Finite Group Databasebeta

← #29689862

Subgroups — #29689862

Order 15616.

1076 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840297C211yesmaximal, characteristic
2#21454450C211yesmaximal, characteristic, complemented
3#423×D61C211yesmaximal, characteristic, complemented
4#21446813C211yesmaximal, characteristic, complemented
5#21446668C211yesmaximal, characteristic, complemented
6#21446666C211yesmaximal, characteristic, complemented
7#21446286C212yesmaximal, complemented
8#21446281C212yesmaximal
9#21445814C212yesmaximal, complemented
10#21445810C212yesmaximal, complemented
11C61×#1998C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13#16840297—21no—
14#16840297—21no—
15#16840297—21no—
16#16840297—21no—
17#16840297—21no—
18#16840297—21no—
19#16840297—21no—
20#16840297—21no—
21C2×#14519772C412yes—
22C2×#14519765C2211yescharacteristic
23C2×#14519762C412yes—
24C2×#14519757C2211yescharacteristic
25C2×#14519752C2211yescharacteristic
26C2×#14519742C2211yescharacteristic
27C2×#14519739C412yes—
28C2×#14519739C2211yescharacteristic
29C2×#14519735C412yes—
30C2×C42×D61C2211yescharacteristic
31#16839817—21no—
32#16839817—21no—
33#16839813—21no—
34#16839813—21no—
35#16839786C2211yescharacteristic
36#16839785C2212yes—
37#16839785C2212yes—
38#16839784—21no—
39#16839784—21no—
40C4⋊C8×D61—21no—
41C4⋊C8×D61—21no—
42C2×#14519695C2212yes—
43C2×#14519695C2212yes—
44C2×#14519693C2212yes—
45C2×#14519693C2212yes—
46#16839725C2212yes—
47C4×#11758194C2211yescharacteristic
48#16839531C2212yes—
49#16839531C2212yes—
50C2×#125×C61C2211yescharacteristic
51C61×#423C2211yescharacteristic
52C4×M16×C61C2211yescharacteristic
53C61×#343C2212yes—
54C61×#343C2212yes—
55C22×Dic122C2311yescharacteristic
56C22×D244C2×C412yes—
57C22×D244C2311yescharacteristic
58C22×C4×D61C2×C412yes—
59C22×C4×D61C2×C411yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C2×#11758194C2312yes—
63C2×#11758194C2312yes—
64C2×#11758189—22no—
65M16×D61—42no—
66M16×D61—42no—
67C2×C8×D61—22no—
68#14519772—22no—
69#14519772—22no—
70#14519772—22no—
71#14519772—22no—
72#14519765—21no—
73#14519765—21no—
74#14519765—21no—
75#14519765—21no—
76#14519762—22no—
77#14519762—22no—
78#14519762—22no—
79#14519762—22no—
80#14519757—21no—
81#14519757—21no—
82#14519757—21no—
83#14519757—21no—
84#14519752—21no—
85#14519752—21no—
86#14519752—21no—
87#14519752—21no—
88#14519742—21no—
89#14519742—21no—
90#14519742—21no—
91#14519742—21no—
92#14519739—22no—
93#14519739—22no—
94#14519739—22no—
95#14519739—22no—
96#14519739—21no—
97#14519739—21no—
98#14519739—21no—
99#14519739—21no—
100#14519735—22no—
101#14519735—22no—
102#14519735—22no—
103#14519735—22no—
104C2×C4⋊Dic61C2×C412yes—
105C2×#11758196C2×C412yes—
106C2×#11758198C2×C412yes—
107C2×#11758198C2×C412yes—
108C2×#11758198C2×C412yes—
109C2×#11758198C2×C412yes—
110C2×#11758198C2×C412yes—
111C2×C4×Dic61C2×C412yes—
112C2×C4×Dic61C2×C411yescharacteristic
