Finite Group Databasebeta

← #29714273

Subgroups — #29714273

Order 15616.

1006 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455620C211yesmaximal, characteristic, complemented
2C4×#14519807C211yesmaximal, characteristic
3#21450808C211yesmaximal, characteristic, complemented
4#21447033C212yesmaximal
5#21446853C212yesmaximal
6#21446764C211yesmaximal, characteristic, complemented
7#21446664C211yesmaximal, characteristic, complemented
8#21446212C212yesmaximal, complemented
9#21446034C212yesmaximal, complemented
10#21445839C211yesmaximal, characteristic
11C61×#3509C211yesmaximal, characteristic
12C2×#14519807C2211yescharacteristic
13C2×#14519775C2212yes—
14C2×C4×Dic122C2211yescharacteristic
15#16840066—22no—
16#16840054—21no—
17#16840044—21no—
18#16839867—21no—
19#16839862—21no—
20C4×#11758202—21no—
21C4×#11758200—21no—
22#16839805C2212yes—
23#16839805C2211yescharacteristic
24#16839786C2211yescharacteristic
25C2×#14519704C2212yes—
26#16839770C2212yes—
27#16839770C2211yescharacteristic
28#16839763C2211yescharacteristic
29C2×#14519696C2212yes—
30C2×#14519696C2212yes—
31C4×#11758194C2211yescharacteristic
32#16839693—22no—
33#16839669C2212yes—
34#16839658—22no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839521C2212yes—
37#16839521C2212yes—
38#16839521C2212yes—
39C61×#557—21no—
40C61×#557—21no—
41C61×#556—21no—
42C61×#556—21no—
43C61×#552—22no—
44C61×#552—22no—
45C61×#541—22no—
46C61×#536C2212yes—
47C61×#533—21no—
48C61×#533—21no—
49C61×#533C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C61×#519C2212yes—
52C4×#41×C61C2211yescharacteristic
53C22×Dic122C2311yescharacteristic
54#14519807—22no—
55#14519807—22no—
56C2×#11758202—21no—
57C2×#11758200—21no—
58C2×#11758194C2311yescharacteristic
59C2×#11758194C2312yes—
60C2×#11758194C2311yescharacteristic
61#14519775—24no—
62#14519775—42no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65C4×Dic122—21no—
66C4×Dic122—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—21no—
70C2×#11758196—24no—
71C2×C4×Dic61—21no—
72C2×C4×Dic61C2312yes—
73#14519710—22no—
74#14519710—21no—
75#14519708—22no—
76#14519708—21no—
77C4⋊#92418—21no—
78C4⋊#92418—21no—
79#14519704—42no—
80#14519701—21no—
81#14519700—22no—
82#14519700—22no—
83#14519700—21no—
84#14519700—21no—
85#14519698—21no—
86#14519696—24no—
87#14519696—42no—
88#14519688—21no—
89C4×#92418—21no—
90#14519684—21no—
91#14519684—24no—
92C2×#41×C61C2311yescharacteristic
93C22×Q8×C61C2312yes—
94C4⋊Q8×C61—22no—
95C4⋊Q8×C61—21no—
96C4⋊Q8×C61—21no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61D411yescharacteristic
100C4⋊Q8×C61D411yescharacteristic
101#126×C61—24no—
102#126×C61—24no—
103#125×C61—22no—
104#125×C61—22no—
105#125×C61—24no—
106#125×C61—24no—
107#125×C61D412yes—
108#124×C61—22no—
109#123×C61—24no—
110C22⋊Q8×C61—22no—
111C22⋊Q8×C61—22no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61—21no—
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61—22no—
116C4×Q8×C61—22no—
117C4×Q8×C61—22no—
118C4×Q8×C61—22no—
119C4×Q8×C61—22no—
120C4×Q8×C61—22no—
121C4×Q8×C61—22no—
122C4×Q8×C61—21no—
123C4×Q8×C61—21no—
124C4×Q8×C61D411yescharacteristic
125C4×Q8×C61D411yescharacteristic
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61D411yescharacteristic
129C4×D4×C61D411yescharacteristic
130#117×C61—22no—
131#117×C61—22no—
132#117×C61—24no—
133#117×C61—21no—
134#117×C61C2312yes—
135#117×C61C2311yescharacteristic
136C2×C4⋊C4×C61D412yes—
137C2×C4⋊C4×C61D412yes—
138C2×C42×C61—21no—
139C2×C42×C61C2311yescharacteristic
140#41×C61—21no—
141#41×C61—21no—
142#41×C61—21no—
143#41×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—22no—
147C2×Q8×C61—22no—
148C2×Q8×C61—22no—
149C2×Q8×C61—22no—
150C2×Q8×C61—22no—
151C2×Q8×C61—22no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—21no—
154C2×D4×C61C2×D411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61C2×D414yes—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—21no—
163C22×Dic61—22no—
164C22×Dic61—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122#4112yes—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D412yes—
