×D4—2442no—
436C2×D4—2442no—
437C2×D4—2442no—
438C2×D4—2442no—
439C2×D4—2442no—
440C2×D4—2442no—
441C2×D4—2442no—
442C2×D4—2442no—
443C2×D4—2442no—
444C2×D4—2442no—
445C2×D4—2442no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—614no—
452C22×C4—614no—
453C22×C4—614no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1228no—
459C22×C4—1222no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1222no—
463C22×C4—1221no—
464C22×C4—2444no—
465C22×C4—2444no—
466C22×C4D4×D6111yescharacteristic
467C22×C4D4×D6111yescharacteristic
468C22×C4C23×D6111yescharacteristic
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1222no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—614no—
479C4⋊C4—1228no—
480C4⋊C4—1228no—
481C4⋊C4—1224no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C4⋊C4D4×D6114yes—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—2442no—
490C22⋊C4—2442no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C42—22no—
494C42—614no—
495C42—1222no—
496C42—1222no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—612no—
505C23—612no—
506C23—612no—
507C23—1221no—
508C23—1221no—
509C23—1221no—
510C23—1221no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23—2441no—
517C23—2441no—
518C23C2×D4×D6111yescharacteristic
519Q8—48no—
520Q8—2448no—
521Q8—24416no—
522D4—42no—
523D4—42no—
524D4—4884no—
525D4—4884no—
526D4—4884no—
527D4—4884no—
528D4—2442no—
529D4—2442no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—612no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—1228no—
548C2×C4—1228no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—24416no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4D8×D6112yes—
561C2×C4SD16×D6112yes—
562C2×C4C2×D4×D6112yes—
563C2×C4#41×D6112yes—
564C2×C4#1451982411yescharacteristic
565C2×C4#1451982511yescharacteristic
566C8—42no—
567C8—2442no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—614no—
576C22—1222no—
577C22—1222no—
578C22—1222no—
579C22—1222no—
580C22—1222no—
581C22—1222no—
582C22—4881no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22—4881no—
587C22—4881no—
588C22—4881no—
589C22#120×D6111yescharacteristic
590C22#124×D6111yescharacteristic
591C22#1684033511yescharacteristic
592C22C2×D8×D6111yescharacteristic
593C22C2×SD16×D6111yescharacteristic
594C22AGL(1,ℤ/8)×D6111yescharacteristic
595C22#137×D6111yescharacteristic
596C4—22no—
597C4—22no—
598C4—44no—
599C4—81no—
600C4—614no—
601C4—614no—
602C4—1222no—
603C4—4881no—
604C4—2444no—
605C4#1684042012yes—
606C4#1684042212yes—
607C2—161no—
608C2—1224no—
609C2—9761no—
610C2#2144947211yescharacteristic
611C2#447×D6111yescharacteristic
612C2#448×D6111yescharacteristic
613C2#486×D6111yescharacteristic
614C2#2145621111yescharacteristic
615C2#2145693911yescharacteristic
616C2#2145694611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2023C2
socleC23×C61C22×D4
prime radical#21454163C2
hypercenter#2023D61
hypocenterC61#25903
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D61C2semidirect
2. C2×#16840107C2semidirect
4. #21446956C2semidirect
5. #21446955C2semidirect
6. #21446952C2semidirect
7. #21446778C2semidirect
8. #21446135C2semidirect
9. #21446133C2semidirect
10. #21445576C2semidirect
11. C61×#2023C2semidirect
63. #14519776D4semidirect
287. C2×D61#486semidirect
305. #2023D61semidirect
353. C61#25903semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
C2×#168401072
#214541632
#214469562
#214469552
#214469522
#214467782
#214461352
#214461332
#214455762
C61×#20232
#2590361