113C2×C4×Dic61C2×C411yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61D411yescharacteristic
121C42×D61D411yescharacteristic
122C42×D61D411yescharacteristic
123C4⋊#92418—21no—
124C4⋊#92418—21no—
125#14519706—22no—
126#14519706—22no—
127#14519706—22no—
128#14519706—22no—
129#14519695—42no—
130#14519695—42no—
131#14519693—42no—
132#14519693—42no—
133#14519689—22no—
134#14519689—22no—
135#14519688—21no—
136C4×#92418—21no—
137C2×M16×C61C2312yes—
138C2×M16×C61C2312yes—
139#125×C61—21no—
140#125×C61—21no—
141#125×C61—21no—
142#125×C61—21no—
143C2×C4⋊C4×C61C2×C412yes—
144C2×C4⋊C4×C61C2×C412yes—
145C2×C4⋊C4×C61C2311yescharacteristic
146C2×C4⋊C4×C61C2311yescharacteristic
147C2×C42×C61C2311yescharacteristic
148C4⋊C8×C61—21no—
149C4⋊C8×C61—21no—
150#97×C61—21no—
151C4×C8×C61—21no—
152C23×D61C22×C411yescharacteristic
153C23×D61C22×C412yes—
154C22×C4×C61C22×C412yes—
155C22×C4×C61C22×C412yes—
156C22×C4×C61C22×C411yescharacteristic
157C22×C4×C61C22×C411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×Dic61C22×C412yes—
160C22×Dic61C22×C411yescharacteristic
161C22×Dic61C22×C411yescharacteristic
162C22×Dic61C22×C411yescharacteristic
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—22no—
168C2×D244—22no—
169C2×D244—44no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×C4×D61—44no—
177C2×C4×D61C22⋊C411yescharacteristic
178C2×C4×D61C22⋊C411yescharacteristic
179C2×C4×D61C22⋊C411yescharacteristic
180C2×C4×D61C22⋊C411yescharacteristic
181C2×C4×D61C22⋊C411yescharacteristic
182C2×C4×D61C22⋊C411yescharacteristic
183C2×C4×D61C22⋊C411yescharacteristic
184C2×C4×D61C22⋊C411yescharacteristic
185C2×C4×D61C2×D411yescharacteristic
186C2×C4×D61C2×D411yescharacteristic
187C2×C4×D61C2×D411yescharacteristic
188C2×C4×D61C2×D411yescharacteristic
189C2×Dic122—21no—
190C2×Dic122—21no—
191C2×Dic122—22no—
192C2×Dic122—22no—
193M16×C61—22no—
194M16×C61—22no—
195M16×C61—22no—
196M16×C61—22no—
197C2×C8×C61—22no—
198C2×C8×C61—22no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—22no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—22no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C4⋊C4×C61—21no—
206C4⋊C4×C61—21no—
207C42×C61C2×D411yescharacteristic
208C42×C61C2×D411yescharacteristic
209C42×C61C2×D411yescharacteristic
210C42×C61C2×D411yescharacteristic
211#11758198—22no—
212#11758198—22no—
213#11758198—22no—
214#11758198—22no—
215#11758198—22no—
216#11758198—22no—
217#11758198—22no—
218#11758198—22no—
219#11758198—22no—
220#11758198—22no—
221C4⋊Dic61—22no—
222C4⋊Dic61—22no—
223#11758196—22no—
224#11758196—22no—
225C4×Dic61—22no—
226C4×Dic61—22no—
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22⋊C411yescharacteristic
229C4×Dic61C22⋊C411yescharacteristic
230C4×Dic61C22⋊C411yescharacteristic
231C4×Dic61C22⋊C411yescharacteristic
232C4×Dic61C22⋊C411yescharacteristic
233C4×Dic61C22⋊C411yescharacteristic
234C4×Dic61C22⋊C411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237C4×Dic61C2×D411yescharacteristic