188C4⋊C4×C61C2×D412yes—
189C22⋊C4×C61—24no—
190C22⋊C4×C61—24no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C42×C61—24no—
196C42×C61—24no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—22no—
201C42×C61C2×D411yescharacteristic
202C42×C61C2×D411yescharacteristic
203C42×C61C2×D411yescharacteristic
204C42×C61C2×D411yescharacteristic
205#11758202—24no—
206#11758200—24no—
207C4⋊Dic61—24no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210#11758196—48no—
211#11758196—41no—
212#11758196—41no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—41no—
216C4×Dic61—41no—
217#11758194—21no—
218#11758194—22no—
219#11758194—21no—
220#11758194—22no—
221#11758194—21no—
222#11758194#4112yes—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—22no—
226C23×C61—21no—
227C23×C61C22×D411yescharacteristic
228Q8×C61—21no—
229Q8×C61—21no—
230Q8×C61—42no—
231Q8×C61—42no—
232Q8×C61—42no—
233Q8×C61—42no—
234Q8×C61—42no—
235Q8×C61—42no—
236D4×C61—21no—
237D4×C61—21no—
238D4×C61—41no—
239D4×C61—41no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—24no—
246C2×C4×C61—21no—
247C2×C4×C61—24no—
248C2×C4×C61—24no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—42no—
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61#12011yescharacteristic
256C2×C4×C61#12011yescharacteristic
257C2×C4×C61#12011yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×Dic61—24no—
265C2×Dic61—22no—
266C2×Dic61—42no—
267C2×Dic61—41no—
268C2×Dic61—42no—
269C2×Dic61—41no—
270C2×Dic61—42no—
271C2×Dic61—41no—
272Dic122—22no—
273Dic122—22no—
274Dic122—41no—
275Dic122—41no—
276#92418—22no—
277#92418—41no—
278#92418—42no—
279#24147—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61C2×#12011yescharacteristic
286C22×C61#54611yescharacteristic
287C22×C61#54711yescharacteristic
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—44no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—42no—
295C4×C61—41no—
296C4×C61—42no—
297C4×C61#45612yes—
298C4×C61#52511yescharacteristic
299C4×C61#53711yescharacteristic
300C4×C61#54811yescharacteristic
301C4×C61#55011yescharacteristic
302Dic61—44no—
303Dic61—42no—
304Dic61—81no—
305#3509D6111yescharacteristic
306#3078—611no—
307C4×#137—611no—
308#2546—611no—
309#2038—612no—
310#2005—612no—
311#1986—611no—
312#1966—611no—
313#1886—612no—
314#1857—612no—
315#1806—611no—
316C2×C61—21no—
317C2×C61—21no—
318C2×C61—81no—
319C2×C61#240711yescharacteristic
320C2×C61C2×#45611yescharacteristic
321C2×C61#302011yescharacteristic
322C2×#137—611no—
323#557—21no—
324#557—21no—
325#556—21no—
326#556—21no—
327#552—22no—
328#552—22no—
329#541—22no—
330#536C2×D6112yes—
331#533—21no—
332#533—21no—
333#533C2×D6111yescharacteristic
334C2×C4⋊Q8—612no—
335C2×C4⋊Q8C2×D6111yescharacteristic
336#519C2×D6112yes—
337C4×#41C2×D6111yescharacteristic
338C2×C4×Q8—611no—
339#479—1222no—
340#477—1221no—
341#475—1221no—
342#441—1221no—
343#440—1221no—
344C4×Q16—1221no—
345C4×SD16—1221no—
346#428—612no—
347#428—611no—
348#423—611no—
349C2×#104—612no—
350#419—612no—
351#419—611no—
352#417—611no—
353C2×#101—612no—
354C2×#101—612no—
355C4×M16—611no—
356#398—1222no—
357#391—612no—
358#389—1222no—
359C4×C4⋊C4—611no—
360#339—612no—
361#339—612no—
362#339—612no—
363C61#2414711yescharacteristic, complemented, Sylow
364C2×#41C22×D6111yescharacteristic
365C22×Q8—611no—
366C22×Q8C22×D6112yes—
367#137—1222no—
368#137—1222no—
369C2×Q16—1221no—
370C2×SD16—1221no—
371C2×M16—611no—
372C2×M16—611no—
373C2×M16—612no—
374C4⋊Q8—22no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8—21no—
379C4⋊Q8—1224no—
380C4⋊Q8—2442no—
381C4⋊Q8C22⋊D6111yescharacteristic
382C4⋊Q8C22⋊D6111yescharacteristic
383#126—24no—
384#126—24no—
385#125—22no—
386#125—22no—
387#125—24no—
388#125—24no—
389#125C22⋊D6112yes—
390#124—22no—
391#123—24no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—1221no—
407C4×Q8—1221no—
408C4×Q8—1221no—
409C4×Q8—1221no—
410C4×Q8C22⋊D6111yescharacteristic
411C4×Q8C22⋊D6111yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4C22⋊D6111yescharacteristic