238C4×Dic61C2×D411yescharacteristic
239#11758194—22no—
240#11758194—22no—
241#11758194—22no—
242#11758194—22no—
243C2×#92418—22no—
244C2×#92418—22no—
245#11758189—44no—
246C8×D61—44no—
247C23×C61C23×C411yescharacteristic
248C22×D61—22no—
249C22×D61—22no—
250C22×D61—22no—
251C22×D61—24no—
252C22×D61—41no—
253C22×D61—42no—
254C22×D61—42no—
255C22×D61C2×C22⋊C411yescharacteristic
256C22×D61C2×C22⋊C411yescharacteristic
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×C22⋊C411yescharacteristic
265C2×C4×C61C2×C22⋊C411yescharacteristic
266C2×C4×C61C2×C22⋊C411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61H(5,𝔽2)12yes—
270C2×C4×C612−1+412yes—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—24no—
275C2×Dic61—41no—
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282D244—44no—
283D244—44no—
284D244—41no—
285D244—41no—
286D244—42no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—21no—
290C4×D61—21no—
291C4×D61—21no—
292C4×D61—21no—
293C4×D61—22no—
294C4×D61—22no—
295C4×D61—22no—
296C4×D61—22no—
297C4×D61—41no—
298C4×D61—41no—
299C4×D61—41no—
300C4×D61—41no—
301C4×D61—44no—
302C4×D61—44no—
303Dic122—41no—
304Dic122—41no—
305Dic122—42no—
306C8×C61—42no—
307C8×C61—42no—
308#92418—42no—
309#92418—42no—
310#21880—611nomaximal, Sylow
311C22×C61—21no—
312C22×C61—21no—
313C22×C61C22×C22⋊C411yescharacteristic
314C22×C61#52012yes—
315C2×D61—21no—
316C2×D61—21no—
317C2×D61—21no—
318C2×D61—21no—
319C2×D61—21no—
320C2×D61—44no—
321C2×D61—42no—
322C2×D61—42no—
323C2×D61—42no—
324C2×D61—84no—
325C2×D61#41312yes—
326C4×C61—21no—
327C4×C61—21no—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—42no—
331C4×C61—42no—
332C4×C61#53511yescharacteristic
333C4×C61#53611yescharacteristic
334C4×C61#53712yes—
335Dic61—21no—
336Dic61—21no—
337Dic61—21no—
338Dic61—21no—
339Dic61—21no—
340Dic61—42no—
341Dic61#41312yes—
342C2×#529—611no—
343#2920—611no—
344C2×#423—611no—
345#1998—611no—
346#1998D6111yescharacteristic, complemented
347#1967—611no—
348#1967—611no—
349#1901—612no—
350#1900—612no—
351#1801—612no—
352#1800—612no—
353C2×C61—22no—
354C2×C61#231111yescharacteristic
355C2×C61C2×#41311yescharacteristic
356C2×C61C2×#41311yescharacteristic
357D61—22no—
358D61—41no—
359D61—82no—
360C22×M16—612no—
361#529—1221no—
362#529—1221no—
363#529—1221no—
364#529—1221no—
365#529—1221no—
366#529—1221no—
367#529—1221no—
368#529—1221no—
369C2×#126—612no—
370C2×#125C2×D6111yescharacteristic
371C2×#124—611no—
372C2×#123—612no—
373C2×C22⋊Q8—611no—
374C2×#121—611no—
375C2×C4×Q8—611no—
376C2×C4×D4—612no—
377C2×C4×D4—611no—
378C2×#117—612no—
379C22×C42—611no—
380#432—1221no—
381#432—1221no—
382#431—1221no—
383#431—1221no—
384#423—612no—
385#423—612no—
386#423—611no—
387#423—1221no—
388#423—1221no—
389#423C2×D6111yescharacteristic
390C2×C4⋊C8—1221no—
391C2×C4⋊C8—1221no—
392C2×#101—612no—
393C2×#101—612no—
394C2×#100—612no—
395C2×#100—612no—
396#407—612no—
397C4×M16—611no—
398C4×M16C2×D6111yescharacteristic
399#343—612no—
400#343—612no—
401#343C2×D6112yes—
402#343C2×D6112yes—
403C61#2188011yescharacteristic, complemented, Sylow
404C22×Q8—611no—
405C22×D4—612no—
406C22×D4—611no—
407C23×C4—611no—
408C23×C4—611no—
409C23×C4—611no—
410C23×C4—612no—
411C2×M16—612no—
412C2×M16—612no—
413C2×M16—1222no—
414C2×M16—2442no—
415C2×M16—2442no—
416C2×M16C22×D6112yes—
417C2×M16C22×D6112yes—
418C22×C8—1222no—
419#126—1222no—
420#126—1222no—
421#126—1222no—
422#126—1222no—
423#125—21no—
424#125—21no—
425#125—21no—
426#125—21no—
427#124—1221no—
428#124—1221no—
429#124—1221no—
430#124—1221no—
431#123—1222no—
432#123—1222no—
433#123—1222no—
434#123—1222no—
435C22⋊Q8—1221no—
436C22⋊Q8—1221no—
437C22⋊Q8—1221no—
438C22⋊Q8—1221no—
439#121—1221no—
440#121—1221no—
441#121—1221no—
442#121—1221no—
443C4×Q8—1221no—
444C4×Q8—1221no—
445C4×Q8—1221no—
446C4×Q8—1221no—
447C4×D4—1221no—
448C4×D4—1221no—
449C4×D4—1221no—
450C4×D4—1221no—
451C4×D4—1222no—
452C4×D4—1222no—
453C4×D4—1222no—
454C4×D4—1222no—
455#117—1222no—
456#117—1222no—
457#117—1222no—
458#117—1222no—
459C2×C4⋊C4—612no—
460C2×C4⋊C4—612no—
461C2×C4⋊C4C4×D6112yes—
462C2×C4⋊C4C4×D6112yes—
463C2×C4⋊C4C22×D6111yescharacteristic
464C2×C4⋊C4C22×D6111yescharacteristic
465C2×C22⋊C4—612no—
466C2×C22⋊C4—612no—
467C2×C22⋊C4—612no—
468C2×C22⋊C4—612no—
469C2×C22⋊C4—612no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—611no—
473C2×C42—611no—
474C2×C42—611no—
475C2×C42—611no—
476C2×C42—611no—
477C2×C42—611no—
478C2×C42—611no—
479C2×C42—611no—
480C2×C42—611no—
481C2×C42—612no—
482C2×C42C22×D6111yescharacteristic
483C4⋊C8—21no—
484C4⋊C8—21no—
485C4⋊C8—1222no—
486C4⋊C8—1222no—
487C4⋊C8—1222no—
488C4⋊C8—1222no—
489C4⋊C8—1221no—
490C4⋊C8—1221no—
491#101—2442no—
492#101—2442no—
493#100—2442no—
494#100—2442no—
495C22⋊C8—1222no—
496C22⋊C8—1222no—
497#97—21no—
498#97—1221no—
499C4×C8—21no—
500C4×C8—1221no—
501C24—611no—
502C24—612no—
503C2×Q8—1221no—
504C2×Q8—1221no—
505C2×Q8—1222no—
506C2×Q8—1222no—
507C2×D4—1222no—
508C2×D4—1222no—
509C2×D4—1221no—
510C2×D4—1221no—
511C2×D4—1222no—
512C2×D4—1222no—
513C2×D4—2444no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—611no—
528C22×C4—611no—
529C22×C4—612no—
530C22×C4—1222no—
531C22×C4—1222no—
532C22×C4—1222no—
533C22×C4—1222no—
534C22×C4—2442no—
535C22×C4—2442no—
536C22×C4—2444no—
537C22×C4C2×C4×D6112yes—
538C22×C4C2×C4×D6112yes—
539C22×C4C2×C4×D6111yescharacteristic
540C22×C4C2×C4×D6111yescharacteristic
541C22×C4C23×D6111yescharacteristic
542M16—22no—
543M16—22no—
544M16—22no—
545M16—22no—
546M16—1222no—
547M16—1222no—
548M16—1222no—
549M16—1222no—
550M16—2444no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—1222no—
554C2×C8—1222no—
555C2×C8—2444no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—1222no—
565C4⋊C4—1222no—
566C4⋊C4—1222no—
567C4⋊C4—1222no—
568C22⋊C4—1222no—
569C22⋊C4—1222no—
570C22⋊C4—1222no—