415C4×D4C22⋊D6111yescharacteristic
416#117—22no—
417#117—22no—
418#117—24no—
419#117—21no—
420#117C22×D6111yescharacteristic
421#117C22×D6112yes—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1224no—
426C2×C4⋊C4C22⋊D6112yes—
427C2×C4⋊C4C22⋊D6112yes—
428C2×C42—21no—
429C2×C42—612no—
430C2×C42—1221no—
431C2×C42C22×D6111yescharacteristic
432#107—1221no—
433#107—1222no—
434#106—1221no—
435#106—1222no—
436C4⋊C8—1221no—
437C4⋊C8—1221no—
438#104—2442no—
439#103—1221no—
440#103—1221no—
441#103—1221no—
442#103—1222no—
443#103—1222no—
444#102—1221no—
445#101—1224no—
446#101—2442no—
447#97—1221no—
448C4×C8—1221no—
449#95—1221no—
450#95—1224no—
451#41—21no—
452#41—21no—
453#41—21no—
454#41—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—612no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8C2×C22⋊D6111yescharacteristic
470C2×D4—21no—
471C2×D4C2×C22⋊D6111yescharacteristic
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—1222no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4C2×C22⋊D6114yes—
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C23×D6111yescharacteristic
482Q16—1224no—
483SD16—1224no—
484M16—612no—
485M16—1221no—
486M16—1222no—
487M16—1221no—
488M16—1222no—
489M16—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1222no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—1224no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—2448no—
511C4⋊C4—2441no—
512C4⋊C4—2441no—
513C4⋊C4C2×C22⋊D6111yescharacteristic
514C4⋊C4C2×C22⋊D6112yes—
515C4⋊C4C2×C22⋊D6111yescharacteristic
516C4⋊C4C2×C22⋊D6112yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—21no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C42—24no—
524C42—24no—
525C42—21no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—1222no—
530C42—1222no—
531C42—2441no—
532C42—2441no—
533C42C2×C22⋊D6111yescharacteristic
534C42C2×C22⋊D6111yescharacteristic
535C42C2×C22⋊D6111yescharacteristic
536C42C2×C22⋊D6111yescharacteristic
537C23—21no—
538C23C22×C22⋊D6111yescharacteristic
539Q8—21no—
540Q8—21no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—1222no—
548Q8—1222no—
549Q8—2441no—
550Q8—2441no—
551D4—21no—
552D4—21no—
553D4—41no—
554D4—41no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—1224no—
570C2×C4—1222no—
571C2×C4—2442no—
572C2×C4—2441no—
573C2×C4—2442no—
574C2×C4—2441no—
575C2×C4—2442no—
576C2×C4—2441no—
577C2×C4#1451974611yescharacteristic
578C2×C4#1451974611yescharacteristic
579C2×C4#1451974611yescharacteristic
580C2×C4#1451974611yescharacteristic
581C2×C4C22×C22⋊D6111yescharacteristic
582C2×C4C22×C22⋊D6111yescharacteristic
583C2×C4#41×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4#1451982311yescharacteristic
586C2×C4#1451982411yescharacteristic
587C8—1222no—
588C8—2441no—
589C8—2442no—
590C22—21no—
591C22—21no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×#1451974611yescharacteristic
596C22#1684040011yescharacteristic
597C22#1684041111yescharacteristic
598C4—21no—
599C4—21no—
600C4—21no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—44no—
606C4—41no—
607C4—4881no—
608C4—2444no—
609C4—2442no—
610C4#1683994512yes—
611C4#1684026911yescharacteristic
612C4#1684034011yescharacteristic
613C4#1684041911yescharacteristic
614C4#1684043111yescharacteristic
615C2—21no—
616C2—21no—
617C2—81no—
618C2#2144915811yescharacteristic
619C2C2×#1683994511yescharacteristic
620C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3509C2
socleC22×C61C2×#120
prime radicalC2×#41×C61C23
hypercenter#3509D61
hypocenterC61#24147
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3509

Complements

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

NH ≅ G/Ntype
1. #21455620C2semidirect
3. #21450808C2semidirect
6. #21446764C2semidirect
7. #21446664C2semidirect
8. #21446212C2semidirect
9. #21446034C2semidirect
363. C61#24147semidirect

Maximal subgroups

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

subgroupindex
#214556202
C4×#145198072
#214508082
#214470332
#214468532
#214467642
#214466642
#214462122
#214460342
#214458392
C61×#35092
#2414761