571C22⋊C4—1222no—
572C22⋊C4—1222no—
573C22⋊C4—1222no—
574C22⋊C4—1222no—
575C22⋊C4—1222no—
576C22⋊C4—1222no—
577C22⋊C4—1222no—
578C42—611no—
579C42—611no—
580C42—611no—
581C42—611no—
582C42—611no—
583C42—611no—
584C42—611no—
585C42—611no—
586C42—611no—
587C42—611no—
588C42—611no—
589C42—611no—
590C42—1222no—
591C42—1222no—
592C42D4×D6111yescharacteristic
593C42D4×D6111yescharacteristic
594C42D4×D6111yescharacteristic
595C42D4×D6111yescharacteristic
596C23—611no—
597C23—611no—
598C23—1222no—
599C23—1222no—
600C23—1224no—
601C23—1222no—
602C23—2441no—
603C23—2442no—
604C23—2442no—
605C23C22×C4×D6111yescharacteristic
606Q8—2441no—
607Q8—2441no—
608Q8—2442no—
609D4—2444no—
610D4—2444no—
611D4—2441no—
612D4—2441no—
613D4—2442no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—611no—
621C2×C4—611no—
622C2×C4—611no—
623C2×C4—611no—
624C2×C4—611no—
625C2×C4—611no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1221no—
633C2×C4—1221no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1222no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—1224no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2444no—
645C2×C4—2444no—
646C2×C4—2441no—
647C2×C4C22⋊C4×D6111yescharacteristic
648C2×C4C22⋊C4×D6111yescharacteristic
649C2×C4C22⋊C4×D6111yescharacteristic
650C2×C4C22⋊C4×D6111yescharacteristic
651C2×C4C2×D4×D6111yescharacteristic
652C2×C4C2×D4×D6111yescharacteristic
653C2×C4#1451982312yes—
654C2×C4#1451982512yes—
655C8—42no—
656C8—42no—
657C8—2442no—
658C8—2442no—
659C22—21no—
660C22—21no—
661C22—612no—
662C22—1221no—
663C22—1221no—
664C22—1221no—
665C22—1221no—
666C22—1221no—
667C22—4884no—
668C22—2444no—
669C22—2442no—
670C22—2442no—
671C22—2442no—
672C22C2×C22⋊C4×D6111yescharacteristic
673C22#1684024412yes—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—42no—
679C4—42no—
680C4—612no—
681C4—1221no—
682C4—1221no—
683C4—1221no—
684C4—1221no—
685C4—1221no—
686C4—2442no—
687C4#1684033011yescharacteristic
688C4#1684033711yescharacteristic
689C4#1684034212yes—
690C2—22no—
691C2—1222no—
692C2—4882no—
693C2—2441no—
694C2#2144841011yescharacteristic
695C2#413×D6111yescharacteristic
696C2#413×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1998C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1998D61
hypocenterC61#21880
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1998

Complements

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

NH ≅ G/Ntype
2. #21454450C2semidirect
3. #423×D61C2semidirect
4. #21446813C2semidirect
5. #21446668C2semidirect
6. #21446666C2semidirect
7. #21446286C2semidirect
9. #21445814C2semidirect
10. #21445810C2semidirect
11. C61×#1998C2semidirect
346. #1998D61semidirect
403. C61#21880semidirect

Maximal subgroups

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

subgroupindex
C2×#168402972
#214544502
#423×D612
#214468132
#214466682
#214466662
#214462862
#214462812
#214458142
#214458102
C61×#19982
#2